Accurate condensed phase quantum chemistry computation in chemistry

CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2011 by Taylor and Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Printed in the United States of America on acid-free paper
10 9 8 7 6 5 4 3 2 1
International Standard Book Number: 978-1-4398-0836-8 (Hardback)
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts
have been made to publish reliable data and information, but the author and publisher cannot assume
responsibility for the validity of all materials or the consequences of their use. The authors and publishers
have attempted to trace the copyright holders of all material reproduced in this publication and apologize to
copyright holders if permission to publish in this form has not been obtained. If any copyright material has
not been acknowledged please write and let us know so we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented,
including photocopying, microfilming, and recording, or in any information storage or retrieval system,
without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access www.copyright.
com ( or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood
Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and
registration for a variety of users. For organizations that have been granted a photocopy license by the CCC,
a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used
only for identification and explanation without intent to infringe.
Library of Congress Cataloging-in-Publication Data

Accurate condensed-phase quantum chemistry / editor, Frederick R. Manby.
p. cm. -- (Computation in chemistry)
Includes bibliographical references and index.
ISBN 978-1-4398-0836-8 (hardcover : alk. paper)
1. Quantum chemistry. 2. Condensed matter. I. Manby, Frederick R.
QD462.A33 2011
541’.28--dc22
Visit the Taylor & Francis Web site at

and the CRC Press Web site at


2010022634


Contents
Series Preface ....................................................................................................vii
Preface.................................................................................................................ix
Editor ................................................................................................................xiii
Contributors......................................................................................................xv

Chapter 1

Laplace transform second-order Møller–Plesset
methods in the atomic orbital basis for periodic
systems .......................................................................................... 1
Artur F. Izmaylov and Gustavo E. Scuseria

Chapter 2

Density fitting for correlated calculations in periodic
systems ........................................................................................ 29
¨ Denis Usvyat, Marco Lorenz, Cesare Pisani,
Martin Schutz,
Lorenzo Maschio, Silvia Casassa, and Migen Halo

Chapter 3

The method of increments—a wavefunction-based
correlation method for extended systems............................. 57
Beate Paulus and Hermann Stoll

Chapter 4

The hierarchical scheme for electron correlation
in crystalline solids................................................................... 85
Stephen J. Nolan, Peter J. Bygrave, Neil L. Allan,
Michael J. Gillan, Simon Binnie, and Frederick R. Manby

Chapter 5

Electrostatically embedded many-body expansion
for large systems ...................................................................... 105
Erin Dahlke Speetzen, Hannah R. Leverentz, Hai Lin,
and Donald G. Truhlar

Chapter 6

Electron correlation in solids: Delocalized
and localized orbital approaches.......................................... 129

So Hirata, Olaseni Sode, Murat Kec¸eli,
and Tomomi Shimazaki

Chapter 7

Ab initio Monte Carlo simulations of liquid water ......... 163
Darragh P. O’Neill, Neil L. Allan, and Frederick R. Manby

Index ................................................................................................................ 195
v


Series Preface
Computational chemistry is highly interdisciplinary, nestling in the fertile
region where chemistry meets mathematics, physics, biology, and computer science. Its goal is the prediction of chemical structures, bondings,
reactivities, and properties through calculations in silico, rather than experiments in vitro or in vivo. In recent years, it has established a secure place
in the undergraduate curriculum and modern graduates are increasingly
familiar with the theory and practice of this subject. In the twenty-first century, as the prices of chemicals increase, governments enact ever-stricter
safety legislations, and the performance/price ratios of computers increase, it is certain that computational chemistry will become an increasingly attractive and viable partner of experiment.
However, the relatively recent and sudden arrival of this subject has
not been unproblematic. As the technical vocabulary of computational
chemistry has grown and evolved, a serious language barrier has developed between those who prepare new methods and those who use them
to tackle real chemical problems. There are only a few good textbooks;
the subject continues to advance at a prodigious pace and it is clear that
the daily practice of the community as a whole lags many years behind
the state of the art. The field continues to advance and many topics that
require detailed development are unsuitable for publication in a journal
because of space limitations. Recent advances are available within complicated software programs but the average practitioner struggles to find
helpful guidance through the growing maze of such packages.
This has prompted us to develop a series of books entitled Computation in Chemistry that aims to address these pressing issues, presenting

