Linear scaling techniques in computational chemistry and physics methods and applications

The author reviews the AdditiveFuzzy Density Fragmentation AFDF Principle and the two, related linear scalingapproaches based on it: the MEDLA, Molecular Electron Density Loge or LegoAss

CHALLENGES AND ADVANCES IN

COMPUTATIONAL CHEMISTRY AND PHYSICS

Volume 13

Series Editor:

JERZY LESZCZYNSKI

Department of Chemistry, Jackson State University, U.S.A.

For further volumes:

http://www.springer.com/series/6918

Linear-Scaling Techniques

in Computational Chemistry and Physics

Methods and Applications

National Hellenic Research Foundation

48 Vas Constantinou Ave

Athens 116 35Greecempapad@eie.grProf Dr Paul G Mezey

Department of Chemistry and Department

of Physics and Physical Oceanography

Memorial University of Newfoundland

283 Prince Philip Drive

1400 Lynch StreetJackson, MS 39217USA

jerzy@icnanotox.org

ISBN 978-90-481-2852-5 e-ISBN 978-90-481-2853-2

DOI 10.1007/978-90-481-2853-2

Springer Dordrecht Heidelberg London New York

Library of Congress Control Number: 2011922660

© Springer Science+Business Media B.V 2011

No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose

of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

Computational chemistry methods have become increasingly important in recentyears, as manifested by their rapidly extending applications in a large number

of diverse fields (e.g computations of molecular structure, properties, the design

of pharmaceutical drugs and novel materials, etc) In part as a result of thisgeneral trend, the size of the systems which can be studied has also increased,generating even further needs for size increases, since larger molecular systemsshow interesting phenomena, important in modern biochemistry, biotechnology, andnanotechnology Thus, it is of great importance to apply and further develop com-putational methods which provide physically sound answers for large molecules at

a reasonable computational cost An important variety of such approaches is sented by the linear scaling techniques, that is, by methods where the computationalcost scales linearly with the size of the system [O(N)] Over the years, satisfactorylinear scaling computational approaches have been developed which are suitable tostudy a variety of molecular problems However, the latest trends also provide hopethat further, substantial breakthrough in this field may be expected, and one mightanticipate developments for which even the early indications have not yet appeared.This book is a collection of chapters which report the state-of-the-art in many of theimportant questions related to the family of linear scaling methods We hope that itmay give motivation and impetus for more rapid developments in the field

repre-Pulay reviews plane-wave (PW) based methods for the computation of theCoulomb interaction, in HF and DFT methods, introduced in order to decrease thescaling The author notes that PW methods have not been fully utilized in quan-tum chemistry, although several groups have shown their advantages The authordiscusses various technical difficulties regarding the applications of PW methodsand compares PW basis sets with atomic basis sets He further comments on ways

to combine both of them in a single algorithm and discusses reported tions and results as well as some of the important problems to be solved in this area(e.g improvement of the efficiency of other major computational tasks to match theperformance of the Coulomb evaluation)

implementa-Nagata et al review the fragment molecular orbital (FMO) method, proposed

in 1999 and used to reduce the scaling of MO theories from N3–N7 to nearlylinear scaling They discuss the implementation of various methods (e.g RI-MP2, DFT, MCSCF) within the framework of the FMO method, the formulation

of FMO-ROHF, the interface of time-dependent DFT (TDDFT) with FMO Theauthors review the implementation of CIS in multilayer FMO as well as the

v

vi Preface

inclusion of perturbative doubles [CIS(D)] and the inclusion of effective tials (e.g model core potentials) due to the environment or inner-shell electronsinto FMO calculations Application of FMO in molecular dynamics simulations,energy decomposition analysis, and property calculations (e.g chemical shifts) arealso discussed

poten-Saebø reviews some linear scaling approaches based on the second-order Plesset (MP2) methods and briefly comments on other, more accurate electroncorrelation techniques He focuses mainly on methods relying on the local cor-relation method introduced by Pulay and Saebø and developed further by otherco-workers The RI-MP2 method is discussed, demonstrating that it is an order

Møller-of magnitude more efficient than MP2 He reviews the RI-LMP2 method, which

is a combination of the density fitting approach (RI) with the local MP2 method,providing linear scaling with the size of the system A new linearly scaling LMP2approach providing essentially identical results to conventional canonical MP2 isdiscussed and applications are presented

Surján and Szabados review perturbative approaches developed to avoid nalization of large one-electron Hamiltonians, taking into account that diagonaliza-tion of matrices scales with the cube of the matrix dimension The first order density

diago-matrix P is obtained from an iterative formula which preserves the trace and the idempotency of P If P is sparse, then the method leads to a linear scaling method.

It is noted that the procedure is useful for geometry optimization or self-consistenttechniques Electron correlation methods based on the Hartree-Fock density matrixare also discussed

Kobayashi and Nakai report on recent developments in the linear-scaling and-conquer (DC) techniques, that is, the density-matrix-based DC self-consistentfield (SCF) and the DC-based post-SCF electron correlation methods, which theyimplemented in the freely available GAMESS-US package It is shown that the DC-based post-SCF calculation achieves near-linear scaling with respect to the systemsize, while the memory and scratch space are hardly dependent on the system size.The performance of the techniques is shown by examples

divide-Mezey reviews the common principles of linear scaling methods as well as thelocality aspects of these techniques Fundamental relations between local andglobal properties of molecules are discussed The author reviews the AdditiveFuzzy Density Fragmentation (AFDF) Principle and the two, related linear scalingapproaches based on it: the MEDLA, Molecular Electron Density Loge (or Lego)Assembler method and the ADMA, Adjustable Density Matrix Assembler method.Mezey notes that the ADMA provides the basis for the Combinatorial QuantumChemistry technique, with a variety of applications (e.g in the pharmaceuticalindustry)

Szekeres and Mezey review the role of molecular fragmentation schemes invarious linear scaling methods with special emphasis on fragmentation based onthe properties of molecular electron densities They discuss various fragmenta-tion schemes, for example, chemically motivated fragment selection, functionalgroups as primary fragments, delocalized fragments, Procrustes fragmentation,

multi-Procrustes fragmentation with trigonometric weighting The authors reviewcomputational techniques for the efficient implementation of the above schemes.Eckard et al discuss approximations used for the separation of short- and long-range interactions in order to facilitate calculations of large systems They focus onfragment-based (FB) techniques, and review the approximations leading to linearscaling In the FB approaches the molecule is divided into two or more parts and theshort- and long-range interactions as well as the interactions between the subsys-tems are calculated employing different methods (embedding schemes) as it is done

in QM/MM approaches They review techniques to solve the border region problem,which arises upon the division of the molecule into subsystems and the result-ing cutting of covalent bonds Many properties (e.g total energies, partial charges,electrostatic potentials, molecular forces, but also NMR chemical shifts) have beenobtained with the aid of the FB methods Using the fragment-based adjustable den-sity matrix assembler (ADMA) method the advantages and disadvantages of thepresented techniques are discussed for some test systems

Gu et al review the linear scaling elongation method for Hartree-Fock andKohn-Sham electronic structure calculations for quasi-one-dimensional systems.Linear scaling is achieved by (i) regional localization of molecular orbitals, and(ii) a two-electron integral cutoff technique combined with fast multipole evalua-tion of non-negligible long-range integrals The authors describe the construction

of regional localized molecular orbitals with the resulting separation into an activeregion and a frozen region They demonstrate that reduction of the variational spacedoes not lead to any significant loss of accuracy Results for test systems (includingpolyglycine and BN nanotubes) are discussed, which show the accuracy and timing

of the elongation method

Rahalkar et al review the Molecular Tailoring Approach (MTA), which belongs

to the Divide-and-Conquer (DC) type methods MTA is a fragment-based linearscaling technique, developed for the ab initio calculations of spatially extended largemolecules The authors discuss procedures for the fragmentation of the moleculeand how to judge the quality of fragments MTA can be used to evaluate thedensity matrix, one-electron properties such as molecular electrostatic potential,molecular electron density, multipole moments of the charge density, the Hessianmatrix, IR and vibrational spectra and accurate energy estimates, to within 1.5 mH(~1 kcal/mol) of the actual one The authors discuss application of MTA to proper-ties of large organic molecules, biomolecules, molecular clusters and systems withcharged centers This method has been incorporated in a local version of GAMESSpackage and has also been interfaced with GAUSSIAN suite of programs

Neese reviews several algorithms for the exact or approximate calculation of theCoulomb and Hartree-Fock exchange parts of the Fock matrix The central thesis ofthis chapter is that for most current quantum chemistry applications, linear scalingtechniques are not needed, however, the author adds, if a really big system (e.g.involving several hundreds of atoms, or with a spatial extent >20–25 Å) must bestudied by quantum chemical methods, then there is no alternative to a linear scalingtechnique As far as the Coulomb part of the Fock/Kohn-Sham matrix is concerned,

viii Preface

various techniques are discussed including analytical approaches, methods based onmultipole approximation and the resolution of identity or Cholesky decomposition.Similarly, algorithms for calculating the exchange term are reviewed (e.g the semi-numerical and the RI-K approximation) The computations have been performed byemploying the ORCA package

