CRYSTAL2003
1.0
User's Manual
August 24, 2004
Main authors of CRYSTAL 2003:
V.R. Saunders1, R. Dovesi2, C. Roetti2, R. Orlando3,2,
C. M. Zicovich-Wilson 2,4, N.M. Harrison1,5, K. Doll1,6,
B. Civalleri2, I.J. Bush1, Ph. D'Arco7,2, M. Llunell2,8
1
Computational Science & Engineering Department - CLRC Daresbury
Daresbury, Warrington, Cheshire, UK WA4 4AD
2
Theoretical Chemistry Group - University of Turin
Department of Chemistry IFM
Via Giuria 5 - I 10125 Torino - Italy
3
University of Eastern Piedmont
Department of Science and Advanced Technologies
Corso Borsalino 54 - I 15100 Alessandria - Italy
4
Departamento de Fisica, Universidad Autonoma del Estado de Morelos,
Av. Universidad 1001, Col. Chamilpa, 62210 Cuernavaca (Morelos) Mexico
5
Chemistry, Imperial College
South Kensington Campus
London, U.K.
6
Institut f
ur Mathematische Physik
TU Braunschweig - Mendelssohnstrasse 3
Braunschweig D-38106 Germany
7
Laboratoire de P
etrologie, Mod
elisation des Materiaux et Processus
Universit
e Pierre et Marie Curie,
4 Place Jussieu, 75232 Paris CEDEX 05, France
8
Departament de Quimica Fisica, Universitat de Barcelona
Diagonal 647, Barcelona, Spain
1

2

Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Functionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
Typographical Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
Getting Started - Installation and testing . . . . . . . . . . . . . . . . . . . . . . . .
10
Program errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1
Wave function calculation
Basic input route
11
1.1
Geometry and symmetry information . . . . . . . . . . . . . . . . . . . . . . . .
11
Geometry input for crystalline compounds . . . . . . . . . . . . . . . . . . . . .
12
Geometry input for molecules, polymers and slabs
. . . . . . . . . . . . . . . .
12
Geometry input from external geometry editor . . . . . . . . . . . . . . . . . .
13
Comments on geometry input . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.2
Basis set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
1.3
General information, computational parameters,
hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
1.4
SCF input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2
Wave function calculation - Advanced input route
22
2.1
Geometry editing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
Finite periodic electric field perturbation . . . . . . . . . . . . . . . . . . . . . .
41
Geometry optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
Notes - From periodic structure to clusters and molecules . . . . . . . . . . . .
47
Notes - The slab model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
Notes - BSSE correction in periodic systems . . . . . . . . . . . . . . . . . . . .
47
2.2
Basis set input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
Effective core pseudo-potentials . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
Pseudopotential libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
Valence Basis set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
Conversion of Stuttgart-Dresden pseudopotentials . . . . . . . . . . . . . . . . .
53
Conversion of Stevens et al. pseudopotentials . . . . . . . . . . . . . . . . . . .
54
2.3
General information, computational parameters,
hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
2.4
SCF input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
3
Properties
82
3.1
Preliminary calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
3.2
Properties keywords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
3.3
Spontaneous polarization and piezoelectricity . . . . . . . . . . . . . . . . . . . 112
4
Input examples
115
4.1
Standard geometry input
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
CRYSTAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
SLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
POLYMER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3

MOLECULE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.2
Basis set input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Valence only basis set input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.3
SCF options - SPINEDIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.4
Geometry optimization - OPTCOORD . . . . . . . . . . . . . . . . . . . . . . . 127
5
Basis set
133
5.1
Molecular BSs performance in periodic systems . . . . . . . . . . . . . . . . . . 133
5.2
Core functions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.3
Valence functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Molecular crystals
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Covalent crystals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Ionic crystals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
From covalent to ionics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.4
Hints on crystalline basis set optimization . . . . . . . . . . . . . . . . . . . . . 136
5.5
Check on basis-set quasi-linear-dependence
. . . . . . . . . . . . . . . . . . . . 137
6
Theoretical framework
139
6.1
Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.2
Remarks on the evaluation of the integrals . . . . . . . . . . . . . . . . . . . . . 140
6.3
Treatment of the Coulomb series . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.4
The exchange series
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.5
Bipolar expansion approximation of Coulomb and exchange integrals . . . . . . 143
6.6
Exploitation of symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Symmetry-adapted Crystalline Orbitals
. . . . . . . . . . . . . . . . . . . . . . 144
6.7
Reciprocal space integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.8
Electron momentum density and related quantities . . . . . . . . . . . . . . . . 146
6.9
Elastic Moduli of Periodic Systems . . . . . . . . . . . . . . . . . . . . . . . . . 147
Examples of
matrices for cubic systems . . . . . . . . . . . . . . . . . . . . . . 149
Bulk modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.10 Spontaneous polarization through the Berry phase approach . . . . . . . . . . . 152
Spontaneous polarization through the localized crystalline orbitals approach . . 152
6.11 Piezoelectricity through the Berry phase approach . . . . . . . . . . . . . . . . 153
Piezoelectricity through the localized crystalline orbitals approach
. . . . . . . 153
A Symmetry groups
155
A.1 Labels and symbols of the space groups . . . . . . . . . . . . . . . . . . . . . . 155
A.2 Labels of the layer groups (slabs) . . . . . . . . . . . . . . . . . . . . . . . . . . 158
A.3 Labels of the rod groups (polymers) . . . . . . . . . . . . . . . . . . . . . . . . 159
A.4 Labels of the point groups (molecules) . . . . . . . . . . . . . . . . . . . . . . . 162
A.5 From conventional to primitive cells: transforming matrices . . . . . . . . . . . 163
B Summary of input keywords
164
C DFT integration through an auxiliary basis set fitting
172
C.1 DFT input example - fitting method . . . . . . . . . . . . . . . . . . . . . . . . 173
D Reciprocal lattice sampling
176
E Printing options
177
F External format
181
G Utility programs
182
H CRYSTAL2003 versus CRYSTAL98
183
4

I
Relevant strings
186
J
Acronyms
187
Bibliography
188
Subject index
194
5

Introduction
The CRYSTAL package performs ab initio calculations of the ground state energy, energy
gradient, electronic wave function and properties of periodic systems. Hartree-Fock or Kohn-
Sham Hamiltonians (that adopt an Exchange- Correlation potential following the postulates of
Density-Functional theory) can be used. Systems periodic in 0 (molecules, 0D), 1 (polymers,
1D), 2 (slabs, 2D), and 3 dimensions (crystals, 3D) are treated on an equal footing. In each
case the fundamental approximation made is the expansion of the single particle wave functions
('Crystalline Orbital', CO) as a linear combination of Bloch functions (BF) defined in terms
of local functions (hereafter indicated as 'Atomic Orbitals', AOs). See Chapter 6.
The local functions are, in turn, linear combinations of Gaussian type functions (GTF) whose
exponents and coefficients are defined by input (section 1.2). Functions of symmetry s, p, d
and f can be used (see page 18. Also available are sp shells (s and p shells, sharing the same
set of exponents). The use of sp shells can give rise to considerable savings in CPU time.
The program can automatically handle space symmetry: 230 space groups, 80 layer groups,
99 rod groups, 45 point groups are available (Appendix A). In the case of polymers it cannot
treat helical structures (translation followed by a rotation around the periodic axis). However,
when commensurate rotations are involved, a suitably large unit cell can be adopted.
Point symmetries compatible with translation symmetry are provided for molecules.
Input tools allow the generation of slabs (2D system) or clusters (0D system) from a 3D
crystalline structure, the elastic distortion of the lattice, the creation of a supercell with a
defect and a large variety of structure editing. See Section 2.1
Previous releases of the software in 1988 (CRYSTAL88, [1]), 1992 (CRYSTAL92, [2]), 1996
(CRYSTAL95, [3]) and 1998 (CRYSTAL98, [4]), have been used in a wide variety of research
with notable applications in studies of stability of minerals, oxide surface chemistry, and defects
in ionic materials.
The CRYSTAL package has been developed over a number of years. Many of the algorithms
developed and reviews of the applications of the code have been previously published (for
instance, [5, 6, 7, 8, 9, 10, 11, 12, 13]).
The required citation for this work is:
V.R. Saunders, R. Dovesi, C. Roetti, R. Orlando, C. M. Zicovich-Wilson,
N.M. Harrison, K. Doll, B. Civalleri, I. Bush, Ph. D'Arco, M. Llunell
CRYSTAL2003 User's Manual, University of Torino, Torino, 2003
CRYSTAL2003 output will display the references relevant to the property computed, when
necessary.
The following sites will supply updated information on the CRYSTAL code:
http://www.cse.dl.ac.uk/Activity/CRYSTAL
http://www.crystal.unito.it
6

Functionality
The basic functionality of the code is outlined below.
The single particle potential
Restricted Hartree-Fock Theory
Unrestricted Open Shell Hartree-Fock Theory
Density Functional Theory for Exchange and Correlation
Spin Density Functional Theory
Hybryds HF-DFT (B3LYP-B3PW)
Effective Core Pseudopotentials
Finite field perturbation added to the Hamiltonian
*
Algorithms
Parallel processing (replicated data)
Massive Parallel Processing (distributed data)
*
Traditional SCF
Full Direct SCF
*
Structural Editing
Use of space, layer, rod and point group symmetry
Deformation of the crystallographic cell
Removal, insertion, deletion and substitution of atoms
Displacement of atoms
Rotation of groups of atoms
Extraction of surface models from a 3D crystal structure
Cluster generation from a 3D crystal
Cluster of molecules from molecular crystal
Properties
Band structure
Density of states
Electronic charge density maps
Electronic charge density on a 3D grid
Mulliken population analysis
Spherical harmonic atom and shell multipoles
X-ray structure factors
Electron momentum distributions
Compton profiles
First order density matrix
*
Reciprocal form factors
*
Electrostatic potential, field and field gradients
Spin polarized generalization of properties
Hyperfine electron-nuclear spin tensor
A posteriori Density Functional correlation energy
Localization of Crystalline Orbitals
*
Spontaneous polarization through Berry phase approach
Spontaneous polarization through localized orbitals approach *
Piezoelectricity through Berry phase approach
*
Piezoelectricity through localized orbitals approach
*
Optical dielectric constant
*
Analytic nuclear coordinates gradient of the energy
*
Geometry optimizer
*
7

Conventions
In the description of the input data which follows, the following notation is adopted:
-

new record
-

free format record
-
An
alphanumeric datum (first n characters meaningful)
-
atom label
sequence number of a given atom in the primitive cell, as
printed in the output file after reading of the geometry input
-
symmops
symmetry operators
-
, [ ]
default values.
-
italic
optional input
-
optional input records follow
II
-
additional input records follow
II
Part of the code is written in fortran 77. The name of the variables is associated with the type
of data, following the fortran 77 convention: if the first letter of the name is I, J, K, L, M or
N, the type is integer. Otherwise the type is real.
Arrays are read in with a simplified implied DO loop instruction of Fortran 77:
(dslist, i=m1,m2)
where: dslist is an input list; i is the name of an integer variable, whose value ranges from m1
to m2.
Example (page 25): LB(L),L=1,NL
NL integer data are read in and stored in the first NL position of the array LB.
All the keywords are entered with an A format (case insensitive); the keywords must be typed
left-justified, with no leading blanks.
conventional atomic number (usually callet NAT) is used to associate a given basis set
with an atom. The real atomic number is the remainder of the division NAT/100.
8

Acknowledgements
Embodied in the present code are elements of programs distributed by other groups.
In particular: the atomic SCF package of Roos et al. [14], the GAUSS70 gaussian integral
package and STO-nG basis set due to Hehre et al. [15], the code of Burzlaff and Hountas for
space group analysis [16] and Saunders' ATMOL gaussian integral package [17].
We take this opportunity to thank these authors. Our modifications of their programs have
sometimes been considerable. Responsibility for any erroneous use of these programs therefore
remains with the present authors.
It is our pleasure to thank Piero Ugliengo for continous help, useful suggestions, rigorous test-
ing.
We are also in debt with Cesare Pisani, for his constant support of and interest in the devel-
opment of the new version of the CRYSTAL program.
Financial support for this research has been provided by the italian MURST (Ministero della
Universit`
a e della Ricerca Scientifica e Tecnologica), and the United Kingdom CCLRC (Council
for the Central Laboratories of the Research Council).
9

Getting Started
See http://www.crystal.unito.it documentation README.
Program errors
A very large number of tests have been performed by researchers of a few laboratories, that
had access to a test copy of CRYSTAL2003. We tried to check as many options as possible,
but not all the possible combinations of options have been checked. We have no doubts that
errors remain.
The authors would greatly appreciate comments, suggestions and criticisms by the users of
CRYSTAL; in case of errors the user is kindly requested to contact the authors, sending a
copy of both input and output by E-mail to the Torino group (crystal@unito.it) or to the
Daresbury team (crystal@dl.ac.uk).
10

Chapter 1
Wave function calculation
Basic input route
1.1
Geometry and symmetry information
The first record of the geometry definition must contain one of the keywords:
CRYSTAL
3D system
SLAB
2D system
POLYMER
1D system
MOLECULE
0D system
EXTERNAL
geometry from external file
DLVINPUT
geometry from DLV [18] Graphical User Interface.
Three input schemes are used. The first is for crystalline systems, and is specified by the
keyword CRYSTAL. The second is for slabs, polymers and molecules as specified by the key-
words SLAB, POLYMER or MOLECULE respectively. In the third scheme, with keyword
EXTERNAL or DLVINPUT, the unit cell, atomic positions and symmetry operators may
be provided directly (see Appendix F, page 181). Such an input file can be prepared by the
keyword EXTPRT (input block 1, page 31; properties). Sample input decks for a number
of structures are provided in section 4.1, page 115.
11

Geometry input for crystalline compounds
rec
variable
value
meaning

IFLAG
convention for space group identification (Appendix A.1, page 155):
0
space group sequential number(1-230)
1
Hermann-Mauguin alphanumeric code
IFHR
type of cell for rhombohedral groups (meaningless for non-
rhombohedral crystals):
0
hexagonal cell
1
rhombohedral cell
IFSO
setting for the origin of the crystal reference frame:
0
origin derived from the symbol of the space group: where there
are two settings, the second setting of the International Tables is
chosen.
1
standard shift of the origin: when two settings are allowed, the first
setting is chosen
>1
non-standard shift of the origin given as input

space group identification code (following IFLAG value):
IGR
space group sequence number (IFLAG=0)
or
A
AGR
space group alphanumeric symbol (IFLAG=1)
if IFSO > 1 insert
II

IX,IY,IZ
non-standard shift of the origin coordinates (x,y,z) in fractions of
the crystallographic cell lattice vectors times 24 (to obtain integer
values).

a,[b],[c],
minimal set of crystallographic cell parameters:
[],[]
translation vector[s] length [
Angstrom],
[]
crystallographic angle[s] (degrees)

NATR
number of atoms in the asymmetric unit.
insert NATR records
II

NAT
conventional atomic number 1
X,Y,Z
atom coordinates in fractional units of crystallographic lattice vec-
tors
optional keywords terminated by END/ENDGEOM or STOP
II
Geometry input for molecules, polymers and slabs
When the geometrical structure of 2D, 1D and 0D systems has to be defined, attention should
be paid in the input of the atom coordinates, that are expressed in different units, fractionary
(direction with translational symmetry) or
Angstrom (non periodic direction).
translational
unit of measure of coordinates
symmetry
X
Y
Z
3D
fraction
fraction
fraction
2D
fraction
fraction

Angstrom
1D
fraction

Angstrom
Angstrom
0D

Angstrom
Angstrom
Angstrom
12

rec variable
meaning
IGR
point, rod or layer group of the system:
0D - molecules (Appendix A.4, page 162)
1D - polymers (Appendix A.3, page 159)
2D - slabs (Appendix A.2, page 158)
if polymer or slab, insert
II
a,[b],
minimal set of lattice vector(s)- length in
Angstrom
(b for rectangular lattices only)
[]
AB angle (degrees) - triclinic lattices only
NATR
number of non-equivalent atoms in the asymmetric unit
insert NATR records
II
NAT
conventional atomic number
X,Y,Z
atoms coordinates. Unit of measure:
0D - molecules: x,y,z in
Angstrom
1D - polymers : y,z in
Angstrom, x in fractional units of crystallographic
cell translation vector
2D - slabs : z in
Angstrom, x, y in fractional units of crystallographic cell
translation vectors
optional keywords terminated by END or STOP
II
Geometry input from external geometry editor
The keywords EXTERNAL and DLVINPUT select the third input scheme. They work
for molecules, polymers, slabs and crystals. The input data are read from Fortran unit 34.
The unit cell, atomic positions and symmetry operators are provided directly according to the
format described in Appendix F, page 181. Coordinates in
Angstrom. Such an input file is
written when OPTBERNY route for geometry optimization is chosen, and can be prepared by
the keyword EXTPRT (input block 1, page 31; properties).
The geometry so defined can be modified by inserting any geometry editing keyword (page 22)
after EXTERNAL.
Comments on geometry input
1. All coordinates in
Angstrom. In geometry editing the unit of measure of coordinates may
be modified by entering the keywords FRACTION (page 31) or BOHR (page 27).
2. The geometry of a system is defined by the crystal structure ([19], Chapter 1 of ref. [20]).
Reference is made to the International Tables for Crystallography [21] for all definitions.
The crystal structure is determined by the space group, by the shape and size of the unit
cell and by the relative positions of the atoms in the asymmetric unit.
3. The lattice parameters represent the length of the edges of the cell (a,b,c) and the angles
between the edges ( = b c; = a c; = a b). They determine the cell volume and
shape.
4. Minimal set of lattice parameters to be defined in input:
cubic
a
hexagonal
a,c
rhombohedral
hexagonal cell
a,c
rhombohedral cell
a,
tetragonal
a,c
orthorhombic
a,b,c
monoclinic
a,b,c, (b unique)
a,b,c, (c unique)
a,b,c, (a unique - non standard)
triclinic
a,b,c, , ,
13

5. The asymmetric unit is the largest subset of atoms contained in the unit-cell, where
no atom pair related by a symmetry operator can be found. Usually several equivalent
subsets of this kind may be chosen so that the asymmetric unit needs not be unique.
The asymmetric unit of a space group is a part of space from which, by application of
all symmetry operations of the space group, the whole of space is filled exactly.
6. The crystallographic, or conventional cell, is used as the standard option in input. It
may be non-primitive, which means it may not coincide with the cell of minimum volume
(primitive cell) which contains just one lattice point. The matrices which transform the
conventional (as given in input) to the primitive cell (used by CRYSTAL) are given in
Appendix A.5, page 163, and are taken from table 5.1 of the International Tables [21].
Examples. A cell belonging to the face-centred cubic Bravais lattice has a volume four
times larger than that of the corresponding primitive cell, and contains four lattice points
(see page 38, keyword SUPERCEL). A unit cell belonging to the hexagonal Bravais
lattice has a volume three times larger than that of the rhombohedral primitive cell (R
Bravais lattice), and contains three lattice points.
7. The use of the International Tables to identify the symmetry groups requires some prac-
tice. The examples given below may serve as a guide. The printout of geometry informa-
tion (equivalent atoms, fractional and Cartesian atomic coordinates etc.) allows a check
on the correctness of the group selected. To obtain a complete neighborhood analysis
for all the non-equivalent atoms, a complete input deck must be read in (blocks 1-4),
and the keyword TESTPDIM inserted in block 3, to stop execution after the symmetry
analysis.
8. Different settings of the origin may correspond to a different number of symmetry oper-
ators with translational components.
Example: bulk silicon - Space group 227 - 1 irreducible atom per cell.
setting of the origin
Si coordinates
symmops with
translational component
2nd (default)
1/8
1/8
1/8
36
1st
0.
0.
0.
24
NB With 2nd setting, the position 0., 0., 0. has multiplicity 4.
The choice is important when generating a supercell, as the first step is the removal of the
symmops with translational component. The keyword ORIGIN (input block 1, page
34) translates the origin in order to minimize the number of symmops with translational
component.
9. When coordinates are obtained from experimental data or from geometry optimization
with semi-classical methods, atoms in special positions, or related by symmetry are not
correctly identified, as the number of significative digits is lower that the one used by
the program crystal to recognize equivalence or special positions.
In that case the
coordinates must be edited by hand (see FAQ at www.crystal.unito.it).
10. The symbol of the space group for crystals (IFLAG=1) is given precisely as it appears
in the International Tables, with the first letter in column one and a blank separating
operators referring to different symmetry directions. The symbols to be used for the
groups 221-230 correspond to the convention adopted in editions of the International
Tables prior to 1983: the 3 axis is used instead of 3. See Appendix A.1, page 155.
Examples:
Group number
input symbol
137 (tetragonal)
P 42/N M C
10 (monoclinic)
P 1 2/M 1
(unique axis b, standard setting)
P 1 1 2/M
(unique axis c)
14

P 2/M 1 1
(unique axis a)
25 (orthorhombic)
P M M 2
(standard setting)
P 2 M M
P M 2 M
11. In the monoclinic and orthorhombic cases, if the group is identified by its number (3-74),
the conventional setting for the unique axis is adopted. The explicit symbol must be
used in order to define an alternative setting.
12. For the centred lattices (F, I, C, A, B and R) the input cell parameters refer to the
centred conventional cell; the fractional coordinates of the input list of atoms are in a
vector basis relative to the centred conventional cell.
13. It is sufficient to supply the coordinates of only one of a group of atoms equivalent under
centring translations (eg: for space group Fm3m only the parameters of the face-centred
cubic cell, and the coordinates of one of the four atoms at (0,0,0), (0, 1 , 1 ), ( 1 ,0, 1 ) and
2 2
2
2
( 1 , 1 ,0) are required).
2 2
The coordinates of only one atom among the set of atoms linked by centring translations
are printed. The vector basis is relative to the centred conventional cell. However when
Cartesian components of the direct lattice vectors are printed, they are those of the
primitive cell.
14. The conventional atomic number NAT is used to associate a given basis set with an
atom (see Basis Set input, Section 1.2, page 16). The real atomic number is given by the
remainder of the division of the conventional atomic number by 100 (Example: NAT=237,
Z=37; NAT=128, Z=28). Atoms with the same atomic number, but in non-equivalent
positions, can be associated with different basis sets, by using different conventional
atomic numbers: e.g. 6, 106 (all electron basis set for carbon atom); 206, 306 (core
pseudo-potential for carbon atom, Section 2.2, page 51).
If the remainder of the division is 0, a "ghost" atom is identified, to which no nuclear
charge corresponds (it may have electronic charge). This option may be used for enriching
the basis set by adding bond basis function [22], or to allow build up of charge density on
a vacancy. A given atom may be transformed into a ghost after the basis set definition
(input block 2, keyword GHOSTS, page 50).
15. The keyword SLAB (Geometry editing input, page 37) allows the creation of a slab
(2D) of given thickness from the 3D perfect lattice. See for comparison test4-test24;
test5-test25; test6-test26; test7- test27.
16. For slabs (2D), when two settings of the origin are indicated in the International Tables
for Crystallography, setting number 2 is chosen. The setting can not be modified.
17. Conventional orientation of slabs and polymers: Polymers are oriented along the x axis.
Slabs are parallel to the xy plane.
18. The keywords MOLECULE (for molecular crystals only; page 33) and CLUSTER
(for any n-D structure; page 28) allow the creation of a non-periodic system (molecule(s)
or cluster) from a periodic one.
15

1.2
Basis set
rec variable value
meaning
NAT
n
conventional atomic number
<200
all-electron basis set
>200
valence electron basis set. ECP (Effective Core Pseudopotential)
must be defined (page 51)
=99
end of basis set input section
NSHL
n
number of shells
0
end of basis set input (when NAT=99)
if NAT > 200 insert ECP input (page 51)
II
NSHL sets of records - for each shell
ITYB
type of basis set to be used for the specified shell:
0
general BS, given as input
1
Pople standard STO-nG (Z=1-54)
2
Pople standard 3(6)-21G (Z=1-54(18)) Standard polarization func-
tions are included.
LAT
shell type:
0
1 s AO (S shell)
1
1 s + 3 p AOs (SP shell)
2
3 p AOs (P shell)
3
5 d AOs (D shell)
NG
Number of primitive Gaussian Type Functions (GTF) in the con-
traction for the basis functions (AO) in the shell
1NG10 for ITYB=0 and LAT 2
1NG6
for ITYB=0 and LAT = 3
2NG6
for ITYB=1
6
6-21G core shell
3
3-21G core shell
2
n-21G inner valence shell
1
n-21G outer valence shell
CHE
formal electron charge attributed to the shell
SCAL
scale factor (if ITYB=1 and SCAL=0., the standard Pople scale
factor is used for a STO-nG basis set.
if ITYB=0 (general basis set insert NG records
II
EXP
exponent of the normalized primitive GTF
COE1
contraction coefficient of the normalized primitive GTF:
LAT=0,1 s function coefficient
LAT=2 p function coefficient
LAT=3 d function coefficient
COE2
LAT=1 p function coefficient
optional keywords terminated by END/ENDB or STOP
II
The choice of basis set is the most critical step in performing ab initio calculations of periodic
systems, with Hartree-Fock or Kohn-Sham Hamiltonians. Optimization criteria are discussed in
Chapter 3.2. When an effective core pseudo-potential is used, the basis set must be optimized
with reference to that potential (Section 2.2, page 51).
1. A basis set (BS) must be given for each atom with different conventional atomic number
defined in the crystal structure input. If atoms are removed (geometry input, keyword
ATOMREMO, page 25), the corresponding basis set input can remain in the input
stream.
2. The basis set for each atom has NSHL shells whose constituent AO basis functions
are built from a linear combination ('contraction') of individually normalized primitive
Gaussian-type functions (GTF) (Chapter 6, page 139).
3. A conventional atomic number NAT links the basis set with the atoms defined in the
16

crystal structure. The atomic number Z is given by the remainder of the division of the
conventional atomic number by 100 (Example: NAT=108, Z=8, all electron; NAT=228,
Z=28, ECP). See point 5 below.
4. A conventional atomic number 0 defines ghost atoms, that is points in space with an
associated basis set, but lacking a nuclear charge (vacancy). See test 28.
5. Atoms with equal conventional atomic number are associated with the same basis set.
NAT < 200: all electron basis set. A maximum of two different basis sets
may be given for the same chemical species in different positions:
NAT=Z, NAT=Z+100.
NAT > 200: valence electron basis set.
A maximum of two different BS
may be given for the same chemical species in positions not
symmetry-related: NAT=Z+200, NAT=Z+300. A core pseudo-
potential must be defined. See Section 2.2, page 51, for informa-
tion on core pseudo-potentials.
Suppose we have four non-equivalent carbon atoms in the unit cell. Conventional atomic
numbers 6 106 206 306 mean that carbon atoms (real atomic number 6) unrelated by
symmetry are to be associated with different basis sets: the first two (6, 106) all-electron,
the second two (206, 306) valence only, with pseudo-potential.
6. The basis set input ends with the card:
99
0
conventional atomic number 99, 0 shell.
The optional keywords may follow.
In summary:
1. CRYSTAL can use the following all electrons basis sets:
a)
general basis sets, including s,p,d functions (given in input);
b)
standard Pople basis sets [23] (internally stored)
STOnG,
Z=1 to 54
6-21G,
Z=1 to 18
3-21G,
Z=1 to 54
The standard basis sets b) are stored as internal data in the CRYSTAL code. They are
all electron basis sets, and can not be combined with ECP.
2. Warning The standard scale factor is used for STO-nG basis set when the input datum
SCAL is 0.0 in basis set input. All the atoms of the same row are attributed the same
Pople STO-nG basis set when the input scale factor SCAL is 1.
3. Standard polarization functions can be added to 6(3)-21G basis sets of atoms up to Z=18,
by inserting a record describing the polarization shell (ITYB=2, LAT=2, p functions on
hydrogen, or LAT=3, d functions on 2-nd row atoms; see test 12).
H
Polarization functions exponents
He
1.1
1.1
__________
______________________________
Li
Be
B
C
N
O
F
Ne
0.8
0.8
0.8
0.8
0.8
0.8
0.8
--
___________
______________________________
Na
Mg
Al
Si
P
S
Cl
Ar
0.175
0.175
0.325 0.45 0.55 0.65 0.75 0.85
_____________________________________________________________________
The formal electron charge attributed to a polarization function must be zero.
4. The shell types available are :
17

shell shell
n.
order of internal storage
code type
AO
0
S
1
s
1
SP
4
s, x, y, z
2
P
3
x, y, z
3
D
5
2z2 - x2 - y2, xz, yz, x2 - y2, xy
4
F
7
(2z2 - 3x2 - 3y2)z, (4z2 - x2 - y2)x, (4z2 - x2 - y2)y,
(x2 - y2)z, xyz, (x2 - 3y2)x, (3x2 - y2)y
The order of internal storage of the AO basis functions is an information necessary to
read certain quantities calculated by the program properties. See Chapter 3: Mul-
liken population analysis (PPAN, page 78), electrostatic multipoles (POLI, page 105),
projected density of states (DOSS,page 91) and to provide an input for some options
(EIGSHIFT, input block 4, page 74).
5. Spherical harmonics d-shells consisting of 5 AOs are used.
6. The formal shell charges CHE, the number of electrons attributed to each shell, are
assigned to the AO following the rules:
shell
shell
max
rule to assign the shell charges
code
type
CHE
0
S
2.
CHE to S functions
1
SP
8.
if CHE>2, 2 to S and (CHE-2) to P functions,
if CHE2, CHE to S function
2
P
6.
CHE to P functions
3
D
10.
CHE to D functions
4
F
14.
CHE to F functions
5
G
18.
CHE to G functions
7. A maximum of one open shell for each of the s, p and or d atomic symmetries is allowed
in the electronic configuration defined in the input. The atomic energy expression is not
correct for all possible double open shell couplings of the form pmdn. Either m must
equal 3 or n must equal 5 for a correct energy expression in such cases. A warning
will be printed if this is the case. However, the resultant wave function (which is a
superposition of atomic densities) will usually provide a reasonable starting point for the
periodic density matrix.
8. When extended basis sets are used, all the functions corresponding to symmetries (an-
gular quantum numbers) occupied in the isolated atom are added to the atomic basis
set for atomic wave function calculations, even if the formal charge attributed to that
shell is zero. Polarization functions are not included in the atomic basis set; their input
occupation number should be zero.
9. The formal shell charges are used only to define the electronic configuration of the atoms
to compute the atomic wave function. The initial density matrix in the scf step may be
built as a superposition of atomic (or ionic) density matrices (default option, GUESS-
PAT 2.4). When a different guess is required (GUESSF or GUESSP), the shell charges
are not used, but checked for electron neutrality when the basis set is entered.
10. Each atom in the cell may have an ionic configuration, when the sum of formal shell
charges (CHE) is different from the nuclear charge. When the number of electrons in
the cell, that is the sum of the shell charges CHE of all the atoms, is different from the
sum of nuclear charges, the reference cell is non-neutral. This is not allowed for periodic
systems, and in that case the program stops. In order to remove this constraint, it is
necessary to introduce a uniform charged background of opposite sign to neutralize the
system [24]. This is obtained by entering the keyword CHARGED (page 48) after the
standard basis set input.
18

11. It may be useful to allow atoms with the same basis set to have different electronic
configurations (e.g, for an oxygen vacancy in MgO one could use the same basis set for
all the oxygens, but begin with different electronic configuration for those around the
vacancy). The formal shell charges attributed in the basis set input may be modified for
selected atoms by inserting the keyword CHEMOD (input block 2, page 48).
12. The energies given by an atomic wave function calculation with a crystalline basis set
should not be used as a reference to calculate the formation energies of crystals. The
external shells should first be re-optimized in the isolated atom by adding a low-exponent
Gaussian function, in order to provide and adequate description of the tails of the isolated
atom charge density [25] (keyword ATOMHF, input block 3, page 57).
Optimized basis sets for periodic systems used in published papers are available on WWW:
http://www.crystal.unito.it
1.3
General information, computational parameters,
hamiltonian
No input is required if the default values are used. Note however that END/ENDM or
STOP, to close the section, are always needed. If no Hamiltonian is specified, RHF (Restricted
Hartree-Fock Hamiltonian) is assumed.
1.4
SCF input
rec variable value
meaning
if the system is periodic insert
II
IS
Shrinking factor in reciprocal space (Section 6.7, page 145)
IDUM
not used.
ISP
Shrinking factor for a denser k point net (Gilat net) in the For peri-
evaluation of the Fermi energy and density matrix.
if IS = 0 insert
II
IS1,IS2,IS3
Shrinking factors along B1,B2,B3 (reciprocal lattice vectors);
to be used when the unit cell is highly anisotropic
optional keywords terminated by END or STOP
II
odic systems, 1D, 2D, 3D, the mandatory input information is the shrinking factor, IS, to
generate a commensurate grid of k points in reciprocal space, according to Pack-Monkhorst
method. The Hamiltonian matrix computed in direct space, Hg, is Fourier transformed for
each k value, and diagonalized, to obtain eigenvectors and eigenvalues:
Hk =
Hgeigk
g
HkAk = SkAkEk
A second shrinking factor, ISP, defines the sampling of k points, i "Gilat net", used for the cal-
culation of the density matrix and the determination of Fermi energy in the case of conductors,
when bands are not fully occupied.
In 3D crystals, the sampling points belong to a lattice i (called the Pack-Monkhorst net), with
basis vectors:
b1/is1, b2/is2, b3/is3
is1=is2=is3=IS, unless otherwise stated
where b1, b2, b3 are the reciprocal lattice vectors, and is1, is2, is3 are integers "shrinking
factors".
19

In 2D crystals, IS3 is set equal to 1; in 1D crystals both IS2 and IS3 are set equal to 1. Only
points ki of the Pack-Monkhorst net belonging to the irreducible part of the Brillouin Zone
(IBZ) are considered, with associated a geometrical weight, wi. The choice of the reciprocal
space integration parameters to compute the Fermi energy is a delicate step for metals. See
Section 6.7, page 145. Two parameters control the accuracy of reciprocal space integration for
Fermi energy calculation and density matrix reconstruction:
IS shrinking factor of reciprocal lattice vectors.
The value of IS determines the number
of k points at which the Fock/KS matrix is diagonalized. Multiples of 2 or 3 should
be used, according to the point symmetry of the system (order of principal axes). The
k-points net is automatically made anisotropic for 1D and 2D systems.
The figure presents the reciprocal lattice cell of 2D graphite (rhombus), the first
Brillouin zone (hexagon), the irreducible part of Brillouin zone (in gray), and the
coordinates of the ki points according to a Monkhorst-Pack sampling, with shrinking
factor 3 and 6.
ISP shrinking factor of reciprocal lattice vectors in the Gilat net (see [7], Chapter II.6). ISP
is used in the calculation of the Fermi energy and density matrix. Its value can be equal
to IS for insulating systems and equal to 2*IS for conducting systems.
Note. The value used in the calculation is ISP=IS*NINT(MAX(ISP,IS)/IS)
1. When an anisotropic net is user defined (IS=0), the ISP input value is taken as ISP1
(shrinking factor of Gilat net along first reciprocal lattice) and ISP2 and ISP3 are set to:
ISP2=(ISP*IS2)/IS1,
ISP3=(ISP*IS3)/IS1.
2. User defined anisotropic net is not compatible with SABF (Symmetry Adapted Bloch
Functions). See NOSYMADA, page 78.
Some tools for accelerating convergence are given through the keywords LEVSHIFT (page
77 and tests 29, 30, 31, 32, 38), FMIXING (page 75), SMEAR (page 79), BROYDEN
(page 73) and ANDERSON (page 72).
At each SCF cycle the total atomic charges, following a Mulliken population analysis scheme,
and the total energy are printed.
The defaulty value of the parameters to control the exit from the SCF cycle (E < 10-6
hartree, maximum number of SCF cycles: 50) may be modified entering the keywords:
TOLSCF (tolerance on change in eigenvalues and total energy),
TOLENE (tolerance on change in total energy),
TOLDEP (tolerance on SQM in density matrix elements),
MAXCYCLE (maximum number of cycles).
20

Spin-polarized system
By default the orbital occupancies are controlled according to the 'Aufbau' principle.
To obtain a spin polarized solution an open shell Hamiltonian must be defined (block3, UHF
or DFT/SPIN). A spin-polarized solution may then be computed after definition of the (-)
electron occupancy. This can be performed by the keywords SPINLOCK and BETALOCK.
21

Chapter 2
Wave function calculation -
Advanced input route
2.1
Geometry editing
The following keywords allow editing of the crystal structure, printing of extended information,
generation of input data for visualization programs. Processing of the input block 1 only
(geometry input) is allowed by the keyword TESTGEOM.
Each keyword operates on the geometry active when the keyword is entered. For instance, when
a 2D structure is generated from a 3D one through the keyword SLABCUT, all subsequent
geometry editing operates on the 2D structure. When a dimer is extracted from a molecular
crystal through the keyword MOLECULE, all subsequent editing refers to a system without
translational symmetry.
The keywords can be entered in any order: particular attention should be paid to the action
of the keywords KEEPSYMM and BREAKSYM, that allow maintaining or breaking the
symmetry while editing the structure. These keywords behave as a switch, and require no
further data. Under control of the BREAKSYM keyword (the default), subsequent mod-
ifications of the geometry are allowed to alter (reduce: the number of symmetry operators
cannot be increased) the point-group symmetry. The new group is a subgroup of the original
group and is automatically obtained by CRYSTAL. However if a KEEPSYMM keyword
is presented, the program will endeavor to maintain the number of symmetry operators, by
requiring that atoms which are symmetry related remain so after a geometry editing (key-
words:ATOMSUBS, ATOMINSE, ATOMDISP, ATOMREMO).
The space group of the system may be modified after editing.
For 3D systems,
the file FINDSYM.DAT is written.
This file is input to the program findsym
(http://physics.byu.edu/ stokesh/isotropy.html), that finds the space-group symmetry of a
crystal given the coordinates of the atoms.
Geometry keywords
Symmetry information
ATOMSYMM
printing of point symmetry at the atomic positions
27

MAKESAED
printing of symmetry allowed elastic distortions (SAED)
32

PRSYMDIR
printing of displacement directions allowed by symmetry.
35

SYMMDIR
printing of symmetry allowed geom opt directions
40

SYMMOPS
printing of point symmetry operators
40

TENSOR
tensor of physical properties
40
I
Symmetry information and control
22

BREAKSYM
allow symmetry reduction following geometry modifications
28

KEEPSYMM
maintain symmetry following geometry modifications
32

MODISYMM
removal of selected symmetry operators
32
I
PURIFY
cleans atomic positions so that they are fully consistent with the 35

group
SYMMREMO
removal of all symmetry operators
40

TRASREMO
removal of symmetry operators with translational components
40

Modifications without reduction of symmetry
ATOMORDE
reordering of atoms in molecular crystals
25

NOSHIFT
no shift of the origin to minimize the number of symmops with 34

translational components before generating supercell
ORIGIN
shift of the origin to minimize the number of symmetry operators 34

with translational components
PRIMITIV
crystallographic cell forced to be the primitive cell
35

REDEFINE
definition of a new cell, with xy
to a given plane
36
I
Atoms and cell manipulation (possible symmetry reduction (BREAKSYMM)
ATOMDISP
displacement of atoms
25
I
ATOMINSE
addition of atoms
25
I
ATOMREMO
removal of atoms
25
I
ATOMROT
rotation of groups of atoms
26
I
ATOMSUBS
substitution of atoms
27
I
ELASTIC
distortion of the lattice
30
I
USESAED
given symmetry allowed elastic distortions, reads
41
I
SUPERCEL
generation of supercell - input refers to primitive cell
38
I
SUPERCON
generation of supercell - input refers to conventional cell
38
I
From crystals to slabs
SLABCUT
generation of a slab parallel to a given plane (3D2D)
37
I
From periodic structure to clusters
CLUSTER
cutting of a cluster from a periodic structure (3D0D)
28
I
HYDROSUB
border atoms substituted with hydrogens (0D0D)
31
I
Molecular crystals
MOLECULE
extraction of a set of molecules from a molecular crystal 33
I
(3D0D)
MOLEXP
variation of lattice parameters at constant symmetry and molec- 33
I
ular geometry (3D3D)
MOLSPLIT
periodic structure of non interacting molecules (3D3D)
34

RAYCOV
modification of atomic covalent radii
35
I
BSSE correction
MOLEBSSE
counterpoise method for molecules (molecular crystals only) 32
I
(3D0D)
ATOMBSSE
counterpoise method for atoms (3D0D)
25
I
Auxiliary and control keywords
23

ANGSTROM
sets inputs unit to
Angstrom
24

BOHR
sets input units to bohr
27

BOHRANGS
input bohr to
A conversion factor (0.5291772083 default value) 27
I
BOHRCR98
bohr to
A conversion factor is set to 0.529177 (CRYSTAL98
value)
END/ENDG
terminate processing of geometry

FRACTION
sets input unit to fractional
31

NEIGHBOR
number of neighbours in geometry analysis
34
I
PARAMPRT
printing of parameters controlling dimensions of static allocation 35

arrays
PRINTOUT
setting of printing options by keywords
35
I
SETINF
setting of inf array options
36
I
SETPRINT
setting of printing options
37
I
STOP
execution stops immediately
38

TESTGEOM
stop after checking the geometry
40

Output of data on external units
COORPRT
coordinates of all the atoms in the cell
29

EXTPRT
generation of file as CRYSTAL input
31

MOLDRAW
generation of file for the program MOLDRAW
32

STRUCPRT
cell parameters and coordinates of all the atoms in the cell
38

External electric field - modified Hamiltonian
FIELD
external field applied (2D-3D systems only)
41
I
Geometry optimization
OPTCOORD
Atom coordinates optimization
43
I
Initial Hessian
HESGUESS
initial guess for the Hessian
I
Convergence criteria
TOLDEG
RMS of the gradient [0.0003]
I
TOLDEX
RMS of the displacement [0.0012]
I
TOLDEE
energy difference between two steps [10-7]
I
MAXOPTC
max number of optimization steps
I
Optimization control
ATOMFREE
partial geometry optimization
I
RESTART
data from previous run

FINALRUN
Wf single point with optimized geometry
I
Gradient calculation control
NUMGRAD
numerical first deivatives

Printing options
PRINTFORCES atomic gradients

PRINTHESS
Hessian

PRINTOPT
optimization procedure

PRINT
verbose printing

ANGSTROM - unit of measure
The unit of length in geometry editing is set to
Angstrom, (default value).
24

ATOMBSSE - counterpoise for closed shell atoms and ions
rec variable meaning
IAT
label of the atom in the reference cell
NSTAR maximum number of stars of neighbors included in the calculation.
RMAX
maximum distance explored searching the neighbors of the atom.
A cluster is defined including the selected atom and the basis functions belonging to the NSTAR
sets of neighbours, when their distance R from the central atom is smaller than RMAX. The
atomic wave function is not computed by the atomic package, but by the standard CRYSTAL
route for 0D, 1 atom system. UHF and SPINLOCK must be used to define a reasonable
orbital occupancy. It is suggested to compute the atomic wave function using a program
properly handling the electronic configuration of open shell atoms.
Warning. The system is 0D. No reciprocal lattice information is required in the scf input
(Section 1.4, page 19).
ATOMDISP
rec variable
meaning
NDISP
number of atoms to be displaced
insert NDISP records
II
LB
label of the atom to be moved
DX,DY,DZ
increments of the coordinates in the primitive cell [
A].
Selected atoms are displaced in the primitive cell. The point symmetry of the system may be
altered (default value BREAKSYM, page 28). Increments are in
Angstrom, unless otherwise
requested (keyword BOHR, FRACTION, page 24). See tests 17, 20, 37.
ATOMINSE
rec variable
meaning
NINS
number of atoms to be added
insert NINS records
II
NA
conventional atomic number
X,Y,Z
coordinates [
A] of the inserted atom. Coordinates refer to the primitive cell.
New atoms are added to the primitive cell. Coordinates are in
Angstrom, unless otherwise
requested (keyword BOHR, FRACTION, page 24). Remember that the original symmetry
of the system is maintained, applying the symmetry operators to the added atoms if the
keyword KEEPSYMM (page 28) was previously entered. The default is BREAKSYM
(page 28). Attention should be paid to the neutrality of the cell (see CHARGED, page 48).
See tests 16, 35, 36.
ATOMORDE
After processing the standard geometry input, the symmetry equivalent atoms in the reference
cell are grouped. They may be reordered, following a chemical bond criterion. This simplifies
the interpretation of the output when the results of bulk molecular crystals are compared with
those of the isolated molecule. See option MOLECULE (page 33) and MOLSPLIT (page
34). No input data are required.
ATOMREMO
rec variable
meaning
NL
number of atoms to remove
LB(L),L=1,NL label of the atoms to remove
25

Selected atoms, and related basis set, are removed from the primitive cell (see test 16). A
vacancy is created in the lattice. The symmetry can be maintained (KEEPSYMM), by
removing all the atoms symmetry-related to the selected one, or reduced (BREAKSYM).
Attention should be paid to the neutrality of the cell (see CHARGED, page 48).
NB. The keyword GHOSTS (basis set input, page 50) allows removal of selected atoms,
leaving the related basis set.
ATOMROT
rec variable value
meaning
NA
0
all the atoms of the cell are rotated and/or translated
>0
only NA selected atoms are rotated and/or translated.
<0
the atom with label |NA| belongs to the molecule to be rotated. The
program selects all the atoms of the molecule on the base of the sum of
their atomic radii (Table on page 35).
if NA > 0, insert NA data
II

LB(I),I=1,NA label of the atoms to be rotated and/or translated.
ITR
>0
translation performed. The selected NA atoms are translated by -r,
where r is the position of the ITR-th atom. ITR is at the origin after
the translation.
0
a general translation is performed. See below.
=999
no translation.
IRO
> 0
a rotation around a given axis is performed. See below.
< 0
a general rotation is performed. See below.
=999
no rotation.
if ITR<0 insert
II
X,Y,Z
Cartesian components of the translation vector [
A]
if ITR=0 insert
II
N1,N2
label of the atoms defining the axis.
DR
translation along the axis defined by the atoms N1 and N2, in the di-
rection N1 N2 [
A].
if IRO<0 insert
II
A,B,G
Euler rotation angles (degree).
IPAR
defines the origin of the Cartesian system for the rotation
0
the origin is the barycentre of the NAT atoms
>0
the origin is the atom of label IPAR
if IRO>0 insert
II
N1,N2
label of the atoms that define the axis for the rotation
ALPHA = 0.
rotation angle around the N1N2 axis (degrees)
0.
the selected atoms are rotated anti-clockwise in order to orientate the
N1N2 axis parallel to the z axis.
This option allows to rotate and/or translate the specified atoms. When the rotation of a
molecule is required (NA < 0), the value of the atomic radii must be checked, in order to
obtain a correct definition of the molecule. It is useful to study the conformation of a molecule
in a zeolite cavity, or the interaction of a molecule (methane) with a surface (MgO).
The translation of the selected group of atoms can be defined in three different ways:
1. Cartesian components of the translation vector (ITR < 0);
2. modulus of the translation vector along an axis defined by two atoms (ITR = 0);
3. sequence number of the atom to be translated to the origin. All the selected atoms are
subjected to the same translation (ITR > 0).
The rotation can be performed in three different ways:
26

1. by defining the Euler rotation angles , , and the origin of the rotating system (IRO
< 0). The axes of the rotating system are parallel to the axes of the Cartesian reference
system. (The rotation is given by: RzRxRz, where R are the rotation matrices).
2. by defining the rotation angle around an axis defined by two atoms A and B. The
origin is at A, the positive direction AB.
3. by defining a z' axis (identified by two atoms A and B). The selected atoms are rotated,
in such a way that the AB z' axis becomes parallel to the z Cartesian axis. The origin
is at A and the positive rotation anti clockwise (IRO>0, =0).
The selected atoms are rotated according to the defined rules, the cell orientation and the
cartesian reference frame are not modified. The symmetry of the system is checked after the
rotation, as the new geometry may have a different symmetry.
See tests 15, rotation of the N H3 molecule in a zeolite cavity, and 16, rotation of the H2O
molecule in the zeolite cavity.
ATOMSUBS
rec variable
meaning
NSOST
number of atoms to be substituted
insert NSOST records
II
LB
label of the atom to substitute
NA(LB)
conventional atomic number of the new atom
Selected atoms are substituted in the primitive cell (see test 17, 34, 37). The symmetry can be
maintained (KEEPSYMM), by substituting all the atoms symmetry-related to the selected
one, or reduced (BREAKSYM). Attention should be paid to the neutrality of the cell: a non-
neutral cell will cause an error message, unless allowed by entering the keyword CHARGED,
page 48.
ATOMSYMM
The point group associated with each atomic position and the set of symmetry related atoms
are printed. No input data are required. This option is useful to find the internal coordinates
to be relaxed when the unit cell is deformed (see ELASTIC, page 30).
BOHR
The keyword BOHR sets the unit of distance to bohr. When the unit of measure is modified,
the new convention is active for all subsequent geometry editing.
The conversion factor
Angstrom/bohr is 0.5291772083 (CODATA 1998). This value can be
modified by entering the keyword BOHRANGS and the desired value in the record following.
The keyword BOHRCR98 sets the conversion factor to 0.529177, as in the program CRYS-
TAL98.
CRYSTAL88 default value was 0.529167).
BOHRANGS
rec variable
meaning
BOHR
conversion factor
Angstrom/bohr
The conversion factor
Angstrom/bohr can be user-defined.
In CRYSTAL88 the default value was 0.529167.
In CRYSTAL98 the default value was 0.529177.
27

BOHRCR98
The conversion factor
Angstrom/bohr is set to 0.529177, as in CRYSTAL98. No input data
required.
BREAKSYM
Under control of the BREAKSYM keyword (the default), subsequent modifications of the
geometry are allowed to alter (reduce: the number of symmetry operators cannot be increased)
the point-group symmetry. The new group is a subgroup of the original group and is automat-
ically obtained by CRYSTAL.
The symmetry may be broken by attributing different spin (ATOMSPI, block4, page 73) to
atoms symmetry related by geometry.
Example: When a CO molecule is vertically adsorbed on a (001) 3-layer MgO slab, (D4h
symmetry), the symmetry is reduced to C4v, if the BREAKSYM keyword is active. The
symmetry operators related to the h plane are removed. However, if KEEPSYMM is
active, then additional atoms will be added to the underside of the slab so as to maintain the
h plane (see page 25, keyword ATOMINSE).
CLUSTER - a cluster (0D) from a periodic system
The CLUSTER option allows one to cut a finite molecular cluster of atoms from a periodic
lattice. The size of the cluster (which is centred on a specified 'seed point' A) can be controlled
either by including all atoms within a sphere of a given radius centred on A, or by specifying
a maximum number of symmetry-related stars of atoms to be included.
The cluster includes the atoms B (belonging to different cells of the direct lattice) satisfying
the following criteria:
1. those which belong to one of the first N (input data) stars of neighbours of the seed point
of the cluster.
and
2. those at a distance RAB from the seed point which is smaller then RMAX (input datum).
The resulting cluster may not reproduce exactly the desired arrangement of atoms, particularly
in crystals with complex structures such as zeolites, and so it is possible to specify border
modifications to be made after definition of the core cluster.
Specification of the core cluster:
rec variable value
meaning
X, Y, Z
coordinates of the centre of the cluster [
A] (the seed point)
NST
maximum number of stars of neighbours explored in defining the core
cluster
RMAX
radius of a sphere centred at X,Y,Z containing the atoms of the core
cluster
NNA
= 0
print nearest neighbour analysis of cluster atoms (according to a radius
criterion)
NCN
0
testing of coordination number during hydrogen saturation carried out
only for Si (coordination number 4), Al (4) and O(2)
N
N user-defined coordination numbers are to be defined
if NNA = 0 insert 1 record
II
RNNA
radius of sphere in which to search for neighbours of a given atom in
order to print the nearest neighbour analysis
if NCN = 0 insert NCN records
II
L
conventional atomic number of atom
MCONN(L)
coordination number of the atom with conventional atomic number L.
MCONN=0, coordination not tested
28