specific topics in computational chemistry for a wide audience. The scope
of this series is broad, and encompasses all the important topics that constitute “computational chemistry” as generally understood by chemists. The
books’ authors are leading scientists from around the world, chosen on
the basis of their acknowledged expertise and their communication skills.
Where topics overlap with fraternal disciplines—for example, quantum
mechanics (physics) or computer-based drug design (pharmacology)—
the treatment aims primarily to be accessible to, and serve the needs of,
chemists.
This book, the second in the series, brings together recent advances
in the accurate quantum mechanical treatment of condensed systems,
whether periodic or aperiodic, solid or liquid. The authors, who include
leading figures from both sides of the Atlantic, describe methods by which
the established methods of gas-phase quantum chemistry can be modified
vii


viii

Series Preface

and generalized into forms suitable for application to extended systems
and, in doing so, open a range of exciting new possibilities for the subject.
Each chapter exemplifies the overarching principle of this series: These will
not be dusty technical monographs but, rather, books that will sit on every
practitioner’s desk.



Preface
Quantum mechanical calculations on polyatomic molecules are necessarily

approximate. But through the development of hierarchies of approximate
treatments of the electron correlation problem, accuracy can be systematically improved. This book explores several attempts to apply the successful
methods of molecular electronic structure theory to condensed-phase systems, and in particular to molecular liquids and crystalline solids.
The wavefunction-based methods described in this book all begin with
a mean-field calculation to produce the Hartree–Fock energy. Relativistic effects are neglected and the Born–Oppenheimer approximation is assumed. The remaining part of the electronic energy arises from electron
correlation. Neither of these terms is easy to compute in periodic boundary conditions.
The Hartree–Fock theory for crystalline solids has a distinguished history, starting with expansion of the molecular orbitals as a linear combination of (Gaussian-type) atomic orbitals [1], leading, for example, to
the development of the CRYSTAL code [2]. CRYSTAL is perhaps the most extensively tested implementation of periodic Hartree–Fock theory, and the
accuracy of the whole approach has recently been greatly extended by the
development of periodic second-order Møller–Plesset perturbation theory
(MP2) in the CRYSCOR collaboration (see [3] and Chapter 2). This allows for
accurate, correlated treatments of complex materials, and high computational efficiency is achieved through a combination of density fitting and
local treatments of electron correlation.
Periodic Hartree–Fock using Gaussian-type orbitals has also been implemented by the Scuseria group in the GAUSSIAN electronic structure package [4] (a recent paper on the periodic Hartree–Fock implementation can be
found in [5]), and they too have developed periodic MP2 methods, based
on an atomic-orbital-driven Laplace-transform formalism of the theory
(see Chapter 1 and references therein). This has recently been further accelerated through the introduction of density fitting (or resolution of the
identity) techniques, as described in Chapter 1.
The alternative approach for periodic Hartree–Fock theory is to represent the molecular orbitals in a basis set of plane waves. This, in
combination with pseudopotentials for the effective description of core
electrons, has proven extraordinarily successful for periodic density functional theory (DFT), because the Coulomb energy, which is a major challenge for atomic-orbital methods, can be evaluated extremely easily. There
are implementations of various flavors of the approach in various codes,
including VASP [6], CASTEP [7], PWSCF [8] and CP2K [9].
ix


x

Preface

As part of an attempt in Bristol and University College London to
characterize a simple crystalline solid—lithium hydride—as accurately as
possible [10, 11], Gillan et al. invested very considerable effort in determining accurate Hartree–Fock energies for the crystal [12]. This proved
extraordinarily difficult, and it was found that converging atomic-orbitalbased Hartree–Fock calculations to high accuracy was extremely difficult,
because linear dependence problems made it impossible to make the basis
set sufficiently flexible.∗ Instead, the approach used was based on planewave Hartree–Fock calculations with corrections to remove the effect of
˚
the pseudopotentials. This yielded a static cohesive energy at a = 4.084 A
of −131.95 mE h [12]. This test system has subsequently been studied in
three groups, producing Hartree–Fock results with an amazing degree of
agreement (see Section 4.3).
The wavefunction-based treatment of electron correlation for crystalline solids is also an area of very considerable activity. Various approaches have been devised that attack the problem directly: examples
include the periodic atomic-orbital-based MP2 method developed in the
Scuseria group (see Chapter 1); the local and density-fitted MP2 approach
of the Regensburg and Torino groups (see Chapter 2); and the plane-wavebased MP2 code developed in VASP [13]. In fact, in VASP, methods beyond
MP2 are being developed including the random-phase approximation [14]
and even coupled-cluster theory. Hirata describes work in his group on
periodic electronic structure theory using localized and crystal orbital approaches in Chapter 6.
An alternative approach is to treat electron correlation through considering finite clusters. This is the principle at the heart of both the hierarchical
scheme (Chapter 4) and of the various approaches inspired by the manybody expansion (Chapters 3 and 5–7). In the hierarchical scheme the surface
effects that dominate the properties of small clusters are carefully removed
to reveal properties that accurately reflect the bulk solid (Chapter 4). In the
incremental scheme, one of the oldest and best tested approaches for the
wavefunction-based treatment of electron correlation in solids, a periodic
Hartree–Fock calculation is followed by a many-body expansion of the
correlation energy, where the individual units of the expansion are either
atoms or other domains of localized molecular orbitals (Chapter 3).
In the work of Dahlke Speetzen et al. (Chapter 5), the many-body
expansion of the energy of a molecular cluster is made more rapidly
convergent through embedding lower-order contributions in suitable point