Rubensson et al discuss methods to compute electron densities using computerresources that increase linearly with system size They focus on the Hartree-Fockand density functional theories The authors review multipole methods, linearscaling computation of the Hartree-Fock exchange and density functional theoryexchange-correlation matrices, hierarchic representation of sparse matrices, anddensity matrix purification They discuss error control and techniques to avoid theuse of the ad hoc selected parameters and threshold values to reach linear scaling.Benchmark calculations are presented, in order to demonstrate the scaling behaviour

of Kohn-Sham density functional theory calculations performed with the authors’linear scaling program It seems that the error control and the distributed memoryparallelization are currently the most important challenges

Aquilante et al review methods which employ the Cholesky Decomposition(CD) technique A brief introduction to the CD technique is given The authorsdemonstrate that the CD-based approaches may be successfully applied in elec-tronic structure theory The technique, which provides an efficient way of removinglinear dependencies, is shown to be a special type of a resolution-of-identity ordensity-fitting scheme Examples of the Cholesky techniques utilized in variousapplications (e.g in orbital localization, gradient calculations, approximate repre-sentation of two-electron integrals, quartic-scaling MP2) as well as examples ofcalibration of the method with respect to various properties (e.g total energies) arepresented In the authors’opinion the full potential of the Cholesky technique hasnot yet been completely explored

Korona et al discuss local methods which are implemented in MOLPRO tum chemistry package for the description of electron correlation in the groundand electronically excited states of molecules The authors review improvements

quan-in the implementation of the density fittquan-ing method for all electron-repulsion quan-grals It is shown how the linear scaling of CPU time and disc space results fromthe local fitting approximations Extension to open shell systems and the effect ofexplicitly correlated terms is discussed and it is shown that they lead to significantimprovement in accuracy of the local methods They review electron excitations byEOM-CCSD and CC2 theories as well as first and second-order properties withinthe framework of local methods Some applications are reviewed which show theefficiency of the discussed techniques

inte-Authors Panczakiewicz and Anisimov discuss the LocalSCF approach, whichrelies on the variational finite localized molecular orbital (VFL) approximation.VFL gives an approximate variational solution to the Hartree-Fock-Roothaan equa-tions by employing compact molecular orbitals using constrained atomic orbitalexpansion (CMO) A localized solution is attained under gradual release of theexpansion constraints A number of tests have confirmed the agreement of the LocalSCF results with those obtained by using less approximate methods

Niklasson reviews some recursive Fermi operator expansion techniques for thecalculation of the density matrix and its response to perturbations in tight-binding,Hartree-Fock and density functional theory, at zero or finite electronic temperatures.

It is shown that the expansion order increases exponentially with the number of ations and the computational cost scales linearly with the system size for sufficientlylarge sparse matrix representations, due to the recursive formulation Applicationsare presented to demonstrate the efficiency of the methods

iter-Zeller reviews a Green function (GF) linear-scaling technique relying on theKorringa- Kohn-Rostoker (KKR) multiple scattering method for Hohenberg-Kohn-Sham density functional calculations of metallic systems The author shows howlinear scaling is achieved in the framework of this approach The KKR-GF methoddirectly determines the Kohn-Sham Green function by using a reference systemconcept Applications involving metallic systems with thousands of atoms are pre-sented and the exploitation of parallel computers for the applications of the KKR-GFmethod is discussed

We would like to take this opportunity to thank all the authors for devoting theirtime and hard work in enabling us to complete this volume

1 Plane-Wave Based Low-Scaling Electronic Structure Methods

for Molecules 1

Peter Pulay 1.1 Introduction 1

1.2 Calculation of the Coulomb Energy in Plane Wave Basis 5

1.2.1 Technical Difficulties 7

1.3 Plane Wave and Atomic Basis Sets 10

1.3.1 Comparison of PW and Atomic Basis Sets 10

1.3.2 The Best of Both Worlds? 11

1.4 Implementations and Results 11

1.5 Outlook and Perspectives 14

References 15

2 Mathematical Formulation of the Fragment Molecular Orbital Method 17 Takeshi Nagata, Dmitri G Fedorov, and Kazuo Kitaura 2.1 Introduction 17

2.2 Formulation of the Restricted Hartree-Fock Equation 18

2.2.1 Many-Body Expansion 18

2.2.2 Restricted Hartree-Fock Equation for a Fragment 21

2.2.3 Fragment Energy 22

2.2.4 Expression in Terms of Basis Functions 23

2.2.5 Fragmentation at Covalent Bonds 25

2.2.6 Green’s Function 27

2.2.7 Approximations 28

2.3 Second Order Møller-Plesset Perturbation Theory 30

2.3.1 MP2 Implementations 31

2.3.2 Using Resolutions of the Identity in MP2 31

2.4 Coupled-Cluster Theory 32

2.5 Density Functional Theory 32

2.6 Multiconfiguration SCF 33

2.7 Open-Shell Treatment 34

2.8 Multilayer Approach 34

xi

xii Contents

2.9 Excited States 35

2.9.1 Time-Dependent DFT 35

2.9.2 Configuration Interaction 37

2.10 Quantum Monte-Carlo 38

2.11 Energy Gradient 39

2.11.1 Derivatives of the Internal Fragment Energy 39

2.11.2 Differentiation of the Density Matrix 41

2.11.3 Differentiation of the Electrostatic Potential 41

2.11.4 Differentiation of the Approximated Electrostatic Potential Energy 42

2.12 Effective Potential Models 44

2.12.1 Polarizable Continuum Model 44

2.12.2 Effective Fragment Potential 46

2.12.3 Model Core Potential 48

2.13 Scaling 49

2.14 Molecular Dynamics 50

2.15 Energy Decomposition Analyses 51

2.15.1 Pair Interaction Energy Decomposition Analysis 52

2.15.2 Configuration Analysis for Fragment Interaction and Fragment Interaction Analysis Based on Local MP2 52

2.16 Basis Set Superposition Error 53

2.17 Property Calculations 54

2.17.1 Definition of Molecular Orbitals 54

2.17.2 Molecular Electrostatic Potential and Fitted Atomic Charges 55

2.17.3 Nuclear Magnetic Resonance 55

2.17.4 Multipole Moments and Dynamic Polarizabilities 56

2.17.5 Nuclear Wave Function 57

2.17.6 Drug Design 58

2.18 Massively Parallel Computers 59

2.19 Recent Applications 59

2.20 Summary 60

References 60

3 Linear Scaling Second Order Møller Plesset Perturbation Theory 65

Svein Saebø 3.1 Introduction 65

3.2 Orbital Invariant Formulation of Møller Plesset Perturbation Theory 67

3.3 Local Correlation 69

3.3.1 Pair Selection 69

3.3.2 Reduction of the Virtual Space 69

3.4 Recent Linearly Scaling MP2 Methods 71

3.4.1 Full Accuracy Local MP2 71

3.4.2 RI-MP2 Methods 78

3.4.3 Local RI-MP2 80

3.5 Conclusions 80

References 81

4 Perturbative Approximations to Avoid Matrix Diagonalization 83

Péter R Surján and Ágnes Szabados 4.1 Introduction 83

4.2 Perturbative Energy Estimation Using Laplace Transform 84

4.3 Iterative Search for the Density Matrix 89

4.4 Electron Correlation 92

4.4.1 E2[P] Functional 92

4.4.2 The FLMO Approach 93

References 94

5 Divide-and-Conquer Approaches to Quantum Chemistry: Theory and Implementation 97

Masato Kobayashi and Hiromi Nakai 5.1 Introduction: History of Divide-and-Conquer 97

5.2 Theories of Divide-and-Conquer Method 100

5.2.1 DC-HF and DC-DFT Theories 100

5.2.2 DC-Based Correlation Theories 105

5.3 Assessments of Divide-and-Conquer Method 111

5.3.1 Implementation 111

5.3.2 DC SCF 113

5.3.3 DC-Based Post-SCF Correlation Calculation 116

5.4 Conclusions and Perspectives 123

References 125

6 Linear Scaling Methods Using Additive Fuzzy Density Fragmentation 129 Paul G Mezey 6.1 Introduction 129

6.2 Common Principles of Linear Scaling Methods 131

6.3 Locality Aspects of Linear Scaling Methods 131

6.4 Fundamental Relations Between Local and Global Properties of Molecules 132

6.5 A Fuzzy Fragment Approach to Linear Scaling Methods 134

6.6 The Linear Scaling Properties of the Medla and Adma Methods 136

xiv Contents

6.7 Combinatorial Quantum Chemistry Based on Linear

Scaling Fragmentation 141

6.8 Summary 143

References 144

7 Fragmentation Selection Strategies in Linear Scaling Methods 147

Zsolt Szekeres and Paul G Mezey 7.1 Introduction 147

7.2 Chemically Motivated Fragment Selection 148

7.3 Functional Groups as Primary Fragments 151

7.4 Delocalized Fragments 152

7.5 Procrustes Fragmentation 152

7.6 Multi – Procrustes Fragmentation with Trigonometric Weighting 153 7.7 Summary 155

References 155

8 Approximations of Long-Range Interactions in Fragment-Based Quantum Chemical Approaches 157

Simon M Eckard, Andrea Frank, Ionut Onila, and Thomas E Exner 8.1 Introduction 157