Border modification:
rec variable value
meaning
NMO
number of border atoms to be modified
if NMO > 0 insert NMO records
II
IPAD
label of the atom to be modified (cluster sequence)
NVIC
number of stars of neighbours of atom IPAD to be added to the cluster
IPAR
= 0
no hydrogen saturation
= 0
cluster border saturated with hydrogen atoms
BOND
bond length Hydrogen-IPAD atom (direction unchanged).
if NMO < 0 insert
II
IMIN
label of the first atom to be saturated (cluster sequence)
IMAX
label of the last atom to be saturated (cluster sequence)
NVIC
number of stars of neighbours of each atom to be added to the cluster
IPAR
= 0
no hydrogen saturation
= 0
cluster border saturated with hydrogen atoms
BOND
H-cluster atom bond length (direction unchanged).
The two kinds of possible modification of the core cluster are (a) addition of further stars of
neighbours to specified border atoms, and (b) saturation of the border atoms with hydrogen.
This latter option can be essential in minimizing border electric field effects in calculations for
covalently-bonded systems.
(Substitution of atoms with hydrogen is obtained by HYDROSUB).
The hydrogen saturation procedure is carried out in the following way. First, a coordination
number for each atom is assumed (by default 4 for Si, 4 for Al and 2 for O, but these may
be modified in the input deck for any atomic number). The actual number of neighbours of
each specified border atom is then determined (according to a covalent radius criterion) and
compared with the assumed connectivity. If these two numbers differ, additional neighbours are
added. If these atoms are not neighbours of any other existing cluster atoms, they are converted
to hydrogen, otherwise further atoms are added until the connectivity allows complete hydrogen
saturation whilst maintaining correct coordination numbers.
The label of the IPAD atoms refers to the generated cluster, not to the original unit cell. The
preparation of the input thus requires two runs:
1. run using the CLUSTER option with NMO=0, in order to generate the sequence number
of the atoms in the core cluster. The keyword TESTGEOM should be inserted in the
geometry input block. Setting NNA = 0 in the input will print a coordination analysis of
all core cluster atoms, including all neighbours within a distance RNNA (which should
be set slightly greater than the maximum nearest neighbour bond length). This can be
useful in deciding what border modifications are necessary.
2. run using the CLUSTER option with NMO = 0, to perform desired border modifica-
tions.
Note that the standard CRYSTAL geometry editing options may also be used to modify the
cluster (for example by adding or deleting atoms) placing these keywords after the specification
of the CLUSTER input.
Warning. The system is 0D. No reciprocal lattice information is required in the scf input
(Section 1.4, page 19). See test 16.
COORPRT
Geometry information is printed: cell parameters, fractionary coordinates of all atoms in the
reference cell, symmetry operators.
A formatted file is written (in append mode) in fortran unit 33. See Appendix F, page 181.
No input data are required.
Fortran unit 33 has the right format for the program Xmol [26].
Download from http://biotech.icmb.utexas.edu/mime/xmol.html
29

or by the program MOLDEN [27] which can be downloaded from:
www.cmbi.kun.nl/ schaft/molden/molden.html
ELASTIC
An elastic deformation of the lattice may be defined in terms of the Z or
strain tensors defined
in section 6.9, page 147.
rec variable
value
meaning
IDEF
1
deformation through equation 6.36, Z matrix.
2
deformation through equation 6.35:
matrix.
> 0
volume conserving deformation (equation 6.37).
< 0
not volume conserving (equation 6.36 or 6.35).
D11 D12 D13
first row of the matrix.
D21 D22 D23
second row of the matrix.
D31 D32 D33
third row of the matrix.
The elastic constant is V -1 2E |
2
i =0 , where V
is the volume of the primitive unit cell.
i
The symmetry of the system is defined by the symmetry operators in the new crystallographic
cell. The keyword MAKESAED gives information on symmetry allowed elastic distortions.
The calculation of the elastic constants with CRYSTAL requires the following sequence of
steps:
1. select the ij matrix elements to be changed ( for example, 4 23 + 32), and set the
others j to zero;
2. perform calculations with different values of the selected matrix element(s) i: 0.02, 0.01,
0.001, -0.001, -0.01, -0.02, for example, and for each value compute the total energy E;
3. perform a polynomial fit of E as a function of i.
is adimensional, Z in
A(default) or in bohr (page 24). The suggested value for IDEF is
-2 (deformation through equation 6.35, not volume conserving). The examples refer to this
setting.
Example
Geometry input deck to compute one of the energy points used for the evaluation of the C44
(page 150) elastic constants of Li2O [28].
CRYSTAL
0 0 0
3D code
225
3D space group number
4.5733
lattice parameter (
A)
2
2 non equivalent atoms in the primitive cell
8 0.0 0.0 0.0
Z=8, Oxygen; x, y, z
3 .25 .25 .25
Z=3, Lithium; x, y, z
ATOMSYMM
printing of the point group at the atomic positions
ELASTIC
-2
deformation not volume conserving through equation 6.35
0. 0.03 0.03
matrix input by rows
0.03 0. 0.03
0.03 0.03 0.
ATOMSYMM
printing of the point group at the atomic positions after the defor-
mation
. . . . . . .
A rhombohedral deformation is obtained, through the
matrix. The printout gives information
on the crystallographic and the primitive cell, before and after the deformation:
LATTICE PARAMETERS (ANGSTROMS AND DEGREES) OF
(1) ORIGINAL PRIMITIVE CELL
30

(2) ORIGINAL CRYSTALLOGRAPHIC CELL
(3) DEFORMED PRIMITIVE CELL
(4) DEFORMED CRYSTALLOGRAPHIC CELL
A
B
C
ALPHA
BETA
GAMMA
VOLUME
(1)
3.233811
3.233811
3.233811
60.000000
60.000000
60.000000
23.912726
(2)
4.573300
4.573300
4.573300
90.000000
90.000000
90.000000
95.650903
(3)
3.333650
3.333650
3.333650
56.130247
56.130247
56.130247
23.849453
(4)
4.577414
4.577414
4.577414
86.514808
86.514808
86.514808
95.397811
After the deformation of the lattice, the point symmetry of the Li atoms is C3v, where the C3
axis is along the (x,x,x) direction. The Li atoms can be shifted along the principal diagonal,
direction (x,x,x) of the primitive cell without altering the point symmetry, as shown by the
printing of the point group symmetry obtained by the keyword ATOMSYMM (page 27).
See test 20 for complete input deck, including shift of the Li atoms.
END
Terminate processing of block 1, geometry definition, input. Execution continues. Subsequent
input records are processed, if required.
EXTPRT
A formatted input deck with explicit structural/symmetry information is written in fortran
unit 34. If the keyword is entered many times, the data are overwritten. The last geometry is
recorded. The deck may be used as input of the crystal geometry to CRYSTAL through the
EXTERNAL keyword (final optimized geometry, geometry obtained by editing who modified
the original space group). See Appendix F, page 181. No input data are required.
FRACTION
The keyword FRACTION means input coordinates given as fraction of the lattice parameter
in subsequent input, along the direction of translational symmetry:
x,y,z
crystals (3D)
x,y
slabs (2D; z in
Angstrom or bohr)
x
polymers (1D; y,z in
Angstrom or bohr)
no action for 0D.
When the unit of measure is modified, the new convention is active for all subsequent geometry
editing.
HYDROSUB - substitution with hydrogen atoms
rec variable
meaning
NSOST
number of atoms to be substituted with hydrogen
insert NSOST records
II
LA
label of the atom to substitute
LB
label of the atom linked to LA
BH
bond length B-Hydrogen
Selected atoms are substituted with hydrogens, and the bond length is modified. To be used
after CLUSTER.
31

KEEPSYMM
n any subsequent editing of the geometry, the program will endeavour to maintain the number
of symmetry operators, by requiring that atoms which are symmetry related remain so after
geometry editing (keywords:ATOMSUBS, ATOMINSE, ATOMDISP, ATOMREMO)
or the basis set (keywords CHEMOD, GHOSTS).
Example: When a CO molecule is vertically adsorbed on a (001) 3-layer MgO slab, (D4h
symmetry) (see page 25, keyword ATOMINSE), the symmetry is reduced to C4v, if the
BREAKSYM keyword is active. The symmetry operators related to the h plane are re-
moved. However, if KEEPSYMM is active, then additional atoms will be added to the
underside of the slab so as to maintain the h plane.
MAKESAED
This generates symmetry allowed elastic distortions. No input data are required.
MODISYMM
rec variable
meaning
N
number of atoms to be attached a flag
LA,LF(LA),L=1,N atom labels and flags (n couples of integers in 1 record).
The point symmetry of the lattice is lowered by attributing a different "flag" to atoms related
by geometrical symmetry. The symmetry operators linking the two atoms are removed and the
new symmetry of the system is analyzed. For instance, when studying spin-polarized systems, it
may be necessary to apply different spins to atoms which are related by geometrical symmetry.
MOLDRAW
A formatted input deck for the visualization program MOLDRAW [29] is written on fortran
unit 93. If the keyword is entered many times, the data are overwritten. The last geometry
can be visualized.
No input data are required. See:
http://www.moldraw.unito.it .
MOLEBSSE - counterpoise for molecular crystals
rec variable meaning
NMOL
number of molecules to be isolated
insert NMOL records
ISEED
label of one atom in the n-th molecule
J,K,L
integer coordinates (direct lattice) of the primitive cell containing the ISEED
atom
NSTAR maximum number of stars of neighbours included in the calculation
RMAX
maximum distance explored searching the neighbours of the atoms belonging
to the molecule(s)
The counterpoise method is applied to correct the Basis Set Superposition Error in molecular
crystals. A molecular calculation is performed, with a basis set including the basis functions
of the selected molecules and the neighbouring atoms. The program automatically finds all
the atoms of the molecule(s) containing atom(s) ISEED (keyword MOLECULE, page 33).
The molecule is reconstructed on the basis of the covalent radii reported in Table on page 35.
32

They can be modified by running the option RAYCOV, if the reconstruction of the molecule
fails. The radius of the hydrogen atom is very critical when intermolecular hydrogen bonds
are present.
All the functions of the neighbouring atoms in the crystal are added to the basis set of the
selected molecule(s) such that both the following criteria are obeyed:
1. the atom is within a distance R lower than RMAX from at least one atom in the molecule
and
2. the atom is within the NSTAR-th nearest neighbours of at least one atom in the molecule.
Warning. The system obtained is 0D. No reciprocal lattice information is required in the scf
input (Section 1.4, page 19). See test 19.
MOLECULE - Extraction of n molecules from a molecular crystal
rec variable meaning
NMOL
number of molecules to be isolated
insert NMOL records
II
ISEED
label of one atom in the nth molecule
J,K,L
integer coordinates (direct lattice) of the primitive cell containing the
ISEED atom
The option MOLECULE isolates one (or more) molecules from a molecular crystal on the
basis of chemical connectivity, defined by the sum of the covalent radii (Table on page 35).
The program stops after printing full neighbouring analysis of the non-equivalent atoms, up to
n neighbours (default value 3; keyword NEIGHBOR, page 34 to modify it).
The input order of the atoms (atoms symmetry related are grouped) is modified, according
to the chemical connectivity. The same order of the atoms in the bulk crystal is obtained by
entering the keyword ATOMORDE (see Section 2.1, page 25). The total number of electrons
attributed to the molecule is the sum of the shell charges attributed in the basis set input (input
block 2, Section 1.2, page 16).
The keyword GAUSS98, entered in input block 2 (basis set input), writes an input deck to
run Gaussian 98 (see page 49)
Warning. The system is 0D. No reciprocal lattice information is required in the scf input
(Section 1.4, page 19).
Test 18 - Oxalic acid. In the 3D unit cell there are four water and two oxalic acid molecules.
The input of test 18 refers to a cluster containing a central oxalic acid molecule surrounded by
four water molecules.
MOLEXP - Variation of lattice parameters at constant symmetry
and molecular geometry
rec
variable
meaning

a,[b],[c], increments of the minimal set of crystallographic cell parameters:
[],[]
translation vectors length [
Angstrom],
[]
crystallographic angles (degrees)
The cell parameters (the minimum set, see page 13) are modified, according to the increments
given in input. The volume of the cell is then modified. The symmetry of the lattice and the
geometry (bond lengths and bond angles) of the molecules within the cell is kept. The fractional
33

coordinates of the barycentre of the molecules are kept constant, the cartesian coordinates
redefined according to the modification of the lattice parameters. Optimization of the geometry
with reference to the compactness of the lattice is allowed, keeping constant the geometry of
the molecules. When there are very short hydrogen bonds linking the molecules in the lattice,
it may be necessary a modification of the atomic radii to allow proper identification of the
molecules (see option RAYCOV, page 35)
MOLSPLIT - Periodic lattice of non-interacting molecules
In order to compare bulk and molecular properties, it can be useful to build a density ma-
trix as a superposition of the density matrices of the isolated molecules, arranged in the same
geometry as in the crystal. The keyword MOLSPLIT (no additional input required) per-
forms an expansion of the lattice, in such a way that the molecules of the crystal are at an
"infinite" distance from each other. The crystal coordinates are scaled so that the distances
inside the molecule are fixed, and the distances among the molecules are expanded by a factor
100, to avoid molecule-molecule interactions. The 3D translational symmetry is not changed.
Reciprocal lattice information is required in the scf input (Section 1.4, page 19).
A standard wave function calculation of the expanded crystal is performed. The density matrix
refers to the non-interacting subsystems. Before running properties, the lattice is automatically
contracted to the bulk situation given in input. If a charge density or electrostatic potential
map is computed (ECHG, POTM options), it corresponds to the superposition of the charge
densities of the isolated molecules in the bulk geometry.
This option must be used only for molecular crystals only (no charged fragments).
See test 21.
NEIGHBOR/NEIGHPRT
rec
variable meaning

INEIGH number of neighbours of each non-equivalent atom to be printed
The option is active when analyzing the crystal structure (bond lengths and bond angles) and
when printing the bond populations following Mulliken analysis. Full input deck must be given
(block 1-2-3-4),in order to obtain neighbors analysis of all the non-equivalent atoms
For each non-equivalent atom information on the first INEIGH neighbours is printed: number,
type, distance, position (indices of the direct lattice cell).
Warning: the neighbors analysis is performed after the symmetry analysis and the screening
of the integrals. If very soft tolerances for the integrals screening are given in input, it may
happen that the information is not given for all the neighbors requested, as their are not taken
into account when truncation criteria are applied.
NOSHIFT
To be used before SUPERCEL keyword. It avoids shift of the origin in order to minimize the
number of symmetry operators with finite translation component. No input data are required.
ORIGIN
The origin is moved to minimize the number of symmetry operators with finite translation
components. Suggested before cutting a slab from a 3D structure (option SLABCUT, page
37) No input data are required.
34

PARAMPRT - printing of parametrized dimensions
The parameters controlling the dimensions of the static allocation arrays of the program are
printed.
No input data are required.
PRIMITIV
Some properties (XFAC, EMDL, EMDP, PROF) input the oblique coordinates of the k
points in the reciprocal lattice with reference to the conventional cell, though the computation
refers to the primitive one. This option allows entering directly the data with reference to the
primitive cell. The transformation matrix from primitive to crystallographic (Appendix A.5,
page 163) is set to the identity. No effect on the CPU time: CRYSTAL always refers to the
primitive cell. No input data are required.
PRINTOUT - Setting of printing environment
Extended printout can be obtained by entering selected keywords in a printing environment
beginning with the keyword PRINTOUT and ending with the keyword END. The possible
keywords are found in the fifth column of the table on page 179.
Extended printing request can be entered in any input block. Printing requests are not trans-
ferred from wave function to properties calculation.
See Appendix E, page 177.
PRSYMDIR
Printing of displacement directions allowed by symmetry. The printing is done after the neigh-
bor analysis, before computing the wave function. Full input must be supplied (4 blocks). Test
run allowed with the keyword TESTPDIM.
No input data required.
PURIFY
This cleans up the atomic positions so that they are fully consistent with the group (to within
machine rounding error). No input data are required.
RAYCOV - covalent radii modification
rec variable meaning
NCOV
number of atoms for which the covalent radius is redefined
insert NCOV records
II
NAT
atomic number (0 NAT 92)
RAY
covalent radius of the atom with atomic number NAT ([
A], default,
or bohr, if the keyword BOHR precedes in the deck)
The option RAYCOV allows modification of the covalent radius default value for a given
atom.
35

Table of covalent radii (Angstrom)
H
He
0.68
1.47
---------
-----------------------------
Li
Be
B
C
N
O
F
Ne
1.65 1.18
0.93 0.81 0.78 0.78 0.76 1.68
---------
-----------------------------
Na
Mg
Al
Si
P
S
Cl
Ar
2.01 1.57
1.50 1.23 1.15 1.09 1.05 0.97
-----------------------------------------------------------------------------------------
K
Ca
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
Ga
Ge
As
Se
Br
Kr
2.31 2.07 1.68 1.47 1.41 1.47 1.47 1.47 1.41 1.41 1.41 1.41 1.36 1.31 1.21 1.21 1.21 2.10
-----------------------------------------------------------------------------------------
Rb
Sr
Y
Zr
Ni
Mo
Tc
Ru
Rh
Pd
Ag
Cd
In
Sn
Sb
Te
I
Xe
2.31 2.10 1.94 1.60 1.52 1.52 1.42 1.36 1.42 1.47 1.68 1.62 1.62 1.52 1.52 1.47 1.47 2.66
-----------------------------------------------------------------------------------------
Cs
Ba
La
Hf
Ta
W
Re
Os
Ir
Pt
Au
Hg
Tl
Pb
Bi
Po
At
Rn
2.73 2.10 1.94 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.63 1.99 1.89 1.68 1.42 1.42 1.62
-----------------------------------------------------------------------------------------
The choice of the covalent radius of hydrogen may be very critical when extracting a molecule
from a hydrogen bonded molecular crystal. See test 15.
REDEFINE - 3D unit cell redefinition
rec variable meaning
h,k,l
Crystallographic (Miller) indices of the basal layer of the new 3D unit cell
1. A new unit cell is defined, with two lattice vectors perpendicular to the [hkl] direction.
The indices refer to the Bravais lattice of the crystal; the hexagonal lattice is used for
the rhombohedral systems, the cubic lattice for cubic systems (non primitive).
2. A new Cartesian reference system is defined, with the xy plane parallel to the (hkl) plane.
3. The atoms in the reference cell are re-ordered according to their z coordinate, in order
to recognize the layered structure, parallel to the (hkl) plane.
4. The layers of atoms are numbered. This information is necessary for generating the input
data for the SLAB option.
5. After neighboring analysis, the program stops.
6. the keyword ORIGIN can be used to shift the origin after the rotation of the cell, and
minimize the number of symmetry operators with translational component. Useful to
maximize the point group of the 2D system that can be generated from 3D using the
keyword SLAB (page 37).
SETINF - Setting of INF values
rec variable
meaning
NUM
number of INF vector positions to set
J,INF(J),I=1,NUM
position in the vector and corresponding value
The keyword SETINF allows setting of a value in the INF array. It can be entered in any
input section.
36

SETPRINT - Setting of printing options
rec variable
meaning
NPR
number of LPRINT vector positions to set
J,LPRINT(J),I=1,NPR
prtrec; position in the vector and corresponding value
The keyword SETPRINT allows setting of a value in the LPRINT array, according to the
information given in Appendix E, page 179. It can be entered in any input section.
SLABCUT (SLAB)
rec variable meaning
h, k, l
crystallographic (Miller) indices of the plane parallel to the surface
ISUP
label of the surface layer
NL
number of atomic layers in the slab
The SLABCUT option is used to create a slab of given thickness, parallel to the given plane
of the 3D lattice.
A new Cartesian frame, with the z axis orthogonal to the (hkl) plane, is defined. A layer is
defined by a set of atoms with same z coordinate, with reference to the new Cartesian frame.
The thickness of the slab, the 2D system, is defined by the number of layers. No reference is
made to the chemical units in the slab. The neutrality of the slab is checked by the program.
1. The crystallographic (Miller) indices of the plane refer to the conventional cell (cubic and
hexagonal systems).
2. A two-sided layer group is derived from the 3D symmetry group of the original crystal
structure: the origin may be shifted to maximize the order of the layer group (keyword
ORIGIN, page 34).
3. The unit cell is selected with upper and lower surface parallel to the (hkl) plane.
4. The 2D translation vectors a1 and a2 are chosen according to the following criteria:
(a) minimal cell area;
(b) shortest translation vectors;
(c) minimum |cos()|, where is the angle between a1 and a2.
5. The surface layer ISUP may be found from an analysis of the information printed by
the REDEFINE (page 36) option. This information can be obtained by a test run,
inserting in the geometry input block the keyword TESTGEOM (page 40). Only the
geometry input block is processed, then the program stops.
Two separate runs are required in order to get the information to prepare the input for a full
SLAB option run:
1. keyword REDEFINE: Rotation of the 3D cell, to have the z axis perpendicular to the
(hkl) place, with numbering of the atomic layers in the rotated reference cell, according
to their z coordinate of the atoms (insert STOP after REDEFINE to avoid further
processing).
2. keyword SLAB: Definition of the 2D system, a slab of given thickness (NL, number of
atomic layers) parallel to the (hkl) crystallographic plane, with the ISUP-th atom on the
surface layer
37

The SLABCUT option, combined with ATOMINSE (page 25), ATOMDISP (page 25),
etc. can be used to create a slab of given thickness, with an atom (or group of atoms) adsorbed
at given position. This is achieved by adding new atoms to the 2D structure, obtained after
executing the SLAB option.
Test cases 5-6-7 refer to a 2D system; test cases 25-26-27 refer to the same system, but generated
from the related 3D one. See also tests 35, 36, 37.
STOP
Execution stops immediately. Subsequent input records are not processed.
STRUCPRT
A formatted deck with cell parameters and atoms coordinates in cartesian reference is written
in the file STRUC.INCOOR. See appendix F.
SUPERCEL
rec variable meaning
E
expansion matrix E (IDIMxIDIM elements, input by rows: 9 reals (3D); 4 reals
(2D); 1 real (1D)
A supercell is obtained by defining the new unit cell vectors as linear combinations of the
primitive cell unit vectors (use SUPERCON for conventional cell vectors reference). The
point symmetry is defined by the number of symmetry operators in the new cell. It may be
reduced, not increased.
The new translation vectors b , b , b are defined in terms of the old vectors b
1
2
3
1, b2, b3 and of
the matrix E, read in input by rows, as follows:
b =
e
1
11 b1
+
e12 b2
+
e13 b3
b =
e
2
21 b1
+
e22 b2
+
e23 b3
b =
e
3
31 b1
+
e32 b2
+
e33 b3
The symmetry is automatically reduced to the point symmetry operators without translational
components and a further reduction of the symmetry is also possible.
Before building the supercell, the origin is shifted in order to minimize the number of sym-
metry operators with translational components (see page 14). To avoid this operation, insert
NOSHIFT before SUPERCEL
Atoms that are related by translational symmetry in the unit cell are considered inequivalent
in a supercell.
The supercell option is a useful starting point for the study of defective systems, of chemisorp-
tion and anti ferromagnetism, by combining the SUPERCELoption with the options de-
scribed in this chapter: ATOMREMO (page 25), ATOMSUBS (page 27), ATOMINSE
(page 25), ATOMDISP (page 25), SLAB (page 37).
To study anti ferromagnetic (AFM) states, it may be necessary to generate a supercell, and
then attribute different spin to atoms related by translational symmetry (ATOMSPIN, input
block 4, page 73). See tests 17, 30, 31, 34, 37.
Example.
Construction of supercells of face-centred cubic 3D system (a = 5.42
A).
The crystallographic cell is non-primitive, the expansion matrix refers to primitive
cell vectors. The E matrix has 9 elements:
38

PRIMITIVE CELL
DIRECT LATTICE VECTORS COMPONENTS
X
Y
Z
B1
.000
2.710
2.710
B2
2.710
.000
2.710
B3
2.710
2.710
.000
2 UNITS SUPERCELL (a)
EXPANSION MATRIX
DIRECT LATTICE VECTORS
E1
.000
1.000
1.000
B1
5.420
2.710
2.710
E2
1.000
.000
1.000
B2
2.710
5.420
2.710
E3
1.000
1.000
.000
B3
2.710
2.710
5.420
2 UNITS SUPERCELL (b)
EXPANSION MATRIX
DIRECT LATTICE VECTORS
E1
1.000
1.000
-1.000
B1
.000
.000
5.420
E2
.000
.000
1.000
B2
2.710
2.710
.000
E3
1.000
-1.000
.000
B3
-2.710
2.710
.000
4 UNITS SUPERCELL (c) crystallographic cell
EXPANSION MATRIX
DIRECT LATTICE VECTORS
E1
-1.000
1.000
1.000
B1
5.420
.000
.000
E2
1.000
-1.000
1.000
B2
.000
5.420
.000
E3
1.000
1.000
-1.000
B3
.000
.000
5.420
8 UNITS SUPERCELL
EXPANSION MATRIX
DIRECT LATTICE VECTORS
E1
2.000
.000
.000
B1
.000
5.420
5.420
E2
.000
2.000
.000
B2
5.420
.000
5.420
E3
.000
.000
2.000
B3
5.420
5.420
.000
16 UNITS SUPERCELL
EXPANSION MATRIX
DIRECT LATTICE VECTORS
E1
3.000
-1.000
-1.000
B1
-5.420
5.420
5.420
E2
-1.000
3.000
-1.000
B2
5.420
-5.420
5.420
E3
-1.000
-1.000
3.000
B3
5.420
5.420
-5.420
27 UNITS SUPERCELL
EXPANSION MATRIX
DIRECT LATTICE VECTORS
E1
3.000
.000
.000
B1
.000
8.130
8.130
E2
.000
3.000
.000
B2
8.130
.000
8.130
E3
.000
.000
3.000
B3
8.130
8.130
.000
32 UNITS SUPERCELL
EXPANSION MATRIX
DIRECT LATTICE VECTORS
E1
-2.000
2.000
2.000
B1
10.840
.000
.000
E2
2.000
-2.000
2.000
B2
.000
10.840
.000
E3
2.000
2.000
-2.000
B3
.000
.000
10.840
a), b) Different double cells
c) quadruple cell. It corresponds to the crystallographic, non-primitive cell, whose parameters
are given in input (page 14).
Example.
Construction of supercells of hexagonal R
3 (corundum lattice) cubic 3D system.
The crystallographic cell is non-primitive: CRYSTAL refer to the primitive cell, with volume
1/3 of the conventional one. The E matrix has 9 elements:
GEOMETRY INPUT DATA:
LATTICE PARAMETERS
(ANGSTROMS AND DEGREES) - CONVENTIONAL CELL
A
B
C
ALPHA
BETA
GAMMA
4.76020
4.76020
12.99330
90.00000
90.00000
120.00000
TRANSFORMATION WITHIN CRYSTAL CODE FROM CONVENTIONAL TO PRIMITIVE CELL:
LATTICE PARAMETERS
(ANGSTROMS AND DEGREES) - PRIMITIVE CELL
A
B
C
ALPHA
BETA
GAMMA
VOLUME
5.12948
5.12948
5.12948
55.29155
55.29155
55.29155
84.99223
3 UNITS SUPERCELL crystallographic cell
EXPANSION MATRIX
DIRECT LATTICE VECTORS
E1
1.000
-1.000
.000
B1
4.122
-2.380
.000
E2
.000
1.000
-1.000
B2
.000
4.760
.000
E3
1.000
1.000
1.000
B3
.000
.000
12.993
LATTICE PARAMETERS (ANGSTROM AND DEGREES)
39

A
B
C
ALPHA
BETA
GAMMA
VOLUME
4.76020
4.76020 12.99330
90.000
90.000
120.000
254.97670
SUPERCON
A supercell is obtained by defining the new unit cell vectors as linear combinations of the
conventional cell vectors. The point symmetry is defined by the number of symmetry operators
in the new cell. It may be reduced, not increased. See SUPERCEL, page 38 for input
instructions.
SYMMOPS
Point symmetry operator matrices are printed in the Cartesian representation. No input data
are required.
SYMMREMO
All the point group symmetry operators are removed. Only the identity operator is left. The
wave function can be computed. No input data are required.
Warning: the CPU time may increase by a factor MVF (order of point-group), both in the
integral calculation and in the scf step. The size of the bielectronic integral file may increase
by a factor MVF2.
SYMMDIR
The symmetry allowed directions, corresponding to internal degrees of freedom are printed.
No input data are required.
TENSOR
rec
variable meaning

IORD
order of the tensor ( 4)
This option evaluates and prints the non zero elements of the tensor of physical properties up
to order 4.
TESTGEOM
Execution stops after reading the geometry input block and printing the coordinates of the
atoms in the conventional cell. Neighbours analysis, as requested by the keyword NEIGH-
BOR, is not executed. The geometry input block must end with the keyword END or ENDG.
No other input blocks (basis set etc) are required.
TRASREMO
Point symmetry operators with fractional translation components are removed. It is suggested
to previously add the keyword ORIGIN (page 34), in order to minimize the number of sym-
metry operators with finite translation component. No input data are required.
40

USESAED
rec variable
meaning
(i),i=1,nsaed for each distortion
Given the symmetry allowed elastic distortion (SAED), (printed by the keyword MAKE-
SAED, page 32) for the allowed distortion are given in input.
FIELD - Electric field in the hamiltonian
rec variable value
meaning
E0MAX
electric field intensity E0 (in atomic units)
if CRYSTAL 3D calculation insert
II
MUL
number of term in Fourier expansion for triangular electric potential
ISYM
+1
triangular potential is symmetric with respect to the z = 0 plane
-1
triangular potential is anti-symmetric with respect to the z = 0 plane
A perturbation due to a finite periodic electric field (E) is added to the Hamiltonian (Fock or
Kohn-Sham):
^
H = ^
H0 + ^
H1(E)
(2.1)
where ^
H0 is the unperturbed Hamiltonian and ^
H1(E) the electric-dipole interaction.
The crystalline orbitals are relaxed under the effect of the field, leading to a perturbed wave
function and charge density.
In the three-dimensional case, the applied electric field is periodic, in order to maintain the
translational symmetry. The corresponding electric potential has a triangular form ("saw-
tooth").
When the field is along the z direction, the perturbation ^
H1(E) takes the form:
^
H(E
1
z ) = V (z) = -qE0 f (z)
(2.2)
where the f + or the f - function is developed in a Fourier series and is chosen according to
the symmetry of the supercell in the direction of the applied field:
+
2C
1
2(2k + 1)z
f +(z) =
cos
(2.3)
2
(2k + 1)2
C
k=0
+
2C
(-1)k
2(2k + 1)z
f -(z) =
sin
(2.4)
2
(2k + 1)2
C
k=0
C is the cell size in the field direction.
Not implemented fo 0-D (molecules) and 1-D (polymers) systems.
1. A supercell approach (see keywords SUPERCEL/SUPERCON, page 38 allows to
minimize edge effect and obtain a better convergence of results.
2. The direction of electric field is along z axis, In 2D systems (slab) it is always perpendic-
ular to the slab. The symmetry of the system can be reduced due to the anti-symmetric
nature of the field, even if intensity E0 = 0.
41

3. In 3D systems, as the direction of electric field is always along the z axis, the keyword
ROTATE (page 36 allows definition of a new crystallographic cell, where in the new
Cartesian reference system the z axis is perpendicular to a selected (hkl) plane.
4. In 3D-crystal, the electric potential takes a triangular form to maintain translational
symmetry and electric neutrality of cell. The symmetry of triangular potential has two
options:
a) ISYM=+1, triangular potential is symmetric with respect to the center of supercell,
along z axis. Use this option if there is a symmetry plane orthogonal to z axis.
b) ISYM=-1, triangular potential is anti-symmetric. This option can be used when
the supercell does not have a symmetry plane orthogonal to z axis.
5. MUL, the number of term in Fourier expansion, can take values between 1 and 60
(LIM060).
MUL=40 is sufficient to perfectly reproduce the triangular shape of the potential.
Note - CPU time is not MUL-dependent.
6. When |E0| takes high value, the system can reach a non converged conducting state during
the SCF process. The threshold |Emax
0
| value depends on the dielectric susceptibility
of the system and on the gap width. For very narrow gap cases, the eigenvalue level
shifting technique (keyword LEVSHIFT, page 77) can be useful to avoid the SCF fall
in a spurious conducting state
7. When an external field is applied, the system can become conducting during the SCF
procedure. In order to avoid convergence problems, it is advisable to set the shrinking
factor of the Gilat net ISP equal to 2 IS, where IS is the Monkhorst net shrinking factor
(see SCF input, page 19.
Conversion factors for electric field
1 AU = 1.71527E+07 ESUCM-2 = 5.72152E+01 C M-2 = 5.14226E+11 VM-1
42

OPTCOORD - Atom coordinates optimization
A modified conjugate gradient algorithm (H.B. Schlegel, J. Comp. Chem. 3, 214, 1982) is
used [30] to optimize the atomic coordinates and locate minima on the potential energy surface
(PES). A cell fixed unconstrained optimization of the atomic coordinates can be carried out.
Lattice parameters must be optimized numerically point by point.
HF and DFT (pure and hybrid functionals) analytical gradients are used for insulators, both
for all-electron and ECP calculations. For conducting systems numerical first derivatives must
be computed (keyword NUMGRAD).
Geometry optimization can be performed on systems with any periodicity: molecules, poly-
mers, slabs, and crystals.
Gradients are evaluated every time the total energy is computed; the second derivative matrix
is built from the gradients. At each step, a one dimensional minimization using a quadratic
polynomial is carried out, followed by an n-dimensional search using the Hessian matrix.
The OPTCOORD keyword must be the last keyword in the geometry input section. The
optimization input block must be closed by the keyword END (or ENDOPT).
No input data are required, as default values for all parameters controlling the optimization
are supplied. The default value for SCF convergence criteria on total energy is set to 10-7.
Geometry optimization is performed in symmetrized cartesian coordinates, in order to exploit
the point group symmetry of the lattice. The keyword PRSYMDIR (input block 1, page
35), may be used to print the symmetrized directions adopted in the geometry optimization.
If there are no symmetry allowed directions, the program prints a warning message and stops.
The output from the wave function and gradient calculation is printed at the first step only.
The output from the other steps is routed to fortran unit 65.
The fortran unit 33 contains the cartesian coordinates of the atoms in the unit cell for each
geometry optimization step in a simple xyz format (see keyword COORPRT, input block1 1,
page 29). This file is suitable to be read by molecular graphics programs (e.g. Molden, XMol,
...) to display the animation of the geometry optimization run.
Fortran unit 34 contains complete geometry input deck, with the last defined geometry. To
run crystal with that geometry, insert EXTERNAL keyword to define the geometry in the
input stream (see keyword EXTPRT, input block 1, page 31, for explanation of the format).
Fortran unit 68 contains information to restart optimization. (see keyword RESTART in
OPTCOORD input block).
ATOMFREE
Partial geometry optimization (default: global optimization)
NL
number of atoms "free"
LB(L),L=1,NL
label of the atoms to move
This keyword allows partial geometry optimization, limited to an atomic fragment rather than
the whole system. Symmetrized cartesian coordinates are generated according to the list of
atoms allowed to move. Note that no advantage is taken in the gradient calculation to reduce
the number of atoms, i.e. gradients are calculated on the whole system. The symmetrized
forces are then computed by using the new set of symmetrized coordinates.
43

FINALRUN
action after geometry optimization - integrals classification is based on the
last geometry
ICODE
Action type code:
0
the program stops (default)
1
single-point energy calculation
2
single-point energy and gradient calculation
3
single-point energy and gradient calculation - if convergence criteria on
gradients are not satisfied, optimization restarts
Truncation of infinite series (Coulomb, exchange, etc.), based on the overlap between two
atomic functions (see chapter 6.11), depends on the geometry of a crystal.
With default
setting of thresholds slightly different truncation levels correspond to different geometries that
often introduce small discontinuities in the PES, producing artificial noise in the optimization
process. To avoid noise in interpolation of PES, FIXINDEX option is always active during
optimization. The adopted truncation pattern refers to the starting geometry.
If equilibrium geometry is significantly different from starting point, reference truncation pat-
tern may be inappropriate and the use of proper truncation becomes mandatory.
Since both total energy and gradients are affected by the integrals classification, a single-point
energy calculation ought to be run always with the final structure, and integrals classified
according to the new geometry, to calculate correct total energy and gradients.
If the convergence test on the forces is not satisfied, optimization has to be restarted, keeping
the integrals classification based on the new geometry.
The three different options of FINALRUN allow the following actions, after classification of
integrals:
1. single-point energy calculation (correct total energy),
2. single-point energy and gradient calculation (correct total energy and gradients),
3. single-point energy and gradient computation, followed by a new optimization process,
starting from the final geometry of the previous one, (used to classify the integrals), if
the convergence test is not satisfied.
If the starting and final geometry are not far away, the energy and gradient calculated from
the final geometry, but the integrals classification based on the initial geometry, and the values
computed with integrals classification based on the final geometry are not very different. In
some cases (e.g. optimization of the geometry of a surface, with reconstruction) the two ge-
ometries are very different, and a second optimization cycle is almost mandatory (ICODE=3).
HESGUESS
defines the initial guess for the Hessian
ICODE
Initial guess code:
-1
identity matrix (default)
0
numerical estimate (two-points formula)
1
external guess (read from fort.66
By default a unit matrix is adopted as initial Hessian. The Hessian matrix is stored in fortran
unit 66 during the optimization at each step. This may be useful to restart the optimization
from a previous run performed at a lower level of theory (e.g. a smaller basis set).
An initial Hessian can also be obtained as numerical first-derivative, but this process is very
expensive.
MAXOPTC
MAX
maximum number of optimization steps (default 100)
44

NOGUESS scf guess at each geometry point: superposition of atomic densities at each
scf calculation
At each geometry point the default guess for scf is the density matrix calculated at the end
of the previous run. If the solution does not correpond at real convergence, but to an en-
ergy stabilization due to the techiques applied to help convergence (LEVSHIFT, FMIXING,
BROYDEN..) when the final density matrix, obtained from the last cycle eigenvectors, is
used to build the first hamiltonian matrix, the hamiltonian eigenvalues can be unphysical,
no chances to recover the scf process. In those cases it may be better an atomic guess.
NUMGRAD numerical first-derivatives are used during the geometry optimization
The nuclear coordinate gradients of the energy can also be computed numerically. A three-point
numerical derivative formula is adopted. A finite positive (and then negative) displacement
is applied to the desired coordinate and a full SCF calculation is performed. The gradient is
then computed as
Ex - E
g
i
-xi
i =
2 xi
where xi is the finite displacement along the i-coordinate.
Such a computation is very expensive compared to analytical gradients, since the cost is 2N t,
where N is the number of coordinates to be optimized and t the cost of the SCF calculation. Nu-
merical first-derivatives should be avoided whenever possible, but sometimes they are the only
way to obtain gradients (i.e. for metals) and therefore to optimize the atom coordinates.
PRINTFORCES printing atomic gradients
PRINTHESS
printing Hessian inormation
PRINTOPT
prints information on optimization process
PRINT
verbose printing
RESTART restart geometry optimization from a previous run
Full restart of geometry optimization is possible for a job which is abruptly terminated (e.g.
number of steps exceeded, available cpu time exceeded,...). The optimization restarts from the
last complete step of the previous run. Needed information is read from fortran unit 68, and
saved in the same file at each step .
The same input deck as for the initial geometry optimization must be used when the RESTART
keyword is added.
Convergence criteria
A stationary point on the potential energy surface is found when the forces acting on atoms
are zero. So, a geometry optimization is usually completed when the gradients are below a
given threshold.
In Crystal, the optimization convergence is checked on the root-mean-square and the absolute
value of the largest component of both the gradients and the estimated displacements. When
these four conditions are all satisfied at a time, optimization is considered complete.
In the current implementation the following defaults are used (in a.u.):
45

TOLDEG
RMS GRADIENT
0.000300 default value - MAX GRADIENT 1.5*TOLDEG
TOLDEX
RMS DISPLAC.
0.001200 default value - MAX DISPLAC. 1.5*TOLDEX
The maximum gradient and the maximum displacement threshold are defined as 1.5 times
TOLDEG and TOLDEX, respectively.
In case of flat surfaces (e.g. a system formed by two weakly-interacting moieties, a molecule
adsorbed on a surface at large distance) it may happen that the convergence criteria on dis-
placements are not satisfied, though the energy does not change. Additional combined test on
gradient and energy has been adopted:
1. If the gradient criteria are satisfied (but not the displacement criteria) and the energy
difference between two steps is below a given threshold (see TOLDEE), the optimization
stops with a warning message;
2. If both the gradient and displacements criteria are not satisfied, but the energy does not
change (TOLDEE parameter) for four subsequent steps, the optimization stops with a
warning message.
TOLDEE
threshold on the energy change between optimization steps
IG
|E| < 10-IG (default: 7)
The value of IG must be larger or equal to the threshold adopted for the SCF convergence.
The value is checked when input block 4, defining the SCF convergence criteria, is processed.
TOLDEG
convergence criterion on the RMS of the gradient
TG
max RMS of the gradient (default: 0.0003)
TOLDEX
convergence criterion on the RMS of the displacement
TX
max RMS of the displacement (default: 0.0012)
Full optimization: cell parameters and atomic coordinates
A full optimization of a crystalline structure includes also the optimization of the lattice pa-
rameters. However, analytical gradients for the cell constants are not yet implemented; thus,
a two-step iterative process must be adopted:
1. the cell parameters are optimized at fixed atomic positions. This must be accomplished
by hand or by means of external drivers (e.g. the LoptCG script).
2. atomic positions are relaxed by keeping the lattice parameters fixed at the previously
optimum values.
The process is then iterated until convergence is attained, that is cell constants no longer
change and convergence criteria on nuclear forces are satisfied.
46

Notes - From periodic structure to molecules and clusters
The geometry editing described in this section allow the generation of finite (non-periodic)
systems, derived from periodic structures: clusters, molecules, groups of molecules (for exam-
ple, dimers), functional groups. The atoms in a molecular crystal are assigned to the same
molecule when their distance is shorter than the sum of the covalent radii, according to table
at page 35.
The possibility of computing the wave function of a finite cluster allows the comparison of
data with those computed for the original periodic structure. In this way the exact nature of
the 'border effect' introduced by the cluster approximation in solid-state calculations may be
analyzed [31, 32].
The default value of covalent radii can be modified for selected atoms by the option RAYCOV
(page 35). This modification may be necessary to force the correct definition of a fragment in
the MOLECULE and CLUSTER options, as well as in the MOLEBSSE option (page 47).
Warning
The result of the geometry editing:
CLUSTER, MOLECULE, ATOMBSSE and
MOLEBSSE is a 0D system. No reciprocal lattice information is required in the scf in-
put (Section 1.4, page 19).
ATOMORDE can be applied to molecular crystals only. The molecules ordered in the lattice
are defined as groups of atoms with a separation less than the sum of their covalent radii (Table
on page 35). When there are very short hydrogen bonds linking the molecules in the lattice,
it may be necessary a modification of the atomic radii, to allow proper identification of the
molecules (see option RAYCOV, page 35)
Notes - From 3D periodic structure to 2D
Slab model for surfaces
to be written
at least:
test on thickness of the slab
non-polar surfaces,
calculation of surface energy
Notes - BSSE correction - counterpoise scheme
The counterpoise method [33] is applied to correct the Basis Set Superposition Error for atoms,
ions or molecules. An atom or cluster of atoms surrounded by the basis functions of the
neighbours are extracted from the periodic structure.
A BSSE estimate, maintaining the periodic symmetry of the system, can be obtained trans-
forming into ghosts selected atoms in the lattice (keyword GHOSTS, page 50, test 36).
example: urea bulk
47

2.2
Basis set input
Symmetry control
ATOMSYMM
printing of point symmetry at the atomic positions
27

Basis set modification
CHEMOD
modification of the electronic configuration
48
I
GHOSTS
eliminates nuclei and electrons, leaving BS
50
I
Auxiliary and control keywords
CHARGED
allows non-neutral cell
48

PARAMPRT
printing of parameters controlling code dimensions
35

PRINTOUT
setting of printing options
35
I
SETINF
setting of inf array options
36
I
SETPRINT
setting of printing options
37
I
STOP
execution stops immediately
38

SYMMOPS
printing of point symmetry operators
40

END/ENDB
terminate processing of basis set definition keywords

Output of data on external units
GAUSS98
printing of an input file for the GAUSS94/98 package
49

ATOMSYMM
See input block 1, page 27
CHARGED - charged reference cell
The unit cell of a periodic system must be neutral. This option forces the overall system to
be neutral even when the number of electrons in the reference cell is different from the sum
of nuclear charges, by adding a uniform background charge density to neutralize the charge in
the reference cell.
CHEMOD - modification of electronic configuration
rec variable
meaning
NC
number of configurations to modify
LA
label of the atom with new configuration
CH(L),L=1,NS shell charges of the LA-th atom. The number NS of shells must coincide
with that defined in the basis set input.
The CHEMOD keyword allows modifications of the shell charges given in the basis set input,
which are used in the atomic wave function routines. The original geometric symmetry is
checked, taking the new electronic configuration of the atoms into account. If the number of
symmetry operators should be reduced, information on the new symmetry is printed, and the
program stops. No automatic reduction of the symmetry is allowed. Using the information
printed, the symmetry must be reduced by the keyword MODISYMM (input block 1, page
32).
See test 37. MgO supercell, with a Li defect. The electronic configuration of the oxygen nearest
to Li corresponds to O-, while the electronic configuration of those in bulk MgO is O2-. The
48

basis set of oxygen is unique, while the contribution of the two types of oxygen to the initial
density matrix is different.
END
Terminate processing of block 2, basis set, input. Execution continues. Subsequent input
records are processed, if required.
GAUSS98 - Printing of input file for GAUSS98 package
The keyword GAUSS98 writes on fortran unit 92 an input deck to run Gaussian 94 (or
Gaussian 98) [34, 35]. The deck can be prepared without the calculation of the wave function
by entering the keyword TESTPDIM in input block 3 (page 69). For periodic systems,
coordinates and basis set for all the atoms in the reference cell are written.
If the keyword is entered many times, the data are overwritten. The fortran file fort.92 contains
the data corresponding to the last call.
No input data required.
1. The route card specifies:
method
HF
basis set
GEN
type of job
SP
geometry
UNITS=AU
2. The title card is the same as in CRYSTAL input.
3. The molecule specification defines the molecular charge as the net charge in the reference
cell. If the system is not closed shell, the spin multiplicity is indicated with a string "??",
and must be defined by the user.
4. Input for effective core pseudopotentials is not written. In the route card PSEUDO =
CARDS is specified; the pseudopotential parameters used for the crystal calculation are
printed in the crystal output.
5. The scale factors of the exponents are all set to 1., as the exponents are already scaled.
6. the input must be edited when different basis sets are used for atoms with the same
atomic number (e.g., CO on MgO, when the Oxygen basis set is different in CO and in
MgO)
To compare molecular energies obtained with GAUSSIAN and CRYSTAL, default computa-
tional parameters of CRYSTAL must be modife by inserting the keywords:
NOBIPOLA remove bipolar exoansio to compute coulomb interaction of non overlapping
pseudo-charge distributions;
BOHRANGS follwed by the value adopteb by GAUSSIAN for the conversion factor bohr/

Angstrom
Warning: Only for 0D systems! The programs does not stop when the keyword GAUSS94
is entered for 1-2-3D systems. Coordinates and basis set of all the atoms in the primitive cell
are written, formatted, on fortran unit 92, following Gaussian 94 scheme.
49

GHOSTS
rec variable
meaning
NA
number of atoms to be transformed into ghosts
LA(L),L=1,NA
label of the atoms to be transformed.
Selected atoms may be transformed into ghosts, by deleting the nuclear charge and the shell
electron charges, but leaving the basis set centred at the atomic position. The conventional
atomic number is set to zero.
If the system is forced to maintain the original symmetry (KEEPSYMM), all the atoms
symmetry related to the given one are transformed into ghosts.
Useful to create a vacancy (Test 37), leaving the variational freedom to the defective region
and to evaluate the basis set superposition error (BSSE), in a periodic system. The periodic
structure is maintained, and the energy of the isolated components computed, leaving the basis
set of the other one(s) unaltered. For instance, the energy of a mono-layer of CO molecules on
top of a MgO surface can be evaluated including the basis functions of the first layer of MgO,
or, vice-versa, the energy of the MgO slab including the CO ad-layer basis functions. See test
36.
Warning Do not use with ECP
Warning The keyword ATOMREMO (input block 1, page 25) creates a vacancy, removing
nuclear charge, electron charge, and basis functions. The keyword GHOSTS creates a vacancy,
but leaves the basis functions at the site, so allowing better description of the electron density
in the vacancy.
PARAMPRT - Printing of parametrized dimensions
See input block 1, page 35.
PRINTOUT - Setting of printing environment
See input block 1, page 35.
SETINF - Setting of INF values
See input block 1, page 36.
SETPRINT - Setting of printing options
See input block 1, page 37.
STOP
Execution stops immediately. Subsequent input records are not processed.
SYMMOPS
See input block 1, page 40
50

Effective core pseudo-potentials.
rec
variable value
meaning
A
PSN
pseudo-potential keyword:
HAYWLC
Hay and Wadt large core ECP.
HAYWSC
Hay and Wadt small core ECP.
BARTHE
Durand and Barthelat ECP.
DURAND
Durand and Barthelat ECP.
INPUT
free ECP - input follows.
if PSN = INPUT insert
II

ZNUC
effective core charge (ZN in eq. 2.6).
M
Number of terms in eq. 2.7
M0
Number of terms in eq. 2.8 for
=0.
M1
Number of terms in eq. 2.8 for
=1.
M2
Number of terms in eq. 2.8 for
=2.
M3
Number of terms in eq. 2.8 for
=3.
insert M+M0+M1+M2+M3 records
II

ALFKL
Exponents of the Gaussians: k .
CGKL
Coefficient of the Gaussians: Ck .
NKL
Exponent of the r factors: nk .
Valence-electron only calculations can be performed with the aid of effective core pseudo-
potentials (ECP). The ECP input must be inserted into the basis set input of the atoms with
conventional atomic number > 200.
The form of pseudo-potential Wps implemented in CRYSTAL is a sum of three terms: a
Coulomb term (C), a local term (W0) and a semi-local term (SL):
Wps = C + W 0 + SL
(2.5)
where:
C = -ZN /r
(2.6)
M
W 0 =
rnk Cke-kr2
(2.7)
k=1
3
M
SL =
[
rnk Ck e-k r2 ]P
(2.8)
=0 k=1
ZN is the effective nuclear charge, equal to total nuclear charge minus the number of electrons
represented by the ECP, P is the projection operator related to the
angular quantum number,
and M, nk, k, M , nk , Ck , k are atomic pseudo-potential parameters.
1. Hay and Wadt (HW) ECP ([36, 37]) are of the general form 2.5. In this case, the NKL
value given in the tables of ref. [36, 37] must be decreased by 2 (2 0, 1 -1, 0 -2).
2. Durand and Barthelat (DB) ([38] - [39], [40], [41]), and Stuttgart-Dresden [42] ECPs
contain only the Coulomb term C and the semi-local SL term.
3. In Durand and Barthelat ECP the exponential coefficient in SL depends only on
(i.e.
it is the same for all the Mk terms).
3
M
SL =
e- r2 [
rnk Ck ]P
(2.9)
=0
k=1
51