charge representations of the remaining molecules. These authors also
explore the feasibility of applying their methodology to Monte Carlo
∗

It is only fair to note that this problem can be avoided and highly converged orbital-based
Hartree–Fock theory is certainly possible—see, for example, [5].


Preface

xi

simulations of bulk molecular liquids. Hirata describes another many-body
approach, aimed at periodic systems, in which the fragments are embedded
in an electrostatic representation of the remainder of the system, which is
self-consistently optimized (Chapter 6). Through development of analytic
derivatives for this scheme, Hirata has been able to compute optimized
structures, phonon dispersion curves, and Raman spectra for extended
systems, and an overview of this work is presented here. Finally, O’Neill
et al. describe a many-body expansion technique aimed directly at the
simulation of molecular liquids, and present MP2-level radial distribution
functions for liquid water (Chapter 7).
Overall, a trend is emerging: where previously Hartree–Fock and DFT
calculations (and perhaps quantum Monte Carlo) were the only feasible
options for treating the electronic structure of condensed-phase systems,
it is now possible to treat crystals with MP2 and coupled-cluster theory,
and it is becoming possible to simulate liquids using wavefunction-based
electronic structure theory. The chapters gathered in this volume cover
a wide range of exciting and novel approaches for theoretical treatment
of solids and liquids, and constitute some of the first steps toward accurate, and systematically improvable, quantum chemistry for condensed

phases.
Frederick R. Manby
Centre for Computational Chemistry, School of Chemistry
University of Bristol, Bristol, U.K.

References
[1] M. Caus`a, R. Dovesi, C. Pisani, and C. Roetti. Electronic structure and stability
of different crystal phases of magnesium oxide. Phys. Rev. B 33, 1308 (1986).
[2] C. Pisani, R. Dovesi, C. Roetti, M. Caus`a, R. Orlando, S. Casassa, and V. R.
Saunders. CRYSTAL and EMBED, two computational tools for the ab initio
study of electronic properties of crystals. Int. J. Quantum Chem. 77, 1032 (2000).
[3] L. Maschio, D. Usvyat, F. R. Manby, S. Cassassa, C. Pisani, and M. Schutz.
¨ Fast
local-MP2 method with density-fitting for crystals. I. Theory. Phys. Rev. B 76,
075101 (2007).
[4] M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb,
J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson,
H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino,
G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J.
Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven,
Montgomery, Jr., J. A., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd,
E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K.
Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N.
Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J.
Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi,


xii

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]
[13]

[14]

Preface
C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G.
A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas,
J. B. Foresman, J. V. Ortiz, J. Cioslowski, and D. J. Fox. GAUSSIAN 09 Revision
A.1, (Gaussian Inc., Wallingford CT, 2009).
J. Paier, C. V. Diaconu, G. E. Scuseria, M. Guidon, J. VandeVondele, and
J. Hutter. Accurate Hartree–Fock energy of extended systems using large
Gaussian basis sets. Phys. Rev. B 80, 174114 (2009).
J. Paier, R. Hirschl, M. Marsman, and G. Kresse. The Perdew–Burke–Ernzerhof
exchange-correlation functional applied to the G2-1 test set using a planewave basis set. J. Chem. Phys. 122, 234102 (2005).
S. J. Clark, M. D. Segall, C. J. Pickard, P. J. Hasnip, M. I. J. Probert, K. Refson,
and M. C. Payne. First principles methods using CASTEP. Z. Kristallogr. 220,