8.2 Short-Range and Long-Range Interactions 158

8.3 Fragment-Based Quantum Chemical Approaches 159

8.4 Embedding Schemes 163

8.5 Border Region 164

8.6 Results of Fa-Adma and Gho-Fa-Adma 167

8.7 Conclusion 170

References 170

9 Elongation Method: Towards Linear Scaling for Electronic Structure of Random Polymers and other Quasilinear Materials 175

Feng Long Gu, Bernard Kirtman, and Yuriko Aoki 9.1 Introduction 175

9.2 The Key Steps of the Elongation Method 177

9.2.1 Construction of RLMOs 178

9.2.2 SCF Elongation Step 181

9.3 Tests of the Accuracy of the Elongation Method: Polyglycine and Cationic Cyanine Chains 182

9.4 Integral Evaluation Techniques for Linear Scaling Construction of Fock Matrix 190

9.4.1 Reducing the Number of ERIs 190

9.4.2 Combination of ERI Cutoff with QFMM Evaluation of Remaining Small Integrals 191

9.5 Illustrative Linear Scaling Calculations for the

Elongation Method with ERI Cutoff and QFMM

Evaluation of Remaining Small Integrals 191

9.5.1 Model Linear Water Chain 191

9.5.2 Polyglycine 192

9.5.3 Nanotubes 194

9.6 Summary and Future Prospects 196

References 197

10 Molecular Tailoring: An Art of the Possible for Ab Initio Treatment of Large Molecules and Molecular Clusters 199

Anuja P Rahalkar, Sachin D Yeole, V Ganesh, and Shridhar R Gadre 10.1 Introduction 199

10.2 Computational Details of MTA 206

10.2.1 Outline of Algorithm 206

10.2.2 Fragmentation 207

10.2.3 Assessment of Fragments 209

10.2.4 Cardinality Expressions 211

10.3 Capabilities 212

10.4 Benchmarks and Applications 213

10.4.1 Establishing MTA 213

10.4.2 HF Level 213

10.4.3 MP2 Method 217

10.4.4 DFT Framework 220

10.5 Comment on Scaling of MTA 221

10.6 Concluding Remarks 222

References 224

11 Some Thoughts on the Scope of Linear Scaling Self-Consistent Field Electronic Structure Methods 227

Frank Neese 11.1 Introduction 227

11.2 Linear Scaling Versus Prefactor 230

11.3 Self Consistent Field Algorithms 232

11.3.1 Coulomb Term 233

11.3.2 Exchange Term 246

11.4 Discussion 258

References 259

xvi Contents

12 Methods for Hartree-Fock and Density Functional Theory

Electronic Structure Calculations with Linearly Scaling

Processor Time and Memory Usage 263

Emanuel H Rubensson, Elias Rudberg, and Pawel Salek 12.1 Introduction 263

12.2 The Self-Consistent Field Procedure 264

12.2.1 Overview of a Linearly Scaling Program 265

12.2.2 Erroneous Rotations 268

12.2.3 Controlling Erroneous Rotations 269

12.3 Integral Evaluation 270

12.3.1 Primitive Gaussian Integrals 270

12.3.2 Screening 270

12.3.3 Cauchy-Schwarz Screening 271

12.3.4 The Coulomb and Exchange Matrices 271

12.4 Coulomb Matrix Construction 272

12.4.1 Multipole Approximations 272

12.5 Exchange Matrix Construction 275

12.6 The Exchange-Correlation Matrix 276

12.6.1 Numerical Grids 278

12.6.2 Evaluation of Sparse Exchange-Correlation Potential Matrix 278

12.7 Error Control in Fock and Kohn–Sham Matrix Constructions 280

12.8 Density Matrix Construction 281

12.8.1 Energy Minimization 282

12.8.2 Polynomial Expansions 284

12.8.3 Accuracy 287

12.8.4 Density Matrix Construction in Ergo 289

12.9 Sparse Matrix Representations 289

12.9.1 How to Select Small Matrix Elements for Removal 290

12.9.2 How to Store and Access Only Nonzero Elements 291

12.10 Benchmarks 292

12.11 Concluding Remarks 298

References 298

13 Cholesky Decomposition Techniques in Electronic Structure Theory 301 Francesco Aquilante, Linus Boman, Jonas Boström, Henrik Koch, Roland Lindh, Alfredo Sánchez de Merás, and Thomas Bondo Pedersen 13.1 Introduction 302

13.2 Mathematical Background 303

13.3 Applications 308

13.3.1 Connection Between Density Fitting and Cholesky Decomposition 308

13.3.2 One-Center CD Auxiliary Basis Sets 309

13.3.3 Orbital Localization Using Cholesky Decomposition 313

13.3.4 The LK Algorithm 315

13.3.5 Quartic-Scaling MP2 320

13.3.6 Calculation of Molecular Gradients 322

13.3.7 Method Specific Cholesky Decomposition 323

13.4 Calibration of Accuracy 329

13.4.1 Accuracy of Total Energies 330

13.4.2 Accuracy of Vertical Transition Energies 331

13.4.3 Auxiliary Basis Set Pruning 332

13.5 Implementational Aspects 333

13.6 Outlook and Perspectives 338

13.7 Summary and Conclusions 339

References 341

14 Local Approximations for an Efficient and Accurate Treatment of Electron Correlation and Electron Excitations in Molecules 345

Tatiana Korona, Daniel Kats, Martin Schütz, Thomas B Adler, Yu Liu, and Hans-Joachim Werner 14.1 Introduction 345