The core orbitals replaced by Hay and Wadt large core and Durand-Barthelat ECPs are as
follows:
Li-Ne
= [He]
Na-Ar
= [Ne]
first series
= [Ar]
second series
= [Kr]
third series
= [Xe]4f 14.
The core orbitals replaced by Hay and Wadt small core ECPs are as follows:
K-Cu
= [Ne]
Rb-Ag
= [Ar] 3d10
Cs-Au
= [Kr] 4d10 .
The program evaluates only those integrals for which the overlap between the charge distri-
bution 0 g (page 139) and the most diffuse Gaussian defining the pseudopotential is larger


than a given threshold Tps (the default value is 10-5). See also TOLPSEUD (Section 1.3).
Pseudopotential libraries
The following periodic tables show the effective core pseudo-potentials included as internal
data in the CRYSTAL code. The f terms have not yet been implemented. The program stops
when an ECP including f terms ( =3, M=3) is entered.
HAY AND WADT LARGE CORE ECP. CRYSTAL92 DATA
-------
------------------
Na Mg
Al Si P
S
Cl Ar
------------------------------------------------------
K
Ca Sc Ti V
Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr
------------------------------------------------------
Rb Sr Y
Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I
Xe
------------------------------------------------------
Cs Ba
Hf Ta W
Re Os Ir Pt Au Hg Tl Pb Bi
------------------------------------------------------
HAY AND WADT SMALL CORE ECP. CRYSTAL92 DATA
-------------------------------------------------------
K
Ca Sc Ti V
Cr Mn Fe Co Ni Cu
-------------------------------------------------------
Rb Sr Y
Zr Nb Mo Tc Ru Rh Pd Ag
-------------------------------------------------------
Cs Ba
Hf Ta W
Re Os Ir Pt Au
-------------------------------------------------------
DURAND AND BARTHELAT'S LARGE CORE ECP - CRYSTAL92 DATA
------
------------------
Li Be
B
C
N
O
F
Ne
------
------------------
Na Mg
Al Si P
S
Cl Ar
-------------------------------------------------------
K
Ca Sc Ti V
Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr
-------------------------------------------------------
Rb
Y
Ag
In Sn Sb
I
-------------------------------------------------------
Tl Pb Bi
-------------------------------------------------------
52

BARTHE, HAYWSC and HAYWLC pseudopotential coefficients and exponents are in-
serted as data in the CRYSTAL code. The data defining the pseudo-potentials where in-
cluded in CRYSTAL92, and never modified. The keyword INPUT allows entering updated
pseudo-potentials, when available. An a posteriori check has been possible for HAYWLC
and HAYWSC only, as the total energy of the atoms for the suggested configuration and
basis set has been published [36, 43]. Agreement with published atomic energies data is satis-
factory (checked from Na to Ba) for Hay and Wadt small core and large core pseudo-potentials,
when using the suggested basis sets. The largest difference is of the order of 10-3 hartree.
For Durand and Barthelat the atomic energies are not published, therefore no check has been
performed. The printed data should be carefully compared with those in the original papers.
The authors of the ECP should be contacted in doubtful cases.
Valence Basis set and pseudopotentials
Hay and Wadt ([36, 43]) have published basis sets suitable for use with their small and large core
pseudopotentials. and in those basis set the s and p gaussian functions with the same quantum
number have different exponent. It is common in CRYSTAL to use sp shells, where basis
functions of s and p symmetry share the same set of Gaussian exponents, with a consequent
considerable decrease in CPU time. The computational advantage of pseudopotentials over
all-electron sets may thus be considerably reduced.
Basis set equivalent to those suggested by Hay and Wadt can be optimized by using CRYSTAL
as an atomic package (page 57), or any atomic package with effective core pseudopotentials.
See Chapter 3.2 for general comments on atomic basis function optimization. Bouteiller et al
[44] have published a series of basis sets optimized for Durand and Barthelat ECPs.
Stuttgart-Dresden ECP (formerly STOLL and PREUSS ECP)
These pseudopotentials are under constant development and the CRYSTAL database is not up
to date in all cases (for example, improved pseudopotentials exist for many of the main group
elements, and pseudopotentials are also available for 5d and other heavier elements). The most
recent pseudopotential parameters, optimized basis sets, a list of references and guidelines for
the choice of the pseudopotentials can be found at http://www.theochem.uni-stuttgart.de/
These can be used in CRYSTAL via the INPUT keyword (see below).
The format of the pseudopotential data in the database is as follows:
The local term
M
W 0 =
rnk Cke-kr2
k=1
always vanishes (Ck = 0.000000), leaving only the semilocal term:
3
M
SL =
[
rnk -2Ck e-k r2 ]P
=0 k=1
Note the different convention for the factor rnk -2 compared to equation 2.8
The pseudopotential parameters are given in a table like this:
53

Cl ECP ECP10MWB : 10 3 0 22
Q=7., MEFIT, WB, Ref 17.
nuclear charge 7
1 2 1.000000
0.000000
one local term, n1=2, 1
=
1.000000,
C1=0.000000
2 2 6.394300
33.136632
2 3.197100
16.270728
2
s-projectors
(M0=2),
n1,0=2,
1,0=6.394300, C1,0=33.136632 and n2,0=2,
2,0=3.197100, C2,0=16.270728
2 2 5.620700
24.416993
2 2.810300
7.683050
2
p-projectors
(M1=2),
n1,1=2,
1,1=5.620700, C1,1=24.416993, and n2,1=2,
2,1=2.810300, C2,1=7.683050
1 2 5.338100
-8.587649
1
d-projector
(M2=1),
n1,2=2,
1,2=5.338100, C1,2=-8.587649
The corresponding basis set (optimized for the free atom) looks like this:
Cl s ECP10MWB : 4 1 1.3
4 primitives, 1 group of them contracted, contracted
are exponents 1-3
(4s5p)/[2s3p]-Basissatz fuer PP. von Ref 17.
14.0730760 2.3315650
0.5071000
0.1824330
4 s-exponents
0.0203450
-0.2892230 0.6303670
contraction coefficients
and
Cl p ECP10MWB : 5 1 1.3
5 primitives, 1 group of them contracted, con-
tracted exponents are 1-3
(4s5p)/[2s3p]-Basissatz fuer PP. von Ref 17.
3.3531290
0.7856860
0.2674540
0.0782750
4 p-exponents
0.0154770
1 p-exponent
-0.0415520 0.3997480
0.5918290
contraction coefficients
In CRYSTAL, this pseudopotential and basis set is entered as follows:
Free input
217 5
Z=17, chlorine basis set - 5 shells (valence only)
INPUT
keyword: free ECP - input follows
7.
0 2 2 1 0
nuclear charge 7; number of terms in eq. 2.7 and 2.8
6.394300
33.136632
0
eq. 2.8, 5 records:
3.197100
16.270728
0
, C, n
5.620700
24.416993
0
2.810300
7.683050
0
5.338100
-8.587649
0
basis set input follows - valence only
0 0 3 2.0 1
1st shell: s type; 3 GTF; CHE=2; scale fact.=1
14.0730760
0.0203450
2.3315650
-0.2892230
0.5071000
0.6303670
0 0 1 0.0 1
2nd shell: s type; 1 GTF; CHE=0; scale fact.=1
0.1824330
1.000000
0 2 3 5.0 1
3rd shell: p type; 3 GTF; CHE=5; scale fact.=1
3.3531290
-0.0415520
0.7856860
0.3997480
0.2674540
0.5918290
0 2 1 0.0 1
4th shell: p type; 1 GTF; CHE=0; scale fact.=1
0.0782750
1.000000
0 2 1 0.0 1
5th shell: p type; 1 GTF; CHE=0; scale fact.=1
0.0154770
1.000000
The Hartree-Fock energy of the atomic ground state with this choice of pseudopotential and
basis set is -1.472982977E+01 hartree.
RCEP Stevens et al.
An other important family of pseudopotentials for the first-, second-, third-, fourth and fifth-
row atoms of the periodic Table (excluding the lanthanide series) is given by Stevens et al.
[45, 46]. Analytic Relativistic Compact Effective Potential (RCEP) are generated in order
to reproduce the "exact" pseudo-orbitals and eigenvalues as closely as possible. The analytic
RCEP expansions are given by:
54

r2Vl(r) =
Alkrnl,k e-Blkr2
k
An example of data for Ga atom (Table 1, page 616 of the second paper) is:
Alk nlk
Blk
Vd
-3.87363 1 26.74302
Vs-d
4.12472 0
3.46530
260.73263 2
9.11130
-223.96003 2
7.89329
Vp-d
4.20033 0 79.99353
127.99139 2 17.39114
The corresponding Input file for the CRYSTAL program will be as follows:
INPUT
21. 1 3 2 0 0
26.74302
-3.87363 -1
3.46530
4.12472 -2
9.11130 260.73263 0
7.89329 -223.96003 0
79.99353
4.20033 -2
17.39114 127.99139 0
Note that for the r-exponent (nlk), -2 has been subtracted to the value given in their papers,
as in the case of Hay and Wadt pseudopotentials.
55

2.3
General information, computational parameters,
hamiltonian
Single particle Hamiltonian
RHF
Restricted Closed Shell
68

UHF
Unrestricted Open Shell
70

ROHF
Restricted Open Shell
70

DFT
DFT Hamiltonian
70

SPIN
spin-polarized solution
59

Choice of the exchange-correlation functionals
EXCHANGE exchange functional
59
I
LDA
Dirac-Slater [47] (LDA)
VBH
von Barth-Hedin [48] (LDA)
BECKE
Becke [49] (GGA)
PWGGA Perdew-Wang 91 (GGA)
PBE
Perdew-Becke-Ernzerhof [50] (GGA)
CORRELAT
correlation functional
59
I
VBH
von Barth-Hedin [48] (LDA)
PWGGA Perdew-Wang 91 (GGA)
PBE
Perdew-Becke-Ernzerhof [50] (GGA)
PZ
Perdew-Zunger [51] (LDA)
PWLSD
Perdew-Wang 92 [52, 53, 54] (GGA)
VWN
Vosko,-Wilk-Nusair [55] (LDA)
P86
Perdew 86 [56] (LDA)
LYP
Lee-Yang-Parr [57] (GGA)
HYBRID
hybrid mixing
60
I
NONLOCAL local term parameterization
60
I
B3PW
B3PW parameterization
60

B3LYP
B3LYP parameterization
60

Integration method
[NUMERICA] numerical integration (default)

RADSAFE
safety radius for grid point screening
I
BATCHPNT grid point grouping for integration
I
FITTING
integration through auxiliary basis set fitting
172
I:
BASIS
Auxiliary basis set input
172
I
TOLLPOT 9

LINEQUAT
173
I
Numerical accuracy control
[BECKE]
selection of Becke weights (default)

SAVIN
selection of Savin weights

RADIAL
definition of radial grid
I
ANGULAR
definition of angular grid
I
LGRID
"large" predefined grid
I
XLGRID
"extra large" predefined grid
I
TOLLDENS
density contribution screening 6
I
TOLLGRID
grid points screening 14
I
Auxiliary and control
PRINT
END
Numerical accuracy and computational parameters control
56

BIPOLAR
Bipolar expansion of bielectronic integrals
58
I
BIPOSIZE
size of coulomb bipolar expansion buffer
58
I
EXCHSIZE
size of exchange bipolar expansion buffer
58
I
INTGPACK
Choice of integrals package 0
67
I
NOBIPOLA
All bielectronic integrals computed exactly
67

POLEORDR
Maximum order of multipolar expansion 4
68
I
TOLINTEG
Truncation criteria for bielectronic integrals
6 6 6 6 12
69
I
TOLPSEUD
Pseudopotential tolerance 6
69
I
Type of run
ATOMHF
Atomic wave functions
57
I
MPP
MPP execution - see makefile
67
I
SCFDIR
SCF direct (mono+biel int computed)
68

SEMIDIR
integrals in memory
68
I
NOMONDIR
SCF semidirect (mono on disk, biel computed)
67

EIGS
S(k) eigenvalues - basis set linear dependence check
64

FIXINDEX
Reference geometry to classify integrals
65

Integral file distribution
BIESPLIT
writing of bielectronic integrals in n files n = 1 ,max=10
58
I
MONSPLIT
writing of mono-electronic integrals in n file n = 1 , max=10
67
I
Auxiliary and control keywords
PARAMPRT
output of parameters controlling code dimensions
35

PRINTOUT
setting of printing options
35
I
SETINF
setting of inf array options
36
I
SETPRINT
setting of printing options
37
I
TESTPDIM
stop after symmetry analysis
69

TESTRUN
stop after integrals classification and disk storage estimate
69

STOP
execution stops immediately
38

END/ENDM
terminate processing of general information keywords
ATOMHF - Atomic wave function calculation
The Hartree-Fock atomic wave functions for the symmetry unique atoms in the cell are com-
puted by the atomic program [14]. Full input (geometry, basis set, general information, SCF)
is processed. No input data are required. The density matrix, constructed from a superpo-
sition of atomic densities, is computed and written on Fortran unit 9, along with the wave
function information. The crystal program then stops. It is then possible to compute charge
density (ECHG) and classical electrostatic potential (CLAS) maps by running the program
properties. This option is an alternative to the keyword PATO in the program properties
(page 104), when the calculation of the periodic wave function is not required. The atomic
wave function, eigenvalues and eigenvectors, can be printed by setting the printing option 71.
1. The atomic basis set may include diffuse functions, as no periodic calculation is carried
out.
2. A maximum of two open shells of different symmetry (s, p, d) are allowed in the electronic
configuration. In the electronic configuration given in input the occupation number of
the shells must follow the rules given in Section 1.2.
57

3. For each electronic configuration, the highest multiplicity state is computed. Multiplicity
cannot be chosen by the user.
Warning: DFT wave function for isolated atoms can not be computed.
BIESPLIT - Splitting of large bielectronic integral files
rec
variable meaning

NFILE
number of files to be used 1 (max 10)
Very compact crystalline systems, and/or very diffuse basis functions and/or very tight toler-
ances can produce billions integrals to be stored. The storage of bielectronic integrals can be
avoided by running the direct SCF code scfdir rather than the standard SCF, at the expenses
of a certain amount of CPU time.
When the standard SCF code is used, distributing the integrals on several disk files can improve
performance.
BIPOLAR - Bipolar expansion approximation control
rec
variable meaning

ITCOUL overlap threshold for Coulomb 14

ITEXCH overlap threshold for exchange 10
The bipolar approximation is applied in the evaluation of the Coulomb and exchange integrals
(page 143). ITCOUL and ITEXCH can be assigned any intermediate value between the default
values (14 and 10) (see page 143) and the values switching off the bipolar expansion (20000
and 20000).
BIPOSIZE -Size of buffer for Coulomb integrals bipolar expansion
rec
variable meaning

ISIZE
size of the buffer in words
Size (words) of the buffer for bipolar expansion of Coulomb integrals (default value 100000.
The size of the buffer is printed in the message:
BIPO BUFFER LENGTH (WORDS) = XXXXXXX
or
COULOMB BIPO BUFFER TOO SMALL - TO AVOID I/O SET BIPOSIZE = XXXXXX
DFT
The Kohn-Sham [58, 59] DFT code is controlled by keywords, that must follow the general
keyword DFT, in any order. These keywords can be classified into four groups:
1 Choice of the exchange-correlation functionals
2 Integration method
3 Integration grid and numerical accuracy control
4 DF energy gradient
58

The DFT input block ends with the keyword END or ENDDFT. Default values are sup-
plied for all computational parameters. Choice of exchange and/or correlation potential is
mandatory.
1. Choice of the exchange-correlation functionals
EXCHANGE and CORRELAT keywords, each followed by an alpha-numeric record, allow
the selection of the exchange and correlation functionals.
If the correlation potential is not set (keyword CORRELAT), an exchange-only potential is
used in the Hamiltonian. If the exchange potential is not set (keyword EXCHANGE), the
Hartree-Fock potential is used.
CORRELAT
Correlation Potential (default: no correlation).
Insert one of the following keywords
II
PZ
LSD. Perdew-Zunger parameterization of the Ceperley-Alder free electron
gas correlation results [51]
PWLSD
LSD. Perdew-Wang parameterization of the Ceperley-Alder free electron
gas correlation results [52, 53, 54]
VWN
LSD. Vosko-Wilk-Nusair parameterization of the Ceperley-Alder free elec-
tron gas correlation results [55]
VBH
LSD. von Barth-Hedin [48]
P86
GGA. Perdew 86 [56]
PWGGA
GGA. Perdew-Wang [60, 52, 53, 54]
LYP
GGA. Lee-Yang-Parr [57]
PBE
GGA. Perdew-Burke-Ernzerhof [50]
EXCHANGE
Exchange potential (default: Hartree-Fock exchange).
Insert one of the following keywords
II
LDA
LSD. Dirac-Slater [47]
VBH
LSD. von Barth-Hedin [48]
BECKE
GGA. Becke [49]
PWGGA
GGA. Perdew-Wang [52, 53, 54]
PBE
GGA. Perdew-Becke-Ernzerhof [50]
All functionals are formulated in terms of total density and spin density. Default is total
density. To use functionals of spin density insert the keyword SPIN.
SPIN unrestricted spin DF calculation (default: restricted)
It is also possible to incorporate part of the exact Hartree-Fock exchange into the exchange
functional through the keyword HYBRID. Any mixing (0-100) of exact Hartree-Fock and
DFT exchange can be used.
NONLOCAL allows modifying the relative weight of the local and non-local part both in
the exchange and the correlation potential with respect to standard definition of GGA type
potentials.
59

HYBRID
Hybrid method - 1 record follows:
A
Fock exchange percentage (default 100.)
NONLOCAL
setting of non-local weighting parameters - 1 record follows:
B
exchange weight of the non-local part of exchange
C
weight of the non-local correlation
B3PW
Becke's 3 parameter functional [61] combined with the non-local correla-
tion PWGGA [60, 52, 53, 54]
B3LYP
Becke's 3 parameter functional [61] combined with the non-local correla-
tion LYP
B3PW and B3LYP are global keywords, defining hybrid exchange-correlation functionals
completely. They replace the following sequences:
B3PW
B3LYP
corresponds to the sequence:
corresponds to the sequence:
EXCHANGE
EXCHANGE
BECKE
BECKE
CORRELAT
CORRELAT
PWGGA
LYP
HYBRID
HYBRID
20
20
NONLOCAL
NONLOCAL
0.9 0.81
0.9 0.81
B3LYP in CRYSTAL is based on the 'exact' form of the Vosko-Wilk-Nusair correlation poten-
tial (corresponds to a fit to the Ceperley-Alder data). In the original paper [55]) it is reported
as functional V, which is used to extract the 'local' part of the LYP correlation potential.
The Becke's 3 parameter functional can be written as:
Exc = (1 - A) (ELDA + B
) + A
+ (1
+ C
x
EBECKE
x
EHF
x
- C) EV WN
c
ELY P
c
A, B, and C are the input data of HYBRYD and NONLOCAL.
Examples of possible selection of the correlation and exchange functionals are:
exchange
correlation
--
PWGGA
Hartree-Fock exchange, non local Perdew-Wang correlation.
LDA
VWN
probably the most popular LDA formulation
VBH
VBH
was the most popular LDA scheme in the early LDA solid state
applications (1975-1985).
PWGGA
PWGGA
- - - - - -
BECKE
LYP
- - - - - -
2. Integration method
The exchange-correlation contribution to the Kohn-Sham matrix and to gradients is computed
through the same quadrature used to compute the exchange-correlation energy [62, 63]. The
alternative method of fitting the exchange-correlation potential to an auxiliary basis set of gaus-
sian functions (see Appendix C) is not available for gradients. Therefore, numerical integration
is the default choice in CRYSTAL2003.
60

3. Integration grid and numerical accuracy control
No input data are required: Becke weights are chosen by default, as well as a set of safe values
for the computational parameters of integration.
The generation of grid points in CRYSTAL is based on an atomic partition method, originally
developed by Becke [64] for molecular systems and then extended to periodic systems [65].
Each atomic grid consists of a radial and an angular distribution of points. Grid points are
generated through a radial and an angular formula: Gauss-Legendre radial quadrature and
Lebedev two-dimensional angular point distribution are used.
Lebedev angular grids are classified according to progressive accuracy levels, as given in the
following table:
LEV CR98
Nang
LEV CR98
Nang
1
1
9
38
16
53 974
2
2
11 50
17
59 1202
Index of Lebedev accuracy levels
3
13 74 *
18
65 1454
4
15 86
19
71 1730
LEV
: Lebedev accuracy level
5
3
17 110
20
77 2030
CR98 : corresponding index in CRYSTAL
6
19 146
21
83 2354
98
7
21 170
22
89 2702
:
maximum quantum number of
8
4
23 194
23
95 3074
spherical harmonics used in Lebedev
9
25 230 *
24
101 3470
derivation
10
5
27 266 *
25
107 389
Nang
: number of angular points generated
11
6
29 302
26
113 4334
per radial point
12
31 350
27
119 4802

: sets with negative weights, to be
13
7
35 434
28
125 5294
avoided
14
41 590
29
131 5810
15
47 770
If one Lebedev accuracy level is associated with the whole radial range, the atomic grid is
called unpruned, or uniform. In order to reduce the grid size and maintain its effectiveness, the
atomic grids of spherical shape can be partitioned into shells, each associated with a different
angular grid. This procedure, called grid pruning, is based on the assumption that core electron
density is usually almost spherically symmetric, and surface to be sampled is small.
Also, points far from the nuclei need lower point density, as associated with relatively small
weights, so that more accurate angular grids are mostly needed within the valence region than
out of it.
The choice of a suitable grid is crucial both for numerical accuracy and need of computer
resources.
Different formulae have been proposed for the definition of grid point weights. In CRYSTAL
Becke and Savin weights are available; Becke weights are default, and provide higher accuracy.
[BECKE] Becke weights [66]. Default choice.
SAVIN
Savin weights [67]
A default grid is available in CRYSTAL, however the user can redefine it by the following
keywords:
61

RADIAL
Radial integration information
rec variable
meaning
NR
number of intervals in the radial integration [default 1]
RL(I),I=1,NR
radial integration intervals limits in increasing sequence [default 4.0]
(last limit is set to )
IL(I),I=1,NR
number of points in the radial quadrature in the I-th interval
[default 55].
ANGULAR
Angular integration information
rec variable
meaning
NI
number of intervals in the angular integration [default 1]
AL(I),I=1,NI
upper limits of the intervals in increasing sequence. The last limit must
be 9999.0 [default 9999.0]
LEV(I),I=1,NI accuracy level in the angular integration over the I-th interval; positive
for Lebedev level (see Lev in page 61) [default 13]
The default grid is a pruned (55,434) grid, having 55 radial points and a maximum number of
434 angular points in regions relevant for chemical bonding. Each atomic grid is split into ten
shells with different angular grids.
This grid is good enough for either single-point energy calculations or medium-accuracy
geometry optimizations. Due to the large pruning, the cost of the calculation is modest.
Default grid - corresponds to the sequence:
RADIAL
Keyword to specify the radial grid
1
Number of intervals in the radial part
4.0
Radial integration limits of the i-th interval
55
Number of radial points in the i-th interval
ANGULAR
Keyword to specify the angular grid
10
Number of intervals in the angular part
0.4 0.6 0.8 0.9 1.1 2.3 2.4 2.6 2.8 9999.0 Angular integration limits of the i-th interval
1 2 5 8 11 13 11 8 5 1
Angular grid accuracy level of the i-th interval
Information on the size of the grid, grid thresholds, and radial (angular) grid is reported in the
CRYSTAL output with the following format:
SIZE OF GRID=
40728
BECKE WEIGHT FUNCTION
RADSAFE =
2.00
TOLERANCES - DENSITY:10**- 6; POTENTIAL:10**- 9; GRID WGT:10**-14
RADIAL INTEGRATION
- INTERVALS (POINTS,UPPER LIMIT):
1( 55,
4.0*R)
ANGULAR INTEGRATION - INTERVALS (ACCURACY LEVEL [N. POINTS] UPPER LIMIT):
1(
1[
38]
0.4)
2(
2[
50]
0.6)
3(
5[ 110]
0.8)
4(
8[ 194]
0.9)
5( 11[ 302]
1.1)
6( 13[ 434]
2.3)
7( 11[ 302]
2.4)
8(
8[ 194]
2.6)
9(
5[ 110]
2.8)
10(
1[
38]9999.0)
Two more pre-defined grids are available which can be selected to improve accuracy by inputing
the following global keywords:
LGRID Large grid
Global keyword to choose a larger grid than default, corresponding to the sequence:
62

RADIAL
1
4.0
75
ANGULAR
5
0.1667 0.5 0.9 3.05 9999.0
2 6 8 13 8
The large grid is a pruned (75,434) grid, having 75 radial points and a maximum number
of 434 angular points in the region relevant for chemical bonding. Five shells with different
angular points are adopted to span the radial range as proposed by Gill et al. [63]. Due to
a larger number of radial points and less aggressive pruning, this grid gives more accurate
results than the default grid. It is recommended for high-accuracy energy calculations and
geometry optimizations. It is also recommended for periodic systems containing second-row
and third-row atoms (transition metals).
XLGRID Extra large grid
global keyword to choose an even larger integration grid, corresponding to the sequence:
RADIAL
1
4.0
75
ANGULAR
5
0.1667 0.5 0.9 3.5 9999.0
4 8 12 16 12
The extra-large grid is a pruned (75,974) grid, consisting of 75 radial points and 974 angular
points in the region of chemical interest. This is a very accurate grid and is recommended
when numerical derivatives of energy or related properties (i.e. spontaneous polarization) and
gradients have to be computed (e.g. bulk modulus, elastic constants, piezoelectric tensor,
ferroelectric transitions). It is also recommended for heavy atoms (fourth-row and heavier).
Unpruned grids
To switch from a pruned grid to the corresponding unpruned grid, only one shell must be
defined in the radial part and the same angular accuracy is used everywhere. The use of
unpruned grids increases the cost of the calculations by about 50-60% with respect to the
pruned grid.
For example, to transform the default grid to the corresponding unpruned grid input the
following data:
ANGULAR
1
9999.0
13
Numerical accuracy and running time are also controlled by the following keywords:
TOLLGRID
IG
DFT grid weight tolerance [default 14]
TOLLDENS
ID
DFT density tolerance [default 6]
63

The DFT density tolerance ID controls the level of accuracy of the integrated charge density
Nel (number of electron per cell):
Nel =
(r)dr =
P g+g
w(r
(r
(r
,
i)g

i)g

i)
cell
,,g,l
i
all contributions where |(ri)| < 10-ID or |(ri)| < 10-ID are neglected (see Chapter 6.11
for notation). The default value of ID is 6.
Grid points with integration weights less than 10-IG are dropped. The default value of IG is
14.
RADSAFE
RAD
[default 2]
BATCHPNT
BATCH
average number of points in a batch for numerical integration [default 100]
Default value of BATCH is 100. In the calculation of the exchange-correlation contribution to
the Kohn-Sham hamiltonian matrix elements and energy gradients, the grid is partitioned into
batches of points as suggested by Ahlrichs [68]. However, in CRYSTAL the number of points
per batch is not constant, as it depends on point density, so that BATCH does not correspond
to the maximun number of points in a batch. As a consequence, in special cases, memory
requirement may become huge and cause problems in dynamic allocation at running time.
When the program runs out of memory, it stops with the following error message:
ERROR *** sub_name *** array_name ALLOCATION
where array_name is one of the following:
DF0 KSXC1 KSXC2 KSXC2Y KSXC2Z DFXX DFYY DFZZ DFXY DFYZ DFXZ
RHO FRHO AXJ,AYJ,AZJ,VGRID GRAZ GRAY GRAZ
In these cases it is recommended that the value of BATCH be reduced, although this may
result in some degree of inefficiency (minimum value: 1).
4. DF energy gradient
[NEWTON]
The current default when computing DFT analytical gradients in CRYSTAL is to include
weight derivatives. Weight derivatives are mandatory when low quality grids are adopted.
NONEWTON
Calculation of DF gradient can also be performed without weight derivatives by specifying
the keyword NONEWTON. In this case the gradients relative to nuclear coordinates do not
rigorously sum to 0, and the accuracy of the gradient is more strongly influenced by the quality
of the integration grid.
EIGS - Check of basis set linear dependence
In order to check the risk of basis set linear dependence, it is possible to calculate the eigenvalues
of the overlap matrix. Full input (geometry, basis set, general information, SCF) is processed.
No input data are required. The overlap matrix in reciprocal space is computed at all the k-
points generated in the irreducible part of the Brillouin zone, and diagonalized. The eigenvalues
are printed.
64

The higher the numerical accuracy obtained by severe computational conditions, the closer
to 0 can be the eigenvalues without risk of numerical instabilities. Negative values indicate
numerical linear dependence. The crystal program stops after the check (even if negative
eigenvalues are not detected). This option is not active when running scfdir.
The Cholesky reduction scheme [69], adopted in the standard SCF route, requires linearly
independent basis functions.
MPP doesn' support EIGS.
END
Terminate processing of block3, general information, input. Execution continues. Subsequent
input records are processed, if required.
EXCHSIZE -Size of buffer for exchange integrals bipolar expansion
rec
variable meaning

ISIZE
size of the buffer in words
Size (words) of the buffer for bipolar expansion of exchange integrals (default value 100000).
The size of the buffer is printed in the message:
EXCH. BIPO BUFFER:
WORDS USED = XXXXXXX
or
EXCH. BIPO BUFFER TOO SMALL - TO AVOID I/O SET EXCHSIZE = XXXXXX
FIXINDEX - Geometry and basis set optimization tools
No input data required.
When the geometrical and/or the basis set parameters of the system are changed, maintaining
the symmetry and the setting, the truncation criteria of the Coulomb and exchange series,
based on overlap (Chapter 6) can lead to the selection of different numbers of bielectronic
integrals. This may be the origin of numerical noise in the optimization curve. When small
changes are made on the lattice parameter or on the Gaussian orbital exponents, the indices of
the integrals to be calculated can be selected for a reference geometry (or basis set), "frozen",
and used to compute the corresponding integrals with the modified geometry (or basis set).
This procedure is recommended only when basis set or geometry modifications are relatively
small. The reference geometry must correspond to the most compact structure, and the reference
basis set must have the lowest outer exponent.
This guards against the loss of significant
contributions after, for example, expansion of the lattice.
If estimate of resource is requested with TESTRUN, the reference geometry is used.
Two sets of input data must be given:
1. complete input (geometry, Section 1.1; basis set, Section 1.2; general information, Section
1.3; SCF, Section 1.4), defining the reference basis set and/or geometry;
2. "restart" option input, selected by one of the following keywords (format A) to be added
after the SCF input:
restart with new geometrical parameters
GEOM
insert geometry input, page 11
or
restart with new basis set
BASE
insert basis set input, page 16
or
65

restart with new basis set and new geometrical parameters
GEBA
insert geometry input, page 11
insert basis set input, page 16
BASE: the only modification of the basis set allowed is the value of the orbital exponent of the
GTFs and the contraction coefficient; the number and type of shells and AOs cannot change.
GEOM: geometry variation must keep the symmetry and the setting unchanged.
The resulting structure of the input deck is as follows:
0
Title
1
standard geometry input (reference geometry). Section 1.1
1b
geometry editing keywords (optional; Section 2.1)
END
2
standard basis set input (reference basis set). Section 1.2
2b
basis set related keywords (optional; Section 2.2)
END
3
General information keywords
3 FIXINDEX
END
4
scf input. Section 1.4
4b scf related keywords (optional; Section 2.4)
END
GEOM
BASE
GEBA



geometry input(p 11)
basis set input(p 16)
geometry input (p 11)
(block 1, 1b)
(block 2, 2b)
(block 1, 1b)
END
END
END
basis set input(p 16)
(block 2, 2b)
END
Warning: The reference geometry and/or basis set is overwritten by the new one, after
symmetry analysis and classification of the integrals. If the reference geometry is edited through
appropriate keywords, the same editing must be performed through the second input. Same
for basis set input.
If the geometry is defined through the keyword EXTERNAL, the reference geometry data
should be in a file linked to fortran unit 34, the wave function geometry in a file linked to
fortran unit 35.
See tests 5 and 20.
66

INTGPACK - Choice of bielectronic integrals package
rec
variable value
meaning

IPACK
[0]
s, sp shells POPLE; p, d shells ATMOL
1
ATMOL for Coulomb integrals;
POPLE for exchange integrals
2
POPLE for Coulomb integrals;
ATMOL for exchange integrals
3
ATMOL for Coulomb integrals;
ATMOL for exchange integrals
By default the bielectronic integrals are computed using a set of routines derived from Pople's
GAUSSIAN 70 package [15], if s and sp shells are involved, and by routines derived from
ATMOL [17] for p and d shells. The value of IPACK allows different choices. Integrals involving
p or d shells are always computed by ATMOL. The ATMOL package can compute integrals over
functions of any quantum number, but the symmetry treatment implemented in the CRYSTAL
package allows usage of s, p and d functions only. The use of sp shells (s and p orbitals sharing
the same exponent) reduces the time required to compute the integrals considerably.
MONSPLIT - Splitting of large monoelectronic integral files
rec
variable meaning

NFILE
number of files to be used [1] (max 10)
Very large basis sets can produce billions monoelectronic integrals to be stored, as the number
of monoelectronic integrals scales as the square of basis set size. The multipolar expansion
technique based on the atoms reduces the disk space up to a factor 3, compared to the value
printed as estimate. The distribution of the integrals over several disk files may be necessary,
if available disk space is limited.
MPP - Massive Parallel Execution
No input data required.
Massive Parallel Libraries are linked, and matrices in K space are distributed over the proces-
sors. MPP doesn't support:
Anderson mixing
ANDERSON
4
Broyden mixing
BROYDEN
4
Symmetry analysis of Bloch Functions
KSYMMPRT
4
Bloch Functions Symmetry Adapted
SYMADAPT
4
Printing of eigenvalues of overlap matrix in k space
EIGS
3
Restricted open shell wave function
ROHF
3
DFT fitting
DFT/FITTING
3
NOBIPOLA - Bipolar expansion approximation suppression
All the bielectronic integrals are evaluated exactly. No input data are required. The CPU time
in the integrals program can increase up to a factor 3.
NOMONDIR - Monoelectronic integrals on disk
No input data required.
In the SCF step bielectronic integrals are computed at each cycle, while monoelectronic inte-
grals are computed once and read from disk at each cycle.
67

PARAMPRT - - printing of parametrized dimensions
See input block 1, page 35.
POLEORDR - Maximum order of multipolar expansion
rec
variable meaning

IDIPO
maximum order of pole [4]
Maximum order of shell multipoles in the long-range zone for the electron-electron Coulomb
interaction. Maximum value = 6. See Section 6.3, page 141.
PRINTOUT - Setting of printing environment
See input block 1, page 35.
RHF [default]
A restricted closed-shell hamiltonian calculation is performed ([70, 7], Chapter 8 of ref. [20]).
Default choice.
ROHF
See UHF
SCFDIR
No input data required.
In the SCF step monoelectronic and bielectronic integrals are evaluated at each cycle. No
screening of the integrals is performed.
SEMIDIR
rec
variable meaning

IBUF
number of integral buffers PER PROCESSOR to store in memory ( a buffer
is about 4 Mbytes )
IWHAT flag telling the code what to do with the rest of the integrals
1 store on disk
0 recalculate at each cycle
This keyword allows Crystal to run in a semidirect mode. Some of the integrals are calculated
only once at the start of the calculation and held in memory, while the others can either be
stored on disk or recalculated each cycle. The code also orders the integral evaluation so
that the most expensive ones are preferentially stored in memory (and this will also help load
balancing in the parallel stuff).
Example
SEMIDIR
5 0
means: store the first 5 buffers worth of integrals in memory and recalculate the rest each
cycle. SEMIDIR should work with both MPP and SCFDIR as well so this means given enough
memory you can do a `conventional' calculation AND use the distributed memory portion of
the MPP code.
It can not be used with BIESPLIT.
68

SETINF - Setting of INF values
See input block 1, page 36
SETPRINT - Setting of printing options
See input block 1, page 37.
STOP
Execution stops immediately. Subsequent input records are not processed.
TESTPDIM
The program stops after processing of the full input (all four input blocks) and performing
symmetry analysis. The size of the Fock/KS and density matrices in direct space is printed.
No input data are required.
It may be useful to obtain information on the neighbourhood of the non equivalent atoms (up
to 3, default value; redefined through the keyword NEIGHBOR, input block 1, page 34).
TESTRUN - Integrals classification and selection
The symmetry analysis is performed, and the monoelectronic and bielectronic integrals classi-
fied and selected, according to the the truncation criteria adopted. The size of the Fock/KS
and density matrices (direct lattice) and the disk space required to store the bielectronic are
printed. The value printed as "disk space for monoelectronic integrals" is an upper limit. The
new technique of atomic multipolar expansion (not shell multipolar expansion as in CRYS-
TAL95) reduces the required space to about 1/3 of the printed value.
Full input (geometry, basis set, general information, SCF) is processed. No input data after
the keyword are required. This type of run is fast, and allows an estimate of the resources to
allocate for the traditional SCF wave function calculation.
TOLINTEG - Truncation criteria for bielectronic integrals
(Coulomb and HF exchange series)
rec
variable meaning

ITOL1
overlap threshold for Coulomb integrals- page 141 6
ITOL2
penetration threshold for Coulomb integrals-page 142 6
ITOL3
overlap threshold for HF exchange integrals-page 142 6
ITOL4
pseudo-overlap (HF exchange series-page 142) 6
ITOL5
pseudo-overlap (HF exchange series-page 142) 12
The five ITOL parameters control the accuracy of the calculation of the bielectronic Coulomb
and exchange series. Selection is performed according to overlap-like criteria: when the overlap
between two Atomic Orbitals is smaller than 10-IT OL, the corresponding integral is disregarded
or evaluated in a less precise way. Criteria for choosing the five tolerances are discussed in
Chapter 6.
TOLPSEUD - Truncation criteria for integrals involving ECPs
rec
variable meaning

ITPSE
overlap threshold for ECP integrals 6
The program evaluates only those integrals for which the overlap between the charge distri-
bution 0 g (page 139) and the most diffuse Gaussian defining the pseudopotential is larger


than a given threshold Tps=10-IT P SE (default value 10-6; it was 5 in CRYSTAL98).
69

T
n
n
T
n+1
c
T

n

T
n
n

-1
T

n
c
c
-1
T
n-1
c
3
T
c

3
T


2
2
c
T
c

2

T
1
1
T

c
1
c
Figure 2.1: Molecular Orbitals diagram for the Restricted Open Shell method (ROHF, left)
and for the Unrestricted Open Shell method (UHF, right)
UHF/ROHF - Hamiltonian for Open Shell Systems
For the description of systems containing unpaired electrons (such as molecules with an odd
number of electrons, radicals, ferromagnetic and anti ferromagnetic solids) a single determinant
is not an appropriate wave-function; in order to get the correct spin eigenfunction of these
systems, it is necessary to choose a linear combination of Slater determinants (whereas, in
closed shell systems, a single determinant gives always the appropriate spin eigenfunction)
([7, 71], Chapter 6 of ref. [20]).
In the Restricted Open Shell (ROHF) [70] Hamiltonian, the same set of molecular (i.e. crys-
talline) orbitals describes alpha and beta electrons; levels can be doubly occupied (by one
alpha and one beta electron, as in the RHF closed shell approach), singly occupied or left va-
cant. The wave-function is multi-determinantal; in the special case of half-closed shell systems,
where we can define a set of orbitals occupied by paired electrons and a second set occupied
by electrons with parallel spins, the wave-function is formed by a single determinant. This
particular mono-determinantal approach can be used in the open-shell part of CRYSTAL. The
correct spin state must be defined by the keyword SPINLOCK.
Another mono-determinantal approach for the study of open-shell systems is the UHF method
[72]. In this theory, the constraint of double occupancy is absent and electrons are allowed
to populate orbitals other than those occupied by the electrons. Energy levels corresponding
to a ROHF and UHF description are plotted in fig. 2.1.
The double occupancy constraint allows the ROHF approach to obtain solutions that are eigen-
functions of the spin operator, S2, whereas UHF solutions are formed by a mixture of spin states.
The greater variational freedom allows the UHF method to produce wave-functions that are
energetically more stable than the corresponding ROHF ones; another advantage of the UHF
method is that it allows solutions with locally negative spin density (i.e. anti ferromagnetic
systems), a feature that ROHF solutions can never exhibit.
To run ROHF with a conducting system does not make much sense. This is because tha
aufbau principle is only very roughly obeyed, and when one uses it (say for non-conductors
or molecules) it is usual to establish the requisit occupation pattern by carefull use of level
shifting.
To use ROHF for a conductor is extremely risky, as the converged state (if there is one) will
70

rarely be the ground state, and it will converge to different points with the smallest change in
numerical conditions (such as diagonalizer accuracy ........).
MPP doesn' support ROHF.
Related keywords
SPINLOCK definition of (n - n electrons)
BETALOCK definition of n electrons.
71

2.4
SCF input
Numerical accuracy control and convergence tools
ANDERSON
Fock matrix mixing
72
I
BROYDEN
Fock matrix mixing
73
I
FMIXING
Fock/KS matrix (cycle i and i-1) mixing 0
75
I
LEVSHIFT
level shifter no
77
I
MAXCYCLE
maximum number of cycles 50
78
I
SMEAR
Finite temperature smearing of the Fermi surface no
79
I
TOLDEE
convergence on total energy 5
81
I
TOLDEP
convergence on density matrix ?
81
I
TOLSCF
convergence on eigenvalues 6 and total energy 5
81
I
Initial guess
EIGSHIFT
alteration of orbital occupation before SCF no
74
I
EIGSHROT
rotation of the reference frame no
74
I
GUESSF
Fock/KS matrix from previous run
76

GUESSP
density matrix from a previous run
76

GUESSPAT
superposition of atomic densities
76

Spin-polarized system
ATOMSPIN
setting of atomic spin to compute atomic densities
73
I
BETALOCK
beta electrons locking
73
I
SPINLOCK
spin difference locking
80
I
SPINEDIT
editing of the spin density matrix
80
I
Auxiliary and control keywords
END/ENDSCF terminate processing of scf I keywords

KSYMMPRT
printing of Bloch functions symmetry analysis
77

NEIGHBOR
number of neighbours to analyse in PPAN
34
I
PARAMPRT
output of parameters controlling code dimensions
35

PRINTOUT
setting of printing options
35
I
NOSYMADA
No Symmetry Adapted Bloch Functions
78

SYMADAPT
Symmetry Adapted Bloch Functions (default)
81

SETINF
setting of inf array options
36
I
SETPRINT
setting of printing options
37
I
STOP
execution stops immediately
38

Output of data on external units
NOFMWF
wave function formatted output not written to fortran unit 98. 78

KNETOUT
Reciprocal lattice information + eigenvalues
77

SAVEWF
wave function data written every two SCF cycles
79

Post SCF calculations
POSTSCF
post-scf calculations when convergence criteria not satisfied
78

EXCHGENE
exchange energy evaluation (spin polarized only)
75

GRADCAL
analytical gradient of the energy
76

PPAN
population analysis at the end of the SCF no
78
ANDERSON
Anderson's method [73], as proposed by Hamann [74], is applied. No input data are required.
MPP doesn't support Anderson mixing.
72

ATOMSPIN - Setting of atomic spin
rec variable
meaning
NA
number of atoms to attribute a spin
LA,LS(LA),L=1,NA
atom labels and spin (1, 0, -1)
The setting of the atomic spins is used to compute the density matrix as superposition of
atomic densities (GUESSPAT must be SCF initial guess); it does not work with GUESSF
or GUESSP). The symmetry of the lattice may be reduced by attributing a different spin to
geometrically symmetry related atoms. In such cases a previous symmetry reduction should
be performed using the MODISYMM keyword. The program checks the symmetry taking
the spin of the atoms into account. If the spin pattern does not correspond to the symmetry,
the program prints information on the new symmetry, and then stops.
The formal spin values are given as follows:
1
atom spin is taken to be alpha;
0
atom spin is irrelevant;
-1
atom spin is taken to be beta.
In a NiO double-cell (four atoms, Ni1 Ni2 O1 O2) we might use:
atom
Ni1 Ni2
spin
1
1
for starting ferromagnetic solutions;
spin
1
-1
for starting anti ferromagnetic solutions;
SPINLOCK forces a given n - n electrons value: to obtain a correct atomic spin density
to start SCF process, the atomic spins must be set even for the ferromagnetic solution.
See test 30 and 31.
BROYDEN
rec variable
meaning
W0
W0 parameter in Anderson's paper [75]
IMIX
percent of Fock/KS matrices mixing when Broyden method is switched on
ISTART
SCf iteration after which Broyden method is active (minimum 2)
A modified Broyden [76] scheme, following the method proposed by Johnson [75], is applied
after the ISTART SCF iteration, with IMIX percent of Fock/KS matrices simple mixing. The
value of % mixing given in input after the keyword FMIXING is overridden by the new one.
Level shifter should be avoided when Broyden method is applied.
Suggested values:
FMIXING
80
BROYDEN
0.0001 50 2
MPP doesn't support Broyden mixing.
See test 43, a metallic Lithium 5 layers slab, PWGGA Hamiltonian.
BETALOCK - Spin-polarized solutions
rec variable meaning
INF97
n electrons
INF98
number of cycles the n electrons is maintained
The total number of of electrons at all k points can be locked at the input value. The number
of electrons is locked to (N + INF95)/2, where N is the total number of electrons in the
73

unit cell. INF95 must be odd when the number of electrons is odd, even when the number of
electrons is even.
EIGSHIFT - Alteration of orbital occupation before SCF
rec variable meaning
NORB
number of elements to be shifted
> 0 level shift of diagonal elements only
< 0 off-diagonal level shift
insert NORB records
if NORB > 0
IAT
label of the atom
ISH
number of the shell in the selected atom Basis Set
IORB
number of the AO in the selected shell
SHIF1
(or total, if Restricted) Fock/KS matrix shift
[SHIF2
Fock matrix shift - spin polarized only ]
if NORB < 0
IAT
label of the atom
ISH
number of the shell in the selected atom Basis Set
IORB1
number of the AO in the selected shell
IORB2
number of the AO in the selected shell
SHIF1
(or total, if Restricted) Fock/KS matrix shift
[SHIF2
Fock matrix shift - spin polarized only]
notes on irreducible atoms
Selected diagonal Fock/KS matrix elements can be shifted upwards when computing the initial
guess, to force orbital occupation. This option is particularly useful in situations involving d
orbital degeneracies which are not broken by the small distortions due to the crystal field, but
which are broken by some higher-order effects (e.g. spin-orbit coupling). The EIGSHIFT
option may be used to artificially locate the degeneracy in order to allow the system to converge
to a stable non-metallic solution. The eigenvalue shift is removed after the first SCF cycle.
Example: KCoF3 (test 38). In the cubic environment, two electrons would occupy the three-
fold degenerate t2g bands. A state with lower energy is obtained if the degeneracy is removed
by a tetragonal deformation of the cell (keyword ELASTIC), and the dxy orbital (see page
18 for d orbital ordering) is shifted upwards by 0.3 hartree.
Warning EIGSHIFT acts on the atoms as specified in input. If there are atoms symmetry-
related to the chosen one, hamiltonian matrix elements shift is not applied to the others. The
programs checks the symmetry compatibility, and, if not satisfied, stops execution.
The keyword ATOMSYMM prints symmetry information on the atoms in the cell.
EIGSHROT
Consider now the case of CoF2. The first six neighbors of each Co2+ ion form a slightly
distorted octahedron (2 axial and 4 equatorial equivalent distances); also in this case, then, we
are interested in shifting upwards the dxy orbital, in order to drive the solution towards the
following occupation:
:
all five d orbitals
:
dxz and dyz
The principal axis of the CoF6 octahedron, however, is not aligned along the z direction, but lies
in the xy plane, at 450 from the x axis. The cartesian reference frame must then be reoriented
before the shift of the dxy orbital.
To this aim the option EIGSHROT must be used. The reoriented frame can be specified in
two ways, selected by a keyword:
74

rec variable
meaning

MATRIX
keyword - the rotation matrix R is provided
R11 R12 R13
first row of the matrix.
R21 R22 R23
second row of the matrix.
R31 R32 R33
third row of the matrix.
or

ATOMS
keyword - the rotation is defined by three atoms of the crystal
IA
label of first atom in the reference cell
AL,AM,AN
indices (direct lattice, input as reals) of the cell where the first atom
is located
IB
label of second atom in the reference cell
BL,BM,BN
indices (direct lattice, input as reals) of the cell where the second
atom is located
IC
label of third atom in the reference cell
CL,CM,CN
indices (direct lattice, input as reals) of the cell where the third atom
is located
insert EIGSHIFT input records (Section 2.4, page 74)
When the rotation is defined by three atoms, the new reference frame is defined as follows :
Z-axis from atom 2 to atom 1
X-axis in the plane defined by atoms 1-2-3
Y-axis orthogonal to Z- and X-axis
Notice that the wave function calculation is performed in the original frame: the aim of the
rotation is just to permit a shift of a particular orbital. An equivalent rotation of the eigen-
vectors can be obtained in properties by entering the keyword ROTREF, so allowing AO
projected Density of States according to the standard orientation of the octahedron.
Example:
CoF2 example
END
Terminate processing of block 4, SCF, input (last input block). Execution continues. Subse-
quent input records are not processed.
EXCHGENE - Exchange energy calculation
In RHF calculations Coulomb and exchange integrals are summed during their calculation,
and there is no way to separate the exchange contribution to the total energy. In UHF/ROHF
calculations, this option allows the independent calculation and printing of the exchange con-
tribution to the total energy. See equation 6.19, page 142.
No input data are required. See tests 29, 30, 31, 38.
FMIXING - Fock/KS matrix mixing
rec variable meaning
IPMIX
percent of Fock/KS matrices mixing
The Fock/KS matrix at cycle i is defined as:
F = (1
i
- p)Fi + pFi-1
where p, input datum IPMIX, is the % of mixing. Too high a value of p (>50%) causes higher
number of SCF cycles and can force the stabilization of the total energy value, without a real
self consistency.
75

GRADCAL
No input data required.
Analytic calculation of the nuclear coordinates gradient of the HF, UHF, DFT energies after
SCF (all electrons and ECP).
If numerical gradient is requested for the geometry optimization (NUMGRAD, page 45),
analyical gradient is not computed.
GUESSF - Fock/KS matrix from a previous run
The Fock/KS matrix F0 (direct lattice) is read from disk (from fortran unit 20), and diagonal-
ized (after Fourier transformation to the reciprocal lattice), to compute the first cycle density
matrix. The data set containing F0 is written on fortran unit 9 at the end of a previous scf
run. No input data are required. When wave function information are stored formatted on
fortran unit 98, the data must be converted to binary (program convert, page 95, or keyword
RDFMWF, page 95 of the properties program). The two cases, the present one and that
used for the restart, must have the same symmetry, and the same number of atoms, basis func-
tions and shells. Atoms and shells must be in the same order. The program does not check the
1:1 oldnew correspondence. Different geometrical parameters, computational conditions or
exponents of the Gaussian primitives are allowed. In geometry and/or basis set optimization,
this technique will significantly reduce the number of SCF cycles. The following scheme shows
how to proceed.
1. First run to generate the Fock/KS matrix
Program
inp. block section
comments
crystal
0
1
Title
1
1.1
geometry input
2
1.2
basis set input
3
1.3
general information
4
1.4
scf input
save fortran unit 9 (binary wf) or 98 (formatted wf)
2. Second run - the Fock/KS matrix is read in as a guess to start scf
copy fortran unit 9 to unit 20 (or convert unit 98 and then copy)
Program
inp. block section
comments
crystal
0
1
Title
1
1.1
geometry input
2
1.2
basis set input
3
1.3
general information input FIXINDEX
4
1.4
scf input (GUESSF)
1b
1.1
geometry input present case
Warning The modification of the geometry may result in a different order in the storage of
the Fock/KS matrix elements associated to each overlap distribution in the present and the
previous run. To avoid the mismatch it is strongly recommended to classify the integrals of
the present case using the geometry of the previous case (FIXINDEX, page 65).
GUESSP - Density matrix from a previous run
The density matrix P0 (direct lattice) is read from disk (from fortran unit 20) to start the SCF
cycles. Same procedure as for GUESSF. No input data are required.
The denist matrix can be edited to modify the spin state. See SPINEDIT, page 80.
GUESSPAT - Superposition of atomic densities
The standard initial guess to start the SCF cycle is the superposition of atomic (or ionic)
densities. No input data are required. The electronic configuration of the atoms is entered as
76

a shell occupation number in the basis set input (page 16). Different electronic configurations
may be assigned to atoms with the same atomic number and basis set (but not symmetry
related) through the keyword CHEMOD (page 16).
KNETOUT - Reciprocal lattice information - Fock/KS eigenvalues
Reciprocal lattice information and Fock/KS eigenvalues are written on fortran unit 30. No
input data required.
NEWK must be called before KNETOUT to compute hamiltonian eigenvelues and FErmi
energy.
See Appendix D, page 176.
KSYMMPRT
Symmetry Adapted Bloch Functions [12, 13] (page 81)are used as basis for the Fock matrix
diagonalization. The results of the symmetry analysis in reciprocal space are printed. At each
k-point: number of point symmetry operators, number of active IRs, maximum IR dimension
and maximum block dimension in the Fock matrix factorization. TESTRUN stops execution
after this information is printed.
No input data required.
Extended information can be obtained by setting the value N of LPRINT(47) (keyword SET-
PRINT, page 37) before KSYMMPRT.
N
information
0
Basic Symmetry Information - At each k-point: list of point symmetry operators,
IR dimensions and number of Irreducible Sets.
> 0
Symmetry Information - At each k-point N: class structure, character table
and IR information concerning the K-Little Group. For the rest of the k-point
the same information as -1 is printed.
< -1
Full Symmetry Information - At each k-point: the same information as N > 0,
together with the matrix representatives of the point operators.
MPP doesn't support KSYMMPRT.
LEVSHIFT - Eigenvalue level shifting
rec variable value
meaning
ISHIFT
The level shifter is set to ISHIFT *0.1 hartree.
ILOCK
0
no locking
1
causes a lock in a particular state (eg non-conducting) even if the so-
lution during the SCF cycles would normally pass through or even con-
verge to a conducting state.
The eigenvalue level shifting technique is well known in molecular studies [77, 78], and may
also be used for periodic systems. The technique involves the addition of a negative energy
shift to the diagonal Fock/KS matrix elements (in the Crystalline Orbital basis) of the occupied
orbitals and thus reducing their coupling to the "unoccupied" set. This shift may be maintained
(ILOCK=1) or removed (ILOCK=0) after diagonalization. The former case causes a lock in a
particular state (eg non- conducting) even if the solution during the SCF cycles would normally
pass through or even converge to a conducting state. This option provides an alternative
damping mechanism to Fock/KS matrix mixing (FMIXING, page 75). The locking is effective
only if ISHIFT is large enough. If locking is used, the Fermi energy and the eigenvalues are
depressed by the value of the level shifter. Suggested values :
1. Normal cases require no mixing of Fock/KS matrices in successive cycles to converge:
ISHIFT=0 (default).
77

