Table Of ContentSpringer
Berlin
Heidelberg
New York
Barcelona
Budapest
Hong Kong
London
Milan
Paris
Santa Clara
Singapore
Tokyo
Lecture Notes in Chemistry 67
Edited by:
Prof. Dr. Gaston Berthier
Universit~ de Paris
Prof. Dr. Hanns Fischer
Universitllt ZUrich
Prof. Dr. Kenichi Fukui
Kyoto University
Prof. Dr. George G. Hall
University of Nottingham
Prof. Dr. JUrgen Hinze
Universitllt Bielefeld
Prof. Dr. Joshua Jortner
Tel-Aviv University
Prof. Dr. Werner Kutzelnigg
Universitllt Bochum
Prof. Dr. Klaus Ruedenberg
Iowa State University
Prof Dr. Jacopo Tomasi
Universitl di Pisa
c.
Pisani (Ed.)
Quantum-Mechanical
Ab-initio Calculation
of the Properties
of Crystalline Materials
Springer
Editor
Professor Cesare Pisani
University of Torino
Department of Inorganic, Physical and Materials Chemistry
Via Giuria 5, 1-10125 Torino
Italy
Library of Congress Cataloging-In-Publication Data
School of COMputational CheMistry of the Italian CheMical SOCiety (4th
: 1994 : Torino. Italy)
QuantUM-mechanical ab-Inltlo calculation of the properties of
crystalline materials: proceedings of the IV School of
Computational Chemistry of the Italian Chemical Society I C. Pisani.
editor.
p. em.
Fourth School of Computational Chemistry of the Italian Chemical
Society. held In Torino on 19-24 September 1994.--Forward
Includes bibliographical references and Index.
ISBN-13: 978-3-540-61645-0
1. Solid state phySlcs--Congresses. 2. Crystals--Congresses.
3. Quantum chemlstry--Congresses. I. Pisani. C. II. Title.
QC176.A1S33 1994
530.4' 13--DC20 96-36109
CIP
ISSN 0342-4901
ISBN-13: 978-3-540-61645-0 e-ISBN-13: 978-3-642-61478-1
DOl: 10.1007/978-3-642-61478-1
This work is subject to copyright. All rights are reserved, whether the whole or part
of the material is concerned, specifically the rights of translation, reprinting, re-use
of illustrations. recitation, broadcasting. reproduction on microfilms or in any other
way. and storage in data banks. Duplication of this publication or parts thereof is
permitted only under the provisions of the German Copyright Law of September 9,
1965, in its current version, and permission for use must always be obtained from
Springer-Verlag. Violations are liable for prosecution under the German Copyright
Law.
© Springer-Verlag Berlin Heidelberg 1996
Springer-Verlag Berlin Heidelberg New York
a member of BertelsmannSpringer Science+Business Media GmbH
The use of general descriptive names, registered names, trademarks. etc. in this
publication does not imply. even in the absence of a specific statement, that such
names are exempt from the relevant protective laws and regulations and therefore
free for general use.
Typesetting: Camera ready by author/editor
SPIN: 10850203 5213142 -54321 -Printed on acid-free paper
Foreword
Many powerful computer codes exist in the field of molecular chemistry, based on ab initio quantum
mechanical techniques. They can predict many properties of small- and medium-sized molecules with
reasonable accuracy and at comparatively low cost. Their use has become common practice in the last
ten years in many areas of activity, particularly in experimental chemical research, both in academic and
in industrial environments.
The situation is much less advanced in the parallel field of condensed-matter studies. This is not due ~ a
lack of potential interest: the development of new materials is one of the areas where the rate of progress
is most rapid, and the amount of financial investment is largest. Materials for electronic and mechanical
applications are in most cases crystalline structures, perfect or with controlled dosage of defects, whose
properties cannot be understood without reference to an accurate description of the system at an atomic
scale.
The intrinsic difficulty of handling systems of potentially infinite size explains only in part why crystalline
studies are so far behind the current frontier of molecular quantum chemistry, a gap which presently is
about fifteen years. Another possible reason may be that theoretical chemists have developed over many
decades widely accepted interpretative schemes, a relatively standard language, a number of practical
tools, good books and excellent computer codes so that having access to that technology does not require
a very high-level entry point. On the other hand, making the wealth of knowledge embodied in stan
dard solid-state theory compatible with an atomic-scale description of condensed systems is still an open
problem. The question is how to transfer concepts and results from a "quantum-chemical" description of
crystals to the parameterized world of solid-state physicists, and vice-versa.