14.2 Local Treatment of Electron Correlation 348

14.2.1 Local Approximations in the Electronic Ground State 350 14.2.2 Exploiting Localization in Program Algorithms 357

14.2.3 Perturbative Triple Excitations 364

14.2.4 Open-Shell Local Correlation Methods 365

14.2.5 Explicitly Correlated Local Correlation Methods 367

14.3 Density Fitting 369

14.3.1 Local Density Fitting in LMP2 370

14.3.2 Local Density Fitting in LCCSD(T) 372

14.4 Local Properties of First and Second Order 374

14.4.1 Analytical Energy Gradients for Local Wave Functions 375 14.5 Local Methods for Excited States 377

14.5.1 Local EOM-CCSD 378

14.5.2 Local CC2 Response Theory 381

14.6 Example Applications 388

14.6.1 Equilibrium Structures, Vibrational Frequencies, and Other Molecular Properties 388

14.6.2 Reaction Energies and Conformational Energies 389

14.6.3 QM/MM Calculations of Reaction Barriers in Enzymes 390 14.6.4 Intermolecular Interactions 391

14.6.5 Open-Shell Local Coupled Cluster Calculations 392

14.6.6 Explicitly Correlated Local Coupled Cluster Calculations 397

14.6.7 Excited States 398

14.7 Outlook 401

References 403

xviii Contents

15 The Linear Scaling Semiempirical LocalSCF Method

and the Variational Finite LMO Approximation 409

Artur Panczakiewicz and Victor M Anisimov 15.1 Introduction 410

15.2 Theory 411

15.2.1 Linear Scaling Problem 411

15.2.2 VFL Approximation 413

15.2.3 LocalSCF Method 417

15.2.4 SCF Convergence Criteria 420

15.2.5 Quantum-Mechanical Molecular Dynamics 423

15.3 Validation 424

15.3.1 Linear Scaling 424

15.3.2 Accuracy of the Linear Scaling Algorithm 426

15.3.3 Validation of QM Molecular Dynamics 431

15.4 Concluding Remarks 435

References 436

16 Density Matrix Methods in Linear Scaling Electronic Structure Theory 439 Anders M.N Niklasson 16.1 Introduction 439

16.2 The Eigenvalue Problem 442

16.3 Quantum Locality and Disorder 444

16.4 Fermi Operator Expansion 446

16.4.1 Chebyshev Expansion 447

16.4.2 Green’s Function Expansion 448

16.4.3 Recursive Fermi-Operator Expansion at T e> 0 449

16.4.4 Recursive Fermi-Operator Expansion at T e= 0 by Purification 451

16.4.5 Convergence and Accuracy 459

16.4.6 Iterative Refinement Techniques 463

16.5 Linear Scaling Density Matrix Perturbation Theory 464

16.5.1 Density Matrix Response by Recursion 465

16.5.2 Calculating Response Properties From the n + 1 and 2n + 1 Rules 467

16.5.3 Example 468

16.6 Summary 469

References 469

17 Linear Scaling for Metallic Systems by the Korringa-Kohn-Rostoker Multiple-Scattering Method 475

Rudolf Zeller 17.1 Introduction 475

17.2 Preliminaries 476

17.2.1 Density Functional Theory 476

17.2.2 Linear Scaling Strategies 477

17.2.3 Metallic Systems 479

17.3 The KKR-GF Method 480

17.3.1 Properties of the Green Function 481

17.3.2 Calculation of the Green Function 482

17.3.3 Complex Energy Integration 485

17.3.4 Total Energy and Forces 486

17.3.5 Temperature Error 488

17.4 Linear Scaling in the KKR-GF Method 490

17.4.1 Repulsive Reference System 490

17.4.2 Iterative Solution 493

17.4.3 Green Function Truncation 494

17.4.4 Model Study 495

17.4.5 Scaling Behaviour 498

17.5 Conclusions and Outlook 500

References 503

Index 507

PLANE-WAVE BASED LOW-SCALING ELECTRONIC

STRUCTURE METHODS FOR MOLECULES

PETER PULAY

Department of Chemistry and Biochemistry, University of Arkansas, Fayetteville, AR 72701, USA, e-mail: pulay@uark.edu

Abstract: This paper reviews the use of plane-wave based methods to decrease the scaling of the

most time-consuming part in molecular electronic structure calculations, the Coulomb interaction The separability of the inverse distance operator allows the efficient calcu- lation of the Coulomb potential in momentum space Using the Fast Fourier Transform, this can be converted to the real space in essentially linearly scaling time Plane wave expansions are periodic, and are better suited for infinite periodic systems than for molecules Nevertheless, they can be successfully applied to molecules, and lead to large performance gains The open problems in the field are discussed.

Keywords: Basis sets, Density functional theory, Molecular orbitals, Plane waves, Quadrature

In most routine molecular quantum chemistry calculations, for instance in Fock and density functional (DFT) theories, the dominant computational work isthe evaluation of the electron repulsion energy and its matrix elements This chap-ter focuses on the efficient evaluation of these quantities, in particular the Coulombcomponent, using expansions in plane waves This reduces the steep scaling of theelectron repulsion terms with molecular and basis set size drastically Alternativemethods are discussed in other chapters The main advantage of plane wavebased low-scaling methods over competing methods, for instance the fast multi-pole method [1,2] is that they become efficient for modest-sized molecules already;many alternative methods don’t show significant improvement until large systemsizes However, their infinite periodic nature is not a natural fit with molecules.The overwhelming majority of molecular electronic structure calculations useatomic basis sets, corresponding to the chemists’ notion of a molecule consisting ofatoms In this method, the unknown molecular orbitals (MOs)ϕ iare represented aslinear combinations of atomic-like fixed basis functionsχp

Hartree-1

R Zale´sny et al (eds.), Linear-Scaling Techniques in Computational Chemistry and Physics, 1–16.

DOI 10.1007/978-90-481-2853-2_1,  Springer Science+Business Media B.V 2011 C

2 P Pulay

ϕi (r)=

p

where r is the position of the electron The basis functions are often designated

as atomic orbitals (AOs), although they are usually not genuine atomic orbitals The

unknown coefficients C piare determined by minimizing the total energy In the relativistic case, the ultimate building blocks, the spin-orbitals, can be written as theproducts of an orbital and a spin function, usually simplyα or β The main advan-tage of the atomic basis set representation is its compactness By using the atomicnature of the electron distribution, a few hundred basis functions can adequatelydescribe the MOs of a typical drug-sized molecule This advantage was essentialwhen computer memories were measured in kilobytes, rather than Gigabytes but it

non-is less important now

Solid state physicists often use a diametrically opposite starting point where thenatural basis functions are plane waves (For a general reference on plane wavemethods, see [3].) In the first approximation, the presence of atoms in the elementarycell is neglected To preserve electrical neutrality, the positive charge of the nuclei

is smeared out evenly in space In this “jellium” model, the natural basis functionsare plane waves (PWs), conveniently written in complex form as

using the Euler formula, e ix = cos (x) + isin(x) Here g = (g x ,g y ,g z) is a vector of

integers, and a = 2π/L where L is dimension of the elementary cell For simplicity,

it is assumed here that L is the same in all three spatial dimensions, although in actual

calculations this condition is fulfilled only for cubic space groups For molecules,

a box enclosing essentially the whole electron density replaces the elementary cell;

the dimensions are adjusted to the size of the molecule The factor a ensures that

the plane waves are commensurate with the dimensions of the elementary cell Tolimit the number of plane wave basis functions, the magnitude or the maximum

component of g must be restricted A plane wave expansion can describe any orbital

or electron density to arbitrary accuracy if the upper limit on g is sufficiently high.

However, the number of plane waves required to describe the sharply peaked coreorbitals is huge, and in practice core orbitals and core electron densities cannot beadequately represented by plane waves Even the valence shells of some electroneg-ative atoms, for instance oxygen, are too compact to be easily represented by planewaves The quantity characterizing the cutoff is usually given as the kinetic energy

corresponding to the maximum wave vector, E cutoff = 2π2(g max/L)2Eh(atomic unit

of energy; this quantity is often quoted in Rydberg units, 1 Ry= 0.5 Eh) Unlessspecifically noted, we will use atomic units in this chapter, i.e., distances are mea-

sured in units of the Bohr radius a 0 , and energies in Hartrees, E h A quantity more

useful than g max is the grid density, d = 2g max/L, i.e the number of plane wave

basis functions per bohr

Plane wave based methods became popular in the physics community after theintroduction of the Car-Parrinello direct dynamics method [4], for which plane

wave basis sets are particularly appropriate A number of plane-wave based densityfunctional programs have been developed, mainly for the treatment of solid stateproblems: CASTEP [5], VASP [6], and Quantum Espresso [7] are three representa-tive examples Plane wave based electronic structure programs are most appropriate

to “pure” DFT, i.e., no Hartree-Fock exchange, because including exact Fock) exchange increases the computational work very much In addition, to keepthe number of plane wave basis functions reasonable, the core electrons (and insome cases even the inner valence) have to be represented by alternative means,treating explicitly only the smoother valence electron distribution Core charge den-sities can be replaced by pseudopotentials (effective core potentials), or represented

(Hartree-as frozen cores in an atomic b(Hartree-asis set The latter Augmented Plane Wave (APW)methods played an important role in solid-state physics [8] but the severe approxi-mations that they used, and their solid-state orientation made them largely irrelevantfor chemistry which is a science of small energy differences The modern version

of the augmented plane wave method, the Projector Augmented Wave (PAW) nique of Blöchl [9], employs only the frozen-core approximation, and can be cast

tech-in a form closely analogous to the “ultrasoft” pseudopotentials of Vanderbilt [10],

as shown by Kresse and Joubert [11] Even after eliminating the cores, the number

of plane waves required for an accurate representation of the orbitals is large Theactual number depends on the size and description of the core, and the desired accu-racy but, for high accuracy, can easily exceed a few million This causes difficultieswith the optimization of the wavefunction which scales computationally as the cube

of the basis set size

The periodic nature of the PW basis is appropriate only for 3-dimensional tals Lower dimensional systems: layers, polymer chains, and molecules can betreated by the supercell method, i.e placing the system in a box sufficiently large

crys-to eliminate the interaction between the system and its periodic images However,this leads to inefficiencies because in PW methods, empty space is not free compu-tationally An alternative method, based on the truncation of the Coulomb operator,will be described below We will not be able to review physics-based, solid-stateoriented PW methods here; there are a number of excellent reviews in the literature.Rather, we will concentrate on the application of an auxiliary PW basis for the cal-culation of the Coulomb energy in traditional atomic (in practice Gaussian) basis

set calculations Such methods were first suggested by Lippert et al [12], and have

since been implemented in at least three comprehensive programs: Quickstep [13],see Refs [12,14–19]; PQS [20,21], see Refs [22–27]; and Q-Chem [28], see Refs.[29–32] The first implementation uses the acronyms GPW and GAPW; the lattertwo implementations are known as the Fourier Transform Coulomb (FTC) method

In spite of the promising results published by these groups (see the last section), thepotential of plane-wave based methods has not yet been fully utilized in mainstreamquantum chemistry

The principal attraction of PW methods in quantum chemistry is that they allowthe low-scaling calculation of the electron repulsion energy which is traditionallythe most expensive part of routine calculations The electron repulsion energy of

a determinantal (Hartree-Fock) wavefunction can be conveniently written as the

i |ψi|2, and the integrals over r and r’ run over all space.