2. When 20% to 30% mixing of Fock/KS matrices is necessary, an ISHIFT value of between
1 and 3 (giving a level shift of 0.1 to 0.3 hartree) may produce an equivalent or even
superior convergence rate.
3. If serious convergence difficulties are encountered, ISHIFT=10 will normally be adequate,
corresponding to a level shift of 1 hartree. But it may happen that the system moves
towards an excited state, and no convergence is obtained.
See tests 29, 30, 31, 32, 38.
MAXCYCLE
rec variable meaning
NMAX maximum number of SCF cycles [50]
The possibility to modify the maximum number of SCF cycles allows: increasing the number
of cycles in case of very slow convergence (metals, magnetic systems, DFT);
The keyword POSTSCF forces saving wave function data in a file linked to fortran unit 9,
even if SCF ends before reaching convergence for "too many cycles".
NEIGHBOR/NEIGHPRT
See input block 1, page 34
NOSYMADA
The Symmetry Adapted Functions are not used in the Fock matrix diagonalization. No input
data are required. This choice increases the diagonalization CPU time when the system has
symmetry operators.
NOFMWF - Wave function formatted output
CRYSTAL is writing the formatted wave function on fortran unit 98 at the end of SCF by
default. This keyword deletes this feature.
PARAMPRT - - printing of parametrized dimensions
See input block 1, page 35.
POSTSCF
Calculation to be done after scf (gradient, population analysis) are performed even if conver-
gence is not reached. It may be useful when convergence is very slow, and scf ends for "TOO
MANY CYCLES" very close to the convergence criteria required.
No input data are required.
PPAN/MULPOPAN - Mulliken Population Analysis
Mulliken population analysis is performed at the end of SCF process. No input data are
required. Bond populations are analysed for the first n neighbours (n default value 3; see
NEIGHBOR, page 34, to modify the value).
Computed data:
1. a =
P g Sg orbital charges

g


2. sl =
a
l
shell charges
3. qA =
s
lA l atomic charges
78

4. b(A0, Bg) =
P g Sg bond populations between the non-equivalent atoms in
A
B


the unit cell (A0) and their first NVI neighbours (Bg in cell g). The printed values must
be multiplied by 2 when B=A to compare with standard molecular calculations.
PRINTOUT - Setting of printing environment
See input block 1, page 35.
SAVEWF
The wave function is written on fortran unit 79 every two cycles. The format is the same as
the one used for fortran unit 9 at the end of SCF.
No input data required.
SETINF - Setting of INF values
See input block 1, page 36
SETPRINT - Setting of printing options
See input block 1, page 37.
SMEAR
rec variable meaning
WIDTH temperature smearing of Fermi surface
Modifies the occupancy of the eigenvalues (fj) used in reconstructing the density matrix from
the step function, (equation 6.9, page 140) to the Fermi function;
( j - F )
f
k
j = (1 + e
b T
)-1
(2.10)
where F is the Fermi energy and kbT is input as WIDTH in hartree.
The smearing of the Fermi surface surface may be useful when studying metallic systems in
which the sharp cut-off in occupancy at F can cause unphysical oscillations in the charge
density. It may also result in faster convergence of the total energy with respect to k-point
sampling.
In density functional theory the use of Fermi surface smearing finds a formal justification in
the finite temperature DFT approach of Mermin [79]. In this case the "free energy" of the
system may be computed as;
F
=
E(T ) - kbT S(T )
Nstates
=
E - kbT
fi ln fi + (1 - fi) ln(1 - fi)
(2.11)
i
where S is the electronic entropy. Often we wish to compute properties for the athermal limit
(T=0). For the free electron gas the dependencies of the energy and entropy on temperature
are;
E(T )
=
E(0) + T 2
S(T )
=
2T
(2.12)
and so the quantity
79

0.72
Using E(T) kT=0.001H
+

0.7
Using E(T) kT=0.02H
+
Using E0 kT=0.02H
0.68
0.66
+
0.64
+
+
+
+
+
+
+
+
+
+
+
+
+
+
0.62

0.6
0.58











0.56

0
2
4
6
8
10
12
14
16
18
Figure 2.2: The surface energy (J/M2) of Li(100) for various numbers of layers in a slab model
showing the effects of WIDTH (0.02H and 0.001H) and the use of E(T) or E0
F (T ) + E(T )
E0 =
= E(0) + O(T 3)
(2.13)
2
may be used as an estimate of E(0).
Figure 2.2 shows the effect of WIDTH on the convergence of the Li(100) surface energy. Despite
the dense k-space sampling (IS=24, ISP=48) the surface energy is rather unstable at low
temperature (0.001H). There is a significant improvement in the stability of the solution for
higher values of WIDTH (0.02H) but use of E(T) results in a surface energy of 0.643 J/M2
significantly above that obtained by extrapolating E(T) to the T=0 limit (0.573 J/M2). The
use of E0 at WIDTH=0.02H results in an excellent estimate of the surface energy - 0.576 J/M2.
SPINEDIT - Editing of the spin density matrix
rec variable
meaning
N
number of atoms for which spin must be reversed
LB, L=1,N
atom labels
The spin density matrix from a previous run is edited to generate an approximate guess for a
new spin configuration. The sign of the elements of the spin density matrix of selected atoms
is reversed. The keyword SPINEDIT must be combined with UHF (input block 3, page 70)
and GUESSP.
Example: the ferro and anti ferromagnetic solution for the spinel MnCr2O4 can be obtained
by calculating only once the integrals, and using as guess to start the SCF process the density
matrix of the ferromagnetic solution with reversed signs on selected atoms.
SPINLOCK - Spin-polarized solutions
rec variable meaning
INF95
n-n electrons
INF96
number of cycles the difference is maintained
The difference between the number of and electrons at all k points can be locked at the
input value. The number of electrons is locked to (N + INF95)/2, where N is the total
80

number of electrons in the unit cell. INF95 must be odd when the number of electrons is odd,
even when the number of electrons is even.
Example. Bulk NiO. If a anti ferromagnetic solution is required, a double cell containing 2
NiO units must be considered (test 30). The two Ni atoms, related by translational symmetry,
are considered inequivalent. The number of electron is 72, each Ni ion is expected to have two
unpaired electrons.
INF95
type of solution
corresponding to the spin setting
0
anti ferromagnetic

4
ferromagnetic

See tests 29, 30, 32, 33, 37, 38.
STOP
Execution stops immediately. Subsequent input records are not processed.
SYMADAPT
The Symmetry Adapted Functions are used in the Fock matrix diagonalization. No input
data are required. This choice reduces the diagonalization CPU time when the system has
symmetry operators. Default choice.
Not supported by MPP execution.
TOLDEE - SCF convergence threshold on total energy
rec
variable meaning

ITOL
10-IT OL threshold for convergence on total energy
The default value for single point calculation is 5, but 7 in geometry optimization process.
TOLDEP - SCF convergence threshold on density matrix
rec
variable meaning

ITOL
10-IT OL threshold for convergence on P
to be written BC
TOLSCF - threshold total energy and eigenvalues
rec variable meaning
IT1
threshold on eigenvalues RMS 6
IT2
threshold on total energy diffference 5
The SCF ends when the root mean square (RMS) of the change in eigenvalues from cycle
i - 1 to i is less than 10-IT 1 or the change in the absolute value of the total energy is less
than 10-IT 2. The energy converges to 10-6 hartree/cell in 8-15 cycles in RHF calculation of
insulators. The number of cycles may increase for UHF spin polarized systems and can be very
high in DFT calculations. The tolerance values to be used depend on the required precision;
when IT2=x, IT1=x+2.
81

Chapter 3
Properties
One-electron properties and wave function analysis can be computed from the SCF wave func-
tion by running properties. At the end of the SCF process, data on the crystalline system
and its wave function are stored as unformatted sequential data on Fortran unit 9, and as
formatted data on Fortran unit 98. The wave function data can be transferred formatted from
one platform to another (see keyword RDFMWF, page 95).
The data set on fortran unit 9 (or 98) is read when running properties, and cannot be
modified. The data include:
1. Crystal structure, geometry and symmetry operators.
2. Basis set.
3. Reciprocal lattice k-points sampling information.
4. Irreducible Fock/KS matrix in direct space (Unrestricted: F, F).
5. Irreducible density matrix in direct space (Unrestricted: P+ P-).
The properties input deck is terminated by the keyword END. See Appendix E, page 179,
for information on printing.
3.1
Preliminary calculations
In order to compute the one-electron properties it is necessary to access wave function data as
binary data set: if binary data are not available on fortran unit 9, the keyword RDFMWF,
entered as 1st record, will read formatted data on fortran unit 98 and convert them in binary
on fortran unit 9.
Full information on the system is generated: :
a.
symmetry analysis information stored in COMMON areas
b.
reducible Fock/KS matrix
stored on Fortran unit 11
c.
reducible density matrix
c.1
all electron
stored on Fortran unit 13 (1st record)
c.2
core electron
stored on Fortran unit 13 (2nd record)
c.3
valence electron
stored on Fortran unit 13 (3rd record)
d.
reducible overlap matrix
stored on Fortran unit 3
e.
Fock/KS eigenvectors
stored on Fortran unit 10
1. a, b, c1, d, are automatically computed and stored any time you run the properties
program.
2. in unrestricted calculations, the total electron density matrix ( +) and the spin density
matrix ( - ) are written as a unique record on fortran unit 13.
82

3. The core and valence electron density matrices (c.2, c.3) are computed
only by the
NEWK option when IFE=1. They are stored as sequential data set on Fortran unit
13, after the all electron density matrix. Calculation of Compton profiles and related
quantities requires such information.
4. Properties can be calculated using a new density matrix, projected into a selected range
of bands (keyword PBAN,PGEOMW), range of energy (keyword PDIDE), or con-
structed as a superposition of the atomic density matrices relative to the atoms (or ions)
of the lattice (keyword PATO). In the latter case a new basis set can be used.
5. Each property is associated with a keyword. The keywords can be entered in any se-
quence. When a specific density matrix is calculated [band projected (PBAN), energy
projected (PDIDE), atomic superposition (PATO)], all the subsequent properties are
computed from that density matrix. The option PSCF restores the SCF density matrix.
6. The Fock/KS eigenvectors are computed on request by running the NEWK option. The
options that need such information are: PBAN - PDIDE - PGEOMW - ANBD -
BWIDTH - DOSS - EMBE - EMDL - EMDP - PROF.
7. The reciprocal space vectors refer by default to the primitive cell when defining the k-
points coordinates to compute the bands (BAND). The keyword CONVCELL allows
entering conventional cell coordinates.
8. The reciprocal space vectors (input and output) in RODK, EMDL, EMDP and PROF
refer to the conventional cell, not to the primitive cell. See Appendix A.5, page 163.
3.2
Properties keywords
RDFMWF
wave function data conversion formatted-binary (fortran unit 98 9)
Preliminary calculations
NEWK
Eigenvectors calculation
103
I
NOSYMADA
No symmetry Adapted Bloch Functions
78

PATO
Density matrix as superposition of atomic (ionic) densities
104
I
PBAN
Band(s) projected density matrix (preliminary NEWK)
104
I
PGEOMW
Density matrix from geometrical weights (preliminary NEWK) 105
I
PDIDE
Energy range projected density matrix (preliminary NEWK)
105
I
PSCF
Restore SCF density matrix
110

ROTREF
Rotation of the reference frame
110
I
Properties computed from the density matrix
ADFT
Atomic density functional correlation energy
85
I
BAND
Band structure
87
I
CLAS
Electrostatic potential maps (point multipoles approximation)
88
I
ECHG
Charge density maps and charge density gradient
93
I
ECH3
Charge density 3D maps
92
I
EDFT
Density functional correlation energy
93
I
POLI
Atom and shell multipoles evaluation
105
I
POTM
Electrostatic potential maps
107
I
POTC
Electrostatic properties
107
I
PPAN
Mulliken population analysis
78
XFAC
X-ray structure factors
111
I
Properties computed from the density matrix (spin-polarized systems)
ANISOTRO
Hyperfine electron-nuclear spin tensor
86
I
ISOTROPIC
Hyperfine electron-nuclear spin interaction - Fermi contact
96
I
POLSPIN
Atomic spin density multipoles
106
I
83

Properties computed from eigenvectors (after keyword NEWK)
ANBD
Printing of principal AO component of selected CO
85
I
BWIDTH
Printing of bandwidth
88
I
DOSS
Density of states
91
I
EMDL
Electron momentum distribution - line
94
I
EMDP
Electron momentum distribution - plane maps
95
I
PROF
Compton profiles and related quantities
109
I
New properties
POLARI
Berry phase calculations
113
I
SPOLBP
Spontaneous polarization (Berry phase approach)
114

SPOLWF
Spontaneous polarization (localized CO approach)
114

PIEZOBP
Piezoelectricity (Berry phase approach) preliminary
112

PIEZOWF
Piezoelectricity (localized CO approach) - preliminary
113

LOCALWF
Localization of Wannier functions
97
I
DIELEC
Optical dielectric constant
89
I
Auxiliary and control keywords
ANGSTROM
Set input unit of measure to
Angstrom
24

BASISSET
Printing of basis set, Fock/KS, overlap and density matrices
87

BOHR
Set input unit of measure to bohr
27

CHARGED
Non-neutral cell allowed (PATO)
48

CONVCELL
Reference cell for k-points coordinates (BAND)
88

END
Terminate processing of properties input keywords

FRACTION
Set input unit of measure to fractional
31

MAPNET
Generation of coordinates of grid points on a plane
100
I
NEIGHBOR
Number of neighbours to analyse in PPAN
34
I
PRINTOUT
Setting of printing options
35
I
RAYCOV
Modification of atomic covalent radii
35
I
SETINF
Setting of inf array options
36
I
SETPRINT
Setting of printing options
37
I
STOP
Execution stops immediately
38

SYMMOPS
Printing of point symmetry operators
40

Output of data on external units
ATOMIR
Coordinates of the irreducible atoms in the cell
86

ATOMSYMM
Printing of point symmetry at the atomic positions
27

COORPRT
Coordinates of all the atoms in the cell
29

EXTPRT
Explicit structural/symmetry information
31

GAUSS98
Printing of an input file for the GAUSS98 package
49

FMWF
Wave function formatted output. Section 3.2
95

INFOGUI
Generation of file with wf information for visualization
96

KNETOUT
Reciprocal lattice information + eigenvalues
77

MOLDRAW
generation of input file for the program MOLDRAW
32

84

ANBD
- Principal AO components of selected eigenvectors
rec variable value
meaning
NK
n
Number of k points considered.
0
All the k points are considered.
NB
n
Number of bands to analyse
0
All the valence bands + 4 virtual are analysed.
TOL
Threshold to discriminate the important eigenvector coefficients. The
square modulus of each coefficient is compared with TOL.
if NK > 0 insert
II
IK(J),J=1,NK Sequence number of the k points chosen (printed at the top of NEWK
output)
if NB > 0 insert
II
IB(J),J=1,NB
Sequence number of the bands chosen
The largest components of the selected eigenvectors are printed, along with the corresponding
AO type and centre.
ADFT/ACOR - A posteriori Density Functional atomic
correlation energy
The correlation energy of all the atoms not related by symmetry is computed. The charge
density is always computed using an Hartree-Fock Hamiltonian (even when the wave function
is obtained with a Kohn-Shamm Hamiltonian).
The input block ends with the keyword END. Default values are supplied for all the com-
putational parameters (different from the values set for a full system calculation). A new
atomic basis set can be entered. It must be defined for all the atoms labelled with a different
conventional atomic number (not the ones with modified basis set only).
BECKE
Becke weights [default] [66]
or
SAVIN
Savin weights [67]
RADIAL
Radial integration information
rec variable
meaning
NR
number of intervals in the radial integration [1]
RL(I),I=1,NR
radial integration intervals limits in increasing sequence [4.]
IL(I),I=1,NR
number of points in the radial quadrature in the I-th interval [55].
ANGULAR
Angular integration information
rec variable
meaning
NI
number of intervals in the angular integration [default 10]
AL(I),I=1,NI
angular intervals limits in increasing sequence. Last limit is set to 9999.
[default values 0.4 0.6 0.8 0.9 1.1 2.3 2.4 2.6 2.8]
IA(I),I=1,NI
accuracy level in the angular Lebedev integration over the I-th interval
[default values 1 2 3 4 6 7 6 4 3 1].
PRINT
printing of intermediate information - no input
PRINTOUT
printing environment (see page 35)
TOLLDENS
ID
DFT density tolerance [default 9]
TOLLGRID
IG
DFT grid weight tolerance [default 18]
NEWBASIS a new atomic basis set is input
insert complete basis set input, Section 1.2
85

ANGSTROM - unit of measure
Unit of measure of coordinates (ECHG, POTM, CLAS) See input block 1, page 24.
ANISOTRO - anisotropic tensor
rec
variable
meaning
A
keyword
enter one of the following keywords:
A3 ALL
The anisotropic tensor is evaluated for all the atoms in the cell
or
A6 UNIQUE
(alias NOTEQUIV) The anisotropic tensor is evaluated for all the non-
equivalent atoms in the cell
or
A6 SELECT
The anisotropic tensor is evaluated for selected atoms

N
number of atoms where to evaluate the tensor

IA(I),I=1,N
label of the atoms
A
PRINT
extended printing
The anisotropic hyperfine interaction tensor is evaluated. This quantity is given in bohr-3
and is transformed into the hyperfine coupling tensor through the relationship [80]
1000
1
T[mT] =
0NgNT = 3.4066697gNT
(0.529177 10-10)3 4
(see ISOTROPIC for the units and conversion factors). The elements of the T tensor at
nucleus A are defined as follows:
r2 ij - 3rAirAj
TA =
Pspin

A
g (r)dr
ij
g
(r)
r5


g
A
where rA = |r - A| and rAi = (r - A)i (ith component of the vector).
ATOMIR - coordinates of irreducible atoms
Cartesian and fractional coordinates of the irreducible atoms are printed.
No input data
required.
ATOMSYMM
See input block 1, page 27
86

BAND - Band structure
rec
variable value
meaning
A
TITLE
any string (max 72 characters).

NLINE
> 0
number of lines in reciprocal space to be explored (max 20)).
ISS
shrinking factor in terms of which the coordinates of the extremes of
the segments are expressed.
NSUB
total number of k points along the path.
INZB
first band
IFNB
last band
IPLO
0
eigenvalues are not stored on disk.
= 1
formatted output on Fortran unit 25 stored for plotting; see Appendix
F, page 181
LPR66
= 0
printing of eigenvalues
add NLINE records

coordinates of the line extremes in units of |bi|/ISS
I1,I2,I3
first point coordinates.
J1,J2,J3
last point coordinates.
The band structure along a given path and for a given range of bands is computed. The data
are printed on standard output and written on fortran unit 25 as formatted data for plotting
(if IPLO = 1) (see Appendix F, page 181).
1. Warning : does not run for molecules!! (0D)
2. The coordinates of the line extremes refer to the primitive cell. They can be expressed
with reference to the conventional cell by entering the keyword CONVCELL before
BAND.
3. For a correct interpretation of HF band-structure and DOS's, it must be stressed that
the HF eigenvalues are not a good approximation to the optical excitation spectrum of
the crystal. However, as discussed in section III.2 of reference [7] and in Chapter 2 of
reference [20], the band structures, in conjunction with total and projected DOS's, can
be extremely useful in characterizing the system from a chemical point of view.
4. Note on band extremes coordinates: in two-(one-) dimensional cases I3, J3 (I2,I3,J2,J3)
are formally input as zero (0). See test 3 and 6.
5. The only purpose of ISS is to express the extremes of the segments in integer units (see
tests 8-9). It does not determine the density of k points along the lines, which depends
only on NSUB. The number of k points for each line is computed by the program. The
step is constant along each line. The step is taken as close as possible to a constant along
different lines.
6. If symmetry adapted Bloch functions are used (default option), BAND generates sym-
metry information in k points different from the ones defined by the Monkhosrt net.
Eigenvectors computed by NEWK in k points corresponding to the Monkhosrt net are
not readable any more. To compute density of states and bands, the sequence must be:
BAND - NEWK - DOSS.
See tests 3, 4, 6, 7, 8, 9, 11, 12 and 30.
BASISSET - Printing of basis set and data from SCF
rec variable value
meaning
NPR
number of printing options.
if NPR = 0 insert prtrec (see page 37)
II
This option allows printing of the basis set and the computational parameters, and, on re-
quest (keyword PRINTOUT before BASISSET), of the Fock/KS matrix (FGRED), the
87

overlap matrix (OVERLAP), and the reducible density matrix (PGRED), in direct lattice
representation.
Printing options:
59 (Density matrix); 60 (Overlap matrix); 64 (Fock/KS matrix).
BOHR - unit of measure
Unit of measure of coordinates (ECHG, POTM, CLAS) See input block 1, page 27.
BWIDTH
- Printing of band width
rec variable meaning
INZB
first band considered
0
analysis from first valence band
IFNB
last band considered
0
analysis up to first 4 virtual bands
The Fock/KS eigenvalues are ordered in bands following their values. Band crossing is not
recognized.
CHARGED - charged reference cell
See input block 2, page 48.
CLAS
- Point charge electrostatic potential maps
rec variable value
meaning
IDER
0
potential evaluation
1
calculation of potential and its first derivatives
IFOR
0
point multipoles have to be evaluated by POLI option
1
point formal charges given as input
if IFOR = 0 insert
II
Q(I),I=1,NAF
formal net charge for all the NAF atoms in the unit cell (equivalent
and non equivalent, following the sequence printed at the top of the
properties printout)
insert MAPNET input records (page 100)
1. When IDER=0, the electrostatic potential is calculated at the nodes of a 2-dimensional
net in a parallelogram-shaped domain defined by the segments AB and BC (see keyword
MAPNET, page 100). The potential values are written formatted on fortran unit 25
(see Appendix F, page 181).
2. When IDER = 0, the electrostatic potential gradient is computed at the nodes of the
same grid. The x, y and z components are printed on the standard output.
3. The potential is generated by an array of point multipoles up to a maximum order IDIPO
defined in the POLI option input, or by atomic point charges given in input (IFOR=1;
IDIPO = 0 is set in that case).
4. The multipoles must be previously computed by running the option POLI when IFOR
is equal to zero.
CONVCELL
When computing the bands, the k-points coordinates are entered with reference to the prim-
itive cell. This option allows entering the coordinates of the k points with reference to the
conventional cell.
88

COORPRT
See input block 1, page 29.
DIEL/DIELECT - Optical dielectric constant
rec variable
meaning
A ENDDIEL
end of DIELECT input block
optional keywords
II
A PRINT
extended output
The electron density must be obtained by applying an electric field (keyword FIELD, page
41). The dielectric constant is calculated by using the concept of macroscopic average of the
total charge density (see for example Fu et al. [81]) and Poisson's equation. The charge density
is first averaged with respect to the (infinite) plane orthogonal to the field
1
(z) =
(z) dA
(3.1)
A
A
where A = |ab|, and a and b are the lattice parameters of the supercell orthogonal to the field
direction. When a Fourier representation of the charge density is used, the previous equation
becomes:
+
1
(z) =
F00 e-i 2 z
C
(3.2)
V =-
F00 are structure factors (note that the two first indices are always zero) calculated analytically
from the SCF crystalline orbitals depending now on the applied field. The quantity is then
averaged with respect to the z coordinate
z+z/2
1
(z) =
(z ) dz
(3.3)
z
z-z/2
that is
+
1
z
(z) =
F00 sinc

e-i 2 z
C
(3.4)
V
C
=-
where the sinc function is the cardinal sinus (sinc(u) = sin(u) ) and z has been chosen equal
u
to c; we can now apply Poisson's equation to (z):
2V (z) = -4(z)
(3.5)
z2
or
E(z) = 4(z)
(3.6)
z
because
V (z) = -E(z)
(3.7)
z
V (z), F (z) and (z) are the mean values of the macroscopic electric potential, electric field
and electron density at z position along the electric field direction.
Structure factors can be separated in a real and an imaginary part:
F00 = F
+ iF
(3.8)
00
00
89

Exploiting the following properties of the structure factors:
F000
=
Ne
(number of electrons in the supercell)
(3.9)
F
=
F
00
00-
F
=
00
-F00-
the real and imaginary parts of take the following form:
+
Ne
2
2 z
2 z
z
(z) =
+
F
cos
+ F
sin
sinc

(3.10)
V
V
00
C
00
C
C
=1
(z) = 0
(3.11)
As expected, the imaginary part is null. The Ne/V term can be disregarded, as it is exactly
canceled by the nuclear charges in the supercell.
According to equation 3.7, the local macroscopic field corresponds to minus the slope of V (z),
it has opposite sign with respect to the imposed outer field, according to the Lenz law, and is
obtained from (z)(eq. 3.6):
+
8
sin 2 z
cos 2 z
z
E(z) =
F
C
- F
C
sinc

(3.12)
V
00
2
00
2
C
=1
C
C
The corresponding macroscopic electric potential can be written as follows:
-
+
8
cos 2 z
sin 2 z
z
V (z) =
F
C
+ F
C
sinc

(3.13)
V
00
00
2
2
2
2
C
=1
C
C
Since -E and E0 have opposite sign, the ratio E0/(E0+E) is larger than one, and characterizes
the relative dielectric constant of the material along the direction of the applied field:
E0
=
(3.14)
E0 + E
The minimal input for the dielectric calculation is:
DIEL
END
for an extended output
DIEL
PRINT
END
In the former case the corresponding output is as follows:
90

*******************************************************************************
DIELECTRIC
MACROSCOPIC AVERAGE OF CHARGE DENSITY, ELECTROSTATIC FIELD AND POTENTIAL
*******************************************************************************
DIRECTION 3
NUMBER OF CONVENTIONAL CELLS (SUPERCELL VOLUME/SLIDING CELL VOL) 5
NUMBER OF FACTORS COMPUTED 300 NUMBER OF SAMPLING POINTS 7592
COORDINATES (A.U.):
SLIDING CELL HEIGHT 7.59103 SLIDING CELL VOLUME 218.71200
MIN SAMPLING POINT -18.97758 MAX SAMPLING POINT 18.97758
SAMPLING INTERVAL 37.95516 SAMPLING STEP 0.00500
MINIMUM MAXIMUM DELTA
DENSITY (A.U.) -0.00010 0.00010 0.00021
FIELD (A.U.) -0.00399 0.00399 0.00799
POTENTIAL (A.U.) -0.02950 0.02949 0.05899
POSITION MINIMUM MAXIMUM DISTANCE
DENSITY -18.58258 -0.39750 18.18508
FIELD -6.71753 6.71753 13.43506
POTENTIAL -18.97758 0.00250 18.98008
APPLIED FIELD 0.0100000000
RESPONCE FIELD -0.0039929327
RESULTANT FIELD 0.0060070673
DIELECTRIC CONSTANT 1.6647058467
DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD
The number of structure factors computed for a Fourier representation of the perturbed charge
density by default is equal to 300, the structure factors from F001 to F00 300.
DOSS
- Density of states
rec variable value meaning
NPRO
0
only total DOS is calculated
> 0
total DOS and NPRO projected densities are calculated. The maximum
number of projections is 15.
NPT
number of uniformly spaced energy values ( LIM019) where DOSs are
calculated, from bottom of band INZB to top of band IFNB.
INZB
first band considered in DOS calculation
IFNB
last band considered in DOS calculation
IPLO
0
DOSs are not stored on disk
1
formatted output on Fortran unit 25 for plotting (Appendix F, page 181).
2
formatted output on Fortran unit 24 for plotting (Appendix F, page 181).
NPOL
number of Legendre polynomials used to expand DOSS ( 25)
NPR
number of printing options to switch on
if INZB and IFNB < 0 insert
II
BMI,BMA
Minimum and maximum energy (hartree) values to span for DOSS. They
must be in a band gap
if NPRO = 0, insert NPRO records
II
N
> 0
DOS projected onto a set of N AOs
< 0
DOS projected onto the set of all AOs of the N atoms.
NDM(J),J=1,N vector NDM identifies the AOs (N>0) or the atoms (N<0) by their sequence
number (basis set order)
if NPR = 0, insert prtrec (see page 37)
II
Following a Mulliken analysis, the orbital (), atom (A) and total (tot) density of states can
91

be defined for a closed shell system as follows:
( ) = 2/VB
dkS(k)aj(k)a (k) eikg [
j
- j(k)]
(3.15)
j

g
BZ
A( ) =
( )
(3.16)
A
tot( ) =
A( )
(3.17)
A
where the last sum extends to all the atoms in the unit cell.
Bond population density of states are not computed.
1. Warning: does not run for molecules!
2. The NEWK option must be executed (to compute Hartree-Fock/KS eigenvectors and
eigenvalues) before running DOSS. The values of the input parameters IS and ISP of
NEWK have a consequent effect on the accuracy of the DOSS calculation. Suggested
values for IS: from 4 to 12 for 3-D systems, from 6 to 18 for 2-D and 1-D systems (Section
6.7, page 145). ISP must be equal or greater than 2*IS; low values of the ratio ISP/IS
can lead to numerical instabilities when high values of NPOL are used.
If BAND is called between NEWK and OSS, and symmetry adapted Bloch functions
are used (default option), the information generated by NEWK is destroyed. To compute
density of states and bands, the sequence must be: BAND - NEWK - DOSS.
3. DOSS are calculated according to the Fourier-Legendre technique described in Chapter
II.6 of reference 1, and in C. Pisani et al, ([82, 83]). Three computational parameters
must be defined: NPOL, IS, ISP. IS and ISP are entered in the NEWK option input.
4. NPOL is the number of Legendre polynomials used for the expansion of the DOS. The
value of NPOL is related to the values of IS and ISP, first and third input data of NEWK
option.
Suggested values for NPOL: 10 to 18.
5. Warning NEWK with IFE=1 must be run when spin-polarized solutions (SPIN-
LOCK, page 80) or level shifter (LEVSHIFT, page 77) were requested in SCF, to
obtain the correct Fermi energy and eigenvalues spectra.
6. Unit of measure: energy: hartree; DOSS: state/hartree/cell.
Printing options: 105 (density of states and integrated density of states); 107 (symmetrized
plane waves).
ECH3
- Electronic charge (spin) density on a 3D grid
rec
variable
meaning

NP
Number of points along the first direction
if non-3D system
keyword to choose the type of grid on the non-periodic direction(s):
92

SCALE
RANGE
length scales for non-periodic dimensions
boundary for non-periodic dimensions (au)
if 2D system
ZSCALE
ZMIN
ZMAX
if 1D system
YSCALE,ZSCALE
YMIN,ZMIN
YMAX,ZMAX
if 0D system
XSCALE,YSCALE,ZSCALE
XMIN,YMIN,ZMIN
XMAX,YMAX,ZMAX
The electronic charge or spin density is computed at a regular 3-dimensional grid of points.
The grid is defined by the lattice vectors of the primitive unit cell and user defined extents in
non-periodic directions. NP is the number of points along the first lattice vector (or XMAX-
XMIN for a molecule). Equally spacing is used along the other vectors. Non-periodic extents
may be specified as an explicit range (RANGE) or by scaling the extent defined by the atomic
coordinates (SCALE).
Formatted data are written in fortran unit 31. See Appendix F, page 181, for description of
the format.
ECHG - Electronic charge density maps and charge density gradient
rec variable value
meaning
IDER
n
order of the derivative - < 2
insert MAPNET input records (Section 3.2, page 100)
1. When IDER=0, the electron charge density (and in sequence the spin density, for
unrestricted wave functions) is calculated at the nodes of a 2-dimensional net in a
parallelogram-shaped domain defined by the segments AB and BC (see keyword MAP-
NET, page 100). The electron density values (electron bohr-3) are written formatted in
fortran unit 25 (see Appendix F, page 181).
2. When IDER = 0, the charge density gradient is computed at the nodes of the same grid.
The x, y and z components are printed on the standard output and written formatted
on fortran unit 25 (see Appendix F, page 181).
3. The electron charge density is computed from the density matrix stored on fortran unit
13. The density matrix computed at the last cycle of SCF is the default.
4. Band projected (keyword PBAN), energy projected (keyword PDIDE) or atomic su-
perposition (keyword PATO) density matrices can be used to compute the charge den-
sity. The sequence of keywords must be: (NEWK-PBAN-ECHG), (NEWK-PDIDE-
ECHG) or (PATO-ECHG).
EDFT/ENECOR -A posteriori Density Functional correlation energy
Estimates a posteriori the correlation energy via a DF of the HF density. It is controlled
by keywords. The input block ends with the keyword END. All the keywords are optional,
as default values for all the integration parameters are supplied by the program, to obtain
reasonably accurate integration of the charge density. Please check the integration error printed
on the output.
93

BECKE
Becke weights [default] [66]
or
SAVIN
Savin weights [67]
RADIAL
Radial integration information
rec variable
meaning
NR
number of intervals in the radial integration [1]
RL(I),I=1,NR
radial integration intervals limits in increasing sequence [4.]
IL(I),I=1,NR
number of points in the radial quadrature in the I-th interval [55].
ANGULAR
Angular integration information
rec variable
meaning
NI
number of intervals in the angular integration [default 10]
AL(I),I=1,NI
angular intervals limits in increasing sequence. Last limit is set to 9999.
[default values 0.4 0.6 0.8 0.9 1.1 2.3 2.4 2.6 2.8]
IA(I),I=1,NI
accuracy level in the angular Lebedev integration over the I-th interval
[default values 1 2 3 4 6 7 6 4 3 1].
PRINT
printing of intermediate information - no input
PRINTOUT
printing environment (see page 35)
TOLLDENS
ID
DFT density tolerance [default 9]
TOLLGRID
IG
DFT grid weight tolerance [default 18]
EMDL - Electron Momentum Density - line maps
rec variable value
meaning
N
number of directions ( 10)
PMAX
maximum momentum value (a.u.) for which the EMD is to be calcu-
lated
STEP
interpolation step for the EMD
IPLO
0
no data stored on disk
1
formatted output on Fortran unit 25 for plotting (Appendix F, page
181).
2
formatted output on Fortran unit 24 for plotting (Appendix F,
page181).
LPR113 = 0
printing of EMD before interpolation
(K(I,J),
directions in oblique coordinates
I=1,3),J=1,N
NPO
number of orbital projections ( 10)
NPB
number of band projections( 10)
if NPO = 0 insert NPO sets of records
II
NO
number of A.O.'s in the I-th projection
IQ(I),I=1,NO
sequence number of the A.O.'s in the I-th projection - basis set se-
quence.
if NPB = 0 insert NPB sets of records
II
NB
number of bands in the I-th projection
IB(I),I=1,NB
sequence number of the bands in the I-th projection
Warning EMDL does not work for UHF wave functions.
The Electron Momentum Density is calculated along given directions (equation 6.23, page
146). The electron momentum distribution, EMD, is a non-periodic function; it falls rapidly
to zero outside the first Brillouin zone. (0) gives the number of electrons at rest. The oblique
coordinates directions given in input refer to the conventional cell, not to the primitive cell,
94

for 3D systems.
Example: in a FCC system the input directions refer to the orthogonal unit cell frame (sides
of the cube) not to the primitive non-orthogonal unit cell frame.
EMDP - Electron Momentum Density - plane maps
rec variable value
meaning
NP
number of planes (< 5)
IS
shrinking factor.
IPLO
0
no data stored on disk.
1
formatted output on Fortran unit 25 for plotting
LPR115
printing of band projections
insert NP set of records
(L1(J),J=1,3),
fractional coordinates of the reciprocal lattice vectors that identify the
(L2(J),J=1,3)
plane
PMX1
maximum p value along the first direction
PMX2
maximum p value along the second direction
NPO
number of orbital projections ( 10)
NPB
number of band projections( 10)
if NPO = 0 insert NPO set of records
II
NO
number of A.O.'s in the I-th projection
IQ(I),I=1,NO
sequence number of the A.O.'s in the I-th projection - basis set order
if NPB = 0 insert NPB set of records
II
NB
number of bands in the I-th projection
IB(I),I=1,NB
sequence number of the bands in the I-th projection
Warning EMDP does not work for UHF wave functions.
Calculation of electron momentum density on definite planes (equation 6.23, page 146).
The fractional coordinates of the reciprocal lattice vectors given in input refer to the conven-
tional cell, not to the primitive cell, for 3D systems.
Example: in a FCC system the input directions refer to the orthogonal unit cell frame (sides
of the cube) not to the primitive non-orthogonal unit cell frame.
END
Terminate processing of properties input. Normal end of the program properties. Subse-
quent input records are not processed.
EXTPRT
See input block 1, page 31
FMWF - Wave function formatted output
The keyword FMWF, entered in properties input, generates formatted wave function data
(same data are written on fortran unit 9, unformatted, at the end of SCF) on fortran unit
98 (LRECL=80). The formatted data can then be transferred to another platform. No input
data required.
The resources requested to compute the wave function for a large system (CPU time, disk
storage) may require a mainframe or a powerful workstation, while running properties is
not so demanding, at least in terms of disk space. It may be convenient computing the wave
function on a given platform, and the properties on a different one.
The keyword RDFMWF, entered in the first record of the properties input deck reads
formatted data from fortran unit 98, and writes unformatted data to fortran unit 9. The key
dimensions of the program computing the wave function and the one computing the properties
are checked. If the dimensions of the arrays are not compatible, the program stops, after
printing the PARAMETER statement used to define the dimension of the arrays in the program
95

which computed the wave function. The sequence of the operations, when transferring data
from one platform to another is the following:
platform
program
input
action
1
properties
FMWF
wave function formatted to fortran unit 98
ftp fortran unit 98 file from platform 1 to platform 2
2
properties
RDFMWF wf read from fortran unit 98, written to fortran unit
9
FRACTION - unit of measure
Unit of measure of coordinates in the periodic direction (ECHG, POTM, CLAS) See input
block 1, page 31.
GAUSS98
See input block 2, page 49
INFOGUI/INFO - output for visualization
Information on the system and the computational parameters are written formatted on fortran
unit 32, in a format suitable for visualization programs. No input data required.
ISOTROPIC - Fermi contact - Hyperfine electron-nuclear
spin interaction isotropic component
rec variable
meaning
A keyword
enter one of the following keywords:
ALL
Fermi contact is evaluated for all the atoms in the cell
or
UNIQUE Fermi contact is evaluated for all the non-equivalent atoms in the cell
or
SELECT Fermi contact is evaluated for selected atoms
N
number of atoms where to evaluate Fermi contact
IA(I),I=1,N label of the atoms
A PRINT
extended printing
The spin density at the nuclei ( spin(rN) ) is evaluated. This quantity is given in bohr-3 and
is transformed into the hyperfine coupling constant aN[mT] through the relationship [80]
1000
2
aN[mT] =
0 N gN spin(rN) = 28.539649 gN spin(rN)
(0.529177 10-10)3 3
where
0 = 4 10-7 = 12.566370614 10-7[T2J-1m3]
(permeability of vacuum)
N = 5.0507866 10-27[JT-1]
(nuclear magneton)
the nuclear gN factors for most of the nuclei of interest are available in the code and are taken
from [80]. Conversion factors:
aN[mT]gee
aN[MHz] =
= 28.02.6 aN[mT]
109h[Js]
aN [MHz]108
aN[cm-1] =
= 0.33356410 10-4 aN[MHz]
c[ms-1]
96

aN[J] = ge e 10-3aN[mT] = 1.856954 10-26aN[mT]
where:
e = 9.2740154 10-24 [JT-1]
(bohr magneton)
ge = 2.002319304386
(free-electron g factor)
c = 2.99792458 108 [ms-1]
(speed of light in vacuum)
h = 6.6260755 10-34 [Js]
(Planck constant)
KNETOUT
See input block 4, page 77
LOCALWF
- Localization of Wannier Functions (WnF)
The localization of WnF is controlled by keywords. The LOCALWF block is finished by the
END keyword. The plotting options (when requested) must be in separated blocks imme-
diately following the first block. Each block should consist of the index number of WnF to
be plotted followed by the corresponding MAPNET block. For UHF calculations two set of
blocks must be inserted for the and electrons. The final WnF are written on unit 80.
Definition of the set of bands considered in the localization process
VALENCE
Valence bands are chosen for localization.
OCCUPIED
All the occupied bands are chosen for localization.(default)
INIFIBND
rec
variable
value
meaning
IBAN
initial band considered for localization
LBAN
last band considered for localization
BANDLIST
rec
variable
meaning
NB
number of bands considered
LB(I),I=1,NB
labels of the bands.
Tolerances for short and large cycles
A short cycle is a sequence of wannierization and Boys localization steps. The accuracies in
both, the calculation of the Dipole Moments (DM) and the definition of the phases assigned
to each periodically irreducible atom, are controlled so that they increase as the localization
process evolves. This results in a significant saving of computational cost. Therefore, each
time a given criterion is fulfilled, the accuracy in the DM evaluation increases and a new large
cycle starts.
CYCTOL
97

rec
variable
value
meaning
ITDP0
> 0
Initial tolerance used to calculate the DM matrix elements:
10-ITDP0 2
ITDP
> 0
Final tolerance used to calculate the DM matrix elements:
10-ITDP 5
ICONV
> 0
Convergence criterion to finish a large cycle: ABS(ADI(N) -
ADI(N-1)) < 10-ICONV, where ADI is the atomic delocaliza-
tion index and N is the short cycle number 5
PHASETOL
rec
variable
value
meaning
ITPH0
> 0
10-ITPH0 is the initial tolerance on the atomic charge popula-
tion to attribute the phase to atoms in the wannierization step
2
ITPH
> 0
10-ITPH is the final tolerance used to attribute this phase 3
ICHTOL
> 0
DM tolerance of the cycle where ITPH0 changes to ITPH.
ITDP0+1
General Keywords:
RESTART
With this option the WnF of a previous job are read from unit 81 (in the same format as output
unit 80). The RESTART option set the same choice of the active bands as the previous job
(and override any other definition) and the tolerances are, by default, the last attained in the
previous calculation. The latter can be changed using CYCTOL and PHASETOL.
MAXCYCLE
rec
variable
value
meaning
NCYC
> 0
maximal number of short cycles for the iterative process 30
PRINTPLO
rec
variable
value
meaning
IPRT
0
Does not print Wannier coefficients
> 0
Prints Wannier coefficients at each cycle up to the IPRT-th
star of direct lattice vectors 0
IPRP
0
Prints population analysis only at the end of the localization.
=0
Print analysis at each W-B cycle 0
ITPOP
Only atomic population larger than 10-ITPOP are printed 2
IPLOT
0
WnFs are not computed for plot
= 0
WnFs are computed in a grid of points, IPLOT being the
number of stars of direct lattice vectors taken into account for
WnF coefficients. Data are written in fortran unit 25 0
BOYSCTRL
Parameters that control the Boys localization step. Convergence of the process is achieved
when the orbital-stability conditions: Bst = 0; Ast > 0, (see Pipek and Mezey 1989 [84]) are
fulfilled for all pairs st of WnFs. Additionally, in order to avoid nearly free rotations (for
instance in core or lone-pair WnFs) those pairs st with Ast close to 0 are not mixed (frozen).
rec
variable
value
meaning
IBTOL
10-IBTOL is the threshold used for the stability condition on
Bst. 4
IBFRZ
If for a pair of WnFs st, |Ast| 10-IBFRZ, then the corre-
sponding WnFs are not mixed. 4
MXBCYC
Maximum number of cycles allowed in the Boys localization
process 500
98

Initial guess options
The iterative localization process of the WnFs needs to start from a reasonable initial guess. By
default the starting functions are obtained automatically from the eigenvectors at the point.
When required (pure covalent bonds that link atoms in different unit cells), a pre-localization
is performed using a scheme similar to that suggested by Magnasco and Perico (1967) [85].
IGSSCTRL
Parameters used to control the pre-localization of the point eigenvectors.
rec
variable
value
meaning
CAPTURE
The capture distance between atoms I and J is given by
CAPTURE (RAYCOV(I) + RAYCOV(J)). An inter-atomic
distance lower than the capture indicates that I and J can
be covalently bonded 1.0 RAYCOV, covalent radius (default
value page 35).
MPMAXIT
Maximum
number
of
iterations
in
the
pre-localization
process 200
ICNVMP
10-ICNVMP is the convergence threshold for the Magnasco-
Perico pre-localization 8
IOVPOP
Just those pairs of atoms whose overlap population are greater
than 10-IOVPOP are considered covalently bonded 4
The initial guess can be given as input in two ways:
IGSSVCTS
The eigenvectors and the phases are given explicitly after the LOCALWF block (and before
the plot parameters if required), in the following format.
rec
variable
value
meaning
NGUES
Number of bands whose phase is pre-assigned such that the
involved atoms are to be located in a given cell.
insert 2 NGUES records
IB
IGAT(I,IB),I=1,NAF
Index of the direct lattice vector corresponding to the cell
where atom I is expected to have the largest charge population
in Wannier IB (NAF is the number of atoms per cell)
insert:
GUESSV(I),I=NDF*NOCC
where NDF is the basis set dimension and NOCC the num-
ber of bands considered. GUESSV is a matrix containing the
initial guess vectors for the iterative Wannier-Boys procedure
(GUESSV is written in free format as a one-dimensional ar-
ray).
IGSSBNDS
Use this option to explicitly indicate the WnFs that are to be assigned to covalent bonds.
rec
variable
value
meaning
NBOND
Number of covalent bonds given as input.
insert NBOND records
NAT1
Label of the first atom of the bond, it is assumed to be located
in the reference cell.
NAT2
Label of the second atom of the covalent bond
IC1,IC2,IC3
Indices of the cell where atom NAT2 is located
NBNDORD
Bond Order
99

Plotting the WnFs
If IPLOT = 0 insert after the keyword block (terminated by END):
rec
variable
value
meaning
NWF
number of WF to plot
insert NWF blocks of data
NUMBWF
sequence number (output order) of the WF to plot
MAPNET input data (Section 3.2, page 100)
The WnFs and the WnF densities (in this order) within the selected regions are given in
fortran unit 25.
1. The NEWK option must be executed before running LOCALWF, to compute the
crystalline orbitals.
2. The number of k points required for a good localization depends on the characteristics
of the bands chosen. For core electrons or valence bands in non-conducting materials, a
IS twice than that used in the SCF part is enough to provide well localized WnFs. For
valence bands in semiconductors or conduction bands the k-point net is required to be
denser, but there are no recipes to determine a priori the optimum IS value. However, a
necessary condition for the WnFs to be well represented, is that the volume in terms of
number of unit cells of the cluster that contains the set of WnFs up to AO coefficients
of 10-IT DP , given as output, should be lower than the number of k-points in the net
(IS**IDIM, being IDIM the dimensionality of the system).
3. The efficiency of the localization can be controlled using the CYCTOL parameters. In
most cases, increasing ITDP and/or ICONV leads to larger and more accurate localiza-
tion of the WnFs.
4. The RESTART option admits MAXCYCLE = 0, then the program just reconstructs
all the information about the WnFs given in fortran unit 81 but does not continue the
localization. This two options together with a IS=1 in NEWK is useful to perform the
analysis of the WnFs after localization by means of the PRINTPLO option.
Bibliography
N. Ashcroft, D. Mermin "Solid State Physics", Holt Rinehart and Winston: New York, 1976.
P.-O. L
owdin (Editor) "Quantum Theory of Atoms, Molecules and the Solid State", Academic:
New York, 1966.
S. F. Boys Rev. Mod. Phys 32 (1960) 296.
J. M. Foster and S. F. Boys Rev. Mod. Phys 32 (1960) 300.
J. Pipek and P. G. Mezey J. Chem. Phys 90 (1989) 4916.
V. Magnasco and A. Perico, J. Chem. Phys. 47 (1967) 971.
MAPNET
- coordinates of grid points on a plane
This is a dummy keyword, to explain the way is generated the grid of points in which is
evaluated a given function F: charge density and spin density (ECHG), electrostatic potential
(CLAS, POTM). The graphic representation of the resulting 2D function is made by external
software.
100

rec variable
meaning
NPY
number of points on the B-A segment.
A keyword
enter a keyword to choose the type of coordinate:
COORDINA
XA,YA,ZA
cartesian coordinates of point A
XB,YB,ZB
cartesian coordinates of point B
XC,YC,ZC
cartesian coordinates of point C
or
ATOMS
IA
label of the atom at point A
AL,AM,AN
indices (direct lattice, input as reals) of the cell where the atom is located
IB
label of the atom at point B
BL,BM,BN
indices (direct lattice, input as reals) of the cell where the atom is located
IC
label of the atom at point C
CL,CM,CN
indices (direct lattice, input as reals) of the cell where the atom is located
optional keyword
II
RECTANGU definition of a new A'B'C'D' rectangular window, with B'C' on BC, A'D'
on AD and diagonals A'C'=B'D'=max(AC,BD) (see Fig 3.1)
optional keyword
II
MARGINS
definition of a new A",B",C",D" window including ABCD (or A'B'C'D')
(see Fig 3.2)
ABM
margins along AB
CDM
margins along CD
ADM
margins along AD
BCM
margins along BC
optional keyword
II
PRINT
printing of the values of the function in the net
ANGSTROM cartesian coordinates in
Angstrom (default)
BOHR
cartesian coordinates in bohr
FRACTION
cartesian coordinates in fractionary units
END
end of MAPNET input block
1. Function F is mapped in a ABCD parallelogram-shaped domain defined by the sides AB
and BC of any ABC angle. F is calculated at the nAB * nBC nodes of a commensurate
net (nAB and nBC integers). If C B, F is calculated along the line AB.
2. nBC is set by the program such that all points in the net are as equally spaced as possible
( AB BC ).
3. Output for plot on formatted (logical record length 80) fortran unit 25
4. The position of the three points A, B and C can be specified in two alternative ways:
COORDINA the cartesian coordinates of the three points are given in bohr /

Angstrom / fractionary units (default
Angstrom; see Section 2.1,
page 24)
ATOMS
A,B,C correspond to the position of 3 nuclei, identified by their
sequence number in the reference cell, and the crystallographic in-
dices of the cell in which they are located (input as real numbers).
5. The symmetry is used to restrict the calculation of the function to the irreducible part of
the parallelogram chosen. To maximize the use of symmetry, the points of the net should
include the low multiplicity positions in the selected plane. For example, B=(0,0,0),
A=(a,0,0), C=(0,b,0) (a,b lattice vectors). Choose NPY=4n+1 for (100) faces of cubic
crystals, or NPY = 6n+1 for (0001) faces of hexagonal crystals.
101