567 (2005).
S. Scandolo, P. Giannozzi, C. Cavazzoni, S. de Gironcoli, A. Pasquarello,
and S. Baroni. First-principles codes for computational crystallography in the
Quantum-ESPRESSO package. Z. Kristallogr. 220, 574 (2005).
M. Guidon, J. Hutter, and J. VandeVondele. Robust periodic Hartree–Fock
exchange for large-scale simulations using Gaussian basis sets. J. Chem. Theo.
Comp. 5, 3010 (2009).
F. R. Manby, D. Alf`e, and M. J. Gillan. Extension of molecular electronic structure methods to the solid state: computation of the cohesive energy of lithium
hydride. Phys. Chem. Chem. Phys. 8, 5178 (2006).
S. J. Nolan, M. J. Gillan, D. Alf`e, N. L. Allan, and F. R. Manby. Comparison of
the incremental and hierarchical methods for crystalline neon. Phys. Rev. B 80,
165109 (2009).
M. J. Gillan, F. R. Manby, D. Alf`e, and S. de Gironcoli. High-precision calculation of Hartree–Fock energy of crystals. J. Comput. Chem. 29, 2098 (2008).
M. Marsman, A. Gruneis,
¨
J. Paier, and G. Kresse. Second-order Møller–Plesset
perturbation theory applied to extended systems. I. Within the projectoraugmented-wave formalism using a plane wave basis set. J. Chem. Phys. 130,
184103 (2009).
J. Harl and G. Kresse. Accurate bulk properties from approximate many-body
techniques. Phys. Rev. Lett. 103, 056401 (2009).


Editor
Frederick R. Manby is a Reader in the
Centre for Computational Chemistry in the
School of Chemistry at the University of
Bristol, and was previously a Royal Society University Research Fellow. His research has focused on two main areas:
first, on development of efficient and accurate electronic structure methods for large
molecules. Second, he has worked on accurate treatment of condensed-phase systems,
including electron correlation in crystalline

solids, and on application of wavefunctionbased electronic structure theories for
molecular liquids, particularly water. He
has been awarded the Annual Medal of
the International Academy of Quantum
Molecular Sciences (2007) and the Marlow Medal of the Royal Society of
Chemistry (2006) for his research in molecular electronic structure theory.

xiii



Contributors
Neil L. Allan
Centre for Computational
Chemistry
School of Chemistry
University of Bristol
Bristol, U.K.

Migen Halo
Dipartimento di Chimica
and Centre of Excellence
Nanostructured Interfaces
and Surfaces
Universit`a di Torino
Torino, Italy

Simon Binnie
London Centre for
Nanotechnology

Department of Physics
and Astronomy
University College London
London, U.K.

So Hirata
Quantum Theory Project
University of Florida
Gainesville, Florida

Peter Bygrave
Centre for Computational
Chemistry
School of Chemistry
University of Bristol
Bristol, U.K.

Silvia Casassa
Dipartimento di Chimica
and Centre of Excellence
Nanostructured Interfaces
and Surfaces
Universit`a di Torino
Torino, Italy

Michael J. Gillan
Department of Physics
and Astronomy
University College London
London, U.K.

Artur F. Izmaylov
Department of Chemistry
Yale University
New Haven, Connecticut
Murat Ke¸celi
Quantum Theory Project
and Center for Macromolecular
Science and Engineering
Department of Chemistry and
Department of Physics
University of Florida
Gainesville, Florida
Hannah R. Leverentz
Department of Chemistry
and Supercomputing
Institute
University of Minnesota
Minneapolis, Minnesota
Hai Lin
Chemistry Department
University of Colorado Denver
Denver, Colorado
xv


xvi
Marco Lorenz
Institute for Physical and
Theoretical Chemistry

Universit¨at Regensburg
Regensburg, Germany
Frederick R. Manby
Centre for Computational
Chemistry
School of Chemistry
University of Bristol
Bristol, U.K.
Lorenzo Maschio
Dipartimento di Chimica
and Centre of Excellence
Nanostructured Interfaces
and Surfaces
Universit`a di Torino
Torino, Italy
Stephen Nolan
Centre for Computational
Chemistry
School of Chemistry
University of Bristol
Bristol, U.K.
Darragh P. O’Neill
Centre for Computational
Chemistry
School of Chemistry
University of Bristol
Bristol, U.K.
Beate Paulus
Institut fur
¨ Chemie und Biochemie