A two-electron integral is defined in general as

antisymmetry of the wavefunction required by the Pauli principle

Common (also called “pure”) DFT replaces the exchange energy K (and usually

also the much smaller correlation energy) by a local or semilocal functional of theelectron density, while in “hybrid” DFT a fraction of the Hartree-Fock exchange,

Eq (1-4), is retained This is important in the present context because most scaling methods can be applied readily only to the classical Coulomb term, and arethus largely restricted to pure DFT methods

low-In traditional Hartree-Fock theory, both J and K are calculated by substituting

the expansion of the orbitals by basis functions, Eq (1-1), in Eqs (1-3) and (1-4)

This gives equations that contain 4-index AO integrals (pq |rs), and scale formally with the fourth power of the AO basis set size N This “integral catastrophy” was

the major hurdle preventing the application of quantum theory to realistic molecularsystems in the early phase of quantum chemistry The problem resolved itself par-tially with the dramatic expansion of computer power from 1980 on, because, using

proper thresholding, the basis function products (usually called charge densities) pq and rs become negligible if the AOs p and q, or r and s are distant This means that,

for large systems, the number of non-vanishing charge densities grows linearly andnot quadratically with the size of the system, assuming that the type of the basis set iskept the same However, the number of necessary two-electron integrals still growsquadratically with the molecular size because the Coulomb operator,|r − r|−1, has

long range Although the steep O(N4) scaling of AO-based MO theory naturally

reduces to O(N2) in the limit of large molecules, it still imposes a stiff limit on thesystem size that can be treated with traditional quantum chemistry programs

Of the two components of the electron repulsion energy, Coulomb and exchange,the scaling of exchange is reduced further naturally (at least in insulators) by the

locality of the density matrix [33] As a consequence, Hartree-Fock exchange shouldscale linearly with system size in the asymptotic limit [34–37] In practice, it is dif-ficult to reach the asymptotic scaling regime, even for linear molecules However,hybrid density functional that use only the short-range component of the electronrepulsion, for instance the HSE functional [38] are comparable in accuracy to func-tional that use the full exact exchange, and converge faster with system size Guidon

et al [19] extended the Quickstep program to include short-range Hartree-Fockexchange The exchange terms are calculated in Gaussian basis as in conventionalHartree-Fock theory PQS [20,21] can also use exact exchange but with a significantloss of efficiency

Natural scaling reduction does not help if the basis set is increased while themolecule size is kept constant in AO basis set methods In this case, the compu-tational effort to generate and process the 2-electron integrals increases roughly

as O(N4)

WAVE BASIS

The main advantage of a plane wave basis is that the Coulomb operator is separable

in plane wave basis The Fourier transform of the inverse distance operator is given

by the Fourier integral

r − r−1= (2π2)−1

k−2exp [ik · (r − r)]dk3 (1-6)

where k = |k|, and k is a vector in the reciprocal (momentum) space The integral is

over the full reciprocal space Representing the infinite periodic charge density by aplane wave expansion (see Eq (1-2)),

 

g D(g)g−2exp (iag · r) (1-8)

where we made use of the the orthogonality of plane waves:

 ∞

−∞e

i(k −k)x

g = |g|, and the factor L2comes from a−2= (2π/L)−2 Associated with the

recip-rocal space grid, there is a corresponding real-space grid with the same number

of grid points, and a spacing h = d−1, i.e., the inverse of the reciprocal

(momen-tum) space grid density introduced in the previous section Using the fast Fourier

6 P Pulay

transform (FFT), one can switch from one representation to the other For quantitiesthat can be represented exactly in the plane-wave basis, the two representations areequivalent The efficiency of the method derives from the ability to perform eachoperation in the appropriate representation, i.e., in direct space or momentum space

Thus the Coulomb potential in the discrete momentum space is simply D(g)g−2(cf.

Eq.1-8); its Fourier transform gives the potential V(r) at the real space gridpoints rp

V(r) is used to calculate the Coulomb energy in real space by numerical quadrature

on the real space grid,

whereχpandχqare (atomic) basis functions

Neglecting some technical difficulties which will be addressed below, the essence

of the plane-wave (or Fourier space) calculation of the Coulomb energy is to use thetwo equivalent representation of the potential and the charge density: on a direct-

space grid with a spacing h = d−1(d = 2gmax/L) in a box of size L, and in

the reciprocal space of the integer vectors g where |g|<gmax Efficient tion between the direct and reciprocal space representations, using the fast Fouriertransform (FFT) technique, is critical to the success of the plane wave method An

transforma-N-point FFT (in one dimension) scales as O(NlogN), i.e., the scaling is only slightly

higher than linear (The base of the logarithm is not specified here as it is not vant for the scaling; it can be assumed to be 2 if the number of grid points is a power

rele-of 2.)

Disregarding the complications caused by the presence of compact orbitals andcharge densities, the calculation of the Coulomb energy and matrix consists of thefollowing steps:

(1) Calculate the electron density on the real-space grid

(2) Use Fast Fourier Transformation (FFT) to obtain a plane wave (reciprocalspace) representation of the charge density, Eq (1-7), on the plane wave grid

defined by the reciprocal vectors g

(3) Divide values of the charge density in the reciprocal space by g2to obtain theCoulomb potential

(4) Transform the potential back to real space, Eq (1-8), using FFT

(5) Evaluate the Coulomb energy in real space by quadrature, Eq (1-10)

(6) For each pair of atomic orbitals p and q, integrate their product with the potential

to obtain the matrix elements of the Coulomb operator, needed to form the Fockmatrix, according to Eq (1-11)

The efficiency and scaling of this procedure is determined by three time-criticalsteps: the calculation of the electron density (1), the Fast Fourier Transformation

(FFT) steps (2) and (4), and the evaluation of the matrix elements of the J operator,

step (6) The first step can be carried out in two different ways: calculating theorbital values first, and forming the sum of their squares, or using a density matrixformalism,ρ(r) =pq Dpqχp (r) χq (r) Without thresholding, both have fairly high

scaling, e.g., the second method scales as O(N3) However, if the basis functions are

local, this scaling reduces to linear As mentioned above, FFT scales as O(NlogN)

where N is the number of grid points This is evident for a one-dimensional FFTbut also holds in the two- and three-dimensional case For a given spatial resolution,

N is proportional to the volume of the system (more precisely, the volume of a box

that contains essentially the whole electron density of the system) In the limit of

large systems, the volume and therefore N is roughly proportional to the number of

atoms, and the scaling is only slightly steeper than linear

The numerical integration for the evaluation of the Coulomb energy is obviouslylinear in the number of the grid points The calculation of the Coulomb matrix ele-ments appears at first to have a steeper scaling However, the usual atomic basisfunctions are highly localized, and, for sufficiently large systems, the number ofCoulomb matrix elements exceeding a given threshold is linear in the system size

If properly implemented, the integration effort is independent of the system size,depending only on the spatial extent of the basis functions and the grid density, giv-ing an overall linear scaling However, this limit is reached only for relatively largesystems

1.2.1 Technical Difficulties

1.2.1.1 Divergencies

The recipe given above appears straightforward However, in actual tion, a number of technical problems arise Chief among these is the fact that theCoulomb potential of an infinite periodic charge density diverges if the lower elec-trical moments (up to quadrupole), of the elementary cell are non-zero This is ofcourse always the case if the electronic charge is considered by itself, without thecancelling nuclear charges (A good discussion of the problems with divergence andconditional convergence, which also arise if the dipole moment of the elementarycell is non-zero, is given in [39], and will not be repeated here.) The problem is

implementa-that the g = 0 component of the density in the reciprocal space, D(0), is non-zero,

resulting in division by 0 in Eq (1-8) Simply omitting D(0) is physically equivalent

to the jellium model, i.e to adding a uniform neutralizing positive charge density tothe system which is far from the actual system Calculating the Coulomb potential

in real space can easily avoid singularities, even for charged systems [40] but thisprocedure has quadratic scaling and its accuracy is limited [23]

8 P Pulay

An alternative approach is to consider the total (nuclear + electronic) charge,which has no net charge for neutral systems The problem here is that the self-repulsion energy of pointlike nuclei, Eq (1-10) diverges to infinity In addition, thepresence of sharp singularities in the potential causes numerical errors in the numer-ical quadratures A possible solution [14,41] is to smear out the nuclear charge, forinstance replacing the pointlike nuclei by positive Gaussian charge distributions,calculate the Coulomb energy and its matrix elements with the modified charge dis-tribution, and correct the result for the difference between the smeared-out and realnuclei This method is viable, although somewhat involved, and the sharp nuclearcharge distributions are a source of numerical errors Neither of these methods canhandle ions This is not a problem in solid-state applications, as a solid shouldobviously be electrically neutral, but is a limitation for molecular calculations