A'
A
DD'
















BB
C
C'
Figure 3.1: Definition of the window where the function F is mapped Effect of optional keyword
RECTANGU.
A'
D'

T



ADM



A
D

c
















ABM

CDM


'
E
'
E




















B
T
C

BCM


c

B'
C'
Figure 3.2: Definition of frame around the original window where the function F is mapped.
Effect of optional keyword MARGINS.
102

MOLDRAW
See input block 1, page 32
NEIGHBOR/NEIGHPRT
See input block 1, page 34
NEWK - Fock/KS eigenvectors
rec variable value
meaning
if system is periodic, insert
II
IS
Shrinking factor for reciprocal space net (Monkhorst net). The num-
ber NKF of k points, where the Fock/KS matrix is diagonalized, is
roughly proportional to ISIDIM /M V F where IDIM denotes the pe-
riodic dimensionality of the system, and MVF denotes the number of
point symmetry operators (see page 19).
ISHF
n
dummy variable
ISP
Shrinking factor of the secondary reciprocal space net (Gilat net) for
the evaluation of the Fermi energy and density matrix.
if system is periodic and IS=0, insert
II

Shrinking factors of reciprocal lattice vectors
IS1
Shrinking factor along B1
IS2
Shrinking factor along B2
IS3
Shrinking factor along B3.
IFE
0
no Fermi energy calculation is performed;
1
Fermi energy is computed, by performing integration on the new k
points net. Total, valence and core density matrices are written on
Fortran unit 13
NPR
number of printing options to switch on
if NPR = 0 insert prtrec (see page 37)
II
The Fock/KS eigenvectors are computed at a number of k points in reciprocal space, defined by
the shrinking factor IS, and written unformatted in fortran unit 10 (in the basis of symmetry
adapted Bloch functions) and in fortran unit 8 (in the basis of AO). Eigenvalues and related
information (coordinates of k points in reciprocal lattice, weights etc) are written on fortran
unit 30 by inserting the keyword KNETOUT (page 77). See Appendix D, page 176.
1. The Fock/KS matrix in direct space is always the SCF step final one. If the SCF con-
vergence was poor, and convergence tools were used, eigenvalues and eigenvectors may
be different from the ones that could be obtained after one more cycle.
2. The shrinking factors IS and ISP (Section 6.7, page 145) can be redefined with respect
to the ones used in the SCF process. If this value is smaller than the one used in the scf
step, numerical inaccuracy may occur in the Fourier transform of the Fock/KS matrix,
Fg Fk (Chapter 6, equation 6.5).
3. A Fermi energy calculation must be performed (IFE=1) to run PROF the Compton
profiles option, PBAN and PDIDE in order to compute the weight associated to each
eigenvalue.
4. Warning NEWK with IFE=1 must be run to obtain the correct Fermi energy and eigen-
values spectra when a shift of eigenvalues was requested in SCF (LEVSHIFT, page 77;
SPINLOCK, page 80; BETALOCK, 73.
A new density matrix is computed. If the convergence of scf was poor, and convergence
tools were used (FMIXING, LEVSHIFT, ..), the density matrix obtained from the eigen-
vectors computed by NEWK may be different from the matrix that could be calculated
with one more scf cycle. Properties depending on the density matrix may be different if
computed before or after NEWK.
103

5. if BAND is called after NEWK, and symmetry adapted Bloch functions are used
(defaul option), the information generated by NEWK is destroyed. For instance, to
compute density of states and bands, the sequence must be: BAND - NEWK - DOSS.
The sequence NEWK BAND DOSS will give the error message:
NEWK MUST BE CALLED BEFORE DOSS
Printing options: 59 (Density matrix - direct lattice); 66 (Hamiltonian eigenvalues); 67 (Hamil-
tonian eigenvectors).
NOSYMADA
See input block 4, page 78
PARAMPRT - - printing of parametrized dimensions
See input block 1, page 35.
PATO
- Density matrix as superposition of atomic densities
rec variable value
meaning
IBN
0
density matrix computed with the same basis set as in the crystal cal-
culation.
= 0
new basis set and/or new electron configuration is given
NPR
= 0
printing of the density matrix for the first NPR direct lattice vectors
if IBN = 0 insert basis set input (page 16)
II
1. The PATO option is used for calculating crystal properties, such as charge density
(ECHG), structure factors (XFAC) with a periodic density matrix obtained as a su-
perposition of atomic solutions (periodic array of non interacting atoms). The density
matrix is written on fortran unit 13.
2. The atomic wave function is computed by the atomic program [14], properly handling
the open shell electronic configuration.
3. If the basis set used for the crystalline calculation (given as input of the integral part)
is not suitable for describing a free- atom or free-ion situation, a new basis set can be
supplied (see Section 1.2). When this option is used (IBN.NE.0) the basis set of all the
atoms with different conventional atomic number has to be provided.
4. The electronic configuration of selected atoms may be modified (CHEMOD in basis set
input). This allows calculation of the density matrix as superposition of atomic densities
or ionic densities, for the same crystal structure.
5. The wave function data stored on fortran unit 9 at the end of the SCF cycles are not
modified. Only the data stored on the temporary data set (reducible density matrix on
fortran unit 13 and overlap matrix on fortran unit 3) are modified. The keyword PSCF
restores the scf density matrix and all the original information (including geometry and
basis set).
6. See also ATOMHF, input block 3, page 57, and CHARGED, input block 2, page 48.
PBAN/PDIBAN
- Band(s) projected density matrix
rec variable
meaning
NB
number of bands to consider.
NPR
printing of the density matrix for the first NPR direct lattice cells.
N(I),I=1,NB sequence number of the bands summed up for the projected density ma-
trix.
104

A density matrix projected onto a given range of bands is computed and stored on fortran unit
13. The properties will subsequently be computed using such a matrix.
For spin polarized systems, two records are written:
first record, total density matrix (N=n + n electrons);
second record, spin density matrix (Ns=n - n electrons).
To be combined only with ECHG and PPAN. Fock/Kohn-Sham eigenvectors and band
weights must be precomputed by running NEWK and setting IFE=1.
PGEOMW - Density matrix from geometrical weights
A density matrix projected onto the range of bands defined in input (see PBAN input instruc-
tions) is computed, using the geometrical weights of the k points in the reciprocal lattice. The
properties will subsequently be computed using such a matrix. All the bands are attributed an
occupation number 1., independently of the position of the Fermi energy. The density matrix
does not have any physical meaning, but the trick allows analysis of the virtual eigenvectors.
For spin polarized systems, two records are written:
first record, total density matrix (N=n + n electrons);
second record, spin density matrix (Ns=n - n electrons).
To be combined only with ECHG and PPAN.
Fock/Kohn-Sham eigenvectors and band weights must be computed by running NEWK and
setting IFE=1. Symmetry adaptation of Bloch functions is not allowed, the keyword NOSY-
MADA must be inserted before NEWK.
PDIDE - Density matrix energy projected
rec variable
meaning
EMI,EMAX lower and upper energy bound (hartree)
A density matrix projected onto a given energy range is computed and stored on fortran unit
13. The properties will subsequently be computed using such a matrix. To be combined only
with DOSS, ECHG and PPAN. Fock/Kohn-Sham eigenvectors and band weights must be
computed by running NEWK and setting IFE=1.
The charge density maps obtained from the density matrix projected onto a given energy range
give the STM topography [86] within the Tersoff-Haman approximation [87].
POLI
- Spherical harmonics multipole moments
rec variable value
meaning
IDIPO
multipole order (maximum order =6)
ITENS
1
the quadrupole cartesian tensor is diagonalized
0
no action
LPR68
maximum pole order for printing:
< 0
atom multipoles up to pole IDIPO
0
atom and shell multipoles up to pole IDIPO
The multipoles of the shells and atoms in the primitive cell are computed according to a
Mulliken partition of the charge density, up to quantum number IDIPO (0 IDIPO 6). The
first nine terms, corresponding to =0,1,2 (for the definition of higher terms, see Appendix A1,
page 170 in reference [7]) are defined as follow:
m
0
0
s
1
0
z
1
1
x
1
-1 y
2
0
z2 - x2/2 - y2/2
105

2
1
3xz
2
-1 3yz
2
2
3(x2 - y2)
2
-2 6xy
3
0
(2z2 - 3x2 - 3y2)z
3
1
(4z2 - x2 - y2)x
3
-1 (4z2 - x2 - y2)y
3
2
(x2 - y2)z
3
-2 xyz
3
3
(x2 - 3y2)x
3
-3 (3x2 - y2)y
If ITENS=1, the cartesian quadrupole tensor is computed, and its eigenvalues and eigen-
vectors are printed after diagonalization.
The components of the cartesian tensor are:
x2, y2, z2, xy, xz, yz
Warning: the shell multipoles are not printed by default. On request (keyword POLIPRT),
they are printed in atomic units (electron charge = +1).
POLSPIN
- Spin multipole moments
rec variable value
meaning
IDIPO
multipole order (maximum order =6)
ITENS
1
the quadrupole cartesian tensor is diagonalized
0
no action
LPR68
maximum pole order for printing:
< 0
atom multipoles up to pole IDIPO
0
atom and shell multipoles up to pole IDIPO
The electron spin density is partitioned in atomic contributions according to the Mulliken
scheme, and the spherical harmonic atomic multipoles up to the IDIPO angular quantum
number are evaluated (see the POLI keyword for definition of the multipoles and references).
The Cartesian tensor Tij =
xixj spin(r) dr is computed and diagonalized, and its eigenvalues
and eigenvectors are printed. This option may be useful in the analysis of the size, shape and
orientation of localized electron holes.
106

POTC
- Electrostatic potential and its derivatives
rec variable
meaning
ICA
0 calculation of potential (V ), its first derivative (E) and second derivatives (E )
in one or more points
1 not implemented
2 calculation of V (z), E(z), E (z) and (z) averaged in the plane at z position
(2D only)
3 calculation of V (z), E(z), E (z) and (z) averaged in the volume between zZD
and z+ZD (2D only)
NPU
n number of points at which these properties are computed
0 these properties are computed at the atomic positions defined by IPA value
IPA
0 calculations are performed at each atomic positions in the cell
1 calculations are performed just for non equivalent atomic positions in the cell
if ICA = 0 and NPU > 0 insert NPU records
II
X,Y,Z
point coordinates (cartesian, bohr)
if ICA = 2 insert
II
ZM,ZP
properties are averaged over NPU planes orthogonal to the z axis from z = ZP
to z = ZM by step of (ZPZM)/(NPU1) (bohr)
if ICA = 3 insert
II
ZM,ZP
properties are averaged over NPU volumes centered on planes orthogonal to
the z axis, same as ICA = 2
ZD
half thickness of the volume (bohr)
The exact electrostatic potential V , its derivatives E (electric field) and E (electric field
gradient) are evaluated for molecules (0D), slabs (2D) and crystals (3D). Plane and volume
averaged properties can be computed for slabs (2D) only. The plane is orthogonal to the z
axis.
For ICA = 3, the volume average is performed around a middle plane at z position, from zZD
to z+ZD, giving a thickness of 2ZD.
According to Poisson's law, the charge density (z) is defined as
1 d2V (z)
-E (z)
(z) = -
=
4
dz2
4
If an electric field of intensity E0 is present (keyword FIELD, see page 2.1, only for slabs),
the total potential Vfield(z) is calculated:
Vfield(z) = V (z) - E0z
where V (z) is the potential of the slab itself and -E0z is the perturbation applied.
Note - When ICA = 0 and NPU = 0, it is possible to enter the cartesian coordinates (bohr)
of the points where the exact values of the properties must be computed. It is useful when
applying fitting procedure to obtain formal point charges.
POTM - Electrostatic potential maps and electric field
rec variable value
meaning
IDER
0
the electrostatic potential is evaluated
1
the potential and its first derivatives are evaluated
ITOL
penetration tolerance (suggested value: 5)
insert MAPNET input records (page 100)
1. When IDER=0, the electrostatic potential is calculated at the nodes of a 2-dimensional
net in a parallelogram-shaped domain defined by the segments AB and BC (see keyword
MAPNET, page 100).
The electrostatic potential values are written formatted on
fortran unit 25 (see Appendix F, page 181).
107

2. When IDER = 0, the electrostatic potential gradient is computed at the nodes of the
same grid. The x, y and z components are printed on the standard output, and written
formatted on fortran unit 25 (see Appendix F, page 181).
3. The electrostatic potential at r is evaluated [10] by partitioning the periodic charge
density (r) in shell contributions h:

(r) =
(r - h)
h

(h translation vector).
4. The long range contributions are evaluated through a multipolar expansion of (r - h)
[11]. The short range contributions are calculated exactly.
5. The separation between long and short range is controlled by ITOL: (r-h) is attributed
to the short range (exact) region if
e-(r-s-h)2 > 10-IT OL
where: = exponent of the adjoined gaussian of shell ; s = internal coordinates of
shell in cell at h.
The difference between the exact and the approximated potential is smaller than 1%
when ITOL (input datum to POTM) =5 and IDIPO (input datum to POLI) =4, and
smaller than 0.01% when ITOL=15 and IDIPO=6 [10, 11].
6. The multipoles of shell charges must be computed by running the POLI option before
POTM.
PPAN/MULPOPAN - Mulliken Population Analysis
See input block 4, page 78.
PRINTOUT - Setting of printing environment
See input block 1, page 35.
108

PROF
- Compton Profiles
rec variable value
meaning
ICORE 1
core plus valence calculation.
2
core only calculation.
3
valence only calculation.
IVIA
0
valence contribution is computed by numerical integration.
1
valence contribution is computed analytically.
NPR
number of printing options to switch on.
IPLO
0
CP related data are not stored on disk
1
formatted CP data stored on Fortran unit 25 (Appendix F, page 181)
2
formatted CP data stored on Fortran unit 24 (Appendix F, page 181)
if NPR = 0 insert prtrec (see page 37)
II
A2
CP
calculation of Compton profiles
(J(q)) along selected directions (eq.
6.27).
ND
number of directions ( 6).
REST
maximum value of q for J(q) calculation (bohr-1).
RINT
internal sphere radius (bohr-1).
IRAP
shrinking factor ratio.
(KD(J,N), J=1,3), directions in oblique coordinates; see note 9
N=1,ND
STPJ
interpolation step (in interpolated Compton profiles calculation).
A4
DIFF
CP difference between all computed directional CPs.
A2
BR
autocorrelation function B(r) calculation (eq. 6.30).
RMAX
maximum r value (bohr) at which B(r) is computed
STBR
step in computation of B(r).
A4
CONV convolution of the data previously computed (CP, DIFF, BR) (eq. 6.29)
FWHM
convolution parameter (a.u.) full width half maximum;
=
(F W HM )2/(8 2log2).
A4
ENDP End of input records for CP data
The keyword PROF starts the calculation of Compton profiles (J(q)) along selected directions
(eq. 6.27). The specific keywords DIFF BR CONV allow the calculation of the related
quantities. The card with the keyword ENDP ends the Compton profiles input section.
1. The input of the options must be given in the order in which they appear in the above
description. To enter this property, the CP option must always be selected after PROF,
while the others are optional.
2. Core and valence contributions are computed by using different algorithms. Core con-
tribution to CP's is always computed analytically via the Pg matrix (direct lattice sum-
mation, equation 6.25); the valence contribution is computed numerically (IVIA=0) by
integrating the EMD (equation 6.23). Valence contribution can be evaluated analytically,
setting IVIA=1.
3. The numerical integration is extended to a sphere (radius RINT) where EMD is sampled
at the points of a commensurate net characterized by a shrinking factor IS (in the IBZ)
and at all the points (with modulus less then RINT) obtained from these by applying
reciprocal lattice translations.
It is possible to define a second sphere (with radius REST); in the volume between
the two spheres a second net is employed with shrinking factor IS1 greater then IS.
IRAP=IS1/IS is given in INPUT card 2; a reasonable value is IRAP=2. The outer
contribution is supposed to be the same for different CP's, and is obtained by integrating
the average EMD.
109

4. If ICORE = 2 (valence electron CP's are required) the NEWK option, with IFE=1,
must be run before the PROF option, in order to generate the eigenvectors required for
the EMD calculation, as well as the weights associated with each k point.
5. If ICORE = 2 and IVIA = 0 the CPs are evaluated at points resulting from the IS
partition of the reciprocal lattice translators. The interpolation is performed at STPJ
intervals (STPJ is given in input).
If ICORE = 2 or IVIA = 1 the CPs are, in any case, evaluated at points resulting from
STPJ intervals.
IVIA=0 (numerical integration) produces much more accurate results;
IVIA=1 (analytical integration) is to be used only for molecular calculations or for non
conducting polymers.
6. Total CP's are always obtained by summing core and valence contributions.
7. Reasonable values of the integration parameters depend on the system under investi-
gation. The normalization integral of the CP's is a good check of the accuracy of the
calculation. For instance, in the case of the valence electron of beryllium (test 9), good
values of RINT and IS are 10. a.u. and 4 respectively. In the case of silicon (test 10),
good values of the same variables are 8. a.u. and 8 respectively. Much greater RINT
values are required in order to include all the core electrons (70. a.u. in the case of
silicon, and 25. a.u. in the case of beryllium).
8. BR (autocorrelation function or reciprocal space form factor) should be calculated only
for valence electrons. All electron BR are reliable when the normalization integral, after
the analytical integration for core electrons contribution, is equal to the number of core
electrons.
9. The oblique coordinates directions given in input refer to the conventional cell, not to
the primitive cell for 3D systems.
Example: in a FCC system the input directions refer to the orthogonal unit cell frame
(sides of the cube) not to the primitive non-orthogonal unit cell frame.
Printing options: 116 (Compton profiles before interpolation); 117 (average EMD before inter-
polation); 118 (printing of core, valence etc. contribution). The LPRINT(118) option should
be used only if ICORE=1, that is, if core plus valence calculation are chosen.
PSCF
- Restore SCF density matrix
The wave function data computed at the last SCF cycle are restored in common areas and
fortran units 3 (overlap matrix), 11 (Fock/KS matrix), 13 (density matrix). The basis set
defined in input block 2 is restored. Any modification in the default settings introduced in
properties is overwritten. No input data required.
RAYCOV - covalent radii modification
See input block 1, page 35
ROTREF Rotation of eigenvectors and density matrix
This option permits the rotation of the cartesian reference frame before the calculation of the
properties.
It is useful, for example, in the population analysis or in the AO projected density of states of
systems containing transition metal atoms with partial d occupation.
Consider for example a d7 occupation as in CoF2, where the main axis of the (slightly distorted)
CoF6 octahedron in the rutile structure makes a 450 angle with the x axis, and lies in the xy
plane, so that the three empty states are a combination of the 5 d orbitals. Re-orienting the
octahedron permits to assign integer occupations to dxz and dyz.
110

Input for the rotation as for EIGSHROT (page 74)
SETINF - Setting of INF values
See input block 1, page 36
SETPRINT - Setting of printing options
See input block 1, page 37.
STOP
Execution stops immediately. Subsequent input records are not processed.
SYMADAPT
See input block 4, page 81
XFAC
- X-ray structure factors
rec variable value
meaning
ISS
> 0
number of reflections whose theoretical structure factors are calculated.
< 0
a set of non-equivalent reflections with indices h,k,l < |ISS| is gener-
ated
if ISS > 0 insert ISS records
II
H,K,L
Miller indices of the reflection (conventional cell) .
The Fourier transform of the ground state charge density of a crystalline system provides the
static structure factors of the crystal, which can be determined experimentally, after taking
into account a number of corrective terms, in particular those related to thermal and zero point
motion of nuclei:
Fk =
(r) eikrdr
where k h b + k b + l b . The Miller indices refer to the conventional cell. The structure
1
2
3
factors are integrated over the primitive cell volume.
111

3.3
Spontaneous polarization and piezoelectricity
Y. Noel, September 2002
PIEZOBP - Piezoelectricity (Berry phase approach)
The calculation of the piezoelectric constants of a system, can be decomposed in few steps. A
preliminary run must be performed for the undistorted system at the equilibrium geometry
(crystal followed by properties with the keyword POLARI). Then, for a distorted system at
equilibrium (all internal parameters must be optimized), a second run (crystal followed by
properties with the keyword POLARI) must be performed, followed by a third run (only
properties with the keyword PIEZOBP that computes l, the polarization vector component
along the reciprocal lattice vector bl (l is a phase). Steps 2 and 3 are repeated for other
strains. Then, the obtained set of data is fitted and the slope ( dl ) at zero strain is computed.
d j
The piezoelectric constant is finally deduced from the following equation:
|e|
(0) dl
eij =
a
(3.18)
2V (0)
li
d j
l
(0)
Where the prefactor
|e|
and a
are given in the output of the keyword PIEZOBP.
2V (0)
li
In summary:
1. First run: calculation at = 0 (undistorted)
Program
Keyword comments
crystal
properties NEWK
additional keywords allowed
POLARI see above
save Fortran unit 27 as undistord.f27 for example.
2. Second run: calculation at = 1 (distorted)
Program
Keyword comments
crystal
Use OPTBERNY to obtain the system at the equilibrium
properties NEWK
same input as in first run
POLARI
save Fortran unit 27 as distord1.f27 for example.
3. Third run: merging previous data.
copy undistord.f27 to Fortran unit 28
copy distord1.f27 to Fortran unit 29
Program
Keyword
comments
properties PIEZOBP
4. Construction of a set of data
Repeat 2. and 3. for several amplitude of the strain.
(Systems labeled distord2 to distordn for example).
5. Evaluation of the slope dl and calculation of the related piezoelectric constant
d j
Use an external program as gnuplot or xmgr to plot and fit the data and
evaluate the slope at zero strain.
Deduce eij from Eq. (3.18).
112

PIEZOWF - Piezoelectricity (localized CO approach)
The calculation the piezoelectric constants of a system, using localized Wannier functions,
follows exactly the same path than in the Berry phase scheme (see PIEZOBP).
In summary:
1. First run: calculation at = 0 (undistorted)
Program
Keyword
comments
crystal
properties NEWK
additional keywords allowed
LOCALWF see above
save Fortran unit 37 as undistord.f37 for example.
2. Second run: calculation at = 1 (distorted)
Program
Keyword
comments
crystal
Use OPTBERNY to obtain the system at the equilibrium
properties NEWK
same input as in first run
LOCALWF
save Fortran unit 37 as distord1.f37 for example.
3. Third run: merging previous data.
copy undistord.f37 to Fortran unit 38
copy distord1.f37 to Fortran unit 39
Program
Keyword
comments
properties PIEZOWF
4. Construction of a set of data
Repeat 2. and 3. for several amplitude of the strain.
(Systems labeled distord2 to distordn for example).
5. Evaluation of the slope dl and calculation of the related piezoelectric constant
d j
Use an external program as gnuplot or xmgr to plot and fit the data and
evaluate the slope at zero strain.
Deduce eij from Eq. (3.18).
POLARI - Spontaneous polarization/ Piezoelectricity
This keyword is used for the preliminary runs of the computation of spontaneous polarization
(see SPOLBP) and the piezoelectric constants (see PIEZOBP)using the Berry phase scheme.
1. This keyword works for 3D systems only.
2. The unit-cell has to contain an even number of electrons (closed shell system).
3. The geometry must be optimized with very high accuracy. The values of the computa-
tional parameters for the calculation of the wave function (TOLINTEG, page 69 and
TOLSCF, page 81) and the optimization of the geometry (XTOL, GTOL and ETOL,
page 43) must have high values and be the same for all runs.
4. The option NEWK (See page 103 for the input) must be used before running POLARI.
The shrinking factor IS has to be the same for the first and the second run, and should
be at least equal to 4. Fermi energy calculation is not necessary, then set IFE=0.
5. Data evaluated with the keyword POLARI in the first two runs do not have any physical
meaning if considered independently. Only the outputs produced choosing the keywords
SPOLBP and PIEZOBP in the third run are significant.
113

SPOLBP - Spontaneous polarization (Berry phase approach)
To calculate the spontaneous polarization, a preliminary with the keyword POLARI run is needed
for each of the two structures ( = 1 and = 0). Then a third run with the keyword SPOLBP
gives the polarization difference between the two systems.
In summary:
1. First run: calculation at = 0
Program
Keyword comments
crystal
see deck 1 for input blocks 1 and 1b
properties NEWK
additional keywords allowed
POLARI
see above
save Fortran unit 27 as sys0.f27
2. Second run: calculation at = 1
Program
Keyword comments
crystal
no input data required
properties NEWK
same input as in first run
POLARI
save Fortran unit 27 as sys1.f27
3. Third run: merging previous data.
copy sys0.f27 to Fortran unit 28
copy sys1.f27 to Fortran unit 29
Program
Keyword comments
properties SPOLBP
SPOLWF - The spontaneous polarization (localized CO approach)
The calculation of the spontaneous polarization of a system, using the localized Wannier func-
tions follows exactly the same procedure than in the Berry phase scheme (see SPOLBP).
It can be summarized as follow:
1. First run: calculation at = 0
Program
Keyword comments
crystal
see deck 1 for input blocks 1 and 1b
properties NEWK
additional keywords allowed
LOCALWF see above
save Fortran unit 37 as sys0.f37
2. Second run: calculation at = 1
Program
Keyword comments
crystal
see deck 1 for input blocks 1 and 1b
properties NEWK
same input as in first run
LOCALWF
save Fortran unit 37 as sys1.f37
3. Third run: merging previous data.
copy sys0.f37 to Fortran unit 38
copy sys1.f37 to Fortran unit 39
Program
Keyword comments
properties SPOLWF
114

Chapter 4
Input examples
4.1
Standard geometry input
3D - Crystalline compounds - 1st input record keyword:
CRYSTAL
Atom coordinates: fractional units of the crystallographic lattice vectors.
Sodium Chloride - NaCl (Rock Salt Structure)
CRYSTAL
0 0 0
IFLAG IFHR IFSO
225
space group, Fm3m, cubic
5.64
a (
A)
2
2 non equivalent atoms
11 .5 .5 .5
Z=11, Sodium, 1/2, 1/2, 1/2
17 .0 .0 .0
Z=17, Chlorine
Diamond - C (2nd Setting - 48 symmops - 36 with translational component)
CRYSTAL
0 0 0
IFLAG IFHR IFSO
227
space group, Fd3m, cubic
3.57
a (
A)
1
1 non equivalent atom
6 .125 .125 .125
Z=6, Carbon, 1/8, 1/8, 1/8 (multiplicity 2)
Diamond - C (1st Setting - 48 symmops - 24 with translational component)
CRYSTAL
0 0 1
IFLAG IFHR IFSO
227
space group 227, Fd3m, cubic
3.57
a (
A)
1
1 non equivalent atom
6 .0 .0 .0
Z=6, Carbon (multiplicity 2)
Zinc Blend - ZnS
CRYSTAL
0 0 0
IFLAG IFHR IFSO
216
space group 216, F
43m, cubic
5.42
a (
A)
2
2 non equivalent atoms
30 .25 .25 .25
Z=30, Zinc, (1/4, 1/4, 1/4)
16 .0 .0 .0
Z=16, Sulphur
Wurtzite - ZnS
CRYSTAL
0 0 0
IFLAG IFHR IFSO
186
space group 186, P63mc, hexagonal
3.81 6.23
a,c (
A)
2
2 non equivalent atoms
30 .6666666667 .3333333333 .0
Zinc, (2/3, 1/3, 0.)
16 .6666666667 .3333333333 .375
Sulphur, (2/3, 1/3, 3/8)
115

Cuprite - Cu2O
CRYSTAL
0 0 0
IFLAG IFHR IFSO
208
space group 208, P4232, cubic
4.27
a (
A)
2
2 non equivalent atoms
8 .0 .0 .0
Z=8, Oxygen
29 .25 .25 .25
Z=29, Copper (1/4, 1/4, 1/4)
Aragonite - CaCO3
CRYSTAL
1 0 0
IFLAG (1, SPGR symbol) IFHR IFSO
P M C N
space group Pmcn, orthorhombic
4.9616 7.9705 5.7394
a,b,c (
A)
4
4 non equivalent atoms
20 .25 .4151 .2103
Z=20, Calcium
6 .25 .7627 .085
Z=6, Carbon
8 .25 .9231 .0952
Z=8, Oxygen
8 .4729 .6801 .087
Z=8, Oxygen
Fluorite - CaF2
CRYSTAL
0 0 0
IFLAG IFHR IFSO
225
space group 225, Fm3m, cubic
5.46
a (
A)
2
2 non equivalent atoms
9 .25 .25 .25
Fluorine
20 .0 .0 .0
Calcium
Cesium chloride - CsCl
CRYSTAL
0 0 0
IFLAG IFHR IFSO
221
space group 221, Pm3m, cubic
4.12
a (
A)
2
2 non equivalent atoms
55 .5 .5 .5
Cesium
17 .0 .0 .0
Chlorine
Rutile - TiO2
CRYSTAL
0 0 0
IFLAG IFHR IFSO
136
space group 136, P42/mnm, tetragonal
4.59 2.96
a, c (
A)
2
2 non equivalent atoms
22 .0 .0 .0
Titanium
8 .305 .305 .0
Oxygen
Graphite - C (Hexagonal)
CRYSTAL
0 0 0
IFLAG IFHR IFSO
194
space group 194, P63/mmc, hexagonal
2.46 6.70
a,c (
A)
2
2 non equivalent atoms
6 .0 .0 .0
Carbon
6 .33333333333 .66666666667 .25
Carbon, 1/3, 2/3, 1/4
Pyrite - FeS2
CRYSTAL
0 0 0
IFLAG IFHR IFSO
205
space group 205, Pa3, cubic
5.40
a (
A)
2
2 non equivalent atoms
26 .0 .0 .0
Iron
16 .386 .386 .386
Sulphur
116

Calcite - CaCO3
CRYSTAL
0 1 0
IFLAG IFHR (=1, rhombohedral representation) IFSO
167
space group 167, R
3c, hexagonal
6.36 46.833
a (
A),
3
3 non equivalent atoms
20 .0 .0 .0
Calcium
6 .25 .25 .25
Carbon
8 .007 .493 .25
Oxygen
Corundum - Al2O3 (hexagonal representation)
CRYSTAL
0 0 0
IFLAG IFHR IFSO
167
space group 167, R
3c, hexagonal
4.7602 12.9933
a,c (
A)
2
2 non equivalent atoms
13 0. 0. 0.35216
Aluminium
8 0.30621 0. 0.25
Oxygen
Corundum - Al2O3 (rhombohedral representation)
CRYSTAL
0 1 0
IFLAG IFHR (=1, rhombohedral cell) IFSO
167
space group 167, R
3c, hexagonal
5.12948 55.29155
a (
A),
2
2 non equivalent atoms
13 0.35216 0.35216 0.35216 Aluminium
8 0.94376 0.25 0.55621
Oxygen
Zirconia - ZrO2 - monoclinic structure
CRYSTAL
0 0 1
IFLAG IFHR IFSO (=1, standard shift of origin)
14
space group 14, P21/c, monoclinic
5.03177 5.03177 5.258 90.0
a,b,c (
A),
3
3 non equivalent atoms
240 0.2500 0.0000 0.25000
Zirconium, Pseudopotential (Z' > 200)
208 0.0000 0.2500 0.07600
Oxygen, Pseudopotential
208 -0.500 -0.250 0.07600
Oxygen, Pseudopotential
Zirconia - ZrO2 - tetragonal structure
CRYSTAL
0 0 1
IFLAG IFHR IFSO (=1, standard shift of origin)
137
space group 137, P42/nmc, tetragonal
3.558 5.258
a,c (
A)
3
3 non equivalent atoms
240 0.0 0.0 0.0
Zirconium, Pseudopotential (Z' > 200)
208 0.0 -0.5 0.174
Oxygen, Pseudopotential
208 0.5 0.0 0.326
Oxygen, Pseudopotential
Zirconia - ZrO2 - cubic structure
CRYSTAL
0 0 1
IFLAG IFHR IFSO (=1, standard shift of origin)
225
space group 225, Fm3m, cubic
5.10
a (
A)
3
3 non equivalent atoms
240 0.00 0.00 0.00
Z=40 Zirconium, Pseudopotential (Z' > 200)
208 0.25 0.25 0.25
Oxygen, Pseudopotential
208 -0.25 -0.25 -0.25
Oxygen, Pseudopotential
117

SiO2, Chabazite
CRYSTAL
0 1 0
IFLAG IFHR (=1,rhombohedral representation) IFSO
166
space group 166 R
3m, hexagonal
9.42 94.47
a (
A),
5
5 non equivalent atoms (36 atoms in the primitive cell)
14 .1045 .334 .8755
Silicon (multiplicity 12)
8 .262 -.262 .0
Oxygen (multiplicity 6)
8 .1580 -.1580 .5000
Oxygen (multiplicity 6)
8 .2520 .2520 .8970
Oxygen (multiplicity 6)
8 .0250 .0250 .3210
Oxygen (multiplicity 6)
SiO2, Siliceous Faujasite
CRYSTAL
0 0 0
IFLAG IFHR IFSO
227
space group 227, Fd3m, cubic
21.53
a (
A)
5
5 non equivalent atoms (144 atoms in the primitive cell)
14 .1265 -.0536 .0370
Silicon (multiplicity 48)
8 .1059 -.1059 .0
Oxygen (multiplicity 24)
8 -.0023 -.0023 .1410
Oxygen (multiplicity 24)
8 .1746 .1746 -.0378
Oxygen (multiplicity 24)
8 .1785 .1785 .3222
Oxygen (multiplicity 24)
SiO2, Siliceous Edingtonite
CRYSTAL
0 0 0
IFLAG IFHR IFSO
115
space group 115, P
4m2, tetragonal
6.955 6.474
a, c (
A)
5
5 non equivalent atoms (15 atoms in the primitive cell)
14 .0 .0 .5000
Silicon (multiplicity 1)
14 .0 .2697 .1200
Silicon (multiplicity 4)
8 .0 .189 .3543
Oxygen (multiplicity 4)
8 .50000 .0 .8779
Oxygen (multiplicity 2)
8 .189 .189 .0
Oxygen (multiplicity 4)
SiO2, Siliceous Sodalite
CRYSTAL
0 0 0
IFLAG IFHR IFSO
218
space group 218, P
43n, cubic
8.950675
a (
A)
3
3 non equivalent atoms (36 atoms in the primitive cell)
14 .25000 .50000 .0
Silicon (multiplicity 6)
14 .25000 .0 .50000
Silicon (multiplicity 6)
8 .14687 .14687 .50000
Oxygen (multiplicity 24)
118

2D - Slabs (surfaces) - 1st input record keyword: SLAB
A 2D structure can either be created by entering directly the 2D cell parameters and
irreducible atoms coordinates to obtain a slab of given thickness (keyword SLAB in the first
record of the geometry input), or it can be derived from the 3D structure through the keyword
SLABCUT (page 37), entered in the geometry editing section of 3D structure input. In that
case the layer group is automatically identified by the program. The input tests 4-24, 5-25,
6-26 and 7-27 show the two different ways to obtain the same 2D structure.
Atom coordinates: z in
Angstrom, x, y in fractional units of the crystallographic cell translation
vectors.
Test05 - graphite 2D (see test 25)
SLAB
77
layer group (hexagonal)
2.47
lattice vector length (
A)
1
1 non equivalent atom
6 -0.33333333333 0.33333333333 0.
Z=6; Carbon; x,y,z
Beryllium - 3 layers slab
SLAB
78
layer group (hexagonal)
2.29
lattice vector length (
A)
2
2 non equivalent atoms
4 0.333333333333 0.666666666667 0.
Z=4, Beryllium; 1/3, 2/3, z
4 0.666666666667 0.333333333333 1.795
Z=4, Beryllium; 2/3, 1/3,z
Test06 - beryllium - 4 layers slab (see test 26)
SLAB
72
layer group (hexagonal)
2.29
lattice vector length (
A)
2
2 non equivalent atoms
4 0.333333333333 0.666666666667 0.897499 Z=4, Beryllium;x,y,z
4 0.666666666667 0.333333333333 2.692499 Z=4, Beryllium;x,y,z
Test04 - Corundum 001 (0001) 2 layers slab (see test 24)
SLAB
66
layer group (hexagonal)
4.7602
lattice vector length (
A)
3
3 non equivalent atoms
13 0. 0. 1.9209
Z=13, Aluminum; x,y,z
8 0.333333333 -0.027093 1.0828
Z=8, Oxygen; x,y,z
13 -0.333333333 0.333333333 0.2446
Z=13, Aluminum; x,y,z
Test07 - Corundum 110 (1010) slab (see test 27)
SLAB
7
layer group (triclinic)
5.129482 6.997933 95.8395
a,b (
A) (degrees)
6
6 non equivalent atoms
8 -0.25 0.5 2.1124
Z=8, Oxygen; x,y,z
8 0.403120 0.153120 1.9189
Z=8, Oxygen; x,y,z
8 0.096880 0.346880 0.4612
Z=8, Oxygen; x,y,z
8 -0.25 0.00 0.2677
Z=8, Oxygen; x,y,z
13 0.454320 0.397840 1.19
Z=13, Aluminum; x,y,z
13 0.045680 0.102160 1.19
Z=13, Aluminum; x,y,z
MgO (110) 2 layers slab
SLAB
40
layer group
4.21 2.97692
lattice vectors length (
A)
2
2 non equivalent atoms
12 0.25 0.25 0.74423
Z=12, Magnesium; x,y,z
8 0.75 0.25 0.74423
Z=8, Oxygen; x,y,z
119

MgO (110) 3 layers slab
SLAB
37
4.21 2.97692
lattice vectors length (
A)
4
4 non equivalent atoms
12 0. 0. 1.48846
Z=12, Magnesium; x,y,z
8 0.5 0. 1.48846
Z=8, Oxygen; x,y,z
12 0.5 0.5 0.
Z=12, Magnesium; x,y,z
8 0. 0.5 0.
Z=8, Oxygen; x,y,z
CO on MgO (001) two layers slab - one-side adsorption
SLAB
55

2.97692
lattice vector length [4.21/
2] (
A)
6
6 non equivalent atoms
108 0. 0. 4.5625
Z=8, Oxygen; x,y,z
6 0. 0. 3.4125
Z=6, Carbon; x,y,z
12 0. 0. 1.0525
Z=12, Magnesium; x,y,z
8 0.5 0.5 1.0525
Z=8, Oxygen; x,y,z
12 0. 0. -1.0525
Z=12, Magnesium; x,y,z
8 0.5 0.5 -1.0525
Z=8, Oxygen; x,y,z
Two different conventional atomic numbers (8 and 108) are attributed to the Oxygen in CO and to the Oxygen
in MgO. Two different basis sets will be associated to the two type of atoms (see Basis Set input, page 16, and
test 36).
CO on MgO (001) two layers slab - two-side adsorption
SLAB
64
2.97692
lattice vector length (
A)
4
4 non equivalent atoms
108 0.25 0.25 4.5625
Z=8, Oxygen; x,y,z
6 0.25 0.25 3.4125
Z=6, Carbon; x,y,z
12 0.25 0.25 1.0525
Z=12, Magnesium; x,y,z
8 0.75 0.75 1.0525
Z=8, Oxygen; x,y,z
Two different conventional atomic numbers (8 and 108) are attributed to the Oxygen in CO and to the Oxygen
in MgO.
Diamond slab parallel to (100) face - nine layers slab
SLAB
59
2.52437
lattice vector length (
A)
5
5 non equivalent atoms
6 0. 0. 0.
Z=6, Carbon; x,y,z
6 0.5 0. 0.8925
Z=6, Carbon; x,y,z
6 0.5 0.5 1.785
Z=6, Carbon; x,y,z
6 0. 0.5 2.6775
Z=6, Carbon; x,y,z
6 0. 0. 3.57
Z=6, Carbon; x,y,z
Diamond slab parallel to (100) face - ten layers slab
SLAB
39
layer group
2.52437 2.52437
lattice vectors length (
A)
5
5 non equivalent atoms
6 0.25 0. 0.44625
Z=6, Carbon; x,y,z
6 0.25 0.5 1.33875
Z=6, Carbon; x,y,z
6 0.75 0.5 2.23125
Z=6, Carbon; x,y,z
6 0.75 0 3.12375
Z=6, Carbon; x,y,z
6 0.25 0. 4.01625
Z=6, Carbon; x,y,z
120

1D - Polymers - 1st input record keyword: POLYMER
Atom coordinates: y,z in
Angstrom, x in fractional units of the crystallographic cell translation
vector.
Test03 - (SN)x polymer
POLYMER
4
rod group
4.431
lattice vector length (
A)
2
2 non equivalent atoms
16 0.0 -0.844969 0.0
Z=16, Sulphur; x, y, z
7 0.141600540 0.667077 -0.00093
Z=7, Nitrogen; x, y, z
Water polymer
POLYMER
1
4.965635
lattice vector length (
A)
6
6 non equivalent atoms
8 0. 0. 0.
Z=8, Oxygen; x, y, z
1 0.032558 0.836088 -0.400375
Z=1, Hydrogen; x, y, z
1 0.168195 -0.461051 0.
Z=1, Hydrogen; x, y, z
8 0.5 -1.370589 0.
Z=8, Oxygen; x, y, z
1 0.532558 -2.206677 0.400375
Z=1, Hydrogen; x, y, z
1 0.668195 -0.909537 0.
Z=1, Hydrogen; x, y, z
Formamide chain - test40 DFT
POLYMER
4
rod group
8.774
lattice vector length (
A)
6
6 non equivalent atoms
8 -7.548E-2 5.302E-3 0.7665
Z=8, Oxygen; x, y, z
7 0.1590 -0.8838 0.3073
Z=7, Nitrogen; x, y, z
6 5.627E-2 7.051E-2 0.2558
Z=6, Oxygen; x, y, z
1 0.2677 -0.6952 -9.1548E-2
Z=1, Hydrogen; x, y, z
1 0.1310 -1.8019 0.7544
Z=1, Hydrogen; x, y, z
1 9.244E-2 0.9973 -0.2795
Z=1, Hydrogen; x, y, z
121

0D - Molecules - 1st input record keyword: MOLECULE
Atom coordinates: x,y,z in
Angstrom.
Test00 - CO molecule
MOLECULE
1
point group
2
2 non equivalent atoms
6 0. 0. 0.
Z=6, Carbon; x, y, z
8 0.8 0.5 0.4
Z=8, Oxygen; x, y, z
Test01 - CH4 Methane molecule
MOLECULE
44
point group
2
2 non equivalent atoms
6 0. 0. 0.
Z=6, Carbon; x, y, z
1 0.629 0.629 0.629
Z=1, Hydrogen; x, y, z
Test02 - CO(NH2)2 Urea molecule
MOLECULE
15
point group
5
5 non equivalent atoms
6 0. 0. 0.
Z=6, Carbon; x, y, z
8 0. 0. 1.261401
Z=8, Oxygen; x, y, z
7 0. 1.14824666034 -0.69979
Z=7, Nitrogen; x, y, z
1 0. 2.0265496501 -0.202817
Z=1, Hydrogen; x, y, z
1 0. 1.13408048308 -1.704975
Z=1, Hydrogen; x, y, z
4.2
Basis set input
Optimized basis sets for periodic systems used in published papers are available on WWW:
http://www.crystal.unito.it
All electron Basis sets for Silicon atom
STO-3G
14 3
Z=14, Silicon; 3 shells
1 0 3 2. 0.
Pople BS; s shell; 3G; CHE=2; standard scale factor
1 1 3 8. 0.
Pople BS; sp shell; 3G; CHE=8; standard scale factor
1 1 3 4. 0.
Pople BS; sp shell; 3G; CHE=4; standard scale factor
6-21G
14 4
Z=14, Silicon; 4 shells
2 0 6 2. 1.
Pople 6-21 BS; s shell; 6G; CHE=2; scale factor 1 (core AO).
2 1 6 8. 1.
Pople 6-21 BS; sp shell; 6G; CHE=8; scale factor 1 (core AOs).
2 1 2 4. 1.
Pople 6-21 BS; sp shell; 2G; CHE=4; scale factor 1 (inner valence).
2 1 1 0. 1.
Pople 6-21 BS; sp shell; 1G; CHE=0; scale factor 1 (outer valence).
NB. The 4th shell has electron charge 0. The basis functions of that shell are included in the basis set to
compute the atomic wave functions, as they correspond to symmetries (angular quantum numbers) occupied
in the ground state of the atom. The atomic basis set is: 4s, 3p.
6-21G modified
14 4
Z=14, Silicon; 4 shells
2 0 6 2. 1.
Pople 6-21 BS; s shell; 6G; CHE=2; scale factor 1.
2 1 6 8. 1.
Pople 6-21 BS; sp shell; 6G; CHE=8; scale factor 1.
2 1 2 4. 1.
Pople 6-21 BS; sp shell; 2G; CHE=4; scale factor 1.
0 1 1 0. 1.
free BS; sp shell; 1G; CHE=0; scale factor 1.
0.16 1. 1.
gaussian exponent; s coefficient; p coefficient
122

3-21G
14 4
Z=14, Silicon; 4 shells
2 0 3 2. 1.
Pople 3-21 BS; s shell; 3G; CHE=2; scale factor 1.
2 1 3 8. 1.
Pople 3-21 BS; sp shell; 3G; CHE=8; scale factor 1.
2 1 2 4. 1.
Pople 3-21 BS; sp shell; 2G; CHE=4; scale factor 1.
2 1 1 0. 1.
Pople 3-21 BS; sp shell; 1G; CHE=0; scale factor 1.
3-21G*
14 5
Z=14, Silicon; 5 shells
2 0 3 2. 1.
Pople 3-21 BS; s shell; 3G; CHE=2; scale factor 1.
2 1 3 8. 1.
Pople 3-21 BS; sp shell; 3G; CHE=8; scale factor 1.
2 1 2 4. 1.
Pople 3-21 BS; sp shell; 2G; CHE=4; scale factor 1.
2 1 1 0. 1.
Pople 3-21 BS; sp shell; 1G; CHE=0; scale factor 1.
2 3 1 0. 1.
Pople 3-21 BS; d shell; 1G; CHE=0; scale factor 1.
NB. The basis functions of the 5th shell, d symmetry, unoccupied in the ground state of Silicon atom, is not
included in the atomic wave function calculation.
3-21G modified+polarization
14 5
Z=14, Silicon; 5 shells
2 0 3 2. 1.
Pople 3-21 BS; s shell; 3G; CHE=2; scale factor 1.
2 1 3 8. 1.
Pople 3-21 BS; sp shell; 3G; CHE=8; scale factor 1.
2 1 2 4. 1.
Pople 3-21 BS; sp shell; 2G; CHE=4; scale factor 1.
0 1 1 0. 1.
free BS; sp shell; 1G; CHE=0; scale factor 1.
0.16 1. 1.
gaussian exponent; s contraction coefficient; p contr. coeff.
0 3 1 0. 1.
free BS; d shell; 1G; CHE=0; scale factor 1.
0.5 1.
gaussian exponent; d contraction coefficient.
free basis set
14 4
Z=14, Silicon; 4 shells
0 0 6 2. 1.
free BS; s shell; 6 GTF; CHE=2; scale factor 1.
16115.9
0.00195948
1st gaussian exponent; s contraction coefficient
2425.58
0.0149288
2nd gaussian exponent; s contraction coefficient
553.867
0.0728478
3rd gaussian exponent; s contraction coefficient
156.340
0.24613
4th gaussian exponent; s contraction coefficient
50.0683
0.485914
5th gaussian exponent; s contraction coefficient
17.0178
0.325002
6th gaussian exponent; s contraction coefficient
0 1 6 8. 1.
free BS; sp shell; 6 GTF; CHE=8; scale factor 1.
292.718
-0.00278094
0.00443826 1st gaussian exp.; s contr. coeff.; p contr. coeff.
69.8731
-0.0357146
0.0326679
2nd gaussian exp.; s contr. coeff.; p contr. coeff.
22.3363
-0.114985
0.134721
3rd gaussian exp.; s contr. coeff.; p contr. coeff.
8.15039
0.0935634
0.328678
4th gaussian exp.; s contr. coeff.; p contr. coeff.
3.13458
0.603017
0.449640
5th gaussian exp.; s contr. coeff.; p contr. coeff.
1.22543
0.418959
0.261372
6th gaussian exp.; s contr. coeff.; p contr. coeff.
0 1 2 4. 1.
free BS; sp shell; 2 GTF; CHE=4; scale factor 1
1.07913
-0.376108
0.0671030
1st gaussian exp.; s contr. coeff.; p contr. coeff.
0.302422
1.25165
0.956883
2nd gaussian exp.; s contr. coeff.; p contr. coeff.
0 1 1 0. 1.
free BS; sp shell; 1 GTF; CHE=0; scale factor 1.
0.123
1.
1.
gaussian exp.; s contr. coeff.; p contr. coeff.
Examples of ECP and valence only basis set input
Nickel atom. Electronic configuration: [Ar] 4s(2) 3d(8)
123

Durand & Barthelat large core
228 4
Z=28,Nickel; 4 shells valence basis set
BARTHE
keyword; Durand-Barthelat ECP
0 1 2 2. 1.
free BS;sp shell;2 GTF;CHE=2;scale factor 1
1.55
.24985
1.
1st GTF exponent;s coefficient;p coefficient
1.24
-.41636
1.
2nd GTF exponent;s coefficient;p coefficient
0 1 1 0. 1.
free BS; sp shell; 1 GTF; CHE=0; scale factor 1
0.0818
1.0
1.
GTF exponent;s coefficient;p coefficient
0 3 4 8. 1.
free BS; d shell; 4 GTF; CHE=8; scale factor 1
4.3842E+01
.03337
1st GTF exponent; d coefficient
1.2069E+01
.17443
2nd GTF exponent; d coefficient
3.9173E+00
.42273
3rd GTF exponent; d coefficient
1.1997E+00
.48809
4th GTF exponent; d coefficient
0 3 1 0. 1.
free BS; d shell; 1 GTF; CHE=0; scale factor 1
0.333
1.
GTF exponent; d coefficient
Hay & Wadt Large Core - [Ar] 4s(2) 3d(8)
228 4
Z=28,Nickel; 4 shells valence basis set
HAYWLC
keyword; Hay-Wadt large core ECP
0 1 2 2. 1.
free BS; sp shell; 2 GTF; CHE=2; scale factor 1
1.257
1.1300E-01
2.6760E-02
exponent,s coefficient,p coefficient
1.052
-1.7420E-01
-1.9610E-02
0 1 1 0. 1.
second shell,sp type,1 GTF
0.0790
1.0
1.
0 3 4 8. 1.
third shell,d type,4 primitive GTF
4.3580E+01
.03204
1.1997E+01
.17577
3.8938E+00
.41461
1.271
.46122
0 3 1 0. 1.
fourth shell,d type,1 GTF
0.385
1.
Hay & Wadt Small Core - [Ne] 3s(2) 3p(6) 4s(2) 3d(8)
228 6
nickel basis set - 6 shells
HAYWSC
keyword; Hay-Wadt small core ECP
0 1 3 8. 1.
first shell,sp type,3 primitive GTF -
2.5240E+01
-3.7000E-03
-4.0440E-02
exponent,s coefficient,p coefficient
7.2019E+00
-5.3681E-01
-7.6560E-02
3.7803E+00
4.2965E-01
4.8348E-01
0 1 2 2. 1.
second shell,sp type,2 primitive GTF
1.40
.84111
.55922
0.504
.13936
.12528
0 1 1 0. 1.
third shell,sp type,1 GTF
0.0803
1.0
1.
0 3 3 8. 1.
fourth shell,d type,4 primitive GTF
4.1703E+01
3.5300E-02
1.1481E+01
1.8419E-01
3.7262E+00
4.1696E-01
0 3 1 0. 1.
fifth shell,d type,1 GTF
1.212
1.
0 3 1 0. 1.
sixth shell,d type,1 GTF
0.365
1.0
124