Freie Universit¨at Berlin
Berlin, Germany

Contributors
Cesare Pisani
Dipartimento di Chimica
and Centre of Excellence
Nanostructured Interfaces
and Surfaces
Universit`a di Torino
Torino, Italy
Martin Schutz
¨
Institute for Physical and
Theoretical Chemistry
Universit¨at Regensburg
Regensburg, Germany
Gustavo E. Scuseria
Department of Chemistry
Rice University
Houston, Texas
Tomomi Shimazaki
Fracture and Reliability Research
Institute
Graduate School of Engineering
Tohoku University
Sendai, Japan
Olaseni Sode
Quantum Theory Project
University of Florida

Gainesville, Florida
Erin Dahlke Speetzen
Chemistry Program, Division
of Life and Molecular Sciences
Loras College
Dubuque, Iowa
Hermann Stoll
Institut fur
¨ Theoretische Chemie
Universit¨at Stuttgart
Stuttgart, Germany


Contributors
Donald G. Truhlar
Department of Chemistry
and Supercomputing Institute
University of Minnesota
Minneapolis, Minnesota

xvii
Denis Usvyat
Institute for Physical and
Theoretical Chemistry
Universit¨at Regensburg
Regensburg, Germany


chapter one

Laplace transform second-order
Møller–Plesset methods in the
atomic orbital basis for periodic
systems
Artur F. Izmaylov and Gustavo E. Scuseria
Contents
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Implementation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 RI basis extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.2 Basis pair screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.3 Distance screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.4 Laplace quadratures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.5 Relation between quadrature points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.6 Transformation and contraction algorithms . . . . . . . . . . . . . . . . . . . .14
1.3.7 Lattice summations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16
1.3.8 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16
1.4 Benchmark calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
1.4.1 RI approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
1.4.2 AO-LT-MP2 applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24

1.1 Introduction
The electron correlation energy is much smaller than the Hartree–Fock
(HF) energy. However, it is of crucial importance for modeling the electronic structure and properties of molecules and solids. The most popular
approaches for including electron correlation are density functional theory
(DFT) and wavefunction methods. DFT usually yields a very good value
in terms of accuracy over computational cost. Unfortunately, there is no

1


2

Accurate condensed-phase quantum chemistry

straightforward path in DFT to get “the right answer for the right reason.”
The latter should be interpreted as a series of well-controlled approximations leading to the exact answer. Traditional semilocal DFT also has problems accurately describing dispersion interactions and transition states.
On the other hand, wavefunction methods do yield a straightforward and
systematic way of improving accuracy, although their computational cost
is usually much higher than that of DFT.
The simplest wavefunction approach to the electron correlation problem is second-order Møller–Plesset perturbation theory (MP2). MP2 radically improves upon HF for dispersion interactions [1], barrier heights [2],
and nuclear magnetic resonance shifts [3] in molecules, and band gaps and
equilibrium geometries in periodic systems [4, 5].
The formal scaling of MP2 in traditional, delocalized, canonical orbital
bases is O(N5 ), where N is a parameter proportional to the system size [6].
This steep computational cost can be drastically reduced by using local
MP2 method formulations [7–10] (see also Chapter 2), or the atomic orbital Laplace transformed MP2 method (AO-LT-MP2) [11, 12]. While the
former approach uses localized orbitals, the latter exploits the natural locality of atomic orbitals. Both of these formulations provide asymptotic
O(N) computational scaling. The latter has been generalized for periodic
systems [13, 14]. Computational cost is not only determined by scaling but
also by prefactors. The main computational bottleneck in MP2 linear scaling methods is the transformation of two-electron integrals. Two-electron
integrals are essentially four-center terms whose evaluation and transformation have O(N4 ) and O(N5 ) complexity, respectively.
Integral generation can be reduced to quadratic scaling using Cauchy–
Schwarz screening [15], and even to linear scaling if multipole-momentbased thresholding is applied [12, 16–18]. The application of screening
protocols in the local MP2 and AO-LT-MP2 methods reduces the integral transformation step from O(N5 ) complexity to O(N). However, the
prefactor is still large, and to efficiently exploit the local nature of correlation (nearsightedness principle), the system under consideration must
be fairly large. In order to obtain an even smaller prefactor, resolution of

the identity (RI) or density-fitting procedures may be introduced. These
are robust alternatives substituting a pair of basis functions in the bra
or ket part of a two-electron integral by a single fitting function [19].
Application of the RI technique to the MP2 formulation leads to an energy expression with three-center, two-electron integrals. This reduces
the complexity of the integral generation and transformation by one
order of magnitude. Although the RI procedure itself has O(N3 ) scaling, its prefactor is very small. The RI expansion has been introduced
into local MP2 and AO-LT-MP2 procedures for systems with periodic
boundary conditions (PBC). These techniques significantly accelerate the