A simpler and more general solution to the divergence problem is to modifythe Coulomb potential, eliminating its infinite range which is the source of diver-gence In a purely numerical fashion, this was introduced by Hockney [42, 43]

The Coulomb potential, 1/r, can be simply truncated to zero at r > D The limit D

should be chosen to exceed the maximum distance between non-negligible charges

in the system If the periodic repeating box is large enough so that the minimumdistance between non-vanishing charges in neighboring boxes is larger than D thenthe spurious Coulomb effect of neighboring boxes vanishes This requires a boxroughly twice as large as the original box that contains all the charges The trun-

cated 1/r function has an exact Fourier transform: 4 πk−2[1 − cos (kD)] which differs from the Fourier transform of the infinite-range 1/r function by the fac-

tor in the square bracket To the knowledge of the current author, this was firstdescribed by Pollock and Glosli [44] but it was not widely known or used, and wasrediscovered several times [23,45] An alternative method, introduced by Martynaand Tuckerman [46], uses the Ewald decomposition of the Coulomb potential in

a long-range and a short-range term, and retains only the latter While not exact,this method can be made arbitrarily accurate by choosing proper thresholds Ithas the advantage that there is no discontinuity in the modified Coulomb poten-tial which generate high frequency components in the reciprocal space and slowdown convergence Note that there is no strict need for the existence of an analyti-cal Fourier transform of the modified Coulomb interaction For instance, a function

v(r) = 1/r if r<D1, p(r) if D1 < r < D2, and zero if r > D2, could be used, in

connection with a tabulated numerical Fourier transform The function p(r) could

be chosen as a polynomial that makes the function v(r) and its first, second,

deriva-tives continuous A function like this has no simple analytical Fourier transform but

is rigorously zero beyond D 2and is smooth Its numerical Fourier transform must

be determined only once and can be stored in tabulated form

The concept of truncated Coulomb potential can be extended from dimensional molecules (“clusters” in the physics literature) to one- and two-dimensional systems, i.e polymers (or wires) and layers (or slabs) The Coulombinteraction in these systems must be truncated in two dimensions (for polymers andwires), or one dimension (for layers), and retains its infinite range in the periodicdimensions The analytical derivation of the Fourier transform of these potentials

zero-is tedious but the resulting formulas are quite simple, both for layers [44] and forpolymers The Ewald decomposition method can be similarly generalized for thesecases [47].

1.2.1.2 Compact Charge Densities

Another obvious difficulty with plane wave expansions is that they can representonly smooth charge densities at reasonable computational effort The sharply peakedcore regions must be treated by alternative methods In programs which use planewaves directly as basis functions, this usually means that the cores, and sometimeseven the inner valence regions of the atoms must be represented by pseudopotentials(effective core potentials) Modern pseudopotentials provide accurate representation

of the effect of the core on the valence electrons, and in some cases are preferable

to all-electron treatment because they allow the inclusion of the main relativisticeffects in a non-relativistic program However, they are obviously inappropriate forproperties that are strongly dependent on the core orbitals Transition metals are

a particularly difficult problem, as they have incompletely filled compact d or f

shells which must be included in the valence orbital set but are difficult to treatusing plane waves with moderate cutoffs Electronegative first-row elements (O, F)likewise have compact valence shells This problem can be addressed by two tech-niques: ultrasoft pseudopotentials [48], and the Projector Augmented Wave (PAW)method of Blöchl [9, 49,50] In both methods, fixed atomic orbitals are used todescribe the bulk of the core charge, with plane waves supplying a smooth correc-tion Both methods avoid the tedious matching of orbitals on surfaces separating thecores from the valence region

Methods which retain atomic basis sets and use plane waves only for the tion of the Coulomb terms must similarly divide the charge density into a diffuse and

calcula-a compcalcula-act component [12] The Coulomb potenticalcula-al of the diffuse component ccalcula-an becalculated efficiently using a plane wave expansion; for the core part, alternativetechniques must be used For Gaussians, charge density components, i.e., products

of atomic basis functions, can be classified based on the sum of the exponents ofthe Gaussians which largely determines the compactness of the resulting chargedensity (In some programs, three cases are distinguished: products of two diffusebasis functions, the product of a compact and a diffuse function, and the product oftwo compact functions However, there is no advantage in treating the second andthird cases differently: the product of a compact and a diffuse basis function gives

a compact charge density.) It is worth mentioning that treating compact and diffusebasis functions separately runs contrary to the idea of contracted Gaussian basissets, i.e., using fixed linear combinations of primitive Gaussians as basis functions.This problem can be easily taken care of by switching to a decontracted (primitive)representation Decontracting increases the size of the basis set 1.5–3 times and thusrequires significantly more memory but this has ceased to be a problem on moderncomputers

There are several methods for the calculation of the Coulomb potential nating from the compact core (and semicore) charges, apart from pseudopotentials

origi-10 P Pulay

and the Projector Augmented Wave method, which presume a frozen core densityand are thus not appropriate for phenomena that involve the cores The simplestmethod is to use traditional integral-based algorithms Because of the highly local-ized nature of the core orbitals, only a small fraction of the integrals (a few percent

in a moderately sized molecule) involve core orbitals We have found, however, thatthe effort needed to evaluate core Coulomb contributions by this method is compara-ble to the evaluation of the rest of the Coulomb terms because of the inherently lowefficiency of traditional two-electron integral based algorithms Other alternativesare density fitting (DF), also called resolution of identity, and a multipole expan-sion of the core potential The situation is simplified by the fact that the chargesinvolved are sums of largely spherical atomic-like charge densities This simplifiesboth the expansion of the basis set in an auxiliary basis, and the truncation of themultipole expansion While neither method is linearly scaling, they should reducethe computational effort needed for the core electrons sufficiently to make this partcomputationally insignificant except in huge calculations

1.3.1 Comparison of PW and Atomic Basis Sets

In this section, we compare plane wave basis sets with atomic basis sets As we shallsee, both have significant advantages and disadvantages The main advantage ofatomic basis sets is their compactness: a small set of basis functions can adequatelyrepresent an atom in a molecule Their disadvantages are:

(1) The definitions of basis sets are complex and somewhat arbitrary, leading to aprofusion of competing basis sets, and makes comparing calculations difficult(2) The formulas for the evaluation of the integrals are complex and computation-ally expensive

(3) They are not orthogonal, which may lead to near-linear dependence andnumerical problems

(4) The coupling of the basis function centers to the nuclear positions complicatesthe evaluation of the forces on the nuclei

(5) The Basis Set Superposition Error (BSSE) introduces unphysical attractiveforces between atoms

The advantages of plane wave basis sets are

(1) They are simple and regular, controlled by a single parameter, the cutoff energy

or maximum wave vector

(2) They allow a highly efficient evaluation of the Coulomb potential

(3) They are orthogonal, and free from linear dependencies

(4) They are independent of the nuclear positions, simplifying the calculation offorces on the atoms

(5) They are free of the Basis Set Superposition Error

These advantages are accompanied by significant disadvantages:

(1) The large number of plane waves makes the optimization of the wavefunctionexpensive

(2) The calculation of the exact (Hartree-Fock) exchange is also expensive in a PWbasis

(3) Atomic cores and compact charge densities cannot be represented with areasonably sized plane wave basis, and must use auxiliary functions or pseu-dopotentials

1.3.2 The Best of Both Worlds?

The comparison between plane waves and atomic basis sets suggests that a nation of both may be more efficient than either one alone There are two possibleroutes:

combi-The first is to use a genuine augmented plane wave basis set, i.e., a basis setthat consists of both plane waves and Gaussians Plane waves would describe onlythe diffuse part of the electron cloud, and Gaussians the inner valence and coreorbitals In this method, a limited number of plane waves suffices because onlythe diffuse basis functions are represented by the plane waves Most of the prob-lems with atomic basis sets (overcompleteness and basis set superposition error)are caused by diffuse basis functions, and a combined basis should eliminate bothwhile still remaining reasonably compact A combination as described makes mostsense for accurate large basis set calculations, and there is no reason why it should

be restricted to density functional theory Configuration-based electron correlationmethods should be feasible, as the basis set size remains modest As yet, there is

no general implementation of this method except for initial tests for very smallmolecules [22], although all tools are available, and it appears a worthwhile goal

A simpler alternative, introduced by the Parrinello group [12,14,15] and mented in Quickstep [13], PQS [21] and Q-Chem [28] uses simply an atomicGaussian basis set However, the most expensive part of the calculation, the eval-uation of the Coulomb terms, is carried out in an auxiliary plane wave basis Thisresults in a significant speed-up, particularly for large basis sets However, some ofthe undesirable aspects of atomic basis sets, in particular overcompleteness of largeatomic basis sets and its consequence, near linear dependence, and also basis setsuperposition error, reappear The rest of this chapter will deal exclusively with thissecond method