Free input
228 5
Z=28, nickel basis set - 5 shells (valence only)
INPUT
keyword: free ECP (Large Core)- input follows
10.
5 4 5 2 0
nuclear charge; number of terms in eq. 2.7 and 2.8
344.84100
-18.00000
-1
eq. 2.7, 5 records:
64.82281
-117.95937
0
, C, n
14.28477
-29.43970
0
3.82101
-10.38626
0
1.16976
-0.89249
0
18.64238
3.00000
-2
eq. 2.8, 4 records
= 0
4.89161
19.24490
-1
1.16606
23.93060
0
0.95239
-9.35414
0
30.60070
5.00000
-2
eq. 2.8, 5 records
= 1
14.30081
19.81155
-1
15.03304
54.33856
0
4.64601
54.08782
0
0.98106
7.31027
0
4.56008
0.26292
0
eq. 2.8, 2 records
= 2
0.67647
-0.43862
0
basis set input follows - valence only
0 1 1 2. 1.
1st shell: sp type; 1 GTF; CHE=2; scale fact.=1
1.257
1.
1.
exponent, s coefficient, p coefficient
0 1 1 0. 1.
2nd shell: sp type; 1 GTF; CHE=0; scale fact.=1
1.052
1.
1.
0 1 1 0. 1.
3rd shell: sp type; 1 GTF; CHE=0; scale fact.=1
0.0790
1.0
1.
0 3 4 8. 1.
4th shell; d type; 4 GTF; CHE=8; scale fact.=1
4.3580E+01
.03204
1.1997E+01
.17577
3.8938E+00
.41461
1.271
.46122
0 3 1 0. 1.
5th shell; d type; 1 GTF; CHE=0; scale fact.=1
0.385
1.
4.3
SCF options - SPINEDIT
Example of how to edit the density matrix obtained for a given magnetic solution to define a
scf guess with a different magnetic solution.
Deck 1 - ferromagnetic solution
Spinel MnCr2O4
CRYSTAL
0 0 0
227
space group number
8.5985
lattice parameter
3
3 non equivalent atoms (14 atoms in the primitive cell)
24 0.500 0.500 0.500
Chromium - x, y, z - multiplicity 4
25 0.125 0.125 0.125
Manganese - x, y, z - multiplicity 2
8 0.2656 0.2656 0.2656
Oxygen - x, y, z - multiplicity 8
END
end of geometry input records - block 1
basis set input terminated by END
UHF
Unrestricted Hartree Fock
TOLINTEG
the default value of the truncation tolerances is modified
7 7 7 7 14
new values for ITOL1-ITOl2-ITOL3-ITOL4-ITOL5
END
end of input block 3
4 0 4
reciprocal lattice sampling (page 19)
SPINLOCK
n - n is locked to be 22 for 50 cycles.
22 50
All the d electrons are forced to be parallel
LEVSHIFT
a level shifter of 0.3 hartree, maintained after diagonalization,
3 1
causes a lock in a non-conducting solution
MAXCYCLE
the maximum number of SCF cycles is set to 50
50
PPAN
Mulliken population analysis at the end of SCF cycles
END
Deck 2 (SCF input only)
125

4 0 4
GUESSP
initial guess: density matrix from a previous run
SPINEDIT
elements of the density matrix are modified
2
the diagonal elements corresponding to 2 atoms
5 6
label of the 2 atoms (6 is equivalent to 5)
LEVSHIFT
a level shifter of 0.3 hartree, maintained after diagonalization,
3 1
causes a lock in a non-conducting solution
PPAN
Mulliken population analysis at the end of SCF cycles
END
=====================================================================
First run - geometry output
=====================================================================
COORDINATES OF THE EQUIVALENT ATOMS (FRACTIONARY UNITS)
N. ATOM
EQUIVALENT
AT. NUMBER
X
Y
Z
1
1
1
24 CR
-5.000E-01 -5.000E-01 -5.000E-01
2
1
2
24 CR
-5.000E-01 -5.000E-01
0.000E+00
3
1
3
24 CR
0.000E+00 -5.000E-01 -5.000E-01
4
1
4
24 CR
-5.000E-01
0.000E+00 -5.000E-01
5
2
1
25 MN
1.250E-01
1.250E-01
1.250E-01
6
2
2
25 MN
-1.250E-01 -1.250E-01 -1.250E-01
7
3
1
8 O
2.656E-01
2.656E-01
2.656E-01
8
3
2
8 O
2.656E-01
2.656E-01 -2.968E-01
9
3
3
8 O
-2.968E-01
2.656E-01
2.656E-01
10
3
4
8 O
2.656E-01 -2.968E-01
2.656E-01
11
3
5
8 O
-2.656E-01 -2.656E-01 -2.656E-01
12
3
6
8 O
-2.656E-01 -2.656E-01
2.968E-01
13
3
7
8 O
-2.656E-01
2.968E-01 -2.656E-01
14
3
8
8 O
2.968E-01 -2.656E-01 -2.656E-01
=====================================================================
Ferromagnetic solution: all unpaired electrons with the same spin
=====================================================================
SPIN POLARIZATION - ALPHA-BETA = 22 FOR
50 CYCLES
=====================================================================
Convergence on total energy reached in 33 cycles (level shifter active)
=====================================================================
CYCLE
33 ETOT(AU) -7.072805900367E+03 DETOT -8.168E-07 DE(K)
9.487E+00
=====================================================================
Population analysis
-
ferromagnetic solution
=====================================================================
MULLIKEN POPULATION ANALYSIS
ALPHA+BETA ELECTRONS - NO. OF ELECTRONS
210.000000
ATOM
Z CHARGE
SHELL POPULATION
s
sp
sp
sp
sp
d
d
1 CR
24 21.884
2.000
8.047
2.251
4.487
1.331
3.078
.690
5 MN
25 23.147
2.000
8.081
2.170
4.299
1.489
4.478
.629
7 O
8
9.521
1.996
2.644
2.467
2.414
MULLIKEN POPULATION ANALYSIS
ALPHA-BETA ELECTRONS - NO. OF ELECTRONS
22.000000
ATOM
Z CHARGE
SHELL POPULATION
s
sp
sp
sp
sp
d
d
1 CR
24
3.057
.000
-.002
.011
.027
-.011
2.790
.242
5 MN
25
4.925
.000
-.003
.019
.055
-.052
4.408
.498
7 O
8
-.010
.000
.003
-.014
.002
================================================================================
================================================================================
Second
run
- Anti ferromagnetic solution:
Integrals calculation not affected by the spin state
Cr (atoms 1-2-3-4) unpaired electrons spin alpha;
Mn (atoms 5 and 6) unpaired electrons spin beta
================================================================================
RESTART FROM A PREVIOUS RUN DENSITY MATRIX
SPIN INVERSION IN SPIN DENSITY MATRIX FOR ATOMS:
5
6
=====================================================================
Convergence on total energy reached in 15 cycles
126

=====================================================================
CYCLE
15 ETOT(AU) -7.072808080821E+03 DETOT -4.930E-07 DE(K)
6.694E-06
======================================uuuu============================
Population analysis
-
anti ferromagnetic solution
=====================================================================
MULLIKEN POPULATION ANALYSIS
ALPHA+BETA ELECTRONS - NO. OF ELECTRONS
210.000000
ATOM
Z CHARGE
SHELL POPULATION
s
sp
sp
sp
sp
d
d
1 CR
24 21.884
2.000
8.047
2.251
4.487
1.331
3.078
.690
5 MN
25 23.149
2.000
8.081
2.170
4.299
1.489
4.479
.631
7 O
8
9.521
1.997
2.644
2.467
2.414
MULLIKEN POPULATION ANALYSIS
ALPHA-BETA ELECTRONS - NO. OF ELECTRONS
2.000000
ATOM
Z CHARGE
SHELL POPULATION
s
sp
sp
sp
sp
d
d
1 CR
24
3.049
.000
-.002
.011
.027
-.012
2.785
.240
5 MN
25 -4.917
.000
.003
-.018
-.055
.054 -4.406
-.495
7 O
8
-.045
.000
-.024
-.013
-.008
================================================================================
4.4
Geometry optimization - OPTCOORD
Crystal geometry input section (block1) for the geometry optimization of the urea molecule:
Example
Urea Molecule
Title
MOLECULE
Dimension of the system
15
Point group (C2v)
5
Number of non equivalent atoms
6 0.
0.
0.
Atomic number and cartesian coordinates
8 0.
0.
1.261401
7 0.
1.148247 -0.699790
1 0.
2.026550 -0.202817
1 0.
1.134080 -1.704975
OPTCOORD
Keyword to perform a geometry optimization
ENDOPT
End of the geometry optimization input block
END
End of the geometry input section
The Crystal output contains additional information on the optimization run after the initial
part of the geometry output:
BERNY OPTIMIZATION CONTROL
MAXIMUM GRADIENT COMPONENT
0.00045 MAXIMUM DISPLACEMENT COMPONENT
0.00030
R.M.S. OF GRADIENT COMPONENT 0.00180 R.M.S. OF DISPLACEMENT COMPONENTS
0.00120
THRESHOLD ON ENERGY CHANGE 0.100E-06 EXTRAPOLATING POLYNOMIAL ORDER
2
MAXIMUM ALLOWED NUMBER OF STEPS
100 SORTING OF ENERGY POINTS:
NO
ANALYTICAL GRADIENTS
On the first step of the optimization, the Crystal output contains both energy (complete
SCF cycle) and gradient parts. At the end of the first step, a convergence check is performed
on the initial forces and the optimization stops if the criteria are already satisfied.
For the subsequent steps, only a few lines on the optimization process are reported, namely:
current geometry, total energy and gradients, and convergence test. The output for the urea
molecule geometry optimization looks as follows:
127

*******************************************************************************
POINT CALCULATION
2
*******************************************************************************
ATOMS IN THE ASYMMETRIC UNIT
5 - ATOMS IN THE UNIT CELL:
8
ATOM
X(ANGSTROM)
Y(ANGSTROM)
Z(ANGSTROM)
*******************************************************************************
1 T
6 C
2.952109148728E-18
2.952109148728E-18
1.112498507543E-01
2 T
8 O
3.738040356496E-17
3.738040356496E-17
1.408675664831E+00
3 T
7 N
1.643486040825E-01
1.142475701821E+00 -6.110581854722E-01
4 F
7 N
-1.643486040825E-01 -1.142475701821E+00 -6.110581854722E-01
5 T
1 H
-2.742826212941E-01
1.173433830045E+00 -1.525846151428E+00
6 F
1 H
2.742826212941E-01 -1.173433830045E+00 -1.525846151428E+00
7 T
1 H
7.072543295810E-02
1.995428931213E+00 -5.916392089218E-02
8 F
1 H
-7.072543295810E-02 -1.995428931213E+00 -5.916392089218E-02
T = ATOM BELONGING TO THE ASYMMETRIC UNIT
INTRACELL NUCLEAR REPULSION (A.U.)
1.224347549602E+02 (BOHR = 0.529177249)
TOTAL ENERGY(DFT)(AU)( 29) -2.2514275144821E+02 DE-1.4E-08 DP 7.7E-09
SYMMETRY ALLOWED FORCES (ANALYTICAL) (DIRECTION, FORCE)
1
1.1072258E-01
2 -7.4201763E-02
3 -2.8080699E-03
4
2.3779951E-02
5 -8.3505121E-03
6
2.6313459E-03
7
3.1574557E-03
8
6.4585908E-03
9
1.1481013E-03
10 -1.1940670E-02
GRADIENT NORM
0.136422
GRADIENT THRESHOLD
0.500000
MAX GRADIENT
0.110723
THRESHOLD
0.000450 CONVERGED NO
RMS GRADIENT
0.043140
THRESHOLD
0.000300 CONVERGED NO
MAX DISPLAC.
0.081245
THRESHOLD
0.001800 CONVERGED NO
RMS DISPLAC.
0.035270
THRESHOLD
0.001200 CONVERGED NO
TTTTTTTTTTTTTTTTTTTTTTTTTTTTTT BERNY
TELAPSE
833.18 TCPU
826.21
After the first step, output for SCF and gradient calculations is routed to an external fortran
unit fort.65.
When all four convergence tests are satisfied, optimization is completed. The final energy and
the optimized structure are printed after the final convergence tests.
******************************************************************
* OPT END - CONVERGED * E(AU):
-2.251504436450E+02
POINTS
10 *
******************************************************************
...
FINAL OPTIMIZED GEOMETRY - DIMENSIONALITY OF THE SYSTEM
0
(NON PERIODIC DIRECTION: LATTICE PARAMETER FORMALLY SET TO 500)
*******************************************************************************
ATOMS IN THE ASYMMETRIC UNIT
5 - ATOMS IN THE UNIT CELL:
8
ATOM
X(ANGSTROM)
Y(ANGSTROM)
Z(ANGSTROM)
*******************************************************************************
1 T
6 C
3.947486080494E-18
3.947486080494E-18
1.487605014533E-01
2 T
8 O
3.633779934042E-17
3.633779934042E-17
1.369385259723E+00
3 T
7 N
1.721504570772E-01
1.150654062355E+00 -6.120839819527E-01
128

4 F
7 N
-1.721504570772E-01 -1.150654062355E+00 -6.120839819527E-01
5 T
1 H
-2.789616513042E-01
1.183857618602E+00 -1.516451281243E+00
6 F
1 H
2.789616513042E-01 -1.183857618602E+00 -1.516451281243E+00
7 T
1 H
6.793388863152E-02
1.994529879710E+00 -6.664311739212E-02
8 F
1 H
-6.793388863152E-02 -1.994529879710E+00 -6.664311739212E-02
T = ATOM BELONGING TO THE ASYMMETRIC UNIT
INTRACELL NUCLEAR REPULSION (A.U.)
1.233193031445E+02 (BOHR = 0.529177249)
SYMMETRY OPERATORS:
2
IV
INV
ROTATION MATRIX
TRANSLATOR (ANGSTROM)
1
1
1.
0.
0.
0.
1.
0.
0.
0.
1.
0.00000000
0.00000000
0.00000000
2
2 -1.
0.
0.
0. -1.
0.
0.
0.
1.
0.00000000
0.00000000
0.00000000
The final geometry is both printed in the Crystal output and written on the fortran unit
34. Thus, it can be recovered by reading the geometry from the external unit 34 (keyword
EXTERNAL, input block1, page 13).
Example
Urea Molecule
Title
EXTERNAL
Geometry read from the external fortran unit 34
optional keywords
END
End of the geometry input section
To restart an optimization, the same input deck used for the first steps has to be modified by
adding the keyword RESTART. The file with information on the previous optimization must
be defined as fortran unit 68
Example
Initial input
Restart input
Urea Molecule
Urea Molecule
MOLECULE
MOLECULE
15
15
5
5
6 0.
0.
0.
6 0.
0.
0.
8 0.
0.
1.261401
8 0.
0.
1.261401
7 0.
1.148247 -0.699790
7 0.
1.148247 -0.699790
1 0.
2.026550 -0.202817
1 0.
2.026550 -0.202817
1 0.
1.134080 -1.704975
1 0.
1.134080 -1.704975
OPTCOORD
OPTCOORD
ENDOPT
RESTART
END
ENDOPT
END
Partial optimization
In order to optimize the coordinates of the hydrogens in the molecular crystal of urea, the
following input must be entered:
129

Example
Urea Molecule
Title
MOLECULE
Dimension of the system
15
Point group (C2v)
5
Number of non equivalent atoms
6 0.
0.
0.
Atomic number and cartesian coordinates
8 0.
0.
1.261401
7 0.
1.148247 -0.699790
1 0.
2.026550 -0.202817
1 0.
1.134080 -1.704975
OPTCOORD
Keyword to perform a geometry optimization
ATOMFREE
Keyword for a partial optimization
4
Number of atoms to be optimized
5 6 7 8
Label of the atoms to be optimized
ENDOPT
End of the geometry optimization input block
END
End of the geometry input section
All the hydrogens are given, the program will detect the symmetry relationship between them.
Final run
In some cases, when the optimized geometry is far from the original one, the series truncation
defined with reference to the starting geometry may be inhomogeneous if applied to the final
geometry (see keyword FIXINDEX for explanation). As a consequence, the total energy com-
puted is not correctly defined, according to the numerical methods applied in CRYSTAL. The
correct energy of the optimized geometry, computed with true truncation series, is different in
those cases from the one given at the end of the optimization. A second optimization process,
with truncation series defined according to the geometry obtained in the first optimization,
must be performed to check if that geometry corresponds to a real energy minimum. The
keyword FINALRUN starts the process automatically.
A typical example is the geometry optimization of a surface, described with a slab
model.
The optimization process may lead to a structure significantly different
from the one cut from the bulk, when there is surface relaxation.
As an exam-
ple, the geometry optimization of the surface (001) of the -Al2O3 is reported.
Example - Optimization of surface
-Al2O3 - (001) surface title
CRYSTAL
dimension of the system
0 0 0
167
space group
4.7602 12.9933
lattice parameters
2
number of irreducible atoms
13 0.
0.
0.35216
fractional coordinates of first atom
8 0.30624 0.
0.25
fractional coordinates of second atom
SLABCUT
3D 2D
0 0 1
(h, k, l) Miller indices of the surface
1 6
number of layers, starting from the first classified
OPTCOORD
Keyword to perform a geometry optimization
FINALRUN
keyword to check gradients vs true series truncation
3
new optimization if convergence criteria are not satisfied
ENDOPT
end of the geometry optimization input block
END
end of the geometry input section
130

Neighbors analysis on the initial geometry obtained with SLABCUT
N = NUMBER OF NEIGHBORS AT DISTANCE R
11 cycles
ATOM
N
R/ANG
R/AU
NEIGHBORS (ATOM LABELS AND CELL INDICES)
1 AL
3
1.8551
3.5057
2 O
0 0 0
3 O
0 0 0
4 O
0 1 0
1 AL
3
3.2192
6.0834
5 AL
0 0 0
5 AL
1 1 0
5 AL
0 1 0
1 AL
3
3.2219
6.0885
2 O
-1 0 0
3 O
1 1 0
4 O
0 0 0
1 AL
3
3.4295
6.4808
7 O
0 1 0
8 O
0 1 0
9 O
0 0 0
1 AL
3
3.4990
6.6121
6 AL
0 0 0
6 AL
-1 0 0
6 AL
0 1 0
1 AL
1
3.8419
7.2601
10 AL
0 0 0
Total energy
E = -1399.7999027 hartree
Series truncation is defined with reference to that geometry.
Optimization begins. After 11 cycles convergence on gradients and displacements
is satisfied.
Neighbors analysis on the optimized geometry:
N = NUMBER OF NEIGHBORS AT DISTANCE Ra
6 cycles
ATOM
N
R/ANG
R/AU
NEIGHBORS (ATOM LABELS AND CELL INDICES)
1 AL
3
1.6886
3.1911
2 O
0 0 0
3 O
0 0 0
4 O
0 1 0
1 AL
1
2.6116
4.9351
10 AL
0 0 0
1 AL
3
2.8198
5.3286
7 O
0 1 0
8 O
0 1 0
9 O
0 0 0
1 AL
3
3.0425
5.7494
5 AL
0 0 0
5 AL
1 1 0
5 AL
0 1 0
1 AL
3
3.0430
5.7504
6 AL
0 0 0
6 AL
-1 0 0
6 AL
0 1 0
1 AL
3
3.1214
5.8987
2 O
-1 0 0
3 O
1 1 0
4 O
0 0 0
Total energy
E = -1400.1148194 hartree
Extensive relaxation
occurred, the Aluminum atoms penetrate into the
slab. Total energy and gradients are
computed after definition of
the truncation series
with reference to the new geometry (the process is authomatic if keyword
FINALRUN is specified in the input stream). Gradients do not match
the convergence criteria.
Total energy
E = -1400.1193593 hartree
The keyword FINALRUN was entered with ICODE=3, so geometry optimization restarts
from the first step final geometry, with truncation series defined relative to it.
After 6 optimization cycles, convergence criteria are satisfied.
Neighbors analysis on the final run optimized geometry
N = NUMBER OF NEIGHBORS AT DISTANCE R
ATOM
N
R/ANG
R/AU
NEIGHBORS (ATOM LABELS AND CELL INDICES)
1 AL
3
1.6863
3.1867
2 O
0 0 0
3 O
0 0 0
4 O
0 1 0
1 AL
1
2.5917
4.8976
10 AL
0 0 0
1 AL
3
2.8095
5.3092
7 O
0 1 0
8 O
0 1 0
9 O
0 0 0
1 AL
3
3.0382
5.7414
5 AL
0 0 0
5 AL
1 1 0
5 AL
0 1 0
1 AL
3
3.0387
5.7424
6 AL
0 0 0
6 AL
-1 0 0
6 AL
0 1 0
1 AL
3
3.1215
5.8987
2 O
-1 0 0
3 O
1 1 0
4 O
0 0 0
Total energy
E = -1400.1194545 hartree
131

The final geometry is printed, and written in fortran unit 34.
A final check on geometry can be done with the following input:
alpha-Al2O3 (corundum) 001 2 LAYERS (3D-->2D)
EXTERNAL
OPTCOORD
ENDOPT
END
The keyword EXTERNAL route the basic geometry input stream to fortran unit 34.
The file to be read is written at the end of the first optimization run.
No optimization starts, convergence criteria are already satisfied.
Total energy
E = -1400.1194544 hartree
132

Chapter 5
Basis set
The most common source of problems with CRYSTAL is probably connected with the basis set.
It should never be forgotten that ultimately the basis functions are Bloch functions, modulated
over the infinite lattice: any attempt to use large uncontracted molecular or atomic basis sets,
with very diffuse functions can result in the wasting of computational resources. The densely
packed nature of many crystalline structures gives rise to a large overlap between the basis
functions, and a quasi-linear dependence can occur, due to numerical limitations.
The choice of the basis set (BS) is one of the critical points, due to the large variety of
chemical bonding that can be found in a periodic system. For example, carbon can be involved
in covalent bonds (polyacetylene, diamond) as well as in strongly ionic situations (Be2C, where
the Mulliken charge of carbon is close to -4).
Many basis sets for lighter elements and the first row transition metal ions have been developed
for use in periodic systems. A selection of these which have been used in published work are
available on WWW:
http://www.crystal.unito.it
We summarize here some general considerations which can be useful in the construction of a
BS for periodic systems.
It is always useful to refer to some standard basis set; Pople's STO-nG, 3-21G and 6-21G have
proved to be good starting points. A molecular minimal basis set can in some cases be used
as it is; larger basis sets must be re-optimized specifically for the chemical periodic structure
under study.
Let us explore the adequacy of the molecular BS for crystalline compounds and add some
considerations which can be useful when a molecular BS must be modified or when an ex novo
crystalline BS is defined.
5.1
Molecular BSs performance in periodic systems
Two sets of all electron basis sets are included in CRYSTAL (see Chapter 1.2):
1. Minimal STO-nG basis set of Pople and co-workers
obtained by fitting Slater type orbitals with n contracted GTFs (n from 2 to 6, atomic
number from 1 to 54) [88, 89, 90, 91].
The above BSs are still widely used in spite of the poor quality of the resulting wave func-
tion, because they are well documented and as a rule provide quite reasonable optimized
geometries (due to fortuitous cancellation of errors) at low cost.
2. "Split valence" 3-21 and 6-21 BSs.
The core shells are described as a linear combination of 3 (up to atomic number 54)
or 6 (up to atomic number 18) gaussians; the two valence shells contain two and one
gaussians, respectively [92, 93]. Exponents (s and p functions of the same shell share the
same exponent) and contraction coefficients have been optimized variationally for the
isolated atoms.
133

A single set of polarization functions (p,d) can be added without causing numerical problems.
Standard molecular polarization functions are usually also adequate for periodic compounds.
When free basis sets are chosen, two points should be taken into account:
1. From the point of view of CPU time, basis sets with sp shells (s and p functions sharing
the same set of exponents) can give a saving factor as large as 4, in comparison with
basis sets where s and p have different exponents.
2. As a rule, extended atomic BSs, or 'triple zeta' type BSs should be avoided. Many of
the high quality molecular BSs (Roos, Dunning, Huzinaga) cannot be used in CRYSTAL
without modification, because the outer functions are too diffuse. One should not forget
that the real basis functions are Bloch functions.
Let us consider in more detail the possibility of using molecular BS for periodic systems. We
can refer to five different situations:
Core
functions
Valence
functions:
molecular crystals
covalent crystals
ionic crystals
metals.
5.2
Core functions
In this case standard (contracted) molecular BSs can be adopted without modification, be-
cause even when very strong crystal field effects are present, the deformation of inner states
is small, and can be correctly described through the linear variational parameters in SCF cal-
culation. An adequate description of the core states is important in order to avoid large basis
set superposition errors.
5.3
Valence functions
Molecular crystals
Molecular BSs, minimal and split-valence, are perfectly adequate. Tests have been performed
on bulk urea [94] and oxalic acid, where the molecules are at relatively small distances, with
STO-3G, 6-21, 6-21* and 6-21** BSs presenting no problem.
Covalent crystals.
Standard minimal and split valence BSs are usually adequate. In the split valence case the
best exponent of the most diffuse shell is always slightly higher than the one proposed for
molecules; in general it is advisable to re-optimize the exponent of this shell. This produces a
slightly improved basis, while reducing the cost of the calculation. Let us consider for example
the 6-21 basis set for carbon (in diamond) and silicon (bulk).
At an atomic level, the best exponent of the outer shell is 0.196 and 0.093 for C and Si, respec-
tively. Optimization of the valence shell has been repeated in the two crystalline compounds.
The innermost valence shell is essentially unaltered with respect to the atomic solution; for the
outer single-gaussian shell the best exponent is around 0.22 and 0.11 bohr-2 for carbon and
silicon, as shown in Table 5.1. The last entry of Table 5.1 refers to "catastrophic" behaviour:
the low value of the exponent generates unphysical states.
A set of 5 polarization single-gaussian d functions can be added to the 6-21G basis (6-21G*
BS); the best exponents for the solid are very close to those resulting from the optimization in
molecular contexts: 0.8 for diamond [95] and 0.45 for silicon.
Basis sets for III-V and IV-IV semiconductors (all electron and valence electron (to be associ-
ated with effective core pseudopotentials) are given in references [96, 97].
134

Table 5.1: Total energy per cell and number of computed bielectronic integrals in 106 units
(N), as a function of the exponent (bohr-2) of the most diffuse shell for carbon and silicon.
____________________________________________________________________
Diamond
Silicon
------------------------
--------------------------
a
N
Et
a
N
Et
___________________________________________________________
0.296
58
-75.6633
0.168
46
-577.8099
0.276
74
-75.6728
0.153
53
-577.8181
0.256
83
-75.6779
0.138
72
-577.8231
0.236
109
-75.6800
0.123
104
-577.8268
0.216
148
-75.6802
0.108
151
-577.8276
0.196
241
-75.6783
0.093
250
-577.8266
0.176
349
catastrophe
0.078
462
catastrophe
____________________________________________________________________
Ionic crystals.
Cations
The classification of covalent or ionic crystals is highly conventional, many systems being
midway. Let us first consider totally ionic compounds, such as LiH, MgO, or similar. For these
systems the cation valence shell is completely empty. Therefore, for cations it is convenient
to use a basis set containing the core functions plus an additional sp shell with a relatively
high exponent. For example, we used for Mg in MgO and for Li in LiH ( Li2 O and Li3 N) a
'valence' sp shell with exponent 0.4-0.3 and 0.5-0.6, respectively [28, 25].
The crystalline total energies obtained by using only core functions for Li or Mg and by adding
a valence shell to the cation differ by 0.1 eV/atom, or less. This figure is essentially the same
for a relatively large range of exponents of the valence shell (say 0.5-0.2 for Mg) [25].
It can be difficult (or impossible) to optimize the exponents of nearly empty shells: the en-
ergy decreases almost linearly with the exponent. Very low exponent values can give rise to
numerical instabilities, or require the calculation of an enormous number of integrals (selected
on the basis of overlap criteria). In the latter cases, when the energy gain is small (E 1 m
hartree for = 0.2 bohr-2), it is convenient to use a relatively large exponent.
Anions
Reference to isolated ion solutions is only possible for halides, because in such cases the ions
are stable even at the HF level. For other anions, which are stabilized by the crystalline
field (H-, O2-, N 3- and also C4-), the basis set must be re-designed with reference to the
crystalline environment. For example, let us consider the optimization of the O2- BS in Li2O
[28]. Preliminary tests indicated the fully ionic nature of the compound; the point was then to
allow the valence distribution to relax in the presence of the two extra electrons. We started
from a standard STO-6G BS. Two more gaussians were introduced in the 1s contraction, in
order to improve the virial coefficient and total energy, as a check of wave function quality. The
6 valence gaussians were contracted according to a 411 scheme; the exponents of the two outer
independent gaussians and the coefficients of the four contracted ones were optimized. Whereas
the two most diffuse gaussians are more diffuse than in the neutral isolated atom (=0.45 and
0.15 to be compared with =0.54 and 0.24 respectively), the rest of the O2- valence shell is
unchanged with respect to the atomic situation. The introduction of d functions in the oxygen
basis-set causes only a minor improvement in energy (1 10-4 hartree/cell, with a population
of 0.02 electrons/atom in the cell). Ionic BSs for H and N can be found in reference 1.
For anions, re-optimization of the most diffuse valence shell is mandatory; when starting from
a standard basis set, the most diffuse (or the two most diffuse) gaussians must be allowed to
relax.
135

From covalent to ionics
Intermediate situations must be considered individually, and a certain number of tests must
be performed in order to verify the adequacy of the selected BSs.
Let us consider for example -quartz (SiO2) and corundum (Al2O3). The exponent of the
outer shell for the 2 cations in the 6-21G BS is 0.093 (Si) and 0.064 (Al), respectively; in both
cases this function is too diffuse (in particular in the Al case it causes numerical catastrophes).
For quartz, re-optimization in the bulk gives =0.15 bohr-2 for Si (the dependence of total
energy per Si atom on is much smaller than the one resulting from Table 5.1; note too that
the cost at =0.15 is only 50% of the one at =0.09). On the contrary, the best molecular
and crystalline exponent (=0.37) for oxygen coincide. Corundum is more ionic than quartz,
and about 2 valence electrons are transferred to oxygen. In this case it is better to eliminate
the most diffuse valence shell of Al, and to use as independent functions the two gaussians of
the inner valence shells (=0.94 and 0.20 bohr-2, respectively [98]).
Metals
Very diffuse gaussians are required to reproduce the nearly uniform density characterizing
simple metallic systems, such as lithium and beryllium. This is the worse situation, where a
full optimization of the atomic basis set is probably impossible. Functions which are too diffuse
can create numerical problems, as will be discussed below.
The optimization procedure can start from 6-21 BS; the most diffuse valence shell (exponent
0.028 for Li and 0.077 for Be) can be dropped and the innermost valence shell (exponents 0.54
and 0.10 for Li, and 1.29 and 0.268 for Be) can be split.
Table 5.2: Example of BS for metallic lithium and beryllium derived from the standard 6-21G
BS
.
_______________________________________________________________
Lithium
Beryllium
shell
Exp.
Coeff.
shell
Exp.
Coeff.
_______________________________________________________________
s
642.418
0.00215096
s
1264.50
0.00194336
96.5164
0.0162677
189.930
0.0148251
22.0174
0.0776383
43.1275
0.0720662
6.1764
0.246495
12.0889
0.237022
1.93511 0.467506
3.80790 0.468789
sp
0.640
1.
1.
1.282
1.
1.
sp
0.10
1.
1.
0.27
1.
1.
_______________________________________________________________
At this point the outer gaussian of the 6G core contraction, with very similar exponents (0.64
and 1.28) to those of the innermost valence shell (0.54 and 1.29), can be used as an independent
(sp) function, and the innermost valence shell can be eliminated.
The resulting (reasonable) BS, derived from the split valence standard one, is reported in Table
5.2. Finally, the most diffuse gaussian can be optimized; in the two cases the minimum has
not been found owing to numerical instabilities.
See [99] for a more extensive discussion of the metallic lithium case.
5.4
Hints on crystalline basis set optimization
In the definition of a valence shell BS, each exponent can be varied in a relatively narrow range:
in the direction of higher exponents, large overlaps with the innermost functions may occur
(the rule of thumb is: exponents must be in a ratio not too far from 3; ratios smaller than
136

2 can give linear dependence problems); proceeding towards lower exponents, one must avoid
large overlaps with a high number of neighbours (remember: the basis functions are Bloch
functions).
Diffuse gaussian orbitals play a critical role in HF-LCAO calculations of crystals, especially
the three-dimensional ones; they are expensive, not always useful, in some cases dangerous.
Cost.
The number of integrals to be calculated increases dramatically with decreasing exponents; this
effect is almost absent in molecular calculations. Table 5.1 shows that the cost of the calculation
(number of bielectronic integrals) for silicon (diamond) can increase by a factor 10 (6) simply
by changing the exponent of the most diffuse single-gaussian from 0.168 to 0.078 (0.296 to
0.176). The cost is largely dominated by this shell, despite the fact that large contractions are
used for the 1s, 2sp and the innermost valence shell.
A high number of contracted primitives tremendously increases the integrals computation time.
Usefulness.
In atoms and molecules a large part of the additional variational freedom provided by diffuse
functions is used to describe the tails of the wave function, which are poorly represented by the
e-r2 decay of the gaussian function. On the contrary, in crystalline compounds (in particular
3D non-metallic systems), low exponent functions do not contribute appreciably to the wave
function, due to the large overlap between neighbours in all directions. A small split valence
BS such as the 6-21G one, is nearer to the variational limit in crystals than in molecules.
Numerical accuracy and catastrophic behaviour.
In some conditions, during the SCF (periodic) calculation, the system 'falls' into non-physical
states, characterized by very low single particle and total energies (see for example the last
entry in Table 5.1 and the above discussion on metals).
This behaviour, generically interpreted in our early papers as due to 'linear dependence', is
actually due to poor accuracy in the treatment of the Coulomb and exchange series. The
exchange series is much more delicate, for two reasons: first, long range contributions are not
taken into account (whereas the long range Coulomb contributions are included, although in an
approximate way); second, the "pseudoverlap" criteria associated with the two computational
parameters ITOL4 and ITOL5 mimic only in an approximate way the real behaviour of the
density matrix.
The risks of "numerical catastrophes" increase rapidly with a decreasing exponent; higher
precision is required in order to obtain physical solutions.
For non-metallic systems, and split-valence type BSs, the default computational conditions
given in section 1.3 are adequate for the optimization of the exponents of the valence shell and
for systematic studies of the energy versus volume curves.
For metallic systems, the optimization of the energy versus exponent curve could require ex-
tremely severe conditions for the exchange series and, as a consequence, for the reciprocal
space net. Reasonable values of the valence shell exponent (say 0.23 for beryllium and 0.10
for lithium, see Table 5.2), though not corresponding to a variational minimum, are reason-
ably adequate for the study of the structural and electronic properties of metallic systems (see
reference 1).
5.5
Check on basis-set quasi-linear-dependence
In order to check the risk of linear dependence of Bloch functions, it is possible to calculate
the eigenvalues of the overlap matrix in reciprocal space by running integrals and entering
the keyword EIGS (input block 3, page 64). Full input (general information, geometry, basis
set, SCF) is to be entered.
137

The overlap matrix in direct space is Fourier transformed at all the k points generated in the
irreducible part of the Brillouin zone, and diagonalized. The eigenvalues are printed.
The higher the numerical accuracy obtained by severe computational conditions, the closer
to 0 can be the eigenvalues without risk of numerical instabilities. Negative values indicate
numerical linear dependence. The program stops after the check (even if negative eigenvalues
are not detected).
The Cholesky reduction scheme [69] requires basis functions linearly independent. A symptom
of numerical dependence may produce an error message in RHOLSK or CHOLSK while running
scf.
138

Chapter 6
Theoretical framework
6.1
Basic equations
CRYSTAL is an ab initio Hartree-Fock LCAO program for the treatment of periodic systems.
LCAO, in the present case, means that each Crystalline Orbital , i(r; k), is a linear combina-
tion of Bloch functions (BF), (r; k), defined in terms of local functions, (r) (here referred
to as Atomic Orbitals, AOs).
i(r; k) =
a,i(k)(r; k)
(6.1)

(r; k) =
(r - A - g) eikg
(6.2)
g
A denotes the coordinate of the nucleus in the zero reference cell on which is centred, and
the
is extended to the set of all lattice vectors g.
g
The local functions are expressed as linear combinations of a certain number, nG, of individually
normalized (basis set) Gaussian type functions (GTF) characterized by the same centre, with
fixed coefficients, dj and exponents, j, defined in the input:
nG
(r - A - g) =
dj G(j; r - A - g)
(6.3)
j
The AOs belonging to a given atom are grouped into shells, . The shell can contain all AOs
with the same quantum numbers, n and , (for instance 3s, 2p, 3d shells), or all the AOs with the
same principal quantum number, n, if the number of GTFs and the corresponding exponents
are the same for all of them (mainly sp shells; this is known as the sp shells constraint). These
groupings permit a reduction in the number of auxiliary functions that need to be calculated
in the evaluation of electron integrals and therefore increase the speed of calculation.
A single, normalized, s-type GTF, G, is associated with each shell (the adjoined Gaussian of
shell ). The exponent is the smallest of the j exponents of the Gaussians in the contraction.
The adjoined Gaussian is used to estimate the AO overlap and select the level of approximation
to be adopted for the evaluation of the integrals.
The expansion coefficients of the Bloch functions, a,i(k), are calculated by solving the matrix
equation for each reciprocal lattice vector, k:
F(k)A(k) = S(k)A(k)E(k)
(6.4)
in which S(k) is the overlap matrix over the Bloch functions, E(k) is the diagonal energy
matrix and F(k) is the Fock matrix in reciprocal space:
F(k) =
Fg eikg
(6.5)
g
139

The matrix elements of Fg, the Fock matrix in direct space, can be written as a sum of
one-electron and two-electron contributions in the basis set of the AO:
F g = Hg + Bg
(6.6)
12
12
12
The one electron contribution is the sum of the kinetic and nuclear attraction terms:
Hg = T g + Zg = 0
+ 0
(6.7)
12
12
12
1 | T | g
2
1 | Z | g
2
In core pseudopotential calculations, ^
Z includes the sum of the atomic pseudopotentials.
The two electron term is the sum of the Coulomb and exchange contributions:
Bg = Cg + Xg =
12
12
12
1
P n
[(0g
h+n)
(0h
h+n)]
(6.8)
3,4
1
2 | h
3
4
- 2 1 3 | g2 4
3,4
n
h
The Coulomb interactions, that is, those of electron-nucleus, electron-electron and nucleus-
nucleus, are individually divergent, due to the infinite size of the system. The grouping of
corresponding terms is necessary in order to eliminate this divergence.
The Pn density matrix elements in the AOs basis set are computed by integration over the
volume of the Brillouin zone (BZ),
P n = 2
dkeikn
a (k)a
3,4
3j
4j (k)( F - j (k))
(6.9)
BZ
j
where ain denotes the i-th component of the n-th eigenvector, is the step function, F , the
Fermi energy and n, the n-th eigenvalue.
The total electronic energy per unit cell is given
by:
1
Eelec =
P g (Hg + Bg )
(6.10)
2
12
12
12
1,2
g
A discussion of the different contributions to the total energy is presented in [6, 10] and in
Chapter 11 of reference [20].
1
Ecoul =
P g
P n
[(0g
h+n)]
(6.11)
2
12
3,4
1
2 | h
3
4
1,2
g
3,4
n
h
1
Eexch = -
P g
P n
[(0h
h+n)]
(6.12)
4
12
34
1
3 | g
2
4
1,2
g
34
n
h
6.2
Remarks on the evaluation of the integrals
The approach adopted for the treatment of the Coulomb and exchange series is based on a few
simple ideas and on a few general tools, which can be summarized as follows:
1. Where possible, terms of the Coulomb series are aggregated so as to reduce the number
of integrals to be evaluated;
2. Exchange integrals which will combine with small density matrix elements are disre-
garded;
3. Integrals between non-overlapping distributions are approximated;
4. Approximations for large integrals must be very accurate; for small integrals large per-
centage errors can be accepted;
5. Selection must be very efficient, because a large number of possible terms must be checked
(adjoined Gaussians are very useful from this point of view).
140

6.3
Treatment of the Coulomb series
For the evaluation of the Coulomb contributions to the total energy and Fock matrix, correct
coupling of electron-nucleus and electron-electron interactions is essential. The computational
technique for doing so was presented by Dovesi et al [100] and by Saunders et al. [10]. It may
be summarized as follows.
Consider the Coulomb bielectronic contribution to the Fock matrix (Cg ) and to the total
12
energy :
1
Ecoul =
P g
P n
[(0g
h+n)
(6.13)
ee
2
12
3,4
1
2 | h
3
4
1,2
g
3,4
n
h
Seven indices are involved in equation 6.13; four of them (1, 2, 3 and 4) refer to the AOs of
the unit cell; in principle, the other three (g, n and h) span the infinite set of translation
vectors: for example, g(r) is AO number 2 in cell g. P is the density matrix; the usual
2
notation is used for the bielectronic integrals. Due to the localized nature of the basis set, the
total charges, q1 and q2, associated with the two pseudo-overlap distributions: {G10G2g} and
{G3hG4h+n}, decay exponentially to zero with increasing |g| and |n| (for example, G1 is the
adjoined gaussian of the shell to which 1 belongs).
A Coulomb overlap parameter, Sc, can be defined in such a way that when either q1 or q2 are
smaller than Sc, the bielectronic integral is disregarded, and the sum over g or n truncated.
The ITOL1 input parameter is defined as ITOL1=-log10Sc.
The same parameter value is
used for selecting overlap, kinetic, and multipole integrals.
The problem of the h summation in equation 6.13 is more delicate, h being related to the dis-
tance between the two interacting distributions. The multipolar expansion scheme illustrated
below is particularly effective when large unit cell or low dimensionality systems are considered.
The electron-electron and electron-nuclei series (Cg and Zg ) can be rearranged as follows:
12
12
1. Mulliken shell net charge distributions are defined as :
(r - h) {} {} - Z =
P n
34
3(r - h) 4(r - h - n) - Z
(6.14)
3 4n
where Z is the fraction of nuclear charge formally attributed to shell , and {} is the
electron charge distribution of shell .
2. Z and C contributions are reordered:
Cg + Zg =
dr dr 0(r) g(r)
12
12
1
2
|r - r - h|-1 (r - h)
(6.15)

h
3. For a given shell , there is a finite set B of h vectors for which the two interacting
distributions overlap; in this B zone (bielectronic zone), all the bielectronic integrals are
evaluated explicitly. In the outer, infinite region which we define as M, complementary
to B (the mono-electronic zone), can be expanded in multipoles and the series can be
evaluated to infinity analytically, using Ewald's method combined with recursion formulae
[10].
The resulting expression for the Coulomb contribution to the Fock matrix is:
Cg + Zg =
[
P n (0g
h+n) +
12
12
{ B

h
3
4
n
34
1
2 | h
3
4
-
m(A
,m
; {})m(12g; A + h)] +
(6.16)
+
m(A
h
,m
; {} )m(12g; A + h)}
where:
m(A; {}) =
dr (r - A)N mXm(r - A)
(6.17)
141

m(12g; A + h) =
dr0(r)g(r)Xm(r
1
2
- A - h) |r - A - h|-2 -1
(6.18)
The Ewald term in eq. 6.16 includes zones B + M. The contribution from B is subtracted.
The Xm functions entering in the definition of the multipoles and field terms are real, solid
harmonics, and N m, the corresponding normalization coefficients.
The advantage of using equation 6.16 is that many four-centre (long-range) integrals can be
replaced by fewer three-centre integrals.
The attribution of the interaction between 1 = {10, 2g} and to the exact, short-range or
to the approximate, long-range zone is performed by comparing the penetration between 1
and with the ITOL2 input parameter (if ITOL2> - log S1, then is attributed to the
exact B zone).
The multipolar expansion in the approximate zone is truncated at L = max. The default value
of L is 4; the maximum possible value is 6, the minimum suggested value, 2 (defined via the
input keyword POLEORDR, input block 3, page 68).
6.4
The exchange series
The exchange series does not require particular manipulations of the kind discussed in the
previous section for the Coulomb series, but needs a careful selection of the terms contributing
appreciably to the Fock operator and to the total energy [9]. The exchange contribution to the
total energy can be written as follows:
1
1
Eex =
P g [-
P n
(0h
h+n)]
(6.19)
2
12
2
34
1
3 | g
2
4
12
g
34
n
h
where the term in square brackets is the exchange contribution to the 12g element of the direct
space Fock matrix. Eex has no counterpart of opposite sign as the Coulomb term has; hence,
it must converge by itself.
The h summation can be truncated after a few terms, since the {0h
1
3 } overlap distribution
decays exponentially as h increases. Similar considerations apply to the second charge distri-
bution. In CRYSTAL, the h summation is, therefore, truncated when the charge associated
with either {G10 G3h} or {G2g G4h + n} is smaller than 10-ITOL3.
The situation is more complicated when g and n summations are analysed. Let us consider
the leading terms at large distance, corresponding to 1=3, 2=4, h = 0 and n = g:
eg =
)2(10 10
12
-1/4(P g
12
|2g 2g) = -(pg)2/(4|g|)
(6.20)
(Here pg indicates the dominant P matrix element at long range). Since the number of terms
per unit distance of this kind increases as |g|d-1, where d is the dimensionality of the system,
it is clear that the convergence of the series depends critically on the long range behaviour of
the bond order matrix.
Cancellation effects, associated in particular with the oscillatory behaviour of the density ma-
trix in metallic systems, are not predominant at long range. Even if the actual behaviour of
the P matrix elements cannot be predicted because it depends in a complicated way on the
physical nature of the compound [83], on orthogonality constraints and on basis set quality,
the different range of valence and core elements can be exploited by adopting a pseudoverlap
criterion. This consists in truncating g summations when the
dr0g overlap is smaller
1
2
than a given threshold, defined as P g
(where ITOL4 = -log
)) and also truncating the
ex
10 (P g
ex
n summation when
dr0n overlap is smaller than the threshold, P n
(ITOL5 = -log
3
4
ex
10
(P n )).
ex
Despite its partially arbitrary nature, this criterion presents some advantages with respect to
other more elaborate schemes: it is similar to the other truncation schemes (ITOL1, ITOL2,
ITOL3), and so the same classification tables can be used; it is, in addition, reasonably efficient
in terms of space occupation and computer time.
This truncation scheme is symmetric with respect to the g and n summations. However, if
account is not taken of the different role of the two summations in the SC (Self Consistent)
stage, distortions may be generated in the exchange field as felt by charge distributions 1T ,
2
142

where T labels the largest (in modulus) g vector taken into account according to ITOL4.
This distortion may be variationally exploited, and unphysically large density matrix elements
build up progressively along the SC stage, eventually leading to catastrophic behaviour (see
Chapter II.5 of reference [7] for a discussion of this point). In order to overcome this problem,
the threshold, P n (ITOL5) for n summation must be more severe than that for g summation
ex
(ITOL4). In this way, all the integrals whose second pseudo charge
dr0n is larger than
3
4
P n are taken into account. A difference in the two thresholds ranging from three to eight
ex
orders of magnitude is sufficient to stabilize the SC behaviour in most cases.
6.5
Bipolar expansion approximation of Coulomb and ex-
change integrals
We may now return to the partition of the h summation in the Coulomb series shown in
equation 6.13. Consider one contribution to the charge distribution of electron 1, centred in
the reference cell: 0 = 0g; now consider the charge distribution
1
2
(h) of shell centred
in cell h (equation 6.14). For small |h| values, and 0 overlap, so that all the related
bielectronic integrals must be evaluated exactly, one by one; for larger values of |h|, is
external to 0, so that all the related bielectronic integrals are grouped and evaluated in an
approximate way through the multipolar expansion of .
However, in many instances, although is not external to 0, the two-centre hh+n con-
3
4
tributions to are external to 0 = 0g; in this case, instead of exactly evaluating the
1
2
bielectronic integral, a two-centre truncated bipolar expansion can be used (see Chapter II.4.c
in reference [7] and references therein).
In order to decide to which zone a shell may be ascribed, we proceed as follows: when, for a
given pair of shells 0g, shell h is attributed to the B (bielectronic) zone, the penetration
1
2
3
between the products of adjoined Gaussians G0Gg and GhGh+n is estimated: the default
1
2
3
4
value of the penetration parameter is 14, and the block of bielectronic integrals is attributed
accordingly to the be (exact) or to the bb (bipolar) zone. The set of h vectors defining the B
zone of 0= {12g} and {3} is then split into two subsets, which are specific for each
partner l of
4
3.
A similar scheme is adopted for the selected exchange integrals (see previous section) whose
pseudo charges do not overlap appreciably. The default value of the penetration parameter is
10.
The total energy change due to the bipolar expansion approximation should not be greater
than 10-4 hartree/atom; exact evaluation of all the bielectronic integrals (obtained by setting
the penetration parameter value > 20000) increases the computational cost by a factor of
between 1.3 and 3. Multipolar expansion is very efficient, because the following two conditions
are fulfilled:
1. A general algorithm is available for reaching high
values easily and economically [100,
10]. The maximum allowed value is =6.
2. The multipolar series converges rapidly, either because the interacting distributions are
nearly spherical (shell expansion), or because their functional expression is such that
their multipoles are zero above a certain (low)
value.
6.6
Exploitation of symmetry
Translational symmetry allows the factorization of the eigenvalue problem in periodic calcula-
tions, because the Bloch functions are a basis for irreducible representations of the translational
group.
In periodic calculations, point symmetry is exploited to reduce the number of points for which
the matrix equations are to be solved. Point symmetry is also explicitly used in the reconstruc-
tion of the Hamiltonian, which is totally symmetric with respect to the point group operators
of the system.
143

In the HF-CO-LCAO scheme, the very extensive use of point symmetry allows us to evaluate
bielectronic and mono-electronic integrals with saving factors as large as h in the number of
bielectronic integrals to be computed or h2 in the number of those to be stored for the SCF part
of the calculation, where h is the order of the point group. The main steps of the procedure
[8] can be summarized as follows:
The set of Coulomb and exchange integrals whose 3,4 indices identify translationally
equivalent pairs of AOs, so that the associated element of the density matrix P34 is the
same, are summed together to give D1234 elements:
D1,2T ;3,4Q =
[(0g
h+n)
h
h+n)]
(6.21)
1
2 | h
3
4
- 1/2(01 3 | g2 4
n
The products of AOs 12 ( and 34) are classified in symmetry-related sets; using the
fact that the Fock matrix is totally symmetric, only those quantities are evaluated whose
indices 1, 2 refer to the first member of a symmetry set. The corresponding saving factor
is as large as h.
Using the symmetry properties of the density matrix, D quantities referring to 3, 4, cou-
ples belonging to the same symmetry set (and with the same 1, 2g index) can be combined
after multiplication by appropriate symmetry matrices, so that a single quantity for each
3, 4 symmetry set is to be stored, with a saving factor in storage of the order of h.
The symmetry Pn = P-n is exploited.
34
43
The symmetry Fg = F-g is exploited.
12
21
Symmetry-adapted Crystalline Orbitals
A computational procedure for generating space-symmetry-adapted Bloch functions, when BF
are built from a basis of local functions (AO), is implemented in the CRYSTAL98 code. The
method, that applies to any space group and AOs of any quantum number, is based on the
diagonalization of Dirac characters. For its implementation, it does not require character tables
or related data as an input, since the information is automatically generated starting from the
space group symbol and the AO basis set. Formal aspects of the method, not available in
textbooks, are discussed in:
C. Zicovich-Wilson and R. Dovesi
On the use of Symmetry Adapted Crystalline Orbitals in SCF-LCAO periodic calculations. I.
The construction of the Symmetrized Orbitals
Int. J. Quantum Chem. 67, 299310 (1998)
C. Zicovich-Wilson and R. Dovesi
On the use of Symmetry Adapted Crystalline Orbitals in SCF-LCAO periodic calculations. II.
Implementation of the Self-Consistent-Field scheme and examples
Int. J. Quantum Chem. 67, 311320 (1998).
The following table presents the performance obtained with the new method. In all cases
convergence is reached in ten cycles.
144