Chapter one:

Laplace transform second-order Møller–Plesset methods

3

computational speed with only a minor loss of accuracy [20–22]; see also
Chapter 2.
In this chapter we review the AO-LT-MP2 and RI-AO-LT-MP2 methods
with a special emphasis on algorithmic features that are responsible for the
computational efficiency of these methods. A comparative assessment of
two methods and some illustrative examples of band gap calculations will
be given at the end.

1.2 Method
Our consideration will be restricted to the closed-shell case because its generalization to the open-shell case is quite straightforward. In this chapter
we use Gaussian atomic orbitals (AOs)
µ p (r) = (x − Rx )l (y − Ry )m (z − Rz )n × exp [−η(r − R − p)2 ],

(1.1)

where (l, m, n) are integers determining the orbital angular momentum,
η is the orbital exponent, and R = (Rx , Ry , Rz ) are the coordinates of the
AO center in the unit cell p. In periodic case, HF self-consistent field (SCF)
crystal orbitals (CO) are linear combinations of AOs that satisfy the Bloch
theorem [23]
jk (r) = Nc

−1/2

Nc

N0

µu (r)C(k)µj e iuk ,

(1.2)

u=0 µ=1

where C(k)µj are CO coefficients, N0 is the number of AOs per unit cell,
and Nc is the number of unit cells. Throughout this chapter we use Greek
letters for AOs, Roman letters i, j, . . . for occupied and a , b, . . . for virtual
COs, K , L , . . . and p, q , . . . for the RI basis set and translational vectors.
Using HF COs and the Mulliken integral notation
(i k1 a k3 | jk2 b k4 ) =

dr1

dr2

i k∗1 (r1 )a k3 (r1 ) jk∗2 (r2 )b k4 (r2 )
,
|r1 − r2 |

(1.3)

the MP2 correlation energy per unit cell is
E MP2 
1
= Re 4
Vk

dk1−4
i j,a b


(ik1 a k3 | jk2 b k4 )[2(i k1 a k3 | jk2 b k4 ) − (i k1 b k4 | jk2 a k3 )]∗ 
, (1.4)
i (k1 ) + j (k2 ) − a (k3 ) − b (k4 )

where (k) is an HF orbital energy, and Vk is the volume of the Brillouin
zone. The MP2 band gap expression can be derived by considering the


4

Accurate condensed-phase quantum chemistry

second-order self-energy correction to the g th HF orbital energy [4, 24, 25]

MP2
(k)
g

=

HF
g (k)

+ U(g, k) + V(g, k),

i,a b

(i k1 a k3 |gk b k2 )[2(i k1 a k3 |gk b k2 ) − (i k1 bk2 |gk a k3 )]∗
,
i (k1 ) + g (k) − a (k3 ) − b (k2 )

(1.5)

where
U(g, k) = Re

1
Vk3

dk1−3

(1.6)
V(g, k) = Re

1
Vk3

dk1−3
i,a b

(i k1 a k3 | jk2 gk )[2(i k1 a k3 | jk2 gk ) − (i k1 gk | jk2 a k3 )]∗
.
i (k1 ) + j (k2 ) − a (k3 ) − g (k)
(1.7)

Then the MP2 correction to the HF direct fundamental band gap is the
difference between self-energy corrections to the highest occupied CO
(HOCO) and the lowest unoccupied CO (LUCO)
E gMP2 = U(HOCO, kmin ) − U(LUCO, kmin )
+ V(HOCO, kmin ) − V(LUCO, kmin ),

(1.8)

where k-point kmin minimizes the total band gap. The fundamental gap is
an energy difference between the electron attachment and detachment processes, and it must not be confused with the optical gap, which represents
the lowest electronic excitation [26].
In addition to computational difficulties related to the delocalized character of canonical COs in Equations (1.4), (1.6), (1.7), an entanglement of
different k vectors in orbital energy denominators requires a computationally expensive multidimensional k-integration. A simple and elegant way
to decouple different k vectors is to apply the Laplace transform to the
energy denominators [11, 27]
1
i (k1 ) + j (k2 ) −

a (k3 )

−

b (k4 )

∞

=−

dt e [

i (k1 )+ j (k2 )]t

e −[

a (k3 )+ b (k4 )]t

.