All three implementations (Quickstep, PQS and Q-Chem) of the GAPW/FTC(Gaussian and Plane Wave or Fourier Transform Coulomb) methods show sig-nificant, in some cases spectacular speed-ups compared to conventional Gaussiannon-hybrid DFT calculations, particularly for large basis sets For instance,

12 P Pulay

VandeVondele et al [13] report a DFT pseudopotential calculation on a oside monophosphate crystal with 280 atoms in the unit cell, using a double zetaplus polarization basis set developed specifically for Quickstep (2,712 basis func-tions) A single optimization cycle (energy plus forces) on a single CPU took only

dinucle-6 min, making such large calculations feasible on a desktop computer Calculationusing over 40,000 basis functions for 1,024 water molecules are reported in[13] Unfortunately, our own attempts to use Quickstep were hindered by SCFconvergence difficulties, even on molecules with large bandgaps

The FTC timings obtained by PQS and Q-Chem are less spectacular but still veryencouraging These programs use a more conservative strategy and try to match thetotal energy of conventional programs to high accuracy, say 10–6 Eh/atom Suchhigh accuracy in the total energy is necessary to convince new users about the relia-bility of the method However, it is not needed for most problems, as shown by theaccuracy tests of VandeVondele et al on molecular geometries [13]

All three implementations show clearly that, in order to take full advantage ofthe large speed-ups in the Coulomb contribution, it is necessary to accelerate allother significant parts of the calculation Both the part of the Coulomb operator that

is calculated in the Gaussian basis, and the evaluation of the exchange-correlationcontribution must be accelerated significantly to match the performance gained bythe improved Coulomb algorithm This is particularly important for smaller basissets where the Coulomb evaluation is less dominant computationally For instance,for a series of diamond-like carbon clusters, Füsti-Molnár and Kong [29] obtain afourfold speed-up for the Coulomb term over the already very efficient (J-engine+Continuous Fast Multipoles) algorithm for a diamond-like carbon cluster of 150atoms, using the 6–311G(d,p) basis set However, the overall speed-up is only 1.71because of the overhead from the calculation of the exchange-correlation (69%

of the total time) and diagonalization (7%) The FTC step itself amounts to only5% of the total calculational time The diagonalization step, or its equivalent (e.g.,pseudodiagonalization [51] or Orbital Transformation [52]) becomes important onlyfor calculations over∼5,000 basis functions Such large systems (several hundredatoms) can be frequently treated by alternative methods, for instance a combination

of quantum mechanical techniques and empirical molecular mechanics (QM/MM)methods As the above example shows, the main overhead steps for moderately largecalculations are the exchange-correlation contribution and the Coulomb contribu-tions that FTC cannot calculate (19% in the above example) The timings obtainedusing PQS and shown below agree with this general picture

Significant effort has been undertaken to improve the computational steps thatare responsible for the non-Coulomb overhead Quickstep [13] uses a sophisticatedmultigrid algorithm for the numerical quadrature The authors of Q-Chem have alsodeveloped new grids for integrating the exchange-correlation term [30–32]

A technical aspect of GAPW/FTC calculations that is worth mentioning is thatcontractions (which typically include basis functions with both high and low expo-nents) interfere with FTC since only the lower exponent (more diffuse) functionscan be treated by FTC Either a special decontracted basis set must be used, or thebasis can be decontracted in the program The second alternative is preferable, asstandard basis sets can be used

Table 1-1 Timings (in minutes) and total energies for aspirine, C9H8O4, on a 2 GHz AMD 246 Opteron processor The column ERI shows timings calculated using the traditional electron repulsion integral algorithm The calculations use slightly decontracted versions of the 6-31G(d,p) and 6-311G(2df, 2pd) basis sets Coulomb, XC and Matrix refer to the calculation of the Coulomb operator and energy, the calculation of the exchange-correlation and miscellaneous (mainly matrix) operations, such as diagonal- ization, pseudodiagonalization and DIIS For FTC calculations, the Coulomb timing includes both the plane wave and the traditional integral time, with the latter dominating for large calculations by about a factor of 3

it contains the whole electron density The results in the Tables are similar to theresults obtained by Q-Chem The plane wave method speeds up the calculation ofthe Coulomb term very much, over a factor of 30 for the largest molecule and largebasis set However, the overall speed-up for taxol is only an order of magnitude forthe large basis set, and less than a factor of 4 for the smaller basis because of theoverhead, mainly from the traditional integral calculation and from the evaluation

of the exchange-correlation contributions

Figure1-1shows the scaling of the evaluation of the Coulomb contributions for

polyalanines, n= 2–15 Although the FTC method provides significant speed-upcompared to a traditional calculation, the timing is dominated by the evaluation ofthe small fraction of the electron repulsion integrals for the compact basis functionsthat cannot be treated by FTC Replacing the traditional electron repulsion integralalgorithm by one of the alternatives discussed in Section1.2.1.2, for instance by a

Table 1-2 Timings and energies for sucrose, C12 H 22 O 11 See Table 1-1 for explanation

14 P Pulay Table 1-3 Timings and energies for taxol, C47 H 51 NO 14 See Table 1-1 for explanation

Figure 1-1 The scaling of the Coulomb time with respect to molecular size for a series of polyalanines,

n= 2−15 FTC PW means the plane wave component of the FTC calculation (diffuse densities) FTC

ERI means the remaining (compact) integrals that were calculated using traditional electron repulsion integrals

multipole expansion, should increase the efficiency of the Coulomb part by almost

an order of magnitude

The plane wave parts of both Quickstep [12] and PQS [26, 27] have beenimplemented in parallel The factor limiting parallel scaling for a large number ofprocessors is the parallel three-dimensional Fast Fourier Transform, which includesthe transposition of a large matrix Therefore, the parallel scaling of plane-wavemethods is limited at this time However, the inherently high performance of thesemethods makes up to a certain extent for the limited parallel scalability

Plane wave based methods for the calculation of the Coulomb term in electronicstructure calculations accelerate the calculation of these terms, usually the mostexpensive part of the calculations, by orders of magnitude for large molecules andbasis sets Unlike some of their competitors, they are surprisingly accurate also forrelatively small systems, and deserve to be more widely used in chemistry-centeredprograms

The most important problems to solve in this field are listed below.

1 Improve the efficiency of the other major computational tasks, for instance toformation of the exchange-correlation matrix, to match the performance of theCoulomb evaluation

2 Develop improved methods for the calculation of the exact (Hartree-Fock)exchange

3 Improve the parallel scaling of the method

ACKNOWLEDGMENTS

This work was supported by the National Science Foundation under grant numberCHE-0911541 and by the Mildred B Cooper Chair at the University of Arkansas.Acquisition of the Star of Arkansas supercomputer was supported in part by theNational Science Foundation under award number MRI-0722625

REFERENCES

1 Greengard L, Rokhlin V (1985) J Comp Phys 60:187

2 White CA, Johnson BG, Gill PMW, Head-Gordon M (1994) Chem Phys Lett 230:8

3 Payne MC, Teter MP, Allan DC, Arias TA, Joannopoulos JD (1992) Rev Mod Phys 64:1045

4 Car R, Parrinello M (1985) Phys Rev Lett 55:22

5 Segall MD, Lindan PJD, Probert MJ, Pickard CJ, Hasnip PJ, Clark SJ, Payne MC (2002) J Phys Condens Matter 14:2727

6 See: http://cms.mpi.univie.ac.at/vasp/

7 Giannozzi P, Baroni S, Bonini N, Calandra M, Car R, Cavazzoni C, Ceresoli D, Chiarotti GL, Cococcioni M, Dabo I, Dal Corso A, de Gironcoli S, Fabris S, Fratesi G, Gebauer R, Gerstmann U, Gougoussis C, Kokalj A, Lazzeri M, Martin-Samos L, Marzari N, Mauri F, Mazzarello R, Paolini

S, Pasquarello A, Paulatto L, Sbraccia C, Scandolo S, Sclauzero G, Seitsonen AP, Smogunov A, Umari P, Wentzcovitch RM (2009) J Phys Condens Matter 21:395502 See also: www.quantum- espresso.org

8 Singh DJ (1994) Plane waves, pseudopotentials and the LAPW method Kluger, Norwell, MA

9 Blöchl PE (1994) Phys Rev B 59:1758

10 Vanderbilt D (1990) Phys Rev B 41:7892

11 Kresse G, Joubert J (1999) Phys Rev B 59:1758

12 Lippert G, Hutter J, Parrinello M (1997) Mol Phys 92:477

13 VandeVondele J, Krack M, Mohamed F, Chassaing T, Hutter J (2005) Comp Phys Comm 167:103

14 Lippert G, Hutter J, Parrinello M (1999) Theor Chem Acc 103:124

15 Krack M, Parrinello M (2000) Phys Chem Chem Phys 2:2105

16 Hutter J (2003) J Chem Phys 118:3928

17 Iannuzzi M, Chassaing T, Wallman T, Hutter J (2005) Chimia 59:499

18 VandeVondele J, Iannuzzi M, Hutter J (2006) Lect Notes Phys 703:287

19 Guidon M, Schiffman F, Hutter J, VandeVondele J (2008) J Chem Phys 128:214104

20 Parallel Quantum Solutions, LLC, Fayetteville, AR, USA See: www.pqs-chem.com

21 Baker J, Wolinski K, Malagoli M, Kinghorn DR, Wolinski P, Magyarfalvi G, Saebo S, Janowski T, Pulay P (2009) J Comput Chem 30:317