System
Chabazite
Pyrope
Faujasite
Space Group
R
3m
Ia3d
F d3m
N. of atoms
36
80
144
N. of AOs
432
1200
1728
N. symmetry operators
12
6
3
48
48
CPU time (sec) on IBM RISC-6000/365
integrals
447
900
1945
4286
815
Atomic BF(ABF) scf (total)
1380
2162
4613
24143
50975
Atomic BF scf (diagonalization)
898
898
898
19833
44970
Symmetry Adapted BF (SABF) scf (total)
526
1391
4335
3394
2729
Symmetry Adapted BF scf (diagonalization)
42
97
570
312
523
ABF/SABF scf time
2.62
1.55
1.06
7.11
18.7
6.7
Reciprocal space integration
The integration in reciprocal space is an important aspect of ab initio calculations for periodic
structures. The problem arises at each stage of the self-consistent procedure, when determining
the Fermi energy, F , when reconstructing the one-electron density matrix, and, after self-
consistency is reached, when calculating the density of states (DOS) and a number of observable
quantities. The P matrix in direct space is computed following equation 6.9. The technique
adopted to compute F and the P matrix in the SCF step is described in reference [101].
The Fourier-Legendre technique presented in Chapter II.6 of reference [7] is adopted in the
calculation of total and projected DOS. The Fermi energy and the integral in equation 6.9 are
evaluated starting from the knowledge of the eigenvalues, n(k) and the eigenvectors, an(k),
at a certain set of sampling points, {}. In 3D crystals, the sampling points belong to a lattice
(called the Monkhorst net, [102] ) with basis vectors b1/s1, b2/s2, b3/s3, where b1, b2 and b3
are the ordinary reciprocal lattice vectors; s1, s2 and s3 (input as IS1, IS2 and IS3) are integer
shrinking factors. Unless otherwise specified, IS1=IS2=IS3=IS. In 2D crystals, IS3 is set equal
to 1; in 1D crystals both IS2 and IS3 are set equal to 1. Only points of the Monkhorst net
belonging to the irreducible part of the Brillouin Zone (BZ) are considered, with associated
geometrical weights, wi.
In the selection of the points for non-centrosymmetric crystal, time-reversal symmetry is
exploited ( n() = n(-)).
The number of inequivalent sampling points, i, is asymptotically given by the product of the
shrinking factors divided by the order of the point group. In high symmetry systems and with
small si values, it may be considerably larger because many points lie on symmetry planes or
axes.
Two completely different situations (which are automatically identified by the code) must now
be considered, depending on whether the system is an insulator (or zero gap semiconductor), or
a conductor. In the former case, all bands are either fully occupied or vacant. The identification
of
F
is elementary, and the Fourier transform expressed by equation 6.9 is reduced to a
weighted sum of the integrand function over the set {i} with weights wi, the sum over n being
limited to occupied bands.
The case of conductors is more complicated; an additional parameter, ISP, enter into play.
ISP (or ISP1, ISP2, ISP3) are Gilat shrinking factors which define a net Gilat net [103, 104]
completely analogous to the Monkhorst net. The value of ISP is larger than IS (by up to a
factor of 2), giving a denser net.
In high symmetry systems, it is convenient to assign IS magic values such that all low multi-
plicity (high symmetry) points belong to the Monkhorst lattice. Although this choice does not
correspond to maximum efficiency, it gives a safer estimate of the integral.
The value assigned to ISP is irrelevant for non-conductors. However, a non-conductor may
give rise to a conducting structure at the initial stages of the SCF cycle, owing, for instance,
to a very unbalanced initial guess of the density matrix. The ISP parameter must therefore be
defined in all cases.
145

6.8
Electron momentum density
and related quantities
Three functions may be computed which have the same information content but different use
in the discussion of theoretical and experimental results; the momentum density itself, (p) or
EMD; the Compton profile function, J(p) or CP; the autocorrelation function, or reciprocal
space form factor, B(r) or BR.
With reference to a Crystalline-Orbital (CO)-LCAO wave function, the EMD can be expressed
as the sum of the squared moduli of the occupied COs in a momentum representation, or equiv-
alently, as the diagonal element of the six-dimensional Fourier transform of the one electron
density matrix from configuration to momentum space:
(p)
=
1/VBZ
dk |j(k, p)|2 [ F - j(k)] =
(6.22)
j
BZ
=
e-ip(s -s )


aj(p0)a (p0)
(p) [
j
(p)

F - j (p0)]
(6.23)
j

(p)
=
N -1
drdr e-ip(r -r)(r - r )
(6.24)
g
=
P
-s )
e-ip(g+s

(p)(p)
(6.25)


g
In the above equations p0 is the value of momentum in the Brillouin zone (BZ), which is related
to p by a reciprocal lattice vector K, s is the fractional coordinate of the

centre, and (p)
is the Fourier transform of (r), calculated analytically:
(p) =
dr(r) eipr
(6.26)
Equation 6.25 is used by default to compute the core band contribution, and equation 6.23 the
valence band contribution.
The CPs are obtained by 2D integration of the EMD over a plane through p and perpendicular
to the p direction. After indicating with p
the general vector perpendicular to p, we have:

J (p) =
dp (p + p )
(6.27)


It is customary to make reference to CPs as functions of a single variable p, with reference to
a particular direction < hkl > identified by a vector
e = (ha1 + ka2 + la3)/|(ha1 + ka2 + la3)|
We have:
J<hkl>(p) = J(p e)
(6.28)
The function J<hkl>(p) will be referred to as directional CPs.
The weighted average of the directional CPs over all directions is the average CP.
In the so called impulse approximation, J<hkl>(p) may be related to the experimental CPs,
after correction for the effect of limited resolution as a convolution of the "infinite resolution"
results, J 0
(p), with a normalized function characterized by a given standard deviation :
<hkl>
+
J
(p) =
dp J 0
(p )g
<hkl>
<hkl>
(p - p )
(6.29)
-
146

In CRYSTAL g is a gaussian function with standard deviation . Once the directional CPs are
available, the numerical evaluation of the corresponding autocorrelation function, or reciprocal
space form factor, B(r) is given by the 1D Fourier Transform:
1
+
B<hkl>(r) =
dpJ<hkl>(p)ei pr
(6.30)
2 -
The average Compton profile can be evaluated from the average EMD:
q
J (q) =
(p)pdp
(6.31)
0
and can be used for the evaluation of the kinetic energy:

KE =
p2J (p)dp
(6.32)
0
6.9
Elastic Moduli of Periodic Systems
The elastic constants are second derivatives of the energy density with respect to strain com-
ponents:
2E
Cij = 1/V
(6.33)
i j
where V is the volume of the cell. The energy derivatives must be evaluated numerically
(analytical gradients are not implemented in CRYSTAL 95). Particular care is required in the
selection of the computational parameters and of the points where the energy is evaluated, in
order to avoid large numerical errors in the fitting procedure (FIXINDEX, page 65, page 65).
When the unit cell is deformed, the point group is reduced to a subgroup of the original
point group (see examples below). The new point group is automatically selected by the code.
Off-diagonal (partial derivatives) elastic constants can be computed as linear combinations
of single-variable energy curves. For example, for a cubic system, C12 can be obtained from
B=(C11 + 2C12)/3 and (C11 - C12) (see examples below). Following the deformation of the
unit cell, internal relaxation of the atoms may be necessary (depending on the space group
symmetry) See test 20, referring to Li2 O.
The analysis of the point group at the atomic positions (printed by the ATOMSYMM option,
page 27) is useful in finding the atomic coordinates to be relaxed. Examples of deformation
strategies are discussed in references [28, 105].
In a crystalline system a point r is usually defined in terms of its fractionary components:
r = h Lp
where :
l1 l1x l1y l1z
Lp =
l
=
l
(6.34)
2
2x
l2y
l2z
l


3
l3x
l3y
l3z
V = det(Lp)
l1, l2, l3 are the fundamental vectors of the primitive cell, h is the fractional vector and V the
cell volume.
Lp can be computed from the six cell parameters a, b, c, , , . For instance, the matrix Lp
for a face centred cubic lattice with lattice parameter a has the form:
0 a/2 a/2
Lp =
a/2
0
a/2
a/2 a/2 0
147

Under an elastic strain, any particle at r migrates microscopically to r according to the relation:
r = r (I + )
where
is the symmetric Lagrangian elastic tensor.
In the deformed crystalline system:
r = h Lp
L = (I + )L
p
p
(6.35)
or:
L = L
p
p + Z
(6.36)
where
Z =
Lp
V = det(L )
p
The deformation may be constrained to be volume-conserving, in which case the lattice vectors
of the distorted cell must be scaled as follows:
L

p" = L (V /V )1/3
(6.37)
p
If a non-symmetric Lagrangian elastic tensor, , is used, instead of , the deformation is the
sum of a strain ( ) and a rotation () of the crystal:
= ( + +)/2
= ( - +)/2
The total energy of the crystal is invariant to a pure rotation, which allows non-symmetric
matrices to be employed. However, a non-symmetric deformation will lower the symmetry of
the system, and therefore increase the complexity of the calculation, since the cost required is
roughly inversely proportional to the order of the point group.
The elastic constants of a crystal are defined as the second derivatives of the energy with
respect to the elements of the infinitesimal Lagrangian strain tensor .
Let us define, according to the Voigt convention:
1 =
11
4 =
32 +
23
2 =
22
5 =
13 +
31
3 =
33
6 =
12 +
21
A Taylor expansion of the energy of the unit cell to second order in the strain components
yields:
6
6
E
2E
E( ) = E(0) +
i + 1/2
i j
(6.38)
i
i j
i
i,j
If E(0) refers to the equilibrium configuration the first derivative is zero, since there is no
force on any atom in equilibrium. The elastic constants of the system can be obtained by
evaluating the energy as a function of deformations of the unit cell parameters. The indices
of the non-zero element(s) (in the Voigt convention) of the
matrix give the corresponding
elastic constants.
148

Examples of
matrices for cubic systems
Consider a face-centred cubic system, for example Li2O, with the Fm3m space group.
For cubic systems there are only three independent elastic constants (C11, C12 and C44), as the
symmetry analysis shows that:
C11
= C22
= C33;
C44
= C55
= C66;
C12
= C13
= C23;
Cij
= 0
for i = 1, 6,
j = 4, 6
and i = j.
Calculation of C11
The
matrix for the calculation of C11 is
0 0
=
0
0
0
0 0 0
The energy expression is:
2E
E() = E(0) + 1/2
2 + = a + b2 + c3
21
where a, b, c are the coefficients of a polynomial fit of E versus , usually truncated to fourth
order (see examples below). Then
2E
2b
C11 = 1/V
=
2
V
1
The above distortion reduces the number of point symmetry operators to 12 (tetragonal dis-
tortion).
Calculation of C11 - C12
The
matrix for the calculation of the C11 - C12 combination has the form:
0 0
=
0

- 0
0
0
0
The energy expression is:
2E
2E
2E
E( 1, 2)
=
E(0, 0) + 1/2
2 + 1/2
2 -
2 + =
2
2

1
2
1 2
= E(0, 0) + V (C11 - C12)2 + = a + b2 +
Then C11 - C12 = b/V
With the previous form of the
matrix the number of point symmetry operators is reduced to
8, whereas the following
matrix reduces the number of point symmetry operators to 16:
0 0
=
0

0
0 0 -2
149

E( 1, 2, 3) = E(0, 0, 0) + 3V (C11 - C12)2 + = a + b2 +
and (C11 - C12) = b/3V
Calculation of C44
Monoclinic deformation, 4 point symmetry operators.
The
matrix has the form:
0 0 0
=
0
0
x
0 x 0
The energy expression is ( = 2x) (see Voigt convention and equation 6.38)
2E
2E
E( 4) = E(0) + 1/2
2 + = E(0) + 2
x2 + = a + bx2 +
2
2
4
4
so that C44 = b/2V .
Calculation of C44
Rhombohedral deformation, 12 point symmetry operators.
The
matrix has the form:
0 x x
=
x
0
x
x x 0
The energy expression is ( = 2x, C45 = C46 = C56 = 0)
2E
2E
E( 4, 5, 6) = E(0) + 3/2
2 + = E(0) + 6
x2 + = a + bx2 +
2
2
4
4
so that C44 = b/6V .
Bulk modulus
The bulk modulus can be evaluated simply by varying the lattice constant, (1 in cubic systems)
without the use of the
matrix, and fitting the curve E(V ).
If the
matrix is used, the relation between B and Cij (cubic systems) must be taken into
account:
B = (C11 + 2C12)/3
The
matrix has the form:
150

0 0
=
0

0
0 0
and the energy:
2E
2E
E( )
=
E(0) + 3/2
2 + 3
2 =
(6.39)
2

1
1 2
3V
=
E(0) +
[C11 + 2C12]2
(6.40)
2
so that B = 2 b
9V
N.B. Conversion factors:
1 hartree
A-3 = 4359.74812 GPa
1 GPa = 1 GN m-2 = 1 GJ m-3 = 1010 dyne cm-2 = 10-2 Mbar.
151

6.10
Spontaneous polarization through the Berry phase
approach
(,1)
The electronic phase of a system in the direction 1,
, can be written as:
el
(,1)
1
(,1)

=

(k)
(6.41)
el
s2s3
j1,j2,j3
j2,j3 j1
The electronic contribution to the polarization of a system can be written as :
-1
()
1
()
P
=
B()

(6.42)
el
()
el
()
Where (B())-1 is the reciprocal lattice vectors components inverse matrix and
the elec-
el
(,i)
tronic phase difference vector of a system (which components are
). The nuclear con-
el
()
tribution to the polarization of a system , Pnuc can also be written as:
1
()
P() =
R
nuc
ZA
(6.43)
()
A
A
()
where R
and Z
A
A are the position vector and the nuclear charge of the atom A respectively of
the system . The total polarization is the sum of these two contributions and can be written
as
()
()
P
= P() + P
(6.44)
tot
nuc
el
The spontaneous polarization is the difference between the systems = 1 and = 0
()
()
P = P
(6.45)
tot - Ptot
Spontaneous polarization through the localized crystalline
orbitals approach
()
The electronic contribution to the polarization of a system , P
, can be written as
el
()
e
P
=
r
el

(6.46)
()
Where r is the centroid of the Wannier function .
()
The nuclear contribution to the polarization of a system , Pnuc can also be written as
1
P() =
R
nuc
A ZA
(6.47)
() A
where RA and ZA are the position vector and the nuclear charge of the atom A respectively.
The total polarization is the sum of these two contributions and can be written as
()
()
P
= P() + P
(6.48)
tot
nuc
el
The spontaneous polarization is the difference between the both systems = 1 and = 0:
(1)
(2)
P = P
(6.49)
tot - Ptot
To calculate the spontaneous polarization, a preliminary run is needed for each of the two
systems = 1 and = 0. Then a third run with the keyword SPOLWF gives the difference of
polarization between systems = 1 and = 0.
152

6.11
Piezoelectricity through the Berry phase approach
The electronic phase vector of a system , is given by (2.1). The nuclear phase vector of a
()
system , nuc, can be written as
() = () B()P()
(6.50)
nuc
nuc
Where B() reciprocal lattice vectors components matrix.The last equation can be simplified
thanks to (6.43):
()
() = B()
R
nuc
Z
A
A
(6.51)
A
So the phase vector of a system , () is:
()
() = () +
(6.52)
nuc
el
The proper piezoelectric constants can be obtained by:
1 1
d
~
eijk = -
a,i
(6.53)
2
d jk

Where is projection of the phase along the direction and a,i is the component of a
lattice vector a along the cartesian axis i . To obtain the improper piezoelectric constants,
the following correction must done:
eijk = ~
eijk + ijPk - jkPi
(6.54)
In the piezoelectric constants calculations the d term is evaluated numerically. The calcu-
d jk
lated term is:
(1)
(0)
d

-
=
(6.55)
d
(1)
jk
jk
- (0)
jk
jk
Piezoelectricity through the localized crystalline orbitals
approach
The electronic phase vector of a system , is given by:
()
()

= () B()P
(6.56)
el
el
Where B() reciprocal lattice vectors components matrix. The nuclear phase vector of a system
()
, nuc, can be written as
() = () B()P()
(6.57)
nuc
nuc
The last equation can be simplified thanks to 6.43:
()
() = B()
R
nuc
Z
A
A
(6.58)
A
So the phase vector of a system , () is:
()
() = () +
(6.59)
nuc
el
The proper piezoelectric constants can be obtained by:
1 1
d
~
eijk = -
a,i
(6.60)
2
d jk

153

Where is projection of the phase along the direction and a,i is the component of a
lattice vector a along the cartesian axis i . To obtain the improper piezoelectric constants,
the following correction must done:
eijk = ~
eijk + ijPk - jkPi
(6.61)
In the piezoelectric constants calculations the d term is evaluated numerically. The calcu-
d jk
lated term is:
(1)
(0)
d

-
=
(6.62)
d
(1)
jk
jk
- (0)
jk
jk
154

Appendix A
Symmetry groups
A.1
Labels and symbols of the space groups
The labels are according to the International Tables for Crystallography [21]. The symbols are
derived by the standard SHORT symbols, as shown in the following examples:
Symbol
Input to CRYSTAL
P
6 2 m

P -6 2 M ;
P 63 m

P 63 M.
For the groups 221-230 the symbols are according to the 1952 edition of the International
Tables, not to the 1982 edition. The difference involves the 3 axis: 3 (1952 edition);
3 (1982
edition) (Example group 221: 1952 ed. P m 3 m ; 1982 ed. P m 3 m)
155

IGR
symbol
IGR
symbol
IGR
symbol
Triclinic lattices
37
Ccc2
Tetragonal lattices
1
P 1
38
Amm2
75
P 4
2
P
1
39
Abm2
76
P 41
Monoclinic lattices
40
Ama2
77
P 42
3
P 2
41
Aba2
78
P 43
4
P 21
42
F mm2
79
I4
5
C2
43
F dd2
80
I41
6
P m
44
Imm2
81
P
4
7
P c
45
Iba2
82
I
4
8
Cm
46
Ima2
83
P 4/m
9
Cc
47
P mmm
84
P 42/m
10
P 2/m
48
P nnn
85
P 4/n
11
P 21/m
49
P ccm
86
P 42/n
12
C2/m
50
P ban
87
I4/m
13
P 2/c
51
P mma
88
I41/a
14
P 21/c
52
P nna
89
P 422
15
C2/c
53
P mna
90
P 4212
Orthoorhombic lattices
54
P cca
91
P 4122
16
P 222
55
P bam
92
P 41212
17
P 2221
56
P ccn
93
P 4222
18
P 21212
57
P bcm
94
P 42212
19
P 212121
58
P nnm
95
P 4322
20
C2221
59
P mmn
96
P 43212
21
C222
60
P bcn
97
I422
22
F 222
61
P bca
98
I4122
23
I222
62
P nma
99
P 4mm
24
I212121
63
Cmcm
100
P 4bm
25
P mm2
64
Cmca
101
P 42cm
26
P mc21
65
Cmmm
102
P 42nm
27
P cc2
66
Cccm
103
P 4cc
28
P ma2
67
Cmma
104
P 4nc
29
P ca21
68
Ccca
105
P 42mc
30
P nc2
69
F mmm
106
P 42bc
31
P mn21
70
F ddd
107
I4mm
32
P ba2
71
Immm
108
I4cm
33
P na21
72
Ibam
109
I41md
34
P nn2
73
Ibca
110
I41cd
35
Cmm2
74
Imma
111
P
42m
36
Cmc21
112
P
42c
156

IGR
symbol
IGR
symbol
IGR
symbol
113
P
421m
155
R32
Cubic lattices
114
P
421c
156
P 3m1
196
F 23
115
P
4m2
157
P 31m
197
I23
116
P
4c2
158
P 3c1
198
P 213
117
P
4b2
159
P 31c
199
I213
118
P
4n2
160
R3m
200
P m3
119
I
4m2
161
R3c
201
P n3
120
I
4c2
162
P
31m
202
F m3
121
I
42m
163
P
31c
203
F d3
122
I
42d
164
P
3m1
204
Im3
123
P 4/mmm
165
P
3c1
205
P a3
124
P 4/mcc
166
R
3m
206
Ia3
125
P 4/nbm
167
R
3c
207
P 432
126
P 4/nnc
Hexagonal lattices
208
P 4232
127
P 4/mbm
168
P 6
209
F 432
128
P 4/mnc
169
P 61
210
F 4132
129
P 4/nmm
170
P 65
211
I432
130
P 4/ncc
171
P 62
212
P 4332
131
P 42/mmc
172
P 64
213
P 4132
132
P 42/mcm
173
P 63
214
I4132
133
P 42/nbc
174
P
6
215
P
43m
134
P 42/nnm
175
P 6/m
216
F
43m
135
P 42/mbc
176
P 63/m
217
I
43m
136
P 42/mnm
177
P 622
218
P
43n
137
P 42/nmc
178
P 6122
219
F
43c
138
P 42/ncm
179
P 6522
220
I
43d
139
I4/mmm
180
P 6222
221
P m
3m
140
I4/mcm
181
P 6422
222
P n
3n
141
I41/amd
182
P 6322
223
P m
3n
142
I41/acd
183
P 6mm
224
P n
3m
Trigonal lattices
184
P 6cc
225
F m
3m
143
P 3
185
P 63cm
226
F m
3c
144
P 31
186
P 63mc
227
F d
3m
145
P 32
187
P
6m2
228
F d
3c
146
R3
188
P
6c2
229
Im
3m
147
P
3
189
P
62m
230
Ia
3d
148
R
3
190
P
62c
149
P 312
191
P 6/mmm
150
P 321
192
P 6/mcc
151
P 3112
193
P 63/mcm
152
P 3121
194
P 63/mmc
153
P 3212
195
P 23
154
P 3221
157

A.2
Labels of the layer groups (slabs)
The available layer groups belong to a subset of the 230 space groups. Therefore they can be
identified by the corresponding space group.
The first column gives the label to be used in the input card (IGR variable).
The second column gives the Hermann-Mauguin symbol of the corresponding space group
(generally the short one; the full symbol is adopted when the same short symbol could refer
to different settings). The third column gives the Schoenflies symbol. The fourth column the
number of the corresponding space group, according to the International Tables for Crystal-
lography. The number of the space group is written in parentheses when the orientation of the
symmetry operators does not correspond to the first setting in the I. T.
IGR Hermann
Schoenflies
N
IGR Hermann
Schoenflies
N
Mauguin
Mauguin
triclinic lattices (P)
41 P bam
D92h
55
1
P 1
C1
1
1
42 P maa
D32h
(49)
2
P
1
C1
i
2
43 P man
D72h
(53)
3
P 112
C1
2
(3)
44 P bma
D12h
(57)
4
P 11m
C1
s
(6)
45 P baa
D82h
(54)
5
P 11a
C2
s
(7)
46 P ban
D42h
50
6
P 112/m
C1
47 Cmmm
D9
2h
(10)
2h
65
7
P 112/a
C4
48 Cmma
D2
2h
(13)
2h
67
rectangular lattices (P or C)
square lattices (P)
8
P 211
C1
2
(3)
49 P 4
C1
4
75
9
P 2111
C2
2
(4)
50 P
4
S1
4
81
10 C211
C3
2
(5)
51 P 4/m
C1
4h
83
11 P m11
C1
s
(6)
52 P 4/n
C3
4h
85
12 P b11
C2
s
(7)
53 P 422
D14
89
13 Cm11
C3
s
(8)
54 P 4212
D24
90
14 P 2/m11
C1
55 P 4mm
C1
2h
(10)
4v
99
15 P 21/m11
C2
56 P 4bm
C2
2h
(11)
4v
100
16 C2/m11
C3
57 P
42m
D1
2h
(12)
2d
111
17 P 2/b11
C4
58 P
42
2h
(13)
1m
D32d
113
18 P 2/b11
C5
59 P
4m2
D5
2h
(14)
2d
115
19 P 222
D12
16
60 P
4b2
D72d
117
20 P 222
D22
(17)
61 P 4/mmm D14h
123
21 P 21212
D32
18
62 P 4/nbm
D34h
125
22 C222
D62
21
63 P 4/mbm
D54h
127
23 P mm2
C1
2v
25
64 P 4/nmm
D74h
129
24 P ma2
C4
2v
28
hexagonal lattices (P)
25 P ba2
C8
2v
32
65 P 3
C1
3
143
26 Cmm2
C1
2v
35
66 P
3
C1
3i
147
27 P 2mm
C1
2v
(25)
67 P 312
D13
149
28 P 21am
C2
2v
(26)
68 P 321
D23
150
29 P 21ma
C2
2v
(26)
69 P 3m1
C1
3v
156
30 P 2mb
C4
2v
(28)
70 P 31m
C2
3v
157
31 P 21mn
C7
2v
(31)
71 P
31m
D13d
162
32 P 2aa
C3
2v
(27)
72 P
3m1
D33d
164
33 P 21ab
C5
2v
(29)
73 P 6
C1
6
168
34 P 2an
C6
2v
(30)
74 P
6
C1
3h
174
35 C2mm
C1
2v
(35)
75 P 6/m
C1
6h
175
36 C2mb
C5
2v
(39)
76 P 622
D16
177
37 P mmm
D12h
47
77 P 6mm
C1
6v
183
38 P mam
D52h
(51)
78 P
6m2
D13h
187
39 P mma
D52h
51
79 P
62m
D33h
189
40 P mmn
D32h
59
80 P 6/mmm D16h
191
158

A.3
Labels of the rod groups (polymers)
The available rod groups belong to a subset of the 230 space groups; the symmetry operators
are generated for the space groups (principal axis z) and then rotated by 90 through y, to
have the polymer axis along x (CRYSTAL convention).
In the table, the first column gives the label to be used in the input card for identifying the
rod group (IGR variable).
The second column gives the "polymer" symbol, according to the the following convention: x
is the first symmetry direction, y the second.
The third column gives the Schoenflies symbol.
The fourth column gives the Hermann-Mauguin symbol (generally the short one; the full symbol
is adopted when the same short symbol could refer to different settings) of the corresponding
space group (principal axis z).
The fifth column gives the number of the corresponding space group, according to the Interna-
tional Tables for Crystallography; this number is written in parentheses when the orientation
of the symmetry operators does not correspond to the first setting in the I. T.
"Polymer"
Hermann
Number of
IGR
symbol
Schoenflies Mauguin
space group
(x direction)
(z direction)
1
P 1
C1
1
P 1
1
2
P
1
C1
i
P
1
2
3
P 211
C1
2
P 112
(3)
4
P 2111
C2
2
P 1121
(4)
5
P 121
C1
2
P 121
(3)
6
P 112
C1
2
P 211
(3)
7
P m11
C1
s
P 11m
(6)
8
P 1m1
C1
s
P 1m1
(6)
9
P 1a1
C2
s
P 1c1
(7)
10
P 11m
C1
s
P m11
(6)
11
P 11a
C2
s
P c11
(7)
12
P 2/m11
C1
2h
P 112/m
(10)
13
P 21/m11
C2
2h
P 1121/m
(11)
14
P 12/m1
C1
2h
P 12/m1
(10)
15
P 12/a1
C4
2h
P 12/c1
(13)
16
P 112/m
C1
2h
P 2/m11
(10)
17
P 112/a
C4
2h
P 2/c11
(13)
18
P 222
D12
P 222
16
19
P 2122
D22
P 2221
17
20
P 2mm
C1
2v
P mm2
25
21
P 21am
C2
2v
P mc21
26
22
P 21ma
C2
2v
P cm21
(26)
23
P 2aa
C3
2v
P cc2
27
24
P m2m
C1
2v
P m2m
(25)
25
P m2a
C4
2v
P c2m
(28)
26
P mm2
C1
2v
P 2mm
(25)
27
P ma2
C4
2v
P 2cm
(28)
28
P mmm
D12h
P mmm
47
29
P 2/m2/a2/a
D32h
P ccm
49
30
P 21/m2/m2/a
D52h
P cmm
(51)
31
P 21/m2/a2/m
D52h
P mcm
(51)
159

"Polymer"
Hermann
Number of
IGR
symbol
Schoenflies Mauguin
space group
(x direction)
(z direction)
32
P 4
C1
4
P 4
75
33
P 41
C2
4
P 41
76
34
P 42
C3
4
P 42
77
35
P 43
C4
4
P 43
78
36
P
4
S1
4
P
4
81
37
P 4/m
C1
4h
P 4/m
83
38
P 42/m
C2
4h
P 42/m
84
39
P 422
D14
P 422
89
40
P 4122
D34
P 4122
91
41
P 4222
D54
P 4222
93
42
P 4322
D74
P 4322
95
43
P 4mm
C1
4v
P 4mm
99
44
P 42am
C3
4v
P 42cm
101
45
P 4aa
C5
4v
P 4cc
103
46
P 42ma
C7
4v
P 42mc
105
47
P
42m
D12d
P
42m
111
48
P
42a
D22d
P
42c
112
49
P
4m2
D52d
P
4m2
115
50
P
4a2
D62d
P
4c2
116
51
P 4/mmm
D14h
P 4/mmm
123
52
P 4/m2/a2/a
D24h
P 4/mcc
124
53
P 42/m2/m2/a
D94h
P 42/mmc
131
54
P 42/m2/a2/m
D10
4h
P 42/mcm
132
55
P 3
C1
3
P 3
143
56
P 31
C2
3
P 31
144
57
P 32
C3
3
P 32
145
58
P
3
C1
3i
P
3
147
59
P 312
D13
P 312
149
60
P 3112
D33
P 3112
151
61
P 3212
D53
P 3212
153
62
P 321
D23
P 321
150
63
P 3121
D43
P 3121
152
64
P 3221
D63
P 3221
154
65
P 3m1
C1
3v
P 3m1
156
66
P 3a1
C3
3v
P 3c1
158
67
P 31m
C2
3v
P 31m
157
68
P 31a
C4
3v
P 31c
159
69
P
31m
D13d
P
31m
162
70
P
31a
D23d
P
31c
163
71
P
3m1
D33d
P
3m1
164
72
P
3a1
D43d
P
3c1
165
160

"Polymer"
Hermann
Number of
IGR
symbol
Schoenflies Mauguin
space group
(x direction)
(z direction)
73
P 6
C1
6
P 6
168
74
P 61
C2
6
P 61
169
75
P 65
C3
6
P 65
170
76
P 62
C4
6
P 62
171
77
P 64
C5
6
P 64
172
78
P 63
C6
6
P 66
173
79
P
6
C1
3h
P
6
174
80
P 6/m
C1
6h
P 6/m
175
81
P 63/m
C2
6h
P 63/m
176
82
P 622
D16
P 622
177
83
P 6122
D26
P 6122
178
84
P 6522
D36
P 6522
179
85
P 6222
D46
P 6222
180
86
P 6422
D56
P 6422
181
87
P 6322
D66
P 6322
182
88
P 6mm
C1
6v
P 6mm
183
89
P 6aa
C2
6v
P 6cc
184
90
P 63am
C3
6v
P 63cm
185
91
P 63ma
C4
6v
P 63mc
186
92
P
6m2
D13h
P
6m2
187
93
P
6a2
D23h
P
6c2
188
94
P
62m
D33h
P
62m
189
95
P
62a
D43h
P
62c
190
96
P 6/mmm
D16h
P 6/mmm
191
97
P 6/m2/a2/a
D26h
P 6/mcc
192
98
P 63/m2/a2/m
D36h
P 63/mcm
193
99
P 63/m2/m2/a
D46h
P 63/mmc
194
161

A.4
Labels of the point groups (molecules)
The centre of symmetry is supposed to be at the origin; for the rotation groups the principal
axis is z.
The first column gives the label to be used in the input card for identifying the point group
(IGR variable). The second column gives the short Hermann-Mauguin symbol. The third
column gives the Schoenflies symbol; for the C2 , C2h and Cs groups the C2 direction or the
direction orthogonal to the plane is indicated.
IGR
Hermann
Schoenflies
Mauguin
1
1
C1
2

1
Ci
3
2 (x)
C2 (x)
4
2 (y)
C2 (y)
5
2 (z)
C2 (z)
6
m (x)
Cs (x)
7
m (y)
Cs (y)
8
m (z)
Cs (z)
9
2/m (x)
C2h (x)
10
2/m (y)
C2h (y)
11
2/m (z)
C2h (z)
12
222
D2
13
2mm
C2v (x)
14
m2m
C2v (y)
15
mm2
C2v (z)
16
mmm
D2h
17
4
C4
18

4
S4
19
4/m
C4h
20
422
D4
21
4mm
C4v
22

42m
D2d
(v planes along x+y and x-y)
23

4m2
D2d
(v planes along x and y)
24
4/mmm
D4h
25
3
C3
26

3
C3i
27
321
D3
(one C2 axis along y)
28
312
D3
(one C2 axis along x)
29
3m1
C3v
(one v plane along x)
30
31m
C3v
(one v plane along y)
31

3m1
D3d
(one d plane along x)
32

31m
D3d
(one d plane along y)
33
6
C6
34

6
C3h
35
6/m
C6h
36
622
D6
37
6mm
C6v
38

6m2
D3h
(one C2 axis along x)
39

62m
D3h
(one C2 axis along y)
40
6/mmm
D6h
41
23
T
42
m
3
Th
43
432
O
44

43m
Td
45
m
3m
Oh
162

A.5
From conventional to primitive cells:
transforming matrices
The matrices describing the transformations from conventional (given as input) to primitive
(internally used by CRYSTAL) cells of Bravais lattices are coded in CRYSTAL. A point called x
in the direct lattice has xP coordinates in a primitive cell and xC coordinates in a conventional
cell. The relation between xP and xC is the following:
W xP = xC
(A.1)
Likewise, for a point in the reciprocal space the following equation holds:
~
W -1x = x
(A.2)
P
C
The W transforming matrices adopted in CRYSTAL, and reported below, satisfy the following
relation between the two metric tensors GP and GC :
G
~
P = W GC W
(A.3)
The values of the elements of the metric tensors GP and GC agree with those displayed in
Table 5.1 of the International Tables of Crystallography (1992 edition).
1 0 0
1 0 1
2
2

P A



1

0 1 0
0 1
P B
2
2











1
0
0
1 0 -1
0
1
1

1 0 1
2
2
2
2
A P
0
1
1
B
0
1
0

P
0
-1
1
1 0
1
1 1 0
0
1
1
2
2


2
2

1 1 0
-1
1
1
P C

1 1

1


0
P F
0
1
C P
-1
1
0
F P
1
-1
1
2
2

2
2













0
0
1
1
1
-1
0
0
1
1 1 0
2
2
0 1 1
1
0
1
1 1 1
2

1

1
I P
1
0
1
H R
-1
1
1
2
2
2

3 3 3
1 1 0
0 -1 1
P I

1 1 1
1 1 2

R H
2
2
2

3 3 3








1
1

1

1 1 1
2
2
2
3
3
3
Table A.1: W matrices for the transformation from conventional to primitive and from prim-
itive to conventional cells. P stands for primitive, A, B and C for A-, B- and C-face centred,
I for body centred, F for all-face centred, R for primitive rhombohedral (`rhombohedral axes')
and H for rhombohedrally centred (`hexagonal axes') cell (Table 5.1, ref. [21]).
163

Appendix B
Summary of input keywords
All the keywords are entered with an A format; the keywords must be typed left-justified, with
no leading blanks. The input is not case sensitive.
Geometry (Input block 1)
Symmetry information
ATOMSYMM
printing of point symmetry at the atomic positions
27

MAKESAED
printing of symmetry allowed elastic distortions (SAED)
32

PRSYMDIR
printing of displacement directions allowed by symmetry.
35

SYMMDIR
printing of symmetry allowed geom opt directions
40

SYMMOPS
printing of point symmetry operators
40

TENSOR
tensor of physical properties
40
I
Symmetry information and control
BREAKSYM
allow symmetry reduction following geometry modifications
28

KEEPSYMM
maintain symmetry following geometry modifications
32

MODISYMM
removal of selected symmetry operators
32
I
PURIFY
cleans atomic positions so that they are fully consistent with the 35

group
SYMMREMO
removal of all symmetry operators
40

TRASREMO
removal of symmetry operators with translational components
40

Modifications without reduction of symmetry
ATOMORDE
reordering of atoms in molecular crystals
25

NOSHIFT
no shift of the origin to minimize the number of symmops with 34

translational components before generating supercell
ORIGIN
shift of the origin to minimize the number of symmetry operators 34

with translational components
PRIMITIV
crystallographic cell forced to be the primitive cell
35

REDEFINE
definition of a new cell, with xy
to a given plane
36
I
Atoms and cell manipulation (possible symmetry reduction (BREAKSYMM)
164

ATOMDISP
displacement of atoms
25
I
ATOMINSE
addition of atoms
25
I
ATOMREMO
removal of atoms
25
I
ATOMROT
rotation of groups of atoms
26
I
ATOMSUBS
substitution of atoms
27
I
ELASTIC
distortion of the lattice
30
I
USESAED
given symmetry allowed elastic distortions, reads
41
I
SUPERCEL
generation of supercell - input refers to primitive cell
38
I
SUPERCON
generation of supercell - input refers to conventional cell
38
I
From crystals to slabs
SLABCUT
generation of a slab parallel to a given plane (3D2D)
37
I
From periodic structure to clusters
CLUSTER
cutting of a cluster from a periodic structure (3D0D)
28
I
HYDROSUB
border atoms substituted with hydrogens (0D0D)
31
I
Molecular crystals
MOLECULE
extraction of a set of molecules from a molecular crystal 33
I
(3D0D)
MOLEXP
variation of lattice parameters at constant symmetry and molec- 33
I
ular geometry (3D3D)
MOLSPLIT
periodic structure of non interacting molecules (3D3D)
34

RAYCOV
modification of atomic covalent radii
35
I
BSSE correction
MOLEBSSE
counterpoise method for molecules (molecular crystals only) 32
I
(3D0D)
ATOMBSSE
counterpoise method for atoms (3D0D)
25
I
Auxiliary and control keywords
ANGSTROM
sets inputs unit to
Angstrom
24

BOHR
sets input units to bohr
27

BOHRANGS
input bohr to
A conversion factor (0.5291772083 default value) 27
I
BOHRCR98
bohr to
A conversion factor is set to 0.529177 (CRYSTAL98
value)
END/ENDG
terminate processing of geometry

FRACTION
sets input unit to fractional
31

NEIGHBOR
number of neighbours in geometry analysis
34
I
PARAMPRT
printing of parameters controlling dimensions of static allocation 35

arrays
PRINTOUT
setting of printing options by keywords
35
I
SETINF
setting of inf array options
36
I
SETPRINT
setting of printing options
37
I
STOP
execution stops immediately
38

TESTGEOM
stop after checking the geometry
40

Output of data on external units
COORPRT
coordinates of all the atoms in the cell
29

EXTPRT
generation of file as CRYSTAL input
31

MOLDRAW
generation of file for the program MOLDRAW
32

STRUCPRT
cell parameters and coordinates of all the atoms in the cell
38

External electric field - modified Hamiltonian
165

FIELD
external field applied (2D-3D systems only)
41
I
Geometry optimization
OPTCOORD
Atom coordinates optimization
43
I
Initial Hessian
HESGUESS
initial guess for the Hessian
I
Convergence criteria
TOLDEG
RMS of the gradient [0.0003]
I
TOLDEX
RMS of the displacement [0.0012]
I
TOLDEE
energy difference between two steps [10-7]
I
MAXOPTC
max number of optimization steps
I
Optimization control
ATOMFREE
partial geometry optimization
I
RESTART
data from previous run

FINALRUN
Wf single point with optimized geometry
I
Gradient calculation control
NUMGRAD
numerical first deivatives

Printing options
PRINTFORCES atomic gradients

PRINTHESS
Hessian

PRINTOPT
optimization procedure

PRINT
verbose printing

Basis set input (Input block 2)
Symmetry control
ATOMSYMM
printing of point symmetry at the atomic positions
27

Basis set modification
CHEMOD
modification of the electronic configuration
48
I
GHOSTS
eliminates nuclei and electrons, leaving BS
50
I
Auxiliary and control keywords
CHARGED
allows non-neutral cell
48

PARAMPRT
printing of parameters controlling code dimensions
35

PRINTOUT
setting of printing options
35
I
SETINF
setting of inf array options
36
I
SETPRINT
setting of printing options
37
I
STOP
execution stops immediately
38

SYMMOPS
printing of point symmetry operators
40

END/ENDB
terminate processing of basis set definition keywords

Output of data on external units
GAUSS98
printing of an input file for the GAUSS94/98 package
49

General information, hamiltonian (Input block 3)
All DFT related keyword are collected under the heading "DFT".
166

Single particle Hamiltonian
RHF
Restricted Closed Shell
68

UHF
Unrestricted Open Shell
70

ROHF
Restricted Open Shell
70

DFT
DFT Hamiltonian
70

SPIN
spin-polarized solution
59

Choice of the exchange-correlation functionals
EXCHANGE exchange functional
59
I
LDA
Dirac-Slater [47] (LDA)
VBH
von Barth-Hedin [48] (LDA)
BECKE
Becke [49] (GGA)
PWGGA Perdew-Wang 91 (GGA)
PBE
Perdew-Becke-Ernzerhof [50] (GGA)
CORRELAT
correlation functional
59
I
VBH
von Barth-Hedin [48] (LDA)
PWGGA Perdew-Wang 91 (GGA)
PBE
Perdew-Becke-Ernzerhof [50] (GGA)
PZ
Perdew-Zunger [51] (LDA)
PWLSD
Perdew-Wang 92 [52, 53, 54] (GGA)
VWN
Vosko,-Wilk-Nusair [55] (LDA)
P86
Perdew 86 [56] (LDA)
LYP
Lee-Yang-Parr [57] (GGA)
HYBRID
hybrid mixing
60
I
NONLOCAL local term parameterization
60
I
B3PW
B3PW parameterization
60

B3LYP
B3LYP parameterization
60

Integration method
[NUMERICA] numerical integration (default)

RADSAFE
safety radius for grid point screening
I
BATCHPNT grid point grouping for integration
I
FITTING
integration through auxiliary basis set fitting
172
I:
BASIS
Auxiliary basis set input
172
I
TOLLPOT 9

LINEQUAT
173
I
Numerical accuracy control
[BECKE]
selection of Becke weights (default)

SAVIN
selection of Savin weights

RADIAL
definition of radial grid
I
ANGULAR
definition of angular grid
I
LGRID
"large" predefined grid
I
XLGRID
"extra large" predefined grid
I
TOLLDENS
density contribution screening 6
I
TOLLGRID
grid points screening 14
I
Auxiliary and control
PRINT
END
Numerical accuracy and computational parameters control
167

BIPOLAR
Bipolar expansion of bielectronic integrals
58
I
BIPOSIZE
size of coulomb bipolar expansion buffer
58
I
EXCHSIZE
size of exchange bipolar expansion buffer
58
I
INTGPACK
Choice of integrals package 0
67
I
NOBIPOLA
All bielectronic integrals computed exactly
67

POLEORDR
Maximum order of multipolar expansion 4
68
I
TOLINTEG
Truncation criteria for bielectronic integrals
6 6 6 6 12
69
I
TOLPSEUD
Pseudopotential tolerance 6
69
I
Type of run
ATOMHF
Atomic wave functions
57
I
MPP
MPP execution - see makefile
67
I
SCFDIR
SCF direct (mono+biel int computed)
68

SEMIDIR
integrals in memory
68
I
NOMONDIR
SCF semidirect (mono on disk, biel computed)
67

EIGS
S(k) eigenvalues - basis set linear dependence check
64

FIXINDEX
Reference geometry to classify integrals
65

Integral file distribution
BIESPLIT
writing of bielectronic integrals in n files n = 1 ,max=10
58
I
MONSPLIT
writing of mono-electronic integrals in n file n = 1 , max=10
67
I
Auxiliary and control keywords
PARAMPRT
output of parameters controlling code dimensions
35

PRINTOUT
setting of printing options
35
I
SETINF
setting of inf array options
36
I
SETPRINT
setting of printing options
37
I
TESTPDIM
stop after symmetry analysis
69

TESTRUN
stop after integrals classification and disk storage estimate
69

STOP
execution stops immediately
38

END/ENDM
terminate processing of general information keywords
SCF (Input block 4)
Numerical accuracy control and convergence tools
ANDERSON
Fock matrix mixing
72
I
BROYDEN
Fock matrix mixing
73
I
FMIXING
Fock/KS matrix (cycle i and i-1) mixing 0
75
I
LEVSHIFT
level shifter no
77
I
MAXCYCLE
maximum number of cycles 50
78
I
SMEAR
Finite temperature smearing of the Fermi surface no
79
I
TOLDEE
convergence on total energy 5
81
I
TOLDEP
convergence on density matrix ?
81
I
TOLSCF
convergence on eigenvalues 6 and total energy 5
81
I
Initial guess
EIGSHIFT
alteration of orbital occupation before SCF no
74
I
EIGSHROT
rotation of the reference frame no
74
I
GUESSF
Fock/KS matrix from previous run
76

GUESSP
density matrix from a previous run
76

GUESSPAT
superposition of atomic densities
76

168

Spin-polarized system
ATOMSPIN
setting of atomic spin to compute atomic densities
73
I
BETALOCK
beta electrons locking
73
I
SPINLOCK
spin difference locking
80
I
SPINEDIT
editing of the spin density matrix
80
I
Auxiliary and control keywords
END/ENDSCF terminate processing of scf I keywords

KSYMMPRT
printing of Bloch functions symmetry analysis
77

NEIGHBOR
number of neighbours to analyse in PPAN
34
I
PARAMPRT
output of parameters controlling code dimensions
35

PRINTOUT
setting of printing options
35
I
NOSYMADA
No Symmetry Adapted Bloch Functions
78

SYMADAPT
Symmetry Adapted Bloch Functions (default)
81

SETINF
setting of inf array options
36
I
SETPRINT
setting of printing options
37
I
STOP
execution stops immediately
38

Output of data on external units
NOFMWF
wave function formatted output not written to fortran unit 98. 78

KNETOUT
Reciprocal lattice information + eigenvalues
77

SAVEWF
wave function data written every two SCF cycles
79

Post SCF calculations
POSTSCF
post-scf calculations when convergence criteria not satisfied
78

EXCHGENE
exchange energy evaluation (spin polarized only)
75

GRADCAL
analytical gradient of the energy
76

PPAN
population analysis at the end of the SCF no
78
Properties
RDFMWF
wave function data conversion formatted-binary (fortran unit 98 9)
Preliminary calculations
NEWK
Eigenvectors calculation
103
I
NOSYMADA
No symmetry Adapted Bloch Functions
78

PATO
Density matrix as superposition of atomic (ionic) densities
104
I
PBAN
Band(s) projected density matrix (preliminary NEWK)
104
I
PGEOMW
Density matrix from geometrical weights (preliminary NEWK) 105
I
PDIDE
Energy range projected density matrix (preliminary NEWK)
105
I
PSCF
Restore SCF density matrix
110

ROTREF
Rotation of the reference frame
110
I
Properties computed from the density matrix
169

ADFT
Atomic density functional correlation energy
85
I
BAND
Band structure
87
I
CLAS
Electrostatic potential maps (point multipoles approximation)
88
I
ECHG
Charge density maps and charge density gradient
93
I
ECH3
Charge density 3D maps
92
I
EDFT
Density functional correlation energy
93
I
POLI
Atom and shell multipoles evaluation
105
I
POTM
Electrostatic potential maps
107
I
POTC
Electrostatic properties
107
I
PPAN
Mulliken population analysis
78
XFAC
X-ray structure factors
111
I
Properties computed from the density matrix (spin-polarized systems)
ANISOTRO
Hyperfine electron-nuclear spin tensor
86
I
ISOTROPIC
Hyperfine electron-nuclear spin interaction - Fermi contact
96
I
POLSPIN
Atomic spin density multipoles
106
I
Properties computed from eigenvectors (after keyword NEWK)
ANBD
Printing of principal AO component of selected CO
85
I
BWIDTH
Printing of bandwidth
88
I
DOSS
Density of states
91
I
EMDL
Electron momentum distribution - line
94
I
EMDP
Electron momentum distribution - plane maps
95
I
PROF
Compton profiles and related quantities
109
I
New properties
POLARI
Berry phase calculations
113
I
SPOLBP
Spontaneous polarization (Berry phase approach)
114

SPOLWF
Spontaneous polarization (localized CO approach)
114

PIEZOBP
Piezoelectricity (Berry phase approach) preliminary
112

PIEZOWF
Piezoelectricity (localized CO approach) - preliminary
113

LOCALWF
Localization of Wannier functions
97
I
DIELEC
Optical dielectric constant
89
I
Auxiliary and control keywords
ANGSTROM
Set input unit of measure to
Angstrom
24

BASISSET
Printing of basis set, Fock/KS, overlap and density matrices
87

BOHR
Set input unit of measure to bohr
27

CHARGED
Non-neutral cell allowed (PATO)
48

CONVCELL
Reference cell for k-points coordinates (BAND)
88

END
Terminate processing of properties input keywords

FRACTION
Set input unit of measure to fractional
31

MAPNET
Generation of coordinates of grid points on a plane
100
I
NEIGHBOR
Number of neighbours to analyse in PPAN
34
I
PRINTOUT
Setting of printing options
35
I
RAYCOV
Modification of atomic covalent radii
35
I
SETINF
Setting of inf array options
36
I
SETPRINT
Setting of printing options
37
I
STOP
Execution stops immediately
38

SYMMOPS
Printing of point symmetry operators
40

Output of data on external units
170

ATOMIR
Coordinates of the irreducible atoms in the cell
86

ATOMSYMM
Printing of point symmetry at the atomic positions
27

COORPRT
Coordinates of all the atoms in the cell
29

EXTPRT
Explicit structural/symmetry information
31

GAUSS98
Printing of an input file for the GAUSS98 package
49

FMWF
Wave function formatted output. Section 3.2
95

INFOGUI
Generation of file with wf information for visualization
96

KNETOUT
Reciprocal lattice information + eigenvalues
77

MOLDRAW
generation of input file for the program MOLDRAW
32

Common keywords
Some keywords may be entered in more then one input block, with the same action. They are
summarized in the following table.
input block
keyword
page
1-geometry 2-basis set
3-general
4-scf
properties
ANGSTROM
24
X
X
ATOMSYMM
27
X
X
BOHR
27
X
X
CHARGED
48
X
X
COORPRT
29
X
X
COORXXL
29
X
X
END
ENDG
ENDB
ENDM
ENDSCF
ENDP
EXTPRT
31
X
X
FRACTION
31
X
X
GAUSS98
49
X
X
KNETOUT
77
X
X
MOLDRAW
32
X
X
NEIGHBOR
34
X
X
X
PARAMPRT
35
X
X
X
X
X
PPAN
78
X
X
PRINTOUT
35
X
X
X
X
X
RAYCOV
35
X
X
SETINF
36
X
X
X
X
X
SETPRINT
37
X
X
X
X
X
STOP
38
X
X
X
X
X
SYMMOPS
40
X
X
X
171

Appendix C
DFT integration through an
auxiliary basis set fitting
The calculation of the exchange-correlation contribution to the Kohn-Sham matrix through the
fitting of the exchange-correlation potential to an auxiliary basis set of gaussian functions, as in
CRYSTAL98 is not recommended, as the calculation of energy gradients has been implemented
only with numerical quadrature.
FITTING
analytical integration through an auxiliary basis set fitting.
An even-
tempered auxiliary basis set is generated by default.
user defined fitting basis set (optional)
II
BASIS
Auxiliary basis for the fitting of the exchange correlation potential. Enter
the numerical code in the next record:
3
a predefined eventempered basis set of 12 s-type GTF is used. The largest
exponent is 2000., the smallest 0.1. Default choice.
2
a predefined even-tempered basis set of 8 s-type GTF is used. The largest
exponent is 2000., the smallest 0.1.
1
A predefined even-tempered basis set with 4 s-type GTF is used. The largest
exponent is 2000., the smallest 0.1.
0
general basis set is read from input records as follows:
172

rec
variable value
meaning

NAT
n
conventional atomic number.
=99
end of auxiliary basis set input section
NFIN
n
number of groups of functions
0
end of basis set input (when NAT=99)
insert NFIN records
II

ITYB
0
read general basis set.
1
even-tempered basis set is generated from input data.
LATD
0
s-type GTF.
2
p-type GTF.
3
d-type GTF.
4
f-type GTF.
5
g-type GTF.
6
h-type GTF.
7
i-type GTF.
NFUNC
number of gaussians in the contraction
if ITYB = 1 add on the same record
II
EMIN
minimum exponent of the even-tempered BS
EMAX
maximum exponent of the even-tempered BS
if ITYB = 0 insert NFUNC records
II