0

(1.9)
This identity is valid only when the denominator preserves its sign for all
k vectors. Thus, it can be used within the MP2 method which is applicable only to the systems where i (k1 ) + j (k2 ) < a (k3 ) + b (k4 ). After an
appropriate discretization of the Laplace integral [28]
∞
0

dt e [

i (k1 )+ j (k2 )]t

e −[

Nt
a (k3 )+ b (k4 )]t

→

wt e [

i (k1 )+ j (k2 )]t

e −[

a (k3 )+ b (k4 )]t

,

t=1

(1.10)


Chapter one:

Laplace transform second-order Møller–Plesset methods

5

we can rewrite the MP2 energy and band gap correction (Equations [1.4]
and [1.8]) in the AO form
Nt

ν σ

E MP2 =

Tµ0pλrs (t)[2(µ0 ν p |λr σs ) − (µ0 σs |λr ν p )]
t=1 µνλσ, pr s
Nt

ν σ

=

σν

(µ0 ν p |λr σs ) 2Tµ0pλrs (t) − Tµ0sλrp (t) ,

(1.11)

t=1 µνλσ, pr s

and
Nt

ν σ

E gMP2 =

G µp0 λsr (t)[2(µ0 ν p |λr σs ) − (µ0 σs |λr ν p )]
t=1 µνλσ, pr s
Nt

ν σ

=

σν

(µ0 ν p |λr σs ) 2G µp0 λsr (t) − G µs0 λpr (t) .

(1.12)

t=1 µνλσ, pr s

The tensors T and G in Equations (1.11) and (1.12) are transformed
Coulomb two-electron integrals
ν σ

Tµ0pλrs (t) =
γ δκτ,q uvw

Xµt 0 γq Yνtp δu (γq δu |κv τw )Xλt r κv Yσts τw

= (µ0 ν p |λq σs ),
ν σ
G µp0 λsr (t) =
γ δκτ,q uvw

(1.13)

Wµt 0 γq Yνtp δu

+

Xµt 0 γq

Zνt p δu

(γq δu |κv τw )Xλt r κv Yσts τw ,

(1.14)

where matrices X, Y, W, and Z have a form of weighted densities
1/4

Xµt p γs =

wt
Vk

C(k)∗µj e −

dk

wt
Vk

C(k)∗µa e

dk
C(k)∗µg e −

g (k)t

1/4

C(k)∗µg e

(k)t

Zµt p γs = wt

(1.15)

a (k)t

C(k)γ a e ik(p−s) ,

(1.16)

a

1/4

Wµt p γs = wt

C(k)γ j e ik(p−s) ,

j

1/4

Yµt p γs =

j (k)t

g

C(k)γ g − C(k)∗µg e −

C(k)γ g − C(k)∗µg e

g

(k)t

g (k)t

C(k)γ g e ik(p−s) ,

C(k)γ g e ik(p−s) ,

(1.17)
(1.18)

with g and g , respectively, HOCO and LUCO. Comparison of Equations (1.11) and (1.12) for the MP2 energy and band gap correction to the
canonical reciprocal-space Equations (1.4), (1.6), and (1.7) reveals that the

multidimensional k-integration has been reduced to a series of independent Fourier transforms (Equations [1.15]–[1.18]). The evaluation of the X,


6

Accurate condensed-phase quantum chemistry

Y, W, and Z matrices is the only part of the AO-LT-MP2 calculation that
does depend on the number of k-points (Nk ) employed in the discretization
of the Brillouin zone [29]. Therefore, the computational cost of the AO-LTMP2 method is essentially Nk -independent, because the CPU time for the
X, Y, W, and Z construction is negligible with respect to the total CPU time
of the AO-LT-MP2 calculation.
Generation: (µ0 ν p |λr σs ) O(Nc3 N04 )
Generation: (µ0 ν p ||λr σs ) = 2(µ0 ν p |λr σs ) − (µ0 σs |λr ν p ) O(Nc3 N04 )
Loop over Laplace points t = 1, Nt
Generation: (γq δu |κv τw ) [O(N4 )]
1st transformation: (µ0 δu |κv τw ) =
Xµt 0 γq (γq δu |κv τw ) [O(N0 N4 )]
Yνtq δu (µ0 δu |κv τw ) [O(N0 N4 )]
2nd transformation: (µ0 νq |κv τw ) =
Xλt r κv (µ0 νq |κv τw ) [O(N0 N4 )]
3rd transformation: (µ0 νq |λr τw ) =
4th transformation: (µ0 νq |λr σs ) =
Yσts τw (µ0 νq |λr τw ) [O(N0 N4 )]
Contraction: e t = (µ0 ν p |λq σs )(µ0 ν p ||λq σs ) [O(N0 N3 )]
End loop over t