Furthermore, the research 8lld development effort which is being devoted to the implementation of pow
erful and user-friendly codes tor the study of crystalline properties is curiously enough much less intensive
than that devoted to the production of new molecular codes, and in the improvement of existing ones
whose performance is already excellent. It is thus not surprising that the molecular cluster model, which
can utilize standard codes with only minor modifications, is at present the favourite tool for quantum
mechanical investigations of the properties of crystals and their surfaces.
We now have the opportunity to change this state of affairs. There are a number of general-purpose, rea
sonably accurate and well-tested Qb initio computer codes for crystals which are available to the scientific
community. The rate of their improvement depends, in a sense, on their circulation, on the criticisms
they receive and on the suggestions which are derived from their use. It is also very important that
the various groups active in this field are open to unbiased comparison of the merits and drawbacks of
their proposals, both as concerns basic ideas and approximations, and actual implementation of computer
codes.
Schools and workshops can play a useful role to this effect, by teaching young people active in the field of
Material Science how to exploit these new powerful tools. The present book contains the Proceedings of
the Fourth School of Computational Chemistry, organized by the Interdivisional Computer Chem
istry Group (GICC) of the Italian Chemical Society, and held in Torino on 19-24 September 1994, whose
schedule is reported in Appendix A. Only the morning lectures (with few exceptions) are reproduced
here, the afternoons being devoted to practical exercises performed by the students with use of the three
codes available at the School, and described in Part 3.
VI
The texts of the lectures can give only part of the information which can be obtained at a School through
the possibility of practicing with the different programs, and from the contact ""ith their authors. They
nevertheless represent, on the whole, a useful introduction to the field, a reference for deeper study of
certain specific subjects, and an objective body of information concerning the state-of-the-art in ab-initio
simulations of the quantum-mechanical properties of crystalline materials.
Part One (Chapters 1-3) of the book provides a general introduction to the subject, addressed
particularly to readers with a general knowledge in quantum chemistry, but not much confidence in solid
state theory and its concepts.
Part Two (Chapters 4-1) is intended to give a deeper insight into the special algorithms and compu
tational techniques which are currently adopted in ab initio computer codes for crystals.
Part Three (Chapters 8-10) presents in parallel three different programs which are available to
all interested potential users on request, and based on very different approaches. These presentations
may help newcomers in the field to understand the meaning hidden in the acronyms, and to choose the
most suitable tool for their needs.
Finally, Part Four (Chapters 11-16) is an attempt to show what kind of information on the observ
able properties of condensed systems can be obtained from ab initio quantum-mechanical calculations.
In particular, Resta's contribution demonstrates that important observables have become accessible to
simulation, using quite unconventional new approaches. The last chapter, devoted to the hot topics of
superconductivity, shows the importance of finding the connection between the results of ab initio calcu
lations and high-quality theoretical schemes using parameterized Hamiltonians.
The publishing of these notes and their distribution to all the students of the above mentioned School
has been possible thanks to the support of the Italian CNR (Consiglio Nazionale delle Ricerche).
A preliminary draft of these Proceedings has been distributed among attendants of the 1995 School
organized under contract CHRX-CI'93-0155 of the Human Capital &: Mobility Programme of the Eur<>
pean Community, which has provided additional funding for the work of revision.
I would finally like to thank Dr. Fiona Healy, who has read the manuscripts with patience and in
telligence, and corrected their English, when necessary and when possible.
Cesare Pisani Torino, May 1996
Contents
1 D. Viterbo: Crystal Lattices and Crystal Symmetry 1
2 R. Dovesi: The Language of Band Theory 31
3 C. Pisani: Ab-Initio Approaches to the Quantum-Mechanical Treatment of
Periodic Systems 41
4 A. Dal Corso: Reciprocal Space Integration and Special-Point Techniques 11
5 M. Causl: Numerical Integration in Density Functional Methods with Linear
Combination of Atomic Orbitals 91
6 E. Aprl: Hartree-Fock Treatment of Spin-Polarized Crystals 101
1 N .M. Harrison: The Quantum Theory of Periodic Systems on Modern
Computers 113
8 C. Roetti: The CRYSTAL Code 125
9 K. Schwarz and P. Blaha: Description of an LAPW DF Program (WIEN95) 139
10 A. Dal Corso: A Pseudopotential Plane Waves Program (PWSCF) and some
Case Studies 155
11 R. Dovesi: Total Energy and Related Properties 119
12 M. Catti: Lattice Dynamics and Thermodynamic Properties 209
13 C. Pisani: Loss of Symmetry in Crystals: Surfaces and Local Defects 221
14 W. Weyrich: One-Electron Density Matrices and Related Observables 245
15 R. Resta: Macroscopic Dielectric Polarization: Hartree-Fock Theory 213
16 M. Rasetti: The Hubbard Models and Superconductivity 289
A Schedule of the 1994 GICC School of Computational Chemistry 321
B Subject Index 323
C List of Acronyms 328
Crystal Lattices and Crystal Symmetry
Davide Viterbo
Department of Inorganic, Physical and Materials Chemistry, University of Torino, via Giuria
7,1-10125 Torino, Italy
June 4,1996
Summary. The basic concepts of the geometrical representation of crystalline
solids and of their symmetry are outlined. The combination of periodic trans
lational symmetry (describing crystal lattices) with other symmetry elements
(rotation axes, mirror planes, inversion centers, etc.) is described as the basis of
the space group theory.