EXP
Exponent of the GTF.
CX
Coefficient of the GTF.
PRINT
extended printing
LINEQUAT
keyword - iterative linear equation solver control
MICRO
number of micro-cycles (default 1)
MACRO number of macro-cycles (default 1)
TOLLPOT
The DFT potential tolerance IP controls the level of accuracy for the integration which yields
matrix elements of the Kohn-Sham Hamiltonian. Contributions less than 10-IP will be ignored.
The default value is 9.
MPP does not support DFT fitting.
A basis set for the DF fitting must be defined for each type of atom in the cell
A basis set for the fitting can be defined for ghost atoms, that means centres with AO
basis set, but without nuclear and electron charge. It can be useful when there is a
localized electron in a vacancy
A given atom may have no basis for the DF fitting assigned. This choice must be
explicitely defined by setting the value of NFIN to 0.
C.1
DFT input example - fitting method
The following exemples explain the usage of an auxiliary basis set for the fitting.
Example 1. Urea molecule
geometry input terminated by END (page 11)
basis set input terminated by END (page 16)
input block 3 - general information input (page 19)
173

DFT
DFT Hamiltonian selected
FITTING
fitting technique chosen
BASIS
auxiliary basis for the fitting of XC potential
0
general basis set read from the following input records
1 3
Z=1, Hydrogen; 3 groups of functions
1 0 9 0.06 60.
1st: even temp BS; s type; 9 gaussians; 0.06 min. exp; 60. max. exp.
1 2 3 0.08 0.72
2nd: even temp. BS; p-type; 3 gaussians; 0.08 min. exp.; 0.72 max. exp.
1 3 1 0.10 0.10
3rd: even temp. BS; d-type; 1 gaussian; 0.1 exp.
6 3
Z=6, Carbon; 3 groups of functions
1 0 13 0.07 2000.
1st: even temp BS; s type; 13 gaussians; 0.07 min. exp.; 2000. max. exp.
1 2 3 0.10 0.8
2nd: even temp BS; p type; 3 gaussians; 0.10 min. exp.; 0.8 max. exp.
1 3 2 0.12 0.3
3rd: even temp BS; d type; 2 gaussians; 0.12 min. exp.; 0.3 max. exp.
7 3
Z=7, Nitrogen; 3 groups of functions
1 0 13 0.07 2000.
1st: even temp BS; s type; 13 gaussians; 0.07 min. exp.; 2000. max. exp.
1 2 3 0.10 0.8
2nd: even temp BS; p type; 3 gaussians; 0.10 min. exp.; 0.8 max. exp.
1 3 2 0.12 0.3
3rd: even temp BS; d type; 2 gaussians; 0.12 min. exp.; 0.7 max. exp.
8 3
Z=8, Oxygen; 3 groups of functions
1 0 13 0.07 2000.
1st: even temp BS; s type; 13 gaussians; 0.07 min. exp.; 2000. max. exp.
1 2 3 0.10 0.8
2nd: even temp BS; p type; 3 gaussians; 0.10 min. exp.; 0.8 max. exp.
1 3 2 0.12 0.3
3rd: even temp BS; d type; 2 gaussians; 0.12 min. exp.; 0.3 max. exp.
99 0
end of auxiliary basis input
CORRELAT
selection of correlation potential
PZ
GGA Perdew-Zunger LSD
END
end of DFT input
END
end of general information input, block 3
SCF input terminated by END (page 19)
Example 2. Nickel (II) Oxide
geometry input terminated by END (page 11)
basis set input terminated by END (page 16)
input block 3 - general information input (page 19)
DFT
DFT Hamiltonian selected
FITTING
PRINT
extended printing required
SPIN
spin unrestricted DF calculation
BASIS
auxiliary basis for the fitting of XC potential
0
general basis set read from the following input records
28 2
Z=28, Nickel; 2 groups of functions
1 0 12 0.1 6000.
1st: even temp BS; s type; 12 gaussians; 0.1 min. exp; 6000. max. exp.
1 5 3 0.45 3.3
2nd: even temp. BS; g-type; 3 gaussians; 0.45 min. exp.; 3.3 max. exp.
8 1
Z=8, Oxygen; 1 group of functions
1 0 10 0.1 1000.
even temp BS; s type; 10 gaussians; 0.1 min. exp; 1000. max. exp.
99 0
end of auxiliary basis input
CORRELAT
selection of correlation potential
PWGGA
GGA Perdew-Wang
END
end of DFT input
END
end of general information input, block 3
SCF input terminated by END (page 19)
Example 3. Silicon bulk
geometry input terminated by END (page 11)
basis set input terminated by END (page 16)
input block 3 - general information input (page 19)
174

DFT
DFT Hamiltonian selected
FITTING
RADIAL
radial integration information
1
one integration interval
3.5
limit of the interval
80
number of points in the interval
EXCHANGE
selection of the exchange potential
BECKE
GGA Becke potential
CORRELAT
selection of correlation potential
PZ
LSD Perdew-Zunger potential
BASIS
auxiliary basis for the fitting of XC potential
0
general basis set read from the following input records
14 3
Z=14, Silicon; 3 groups of functions
1 0 14 0.09 4500.
1st: even temp BS; s type; 14 gaussians; 0.09 min. exp; 4500. max. exp.
1 4 1 0.3 0.3
2nd: even temp BS; f type; 1 gaussian; 0.3 exp;
1 5 1 0.3 0.3
3rd: even temp BS; g type; 1 gaussian; 0.3 exp;
99 0
end of auxiliary basis input
END
end of DFT input
END
end of general information input, block 3
SCF input terminated by END (page 19)
Example 4. Quartz
geometry input terminated by END (page 11)
basis set input terminated by END (page 16)
input block 3 - general information input (page 19)
DFT
DFT Hamiltonian selected
FITTING
RADIAL
radial integration information
1
one integration interval
3.5
limit of the interval
44
number of points in the interval
EXCHANGE
selection of the exchange potential
LDA
selection of correlation potential
CORRELAT
selection of correlation potential
PZ
LSD Perdew-Zunger potential
BASIS
auxiliary basis for the fitting of XC potential
0
general basis set read from the following input records
14 6
Z=14, Silicon; 6 groups of functions
1 0 15 0.07 14000.
1st: even temp BS; s type; 15 gaussians; 0.07 min.exp; 14000. max. exp.
1 2 6 0.1 5.
2nd: even temp BS; p type; 6 gaussians; 0.1 min.exp; 5. max. exp.
1 3 4 0.2 1.5
3rd: even temp BS; d type; 4 gaussians; 0.2 min.exp; 1.5 max. exp.
1 4 3 0.3 1.5
4th: even temp BS; f type; 3 gaussians; 0.3 min.exp; 1.5 max. exp.
1 5 2 0.25 1.3
5th: even temp BS; g type; 2 gaussians; 0.25 min.exp; 1.3 . max. exp.
1 6 1 0.5 0.5
6th: even temp BS; h type; 1 gaussian; 0.5 exp.
8 6
Z=14, Oxygen; 6 groups of functions
1 0 14 0.07 700.
1st: even temp BS; s type; 14 gaussians; 0.07 min.exp; 700. max. exp.
1 2 6 0.08 2.5
2nd: even temp BS; p type; 6 gaussians; 0.08 min.exp; 2.5 max. exp.
1 3 4 0.4 2.5
3rd: even temp BS; d type; 4 gaussians; 0.4 min.exp; 2.5 max. exp.
1 4 3 0.4 1.3
4th: even temp BS; f type; 3 gaussians; 0.4 min.exp; 1.3 max. exp.
1 5 2 0.25 1.3
5th: even temp BS; g type; 2 gaussians; 0.25 min.exp; 1.3 max. exp.
1 6 1 0.5 0.5
6th: even temp BS; h type; 1 gaussian; 0.5 exp.
99 0
end of auxiliary basis input
END
end of DFT input
END
end of general information input, block 3
SCF input terminated by END (page 19)
175

Appendix D
Reciprocal lattice sampling
The keyword KNETOUT entered in the program crystal or properties generates an unformatted file on
fortran unit 30. The structure of the file is as follows:
rec
data type
n. data
content
1
3I, F
3+9
ndf, nkf, iuhf, reciprocal lattice vectors cartesian
components (a.u.)
2
I
3*nkf
oblique coordinates of the points in reciprocal lattice
3
I
nkf
k points flag: 0 (complex); 1 (real)
4
I
3x3x48
symmetry operators matrices
5
F
nkf
geometrical weight of k points
6
F
ntot
eigenvalues
7
F
ntot
weight of eigenvalues
where:
ndf
number of basis set functions
nkf
number of k points (Monkhorst sampling)
iuhf
0 (Restricted calculation); 1 (Unrestricted calculation)
ntot
(nkf*ndf*(iuhf+1))
number of eigenvalues
The eigenvectors corresponding to the eigenvalues are written on the fortran unit 10 when computed by prop-
erties (keyword NEWK, page 103), and on fortran unit whose number is printed at the end of the printout,
when computed by crystal. The two sets of eigenvectors may be slightly different, as the ones computed by
properties correspond to one SCF cycle more then the one computed by crystal.
Symmetry adapted Bloch functions generation must be removed (keyword NOSYMADA, page 78),in order
to read eigenvectors as nkf*(iuhf+1) matrices of size ndfxndf.
The reciprocal lattice vectors cartesian components and the oblique coordinates of the points in reciprocal
lattice are printed when the input block 4, SCF input is processed. Printing of the other data may be obtained
by setting the appropriate printing options (see keyword PRINTOUT, page 177):
keyword
input
information
KNETCOOR
reciprocal lattice sampling points coordinates
KWEIGHTS
geometrical weight of k points
EIGENVAL
n
eigenvalues at the first n k points
EIGENVEC
n
eigenvectors at the first n k points
EIGENALL
eigenvalues at all k points
176

Appendix E
Printing options
Extended printing can be obtained by entering the keywords PRINTOUT (page 35) or SETPRINT (page
37).
In the scf (or scfdir) program the printing of quantities computed is done at each cycle if the corresponding
LPRINT value is positive, only at the last cycle if the LPRINT value is negative.
The LPRINT options to obtain intermediate information can be grouped as follows. The following table gives
the correspondence between position number, quantity printed, and keyword.
crystal
Keyword
inp
direct lattice - geometry information: 1
GLATTICE

symmetry operators : 4, 2
SYMMOPS

atomic functions basis set : 72
BASISSET

DF auxiliary basis set for the fitting: 79
DFTBASIS

scale factors and atomic configuration: 75
SCALEFAC

k-points geometrical wheight: 53
KWEIGHTS

shell symmetry analysis : 5, 6, 7, 8, 9
Madelung parameters: 28
multipole integrals: 20
Fock/KS matrix building - direct lattice: 63, 64, 74
FGRED FGIRR N
Total energy contributions: 69
ENECYCLE

crystal - properties
shell and atom multipoles: 68
MULTIPOLE N
reciprocal space integration to compute Fermi energy: 51, 52, 53, 54, 55, 78
density matrix - direct lattice: irreducible (58); reducible (59)
PGRED PGIRR N
Fock/KS eigenvalues : 66
EIGENVAL N
EIGENALL

Fock/KS eigenvectors : 67
EIGENVEC N
symmetry adapted functions : 47
KSYMMPRT

Population analysis: 70, 73, 77
MULLIKEN N
Atomic wave-function: 71
properties
overlap matrix S(g) - direct lattice: 60 (keyword PSIINF)
OVERLAP N
Densities of states: 105, 107
DOSS

Projected DOSS for embedding: 36, 37, 38
DF correlation correction to total energy: 106
Compton profile and related quantities: 116, 117, 118
Fermi contact tensor : 18
FTENSOR

rotated eigenvectors (keyword ROTREF): 67
EIGENVEC

Charge density and electrostatic potential maps: 119
MAPVALUES

Example
To print the eigenvalues at each scf cycle enter:
177

PRINTOUT
EIGENALL
END
To print the eigenvalues at the first 5 k points at the end of scf only, enter in any input block:
SETPRINT
1
66 -5
Eigenvectors printed by default are from the first valence eigenvector up to the first 6 virtual
ones. Core eigenvectors are printed by "adding" 500 to the selected value of LPRINT(67). To
obtain print all the eigenvectors at the end of scf insert in any input block:
SETPRINT
1
66 -505
178

Printing options LPRINT array values
subroutine value
printed information
keyword
input
1
GCALCO
N
up N=6 stars of direct lattice vectors
GLATTICE
CRYSTA
= 0
crystal symmetry operators
SYMMOPS
2
EQUPOS
= 0
equivalent positions in the reference cell
EQUIVAT
3
CRYSTA
= 0
crystal symmops after geometry editing
4
5
GILDA1
N>0
g vector irr- first n set type of couples
N<0
g vector irr- n-th set type of couples
6
GROTA1
= 0
information on shells symmetry related
7
GV
N>0
stars of g associated to the first n couples
7
N<0
stars of g associated to the n-th couple
8
GORDSH
= 0
information on couples of shells symmetry related
9
GSYM11
= 0
intermediates for symmetrized quantities
10
GMFCAL = 0
nstatg, idime, idimf, idimcou
11
MAIN2U
= 0
exchange energy
EXCHGENE
MAIND
EXCHGENE
12
IRRPR
= 0
symmops (reciprocal lattice)
SYMMOPSR
MATVIC
N
n stars of neighbours in cluster definition
13
14
GSLAB
= 0
coordinates of the atoms in the slab
15
symdir
= 0
print symmetry allowed directions
PRSYMDIR
18
TENSOR
= 0
extended printing for hyperfine coupling cost
FTENSOR
19
20
MONIRR
N
multipole integrals up to pole l=n
21
28
MADEL2
= 0
Madelung parameters
29
31
= 0
values of the dimension parameters
PARAMETERS
N > 0 printing of ccartesian coordinates of the atoms
32
33
COOPRT
N > 0 cartesian coordinates of atoms on fortran unit 33
ATCOORDS
FINE2
N > 0
KNETOUT
34
READ2
output of reciprocal space information
KNETOUT
35
N > 0 printing of symmops in short fomr
36
XCBD
=
properties - exchange correlation printing
37
LEGDGS
towards EMBED - see EMBED manual
38
KROT
towards EMBED - see EMBED manual
39
towards EMBED - see EMBED manual
40
towards EMBED - see EMBED manual
41
SHELL*
=0
printing of bipolar expansion parameters
47
KSYMBA n
Symmetry Adapted Bloch Functions printing level
48
KSYMBA =0
Symmetry Adapted Bloch Functions printing active KSYMMPRT
AB
= 0
B functions orthonormality check
51
52
DIF
> 0
Fermi energy - Warning !!!! Huge printout !!!
53
SCFPRT
= 0
k points geometrical weights
KWEIGHTS
CALPES
> 0
k points weights- Fermi energy
54
55
OMEGA
> 0
f0 coefficients for each band
56
57
PDIG
N
p(g) matrices-first n g vectors
PGIRR
N
PROT1
= 0
mvlu, ksh, idp4
58
59
RROTA
N > 0 P(g) matrices - first N vectors - each scf cycle
PGRED
N
N < 0 P(g) matrices - first N vectors - last scf cycle
PGRED
N
NEWK
N
P(g) matrices - first N vectors
PGRED
N
PSIINF
> 0
P(g) matrices - first N vectors
PGRED
N
N <0
P(g) matrix for g=N
PGRED
N
60
PSIINF
> 0
overlap matrix S(g) - first N vectors
OVERLAP
N
N <0
overlap matrix S(g) for g = N
N
61
63
TOTENY = 0
bielectronic contribution to irred. F(g) matrix
64
FROTA
N
F(g) matrix - first N g vectors
FGRED
N
PSIINF
N> 0
FGRED
N
N <0
f(g) matrix - for g = N (N-th g vector only)
FGRED
N
179

subroutine value
printed information
keyword
input
65
66
AOFK
N
e(k)- fock eigenvalues- first N k vectors
EIGENVAL
N
ADIK
N
BANDE
N
DIAG
N
EIGENALL
FDIK
N
FINE2
N
NEWK
N
67
AOFK
N
a(k) - fock eigenvectors - first N k vectors
EIGENVEC
N
ADIK
N
DIAG
N
FINE2
N
NEWK
N
68
POLGEN
N <0
shell and atom multipoles up to pole l=N
MULTIPOL
N
POLGEN
N >0
atom multipoles up to pole l=N
MULTIPOL
N
QGAMMA N
shell multipoles up to pole l=N
MULTIPOL
N
69
TOTENY = 0
contributions to total energy at each cycle
ENECYCLE
FINE2
= 0
Mulliken population analysis
70
NEIGHB
at the end of scf cycles
POPAN
calls PPBOND, to perform Mulliken analysis
PDIBAN
71
PATIRR
= 0
atomic wave function
ATOMICWF
PATIR1
= 0
" "
ATOMICWF
72
INPBAS
= 0
basis set
BASISSET
INPUT2
= 0
basis set
BASISSET
READFG
SET
= 1
73
POPAN
= 0
Mulliken matrix up to N direct lattice vector
MULLIKEN
N
PPBOND
PDIBAN
N
74
TOTENY N
f(g) irreducible up to g=N
FGIRR
N
DFTTT2
N
FGIRR
N
75
INPBAS
= 0
printing of scale factor and
SCALEFAC
atomic configuration
CONFIGAT
76
77
PPBOND
0
printing of neighbouring relationship
= 0
no printing of neighbours relationship
78
FERMI
= 0
informations on Fermi energy calculation
EMIMAN
= 0
79
DFGPRT
= 0
dft auxiliary basis set - default no printing
DFTBASIS
ROTOP
> 0
printing of atoms coord. in rotated ref. frame
ROTREF
80
92
INPBAS
G94 deck on ft92
GAUSS94
MOLDRW
input deck to MOLDRAW
93
105 DENSIM
< 0
DOSS along energy points
DOSS
DFFIT3
> 0
DFT intermediate printout
106
(keyword PRINT in dft input)
107 STARIN
= 0
DOSS information
112 PROFCA
= 0
projected DOSS coefficients
116 PROFI
= 0
Compton profile information
117 PROFI
= 0
118 PROFI
N
119 INTEG
= 0
charge density at grid points
MAPVALUES
JJTEG
= 0
charge density at grid points
MAPVALUES
MAPNET = 0
electrostatic potential at grid points
MAPVALUES
NAPNET
= 0
charge density gradient components
MAPVALUES
120 LIBPHD
= 0
extended printing in berny optimizer
121
reserved for geometry optimizer
122
reserved for geometry optimizer
123
reserved for geometry optimizer
124
reserved for geometry optimizer
125
reserved for geometry optimizer
180

Appendix F
External format
181

Appendix G
Utility programs
182

Appendix H
CRYSTAL2003 versus
CRYSTAL98
CRYSTAL03 consists of 2 programs:
crystal
reads geometry, basis set, computational parameters and Hamiltonian, scf information (4 input
blocks ending with the keyword END), computes the wave function, and writes wave function
information in fortran unit 9 (binary) and 98 (formatted).
A unique program, crystal, controls scf execution technique, traditional, scf direct, scf semidi-
rect (part of bielectronic integrals in memory), and substitutes the programs integrals, scf and
scfdir of CRYSTAL98 package. The choice of SCF technique is controlled by keywords in input
block 3.
properties
reads wave function information from fortran unit 9 (default) or 98 (keyword RDFMWF first
record in the input deck) and computes the properties defined in the input stream.
New features include:
Dynamic allocation of the arrays depending on the size of the system. Few arrays have
fixed dimensions in Release V1.0.
Max dimensions allowed:
N. of atoms
1200
N. of shells
3200
N. of AOs
10000
N. of Monkhorst net k points
400
N. of Gilat net k points
2400
Fully numerical quadrature of the density for DFT methods (default choice). The fitting
technique for the DFT potentials is still present, but not recommended (the choice of the
auxiliary basis set for the fitting is critical, and the analytical gradient is implemented
for numerical integration only);
a very accurate McMurchie-Davidson integration scheme for Effective Core Pseudopo-
tentials (ECP) methods, allowing higher angular quantum numbers (in CRYSTAL98
183

the package for computing integrals involving pseudo potentials was derived from
PSHONDO);
Analytic computation of the nuclear coordinate gradient of the energy: HF, DFT, hybrids
HF-DFT;
Automated geometry optimizer (analytical or numerical gradient);
Improved techniques for finding the ground state in metallic systems;
Localization of the Crystalline Orbitals (Wannier functions);
Dielectric properties.
Main differences between CRYSTAL98 and CRYSTAL03
We tried to maintain compatibility between input deck prepared for CRYSTAL98 and CRYS-
TAL03. The CRYSTAL98 library of Hartree-Fock test cases can be used to test CRYSTAL03.
CRYSTAL03 writes wave function infos at the end of SCF (fortran units 9 and/or 98)
differently from Crystal98.
The data written in fortran unit 25 (formatted data for plotting maps, doss, band)
are different from the data written by CRYSTAL98. The package Crgra2003 can be
downloaded from http://www.crystal.unito.it/Crgra2003 to visualize bands, density of
states, isovalue contour maps of charge and spin density, classic and quantum electrostatic
potential.
Change in physical constant definition
Physical constants values are from "CODATA Recommended values of the fundamental
Physical Constants 1998" as reported in
http://physics.nist.gov/constants .
CRYSTAL03 adopts for the bohr unit 0.5291772083 (it was 0.529177 in CRYSTAL98,
according to "CODATA Recommended values of the fundamental Physical Constants
1992").
The effect on the total energy of 39 tests is presented in the table
http://www.crystal.unito.it/cr03vscr98.html .
The atomic coordinates, as provided in the input file, are re-adjusted to be fully consistent
with the symmetry group of the system.
The effect on the total energy of 39 tests is presented in the table
http://www.crystal.unito.it/cr03vscr98.html .
The symmetry operators in fractionary units are printed in compact form after atomic
coordinates.
SCF
The default value to test convergence on the total energy is 10**-5 as in CRYSTAL98,
for single point calculation. This value leads to a poor convergence, it was maintained
in release V1.0 for compatibility with CRYSTAL98 data. In geometry optimization the
value is set to 10**-7.
184

DFT
The following input deck:
input block 1:
geometry definition
input block 2 :
basis set
input block 3 :
DFT
B3LYP
(Exchange/correlation functional choice)
END
input block 4 :
scf data
submitted to CRYSTAL98 and to CRYSTAL03 gives different results, as the integration
method is different.
CRYSTAL03 uses by default a numerical integration technique, and at each scf cycle the
correct energy is computed. The same integration grid is used for scf and energy calculation.
Analytical energy gradient is implemented for numerical DFT only, so the fitting technique
used in CRYSTAL98 is no longer recommended.
The documentation is available in an
appendix of "CRYSTAL03 Users Manual":
http://www.crystal.unito.it/Manuals/cry2003.pdf .
CRYSTAL98 used by default an auxiliary basis set for fitting the potential; if the basis set
was not supplied in input, an event-tempered basis set was authomatically generated. At each
cycle a DFT pseudo energy was computed, and used to check convergence. The correct energy
was computed a posteriori, by the program properties, from the density matrix obtained at
the end of SCF procedure. The computational parameters used in scf were different from the
parameters (more severe) used to compute energy.
The keyword FITTING allows usage of the fitting technique in CRYSTAL03. In that case
the scf convergence is based on a pseudo energy, as in CRYSTAL98, but the correct energy
is evaluated at the end of scf, by the program crystal, and the grids used in scf and energy
calculation can not be modified independently.
CRYSTAL98 and CRYSTAL03 give the same pseudo energy, if the computational parame-
ters are the same, or the same a posteriori correlation energy (in CRYSTAL03 computed by
properties with an HF hamiltonian only), but there is no way to have both pseudo energy
and correct total energy coincident, as it is not possible input different parameters for scf and
energy calculation, as before.
Properties
The only difference is the input to PPAN, Mulliken population analysis. No input follows the
keyword PPAN. If more neighbors than default (6) are requested, use keyword NEIGHBORS
before PPAN to define a higher number of neighbors.
185

Appendix I
Relevant strings
Selected information can be extracted from CRYSTAL output referring to some strings of
characters uniquely linked to the requested information.
TOTAL ENERGY(
final SCF energy
TOTAL ENERGY(HF
Hartree-Fock
TOTAL ENERGY(DFT
DFT
TTT END
final elapsed and CPU time (crystal/properties)
OPT END
energy after geometry optimization
OPT END - FAILED
failed opt only
OPT END - CONVERGED successful opt only
TTT BERNY
cpu time for each opt cycle
GEOMETRY FOR WAVE
printing of the geometry used for wf calculation
(after editing)
FINAL OPTIMIZED
printing of geometry at the end of optimization
186

Appendix J
Acronyms
Acronyms
AFM Anti ferromagnetic
AO Atomic Orbital
APW Augmented Plane Wave
a.u. atomic units
BF Bloch Function
BS Basis set
BSSE Basis Set Superposition Error
BZ Brillouin Zone (first)
B3PW Becke Perdew Wang
B3LYP Becke - Lee - Yang - Parr
CO Crystalline Orbital
CPU Central Processing Unit
DF(T) Density Functional (Theory)
DM Density Matrix
DOS Density of States
ECP Effective Core Potentials
EFG Electric Field Gradient
EMD Electron Momentum Density
FM Ferromagnetic
GC Gradient-Corrected
GGA Generalised Gradient Approximation
GS(ES) Ground State (Electronic Structure)
GT(O) Gaussian Type (Orbital)
GT(F) Gaussian Type (Function)
GUI Graphical User Interface
KS Kohn and Sham
HF Hartree-Fock
IBZ Irreducible Brillouin zone
IR Irreducible Representation
LAPW Linearized Augmented Plane Wave
LCAO Linear Combination of Atomic Orbitals
LDA Local Density Approximation
LP Local Potential
187

LSDA Local Spin Density Approximation
LYP GGA Lee-Yang-Parr
MO Molecular Orbital
MPP Massive Parallel Processor
MSI Molecular Simulation Inc.
NLP Non-local potential (correlation)
PBE GGA Perdew-Burke-Ernzerhof
PDOS Projected Density of States
PP Pseudopotential
PVM Parallel Virtual Machine
PW Plane Wave
PWGGA GGA. Perdew-Wang
PWLSD LSD Perdew-Wang
PZ Perdew-Zunger
P86 GGA Perdew 86
P91 Perdew 91
QM Quantum Mechanics
RCEP Relativistic Compact Effective Potential
RHF Restricted Hartree-Fock
ROHF Restricted Open-shell Hartree-Fock
SAED Symmetry Allowed Elastic Distortions
SABF Symmetry Adapted Bloch Functions SC Supercell
SCF Self-Consistent-Field
STO Slater Type Orbital
UHF Unrestricted Hartree-Fock
VBH von Barth-Hedin
VWN Vosko-Wilk-Nusair
WnF Wannier Functions 0D no translational symmetry
1D translational symmetry in 1 direction (x, CRYSTAL convention)
2D translational symmetry in 2 directions (x,y, CRYSTAL convention)
3D translational symmetry in 3 directions (x,y,z CRYSTAL convention)
188

References
[1] R. Dovesi, C. Pisani, C. Roetti, M. Caus`
a and V.R. Saunders, CRYSTAL88, An ab initio
all-electron LCAO-Hartree-Fock program for periodic systems. QCPE Pgm N.577, Quan-
tum Chemistry Program Exchange, Indiana University, Bloomington, Indiana (1989).
[2] R. Dovesi, C. Roetti and V.R. Saunders, CRYSTAL92 User's Manual, Universit`
a di
Torino and SERC Daresbury Laboratory (1992).
[3] R. Dovesi, V.R. Saunders, C. Roetti, M. Caus`
a, N. M. Harrison, R. Orlando and E. Apr`
a,
CRYSTAL95 User's Manual, Universit`
a di Torino (1996).
[4] V.R. Saunders, R. Dovesi, C. Roetti, M. Caus`
a, N. M. Harrison, R. Orlando and C.M.
Zicovich-Wilson, CRYSTAL98 User's Manual, Universit`
a di Torino (Torino,1998).
[5] C. Pisani and R. Dovesi, "Exact exchange Hartree-Fock calculations for periodic systems.
I. Illustration of the method", Int. J. Quantum Chem. 17, 501 (1980).
[6] V.R. Saunders, "Ab initio Hartree-Fock calculations for periodic systems",
Faraday
Symp. Chem. Soc. 19, 7984 (1984).
[7] C. Pisani, R. Dovesi and C. Roetti, Hartree-Fock ab initio Treatment of Crystalline
Systems, Lecture Notes in Chemistry, volume 48, Springer Verlag, Heidelberg (1988).
[8] R. Dovesi, "On the role of symmetry in the ab initio Hartree-Fock linear combination of
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194

Index
Keyword Index
BETALOCK, 73
BIESPLIT, 58
ACOR see ADFT, 85
BIPOLAR, 58
ADFT (see EDFT input), 85
BIPOSIZE, 58
ADFT see EDFT
BOHRANGS, 27
NEWBASIS, 85
BOHRCR98, 28
ALL(ANISOTRO), 86
BOHR, 27, 88
ANBD, 85
BOYSCTRL(LOCALWF, 98
ANDERSON, 72
BR(PROF), 109
ANGSTROM, 24, 86
BREAKSYM, 22, 28
ANGULAR(DFT), 61
BROYDEN, 73
ANGULAR(EDFT), 85, 94
BWIDTH, 88
ANISOTRO
CHARGED, 48, 88
ALL, 86
CHEMOD, 48
PRINT, 86
CLAS, 88
SELECT, 86
CLUSTER, 28
UNIQUE, 86
CONV(PROF), 109
ATOMBSSE, 25
CONVCELL, 88
ATOMDISP, 25
COORDINA(MAPNET) , 101
ATOMFREE(OPTCOORD), 43, 130
COORPRT, 29, 89
ATOMHF, 57
CORRELAT(DFT), 59
ATOMINSE, 25
CP(PROF), 109
ATOMIR, 86
CRYSTAL, 11
ATOMORDE, 25
CYCTOL(LOCALWF), 98
ATOMREMO, 25
DFT, 58
ATOMROT, 26
ANGULAR, 61
ATOMS(MAPNET), 101
B3LYP, 60
ATOMSPIN, 73
B3PW, 60
ATOMSUBS, 27
BASIS, 172
ATOMSYMM, 27, 48, 86
BATCHPNT, 64
B3LYP(DFT), 60
BECKE, 59, 61
B3PW(DFT), 60
CORRELAT, 59
BANDLIST(LOCALWF), 97
ENDDFT, 59
BAND, 87
EXCHANGE, 59
BARTHE, 51
FITTING, 172
BASE(FIXINDEX), 65
HYBRID, 60
BASIS(DFT), 172
LDA, 59
BASISSET, 87
LGRID, 62
BATCHPNT(DFT), 64
LINEQUAT, 173
BECKE(DFT), 59, 61
LYP, 59
BECKE(EDFT), 85, 94
195

NEWTON, 64
ENDM general information input, 65
NONEWTON, 64
ENDSCF SCF input, 75
NONLOCAL, 60
properties input, 95
NUMERICA, 60
ENECOR see EDFT, 93
PB86, 59
EXCHANGE(DFT), 59
PBE, 59
EXCHGENE, 75
PRINT, 173
EXCHSIZE, 65
PWGGA, 59
EXTERNAL, 11, 13
PWLSD, 59
EXTPRT, 31, 95
PZ, 59
FIELD, 41
RADIAL, 61
FINALRUN(OPTCOORD), 43, 130
RADSAFE, 64
FITTING(DFT), 172
SAVIN, 61
FIXINDEX, 65
SPIN, 59
BASE, 65
TOLLDENS, 63
GEBA, 66
TOLLGRID, 63
GEOM, 65
TOLLPOT, 63, 173
FMIXING, 75
VBH, 59
FMWF, 95
VWN, 59
FRACTION, 31, 96
XLGRID, 63
GAUSS98, 49, 96
DIEL/DIELECT, 89
GEBA(FIXINDEX), 66
DIFF(PROF), 109
GEOM(FIXINDEX), 65
DLVINPUT, 11, 13
GHOSTS, 50
DOSS, 91
GRADCAL, 76
DURAND, 51
GUESSF, GUESSF0, 76
ECH3, 92
GUESSP, GUESSP0, 76
RANGE, 92
GUESSPAT, 76
SCALE, 92
HAYWLC, 51
ECHG, 93
HAYWSC, 51
EDFT, 93
HESGUESS(OPTCOORD), 44
ANGULAR, 85, 94
HYBRID(DFT), 60
BECKE, 85, 94
HYDROSUB, 31
PRINTOUT, 85, 94
IGSSBNDS(LOCALWF), 99
PRINT, 85, 94
IGSSCTRL(LOCALWF), 99
RADIAL, 85, 94
IGSSVCTS(LOCALWF), 99
SAVIN, 85, 94
INFOGUI, 96
EIGSHIFT, 74
INFO see INFOGUI, 96
EIGSHROT, 74
INIFIBND(LOCALWF), 97
EIGS, 64
INPUT, 51
ELASTIC, 30
INTGPACK, 67
EMDL, 94
ISOTROPIC, 96
EMDP, 95
PRINT, 96
ENDDFT(DFT), 59
KEEPSYMM, 22, 32
ENDOPT(OPTCOORD), 43
KNETOUT, 77, 97
ENDP(PROF), 109
KSYMMPRT, 77
END
LDA(DFT), 59
ENDB basis set input, 49
LEVSHIFT, 77
ENDG geometry input, 31
LGRID(DFT), 62
196

LINEQUAT(DFT), 173
NOSHIFT, 34
LOCALWF, 97
NOSYMADA, 78, 104
BANDLIST, 97
NUMERICA(DFT), 60
BOYSCTRL, 98
NUMGRAD(OPTCOORD), 45
CYCTOL, 98
OCCUPIED(LOCALWF), 97
IGSSBNDS, 99
OPTCOORD, 43
IGSSCTRL, 99
ATOMFREE, 43, 130
IGSSVCTS, 99
ENDOPT, 43
INIFIBND, 97
FINALRUN, 43, 130
MAXCYCLE, 98
HESGUESS, 44
OCCUPIED, 97
MAXOPTC, 44
PHASETOL, 98
NOGUESS, 45
PRINTPLO, 98
NUMGRAD, 45
RESTART, 98
PRINTFORCES, 45
VALENCE, 97
PRINTHESS, 45
LYP(DFT), 59
PRINTOPT, 45
MAKESAED, 32
PRINT, 45
MAPNET, 100
RESTART, 45
ATOMS, 101
TOLDEE, 46
COORDINA, 101
TOLDEG, 46
MARGINS, 101
TOLDEX, 46
PRINT, 101
ORIGIN, 34
RECTANGU, 101
P86(DFT), 59
MARGINS(MAPNET), 101
PARAMPRT, 35, 50, 68, 78, 104
MAXCYCLE(LOCALWF), 98
PATO, 104
MAXCYCLE(scf), 78
PBAN, 104
MAXOPTC(OPTCOORD), 44
PBE(DFT), 59
MODISYMM, 32
PDIBAN see PBAN, 104
MOLDRAW, 32, 103
PDIDE, 105
MOLEBSSE, 32
PGEOMW, 105
MOLECULE, 11
PHASETOL(LOCALWF), 98
from 3D structure, 33
PIEZOBP, 112
MOLEXP, 33
PIEZOWF, 113
MOLSPLIT, 34
POLARI, 113
MONSPLIT, 67
POLEORDR, 68
MPP, 67
POLI, 105
MULPOPAN, 78, 108
POLSPIN, 106
NEIGHBOR, 34, 78, 103
POLYMER, 11
NEIGHPRT see NEIGHBOR, 34
POSTSCF, 78
NEWBASIS(ADFT), 85
POTC, 107
NEWK, 103
POTM, 107
NEWTON(DFT), 64
PPAN, 78, 108
NOBIPOLA, 67
PRIMITIV, 35
NOFMWF, 78
PRINT(ANISOTRO), 86
NOGUESS(OPTCOORD), 45
PRINT(DFT), 173
NOMONDIR, 67
PRINT(EDFT), 85, 94
NONEWTON(DFT), 64
PRINT(ISOTROPIC), 96
NONLOCAL(DFT), 60
PRINT(MAPNET), 101
197

PRINT(OPTCOORD), 45
PWLSD(DFT), 59
PRINTFORCES(OPTCOORD), 45
PZ(DFT), 59
PRINTHESS(OPTCOORD), 45
RADIAL(DFT), 61
PRINTOPT(OPTCOORD), 45
RADIAL(EDFT), 85, 94
PRINTOUT(EDFT), 85, 94
RADSAFE(DFT), 64
PRINTOUT, 35, 50, 68, 79, 108
RANGE (ECH3), 92
ATCOORDS, 179
RAYCOV/RAYC/RCOVFACT, 35, 110
ATOMICWF, 180
RDFMWF, 95
BASISSET, 180
RECTANGU(MAPNET), 101
CONFIGAT, 180
REDEFINE, 36
DFTBASIS, 180
RESTART(LOCALWF), 98
DOSS, 180
RESTART(OPTCOORD), 45
EIGENALL, 180
RHF, 68
EIGENVAL, 180
ROHF, 70
EIGENVEC, 180
ROTATE see REDEFINE, 36
ENECYCLE, 180
ROTREF, 110
EQUIVAT, 179
SAVEWF, 79
EXCHGENE, 179
SAVIN(DFT), 61
FGIRR, 180
SAVIN(EDFT), 85, 94
FGRED, 179
SCALE (ECH3), 92
GAUSS94, 180
SCFDIR, 68
GLATTICE, 179
SELECT(ANISOTRO), 86
KNETOUT, 179
SEMIDIR, 68
KSYMMPRT, 179
SETINF, 36, 50, 69, 79, 111
KWEIGHTS, 179
SETPRINT, 37, 50, 69, 79, 111
MAPVALUES, 180
SLABCUT/SLAB
MULLIKEN, 180
slab from 3D structure, 37
MULTIPOL, 180
SLAB, 11
OVERLAP, 179
SMEAR, 79
PARAMETERS, 179
SPIN(DFT), 59
PGIRR, 179
SPINEDIT, 80
PGRED, 179
SPINLOCK, 80
ROTREF, 180
SPOLBP, 114
SCALEFAC, 180
SPOLWF, 114
SYMMOPSR, 179
STOP, 38, 50, 69, 81, 111
SYMMOPS, 179
STRUCPRT, 38
PRINTPLO(LOCALWF), 98
SUPERCEL, 38
PRINT, 89
SUPERCON, 40
PROF, 109
SYMADAPT, 81, 111
BR, 109
SYMMDIR, 40
CONV, 109
SYMMOPS, 40, 50
CP, 109
SYMMREMO, 40
DIFF, 109
TENSOR, 40
ENDP, 109
TESTGEOM, 40
PRSYMDIR, 35
TESTPDIM, 69
PSCF, 110
TESTRUN, 69
PURIFY, 35
TOLDEE(OPTCOORD), 46
PWGGA(DFT), 59
TOLDEE, 81
198

TOLDEG(OPTCOORD), 46
atomic number conventional, 15
TOLDEP, 81
Atomic Orbital
TOLDEX(OPTCOORD), 46
definition, 139
TOLINTEG, 69
order, 18
TOLLDENS(DFT), 63
atomic units
TOLLGRID(DFT), 63
bohr, 27
TOLLPOT(DFT), 63, 173
charge, 106
TOLPSEUD, 69
conversion factor, 27, 28
TOLSCF, 81
atoms
TRASREMO, 40
(group of) rotation, 26
UHF, 70
addition, 25
UNIQUE(ANISOTRO), 86
displacement, 25
USESAED, 41
removal, 25
VALENCE(LOCALWF), 97
reordering, 25
VBH(DFT), 59
substitution, 27
VWN(DFT), 59
autocorrelation function theory, 146
XFAC, 111
auxiliary basis set
XLGRID(DFT), 63
DFT, 172
ZCOR see EDFT, 93
for ghost atoms, 173
band structure
Subject Index
calculation, 87
band width, 88
0D from 3D, 2D, 1D, 47
basis set, 133, 139
0D systems input, 12
all electron, 16, 17
1D systems input, 12
auxiliary (DFT), 172
2D from 3D, 37, 47
criteria for selection, 133
2D systems input, 12
crystal, 17
3D systems input, 12
Effective Core Pseudopotential, 53
acceleration techniques
input, 16
see SCF acceleration techniques, 20
input examples, 122
adjoined gaussian, 139
libraries, 133
adsorbed molecule rotation, 26
linear dependence check, 64
adsorption of molecules, 25
metals, 136
Anderson method for accelerating conver-
optimization, 65
gence, 72
orbital ordering, 17
angular integration (DFT), 61, 85, 94
Pople, 16
anisotropic tensor, 86
printing, 180
anisotropy shrinking factor, 20
type, 16
anti ferromagnetic systems, 70
valence only, 16, 17
Aragonite, 116
basis set superposition error
asymmetric unit, 14
molecular, 32
ATMOL integral package, 67
atomic, 25
atomic
periodic, 50
density matrix, 57, 104
Beryllium slab, 119
wave function, 57
BF - Bloch Functions, 139
atomic energy
bielectronic integrals
(correlation) a posteriori, 85
file split, 58
199

indexing, 65
cluster from 3D, 28
package, 67
CO - Carbon Monoxide
bipolar expansion
molecule, 122
bielectronic integrals, 58, 143
on MgO (001), 120
Coulomb buffer, 58
CO - Crystalline Orbital, 139
elimination, 67, 143
Compton profile
exchange buffer, 65
average, 147
Bloch Functions
directional, 147
definition, 139
input, 109
Symmetry Adapted, 144
theory, 146
Symmetry Adapted - printing, 77
constraint sp, 139
Boys
contour maps, 100
localization, 97
contraction
Bravais lattice, 14, 36, 163
coefficients, 16
Brillouin zone, 140
of gaussians, 16, 139
sampling, 20, 145
conventional atomic number, 15, 16
Broyden method for accelerating conver-
conventional cell, 14, 88
gence, 73
convergence
buffer
acceleration techniques, 20
Coulomb bipolar expansion, 58
tools
exchange bipolar expansion, 65
Anderson method, 72
bulk modulus, 150
Broyden method, 73
BZ - Brillouin Zone, 140
Fock matrix mixing, 75
level shifter, 77
Calcite, 117
convergence criteria
cell
cycles overflow, 78
centred, 15
density matrix, 81
charged, 18
RMS of eigenvalues, 81
conventional, 14, 88
total energy, 81
conventional/primitive
transforma-
conversion factors, 27
tion, 163
length, 27
crystallographic, 14
conversion factors (CR98), 28
minimum set parameters, 13
conversion wave function data, 95
neutrality, 48
coordinates
non neutral, 48
of equivalent atoms, 15
primitive, 14, 35
output, 29, 89
redefinition, 36
units, 24
cell parameters
bohr, 27
optimization, 46
fraction, 96
Cesium Chloride, 116
fractionary, 31
Chabazite, 118
units of measure, 12
check
unitsi
basis set input, 69
angstrom, 86
complete input deck, 69
bohr, 88
disk storage to allocate, 69
Corundum
geometry input, 40
(0001) surface, 119
chemisorption, 25
(1010) surface, 119
Cholesky reduction, 65, 138
bulk, 117
200

Coulomb energy, 140
integration scheme, 61
Coulomb series, 141
integration technique, 60
bielectronic contribution, 141
numerical integration, 60
Coulomb series threshold, 69
Diamond, 115
counterpoise technique, 47
(100) Surface, 120
covalent radii
dielectric constant (optical), 89
customised, 35, 110
Durand-Barthelat pseudo-potentials, 52
default value, 35, 47
Crystalline Orbital (CO)
ECP
definition, 139
input examples, 123
crystallographic cell, 14
ECP - see Effective Core Pseudopotential,
crystals
51
(3D) input, 12
Edingtonite, 118
molecular, 47
Effective Core Pseudopotential
Cuprite, 116
input, 51
BARTHE, 51
defects
DURAND, 51
displacement, 25
HAYWLC, 51
in supercell, 38
HAYWSC, 51
interstitial, 25
input examples, 123
substitutional, 27
truncation criteria, 69
vacancies, 25
eigenvalues (Hamiltonian), 139
density functional
eigenvalues (Hamiltonian) printing, 177
numerical integration, 60
eigenvectors
see DFT, 58
calculation, 103, 139
density matrix
principal components printout, 85
atomic, 104
printing, 177
band projected, 104
printing (core), 178
behaviour, 143
rotation, 110
core electrons, 83
elastic constant, 30, 147
direct space, 140
elastic distortion, 30
editing, 80
elastic moduli theory, 147
energy projected , 105
elastic strain, 148
from geometrical weights, 105
elastic tensor, 148
initial guess, 76
electric field, 107
restore, 110
in a crystal, 41
rotation, 110
through a slab, 41
valence electrons, 83
electron charge density
density of states
3D maps, 92
calculation, 91
calculation, 93
Fourier-Legendre expansion, 92, 145
gradient, 93
integrated, 92
electron momentum density
DFT
line, 94
auxiliary basis set, 172
plane, 95
functionals, 59
theory, 146
Hamiltonian, 58
electron spin density, 93
input, 58
electronic configuration
input examples, 173
ions, 18
201

open shell atoms, 18
OPTCOORD, 43
electronic properties, 82
ghost atoms
electrostatic potential
atoms converted to, 50
first derivative, 107
auxiliary basis set, 173
maps, 88
input deck, 17
second derivative, 107
Gilat net, 19, 103
with an electric field, 107
Graphite, 116, 119
EMD theory, 146
ground state electronic properties, 82
energy
groups - see symmetry groups, 155
(correlation) a posteriori, 93
GTF
atomic, 19
definition, 139
Coulomb, 140
primitives, 139
exchange (definition), 142
primitives-input, 16
exchange contribution, 75
Fermi, 140
Hamiltonian
total
closed shell, 68
HF, 140
DFT, 58
energy derivatives (elastic constants), 147
open shell, 70
equivalent atoms coordinates, 15
Hay and Wadt pseudo-potentials, 52
exchange energy
hydrogen (border atoms substitution with),
calculation, 75
31
theory, 140
hyperfine electron nucleus interaction
exchange series threshold, 69, 142
anisotropic, 86
expansion coefficients, 139
isotropic, 96
Faujasite, 118
INF
Fermi contact, 96
setting values, 36, 50, 69, 79, 111
Fermi energy, 140, 145
initial guess
smear, 79
atomic densities, 76
findsym, 22
Fock/KS matrix, 76
Fluorite, 116
input density matrix, 76
Fock matrix
input examples
definition in direct space, 140
0D geometry, 122
elements selective shift, 74
1D geometry , 121
Formamide polymer, 121
2D geometry, 120
formatted wave function, 78, 95
3D geometry, 118
functionals
basis set, 122
DFT, 59
DFT, 173
Effective Core Pseudopotential, 123
GAUSS70 integral package, 67
integral evaluation criteria, 140
Gaussian 98 input deck, 49
integration in reciprocal space, 145
gaussian primitives contraction, 139
IS, 20, 145
gaussian type functions definition, 139
ISP, 20, 145
geometry
ITOL1, 69, 141
exported, 38
ITOL2, 69, 142
optimization, 65
ITOL3, 69, 142
visualization, 31, 32, 95, 103
ITOL4, 69, 142
geometry optimization
ITOL5, 69, 142
Berny, 43
202

keywords list, 164
one electron integrals
Kohn - Sham Hamiltonian, 58
kinetic, 140
nuclear, 140
lattice
optimization
centred, 15
cell parameters, 46
definition, 13
orientation convention
vectors, 15
polymer, 15
layer groups, 158
slab, 15
LCAO, 139
origin
Lebedev accuracy levels, 61
moving, 34
level shifter, 77
setting, 15
linear dependence catastrophe, 64, 137
overlap matrix
linear equation solver, 173
definition, 139
localization
printing, 177
Boys, 97
Wannier, 97
Perdew-Zunger, 59
LPRINT, 177
physisorption, 25
piezoelectricity, 113
maps (contour), 100
Berry phase, 112
metals basis set, 136
localized CO approach, 113
Methane molecule, 122
point groups, 162
MgO
polarization functions, 17
(001) surface, 120
polymer
(110) surface, 119
input, 12
molecular crystals
orientation, 15
noninteracting units, 34
population analysis (Mulliken), 78, 108
lattice parameters modification, 33
primitive cell, 14
section, 47
printing
molecules
keywords, 179
from 3D, 33
multipole moments, 106
input, 12
neighbour list, 34, 78, 103
non interacting, 34
parametrized dimensions, 35
Monkhorst net, 103, 145
reciprocal lattice, 77
shrinking factor, 20
setting environment, 35, 50, 68, 79, 108
monoelectronic integral file split, 67
setting options, 37, 50, 69, 79, 111
Mulliken population analysis, 78, 108
properties
multipolar expansion
ground state electronic, 82
definition, 141
pseudopotential
maximum order, 68
Durand-Barthelat, 52
multipole moments
Hay and Wadt, 52
printing, 106
Stevens et al., 54
spin, 106
Stuttgart-Dresden, 53
calculation, 105
Pyrite, 116
ordering, 106
spherical harmonics, 105
radial integration (DFT), 61, 85, 94
RCEP, 54
neighbour printing, 34, 78, 103
reciprocal form factor, 146
NiO anti ferromagnetic - input, 81
reciprocal lattice
output, 77
203

reciprocal space integration, 145
symbol, 14
reference frame rotation, 74
space groups tables, 155
Restricted HF, 68
spherical harmonic multipole moments, 105
Restricted Open Shell HF, 70
spin
Rock Salt structure, 115
DFT, 59
rod groups, 159
multipole moments, 106
rotation
spin configuration
density matrix, 110
locking - electrons, 80
eigenvectors, 110
locking electrons, 73
of adsorbed molecules, 26
setting, 73
reference frame, 74
spin density matrix editing, 80
Rutile, 116
spin polarized systems, 21
Spinel, 125
SAED Symmetry Allowed Elastic Distor-
spontaneous polarization
tions, 32
localized CO approach), 114
scale factor, 16, 17
spontaneous polarization, 113
SCF
Berry phase, 114
acceleration techniques, 20
Stevens et al. pseudopotential, 54
convergence
STM, 105
density matrix, 81
structure factors, 111
total energy, 81
Stuttgart-Dresden pseudopotential, 53
cycles control, 78
supercell
direct bielectronic integrals, 68
creation, 38
input, 19
input examples, 38
level shifter, 77
surfaces
mixing Fock/KS matrices, 75
2D slab model, 12
SCF convergence acceleration
from crystal structures, 37
Anderson, 72
symmetry
Broyden, 73
allowed directions, 40
level shifter, 77
analysis in K space, 77
shell
breaking, 22, 28
definition, 139
electric field, 42
formal charge, 16, 18
maintaining, 22, 32
type, 16, 17
modification, 32
shift of Fock matrix elements, 74
point operators printing, 40
shrinking factor, 20, 145
point symmetry, 144
slab
related atoms printing, 27
input, 12
removal, 40
orientation, 15
translational, 143
slab model, 47
translational components removal, 40
SN polymer, 121
Symmetry Adapted Bloch Functions, 144
Sodalite, 118
symmetry groups
Sodium Chloride, 115
layer, 158
sp constraint, 139
point groups, 162
space group
rod groups, 159
monoclinic input, 15
space, 155
orthorhombic input, 15
setting, 15
tensor of physical properties, 40
204

threshold
tetragonal, 117
Coulomb series, 141
exchange series, 142
tolerances
bipolar expansion, 58
Coulomb series, 141
DFT, 63
Effective Core Pseudopotential, 69
exchange series, 142
integrals, 69
ITOL1, 69, 141
ITOL2, 69, 142
ITOL3, 69, 142
ITOL4, 69, 142, 143
ITOL5, 69, 142
SCF, 81
total energy, 140
transformation matrices in crystallography,
163
two electron
Coulomb contribution, 140
exchange contribution, 140
units

Angstrom, 24
bohr, 27
fractionary, 31
Unrestricted HF, 70
Urea molecule, 122
visualization
geometry, 31, 95
MOLDRAW, 32, 103
Wadt (see Hay), 52
Wannier functions, 97
Water chain, 121
Wurtzite, 115
X-ray structure factors, 111
Zeolites
Chabazite, 118
Edingtonite, 118
Faujasite, 118
Sodalite, 118
Zinc Blend, 115
Zirconia
cubic, 117
monoclinic, 117
205