Scheme 1.1

The general flow of the AO-LT-MP2 algorithm is presented in

Scheme 1.1, where the formal complexity of each step is presented in
square brackets, and N is the product Nc N0 . The limiting steps of the
AO-LT-MP2 formulation are integral transformations (Equations [1.13] and
[1.14]); their formal scaling is O(N0 N4 ). In order to improve upon the formal scaling of each step, various integral screening protocols were added
to Scheme 1.1 [13]. Besides screening of two-electron integrals, which will
be considered in detail later, we also can employ the RI approximation to
two-electron integrals
(µ0 ν p |K s )A−1
(L u |λq σr ),
K L

(µ0 ν p |λq σr ) ≈

s

u

(1.19)

Ks , L u

AK s L u =

dr2

dr1

K s (r1 )L u (r2 )
.
|r1 − r2 |

(1.20)

To obtain a symmetric representation, the matrix A is decomposed and its
parts are used for a transition to the orthonormal RI basis {K , L , . . . }
−1/2
−1/2
AM L (L v |λq σr )
s Mu
u v

(µ0 ν p |K s )A−1
(L u |λq σr ) =
K L
s

(µ0 ν p |K )AK

u

K , L,su

K , L , M,suv

(1.21)
(µ0 ν p |M)(M|λq σr ).

=

(1.22)

M

The orthonormal RI basis functions do not have the unit cell subscripts
because of their delocalized character. After introducing the RI expansion


Chapter one:

Laplace transform second-order Møller–Plesset methods

7

in the transformed integrals we obtain the RI-AO-LT-MP2 analog of Equation (1.11) for the MP2 energy correction
Nt

E MP2 =

(e tSS − 2e tOS ),

(1.23)

t=1

e tOS =

(µ0 ν p |K )(K |λq σs )(µ0 ν p |L)(L|λq σs ),

(1.24)

(µ0 ν p |K )(K |λq σs )(µ0 σs |L)(L|λq ν p ),

(1.25)

µνλσ, pq s, K L

e tSS =
µνλσ, pq s, K L

where OS and SS are the opposite-spin and same-spin terms [30]. The main
steps of the RI-AO-LT-MP2 algorithm are presented in Scheme 1.2, where
NRI is the number of AOs in the RI basis for the whole system. RI
bases optimized for MP2 energy calculations usually contain from four
to five times more basis functions per unit cell than corresponding regular
bases [31]. Scheme 1.2 shows that in the RI-AO-LT-MP2 method the timelimiting step constitutes the contraction of two-electron integrals rather
than their transformations.
−1/2
s Lv

Generation: AK

3
[O(NRI
)]

Generation: (µ p σq |K s ) [O(NRI N2 )]
−1/2
2
N2 )]
0th transformation: (µ p νq |M) = (µ p νq |K r )AK M [O(NRI

r s
Loop over Laplace points t = 1, Nt
1st transformation: (µ p δu |K ) =
Xµt p γv (γv δu |K ) [O(NRI N3 )]
Yνtq δu (µ p δu |K ) [O(NRI N3 )]
2nd transformation: (µ p νq |K ) =
2
OS contraction: e tOS = (µ0 ν p |K )(K |λq σs )(µ0 ν p |L)(L|λq σs ) [O(NRI
N2 )]
SS
SS contraction: e t = (µ0 ν p |K )(K |λq σs )(µ0 σs |L)(L|λq ν p ) [O(NRI N3 N0 )]
End loop over t,

Scheme 1.2

1.3 Implementation details
Our discussion will be oriented toward the RI-AO-LT-MP2 method, because most of the algorithmic features of the AO-LT-MP2 method were
implemented in the version including RI. For the sake of simplicity, cell and
Laplace point indices will be omitted in cases where notation is obvious.

1.3.1

RI basis extension

In order to proceed along Scheme 1.2 we need to decide the extent to
which the RI basis should be replicated. In addition, there is a more fundamental problem of a divergence of the Coulomb metric RI scheme with