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Fragmentation: Toward Accurate Calculations on Complex