Key words: Crystal Lattice - Translational Symmetry - Unit Cell - Crystal
Structure - Crystallographic Rows and Planes - Metric Tensor - Reciprocal Lat
tice - Symmetry Operators - Symmetry Elements - Point Groups - Symmetry
Classes - Laue Classes - Crystal Systems - Bravais Lattices - Space Groups
1. Lattice geometry
1.1. Lattices
Crystalline solids, as confirmed by evidence from several experiments (the
anisotropy of their physical properties, diffraction, etc.), may be described as or
dered repetitions of atoms or groups of atoms in three dimensions. Translational
periodicity in crystals may be conveniently studied by focusing our attention on
the geometry of the repetition rather than on the repeating motif. In an ideal
crystal, all repeating units are identical and we may say that they are related
by translational symmetry operations, corresponding to the set of vectors:
T=ua+vb+wc (1)
where u, v and ware three integers ranging from minus infinity to plus infinity,
zero included, and a, band c are three non-coplanar vectors defining the basis
of the three-dimensional space. Real crystals may present more or less marked
2 Davide Viterbo
Figure 1. A three-dimenaionallattice, showing a unit cell (heavy lines).
deviations from this ideal, perfect order. The set of points at the ends of all the
translation vectors T forms a three-dimensional lattice and the points are called
lattice nodes (Figure 1). The three integers u, v and w defining a given vector,
are the corresponding coordinates of the node in the reference system defined by
a, b and c. The parallelepiped formed by these three basis vectors is called the
unit cell and their directions define the crystallographic axes: X, Y and Z. The
lattice constants are the three moduli a, b and c and the three angles, a, fJ and
'Y between the vectors (a between b and e, fJ between a and e and 'Y between a
and b).
A two-dimensional example will serve to illustrate these concepts. In Figure
2(a), a given "three-atom" motifis repeated at intervals a and b. If we replace
each motif by a point at its centre of gravity we obtain the lattice of Figure
2(b). The same lattice is obtained if the point is located on any other position of
the motif and the position of the lattice with respect to the motif is completely
arbitrary. If any lattice point is chosen as the origin of the lattice, any other
point in Figure 2(b) is uniquely defined by the vector:
T=ua+vb (2)
where u and v are integers and the unit cell is defined by the vectors a and b.
The choice ofthe basis vectors is rather arbitrary, as shown in Figure 2(b), where
four different choices are illustrated, all of which are consistent with relation (2)
with u and v being integers. These cells contain only one lattice point, since the
four points at the corners of each cell are each shared by a total of four cells.
They are called primitive cells. Nevertheless, we are allowed to choose different
types of unit cells, such as those shown in Figure 2(c), which contain two or
more lattice points. Also in this case, each lattice point will satisfy (2) but u and
Crystal Lattices and Crystal Symmetry 3
Figure 2. (a) Repetitionofagraphicalmo Figure 3. Lattice rows and planes.
tif as an example of a two-dimensional crys
tal; (b) the corresponding lattice with some
examples of primitive cells; (c) the same lat
tice with some examples of multiple cells.
v are no longer restricted to integer values (in Figure 2(c), point P is related to
the origin and to the basis vectors a' and h' by a vector with u = ~, v = ~).
These cells are called multiple or centred cells.
1.2. Crystal structure
The periodic repetition of the structural motif (atoms, groups of atoms or
molecules) by the infinite set of vectors (1) yields the crystal structure, which
is completely determined once the lattice constants and the coordinates, x, Y
and z, of all the atoms in the unit cell are known. These coordinates are the
components of the vectors:
rj = Xj a + Yj h + Zj C (j = 1, 2 ... N) (3)
linking the cell origin to the nucleus of the j-th atom. For all N atoms inside
the chosen unit cell, the coordinate values are in the interval 0 to 1. They are
therefore called fractional coordinates and are given by:
x=Xja y=Yjb z=Zjc