22 Füsti-Molnár L, Pulay P (2002) J Chem Phys 116:7795

16 P Pulay

23 Füsti-Molnár L, Pulay P (2002) J Chem Phys 117:7827

24 Füsti-Molnár L, Pulay P (2003) J Mol Struct (THEOCHEM) 666:25

25 Füsti-Molnár L (2003) J Chem Phys 119:11080

26 Baker J, Füsti-Molnár L, Pulay P (2004) J Phys Chem A 108:3040

27 Baker J, Wolinski K, Pulay P (2007) J Comput Chem 28:2581

28 Kong J, White CA, Krylov AI et al (2000) J Comput Chem 21:1532

29 Füsti-Molnár L, Kong J (2005) J Chem Phys 122:074108

30 Kong J, Brown ST, Füsti-Molnár L (2006) J Chem Phys 124:094109

31 Kong J, Brown ST, Füsti-Molnár L (2006) J Chem Phys 124:219904

32 Brown ST, Füsti-Molnár L, Kong J (2006) Chem Phys Lett 418:490

33 Kohn W (1995) Int J Quant Chem 56:229

34 Schwegler E, Challacombe M (1996) J Chem Phys 105:2726

35 Burant JC, Scuseria GE, Frisch M (1996) J Chem Phys 105:8969

36 Schwegler E, Challacombe M, Head-Gordon M (1997) J Chem Phys 106:9708

37 Ochsenfeld C, White CA, Head-Gordon M (1998) J Chem Phys 109:1663

38 Heyd J, Scuseria GE, Ernzerhof M (2003) J Chem Phys 118:8207

39 Harris F (1975) In: Electronic structure of polymers and molecular crystals Plenum Press, New York, NY, p 453

40 Barnett RN, Landman U (1993) Phys Rev B 48:2081

41 DeLeeuw SW, Perram JW, Smit ER (1980) Proc Roy Soc Lond A 373:27

42 Hockney RW In: Alder B, Fernbach S, Rothenberg M (1970) Methods in computational physics, Academic, New York, NY 9:136

43 Hockney RW, Eastwood JW (1988) Computer simulation using particle, Adam Hilger, London

44 Pollock EL, Glosli J (1996) Comp Phys Comm 95:93

45 Bylaska EJ, Valiev M, Kawai R, Weare JH (2002) Comp Phys Comm 143:11

46 Martyna GJ, Tuckerman ME (1999) J Chem Phys 110:2810

47 Mináry P, Moorone JA, Yarne DA, Tuckerman ME, Martyna GJ (2004) J Chem Phys 121:11949

48 Vanderbilt D (1994) Phys Rev B 50:17953

49 Blöchl PE (1990) Phys Rev B 41:5414

50 Blöchl PE, Parrinello M (1992) Phys Rev B 45:9413

51 Stewart JJP, Császár P, Pulay P (1982) J Comp Chem 3:556

52 VandeVondele J, Hutter J (2003) J Chem Phys 118:4365

MATHEMATICAL FORMULATION OF THE FRAGMENT MOLECULAR ORBITAL METHOD

TAKESHI NAGATA1, DMITRI G FEDOROV1, AND KAZUO KITAURA2

1 NRI, National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan, e-mail: takeshi.nagata@aist.go.jp; d.g.fedorov@aist.go.jp

2Graduate School of Pharmaceutical Sciences, Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan; NRI, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki 305-8568, Japan, e-mail: kkitaura@pharm.kyoto-u.ac.jp

Abstract: The fragment molecular orbital (FMO) method is a computational scheme applied to the

conventional molecular orbital theories, which reduces their scaling from N3 N7 to a

nearly linear scaling, where N is the system size FMO provides an accurate treatment

of large molecules such as proteins and molecular clusters, and it can be efficiently allelized to achieve high scaling on massively parallel computers The main purpose of this chapter is to focus on the derivation of the equations and to provide a concise mathe- matical description of FMO A brief summary of the recent applications of FMO is also given.

par-Keywords: FMO, FMO3, ESP, HOP, AFO, Green’s function, ESP-DIM, ESP-PC, MP2, Coupled

cluster, DFT, MCSCF, ROHF, Multilayer, TDDFT, CI, CIS, CIS(D), Quantum Monte-Carlo, Gradient, PCM, EFP, MCP, Linear scaling, Molecular dynam- ics, PIMD, PIE, IFIE, EDA, PIEDA, LMP2, CAFI, BSSE, Counterpoise, FMO-

MO, FMO-LCMO, FMO/F, FMO/FX, FMO/XF, RESP, NMR, GIAO, CSGT, Multipole moment, Dynamic polarizability, MCMO, NEO, Drug design, VISCANA, VLS, QSAR, Parallelization, Protein, Ligand, Enzyme, DNA, Solvent, QSAR, Electrostatic poten- tial, Many-body, Fragment, Fragment molecular orbital, RHF, Excited state, Open- shell, Tessera, Cavity, RDM, MO, FILM, Earth Simulator, Energy decomposition analysis, Atomic charge, Massively parallel

During recent years there has been a considerable progress in the development

of quantum-mechanical methods aimed at computing large molecular systems Inaddition to the traditional ab initio approaches, which frequently rely upon thelocalized molecular orbitals to describe the electron correlation, there has been a

17

R Zale´sny et al (eds.), Linear-Scaling Techniques in Computational Chemistry and Physics, 17–64.

DOI 10.1007/978-90-481-2853-2_2,  Springer Science+Business Media B.V 2011 C

(sometimes referred to as O(N)) The fragment molecular orbital (FMO) method

proposed in 1999 [12] is one of such approaches, which has been considerablydeveloped theoretically as well as applied to a wide variety of systems, includingmolecular clusters, proteins, DNA, enzymes, small molecules explicitly solvated

in water droplets, ionic liquids, molecular crystals, zeolites and nanowires As thesummary of applications and the introductory explanation of FMO have been givenelsewhere [8,13–17], here we focus on the mathematical derivation of the method,followed by a brief summary of recent applications

EQUATION

The basic computational scheme of FMO has been described in detail elsewhere[14,17], and here we only give it very briefly for completeness The system isdivided into pieces (fragments) and each fragment calculation is performed with

an ab initio method (such as restricted Hartree-Fock, RHF), in the presence of theelectrostatic field of the remaining fragments, determined by their atomic nuclei andthe electron density distributions The fragment (monomer) calculations begin withthe field given by some initial guess densities, repeated in a loop self-consistentlywith respect to the field dependent upon the fragment densities This loop is calledthe self-consistent charge (SCC) loop (or monomer self-consistent field, SCF), and itcan be accelerated with the direct inversion in the iterative subspace (DIIS) method[18] Consequently, fragment pairs (dimers), and, optionally, triples (trimers) arecomputed in the fixed field determined in the previous step (see Figure2-1) Thetotal properties are computed from those of fragments and dimers (trimers) as shownbelow

2.2.1 Many-Body Expansion

In the FMO method, a molecular system is divided into N fragments The FMO total

energy is represented as follows [12,19,20]:

DENS 1 DENS 2 DENS 3

Construction of

ESPs from densities

Did all the energies get self-consistent ?

Convergence test of

monomer energies

NOYES

Construction of initial densities

Calculation of the total energy

Figure 2-1 Schematic procedure of the FMO energy calculation for a molecule divided into 3 fragments.

DENS I, MONO I and DIM IJ (I, J = 1,2,3) denote monomer density I, SCF calculation for monomer I,

and the calculation for dimer IJ, respectively The dotted lines represent the electrostatic potential due to

monomer densities Dimer SCF and dimer ES show that some dimers can be computed ab initio (SCF)

or with an approximation (ES)

The first expression gives the two-body expansion (FMO2) of the total energy E,

the second one adds to it the three-body corrections, in terms of the energies of

fragments E I , their pairs E IJ and, optionally, triples E IJK The one-body energy,

if defined in the same manner, would have the electrostatic contribution counted [19], however, in some context (e.g., in time-dependent density functionaltheory, TDDFT) when energy differences are considered, the one-body properties(FMO1) are also useful to consider

double-In many places below we give explicit expressions for FMO2, and it is forward to extend them into FMO3 It is important to realise that FMO2 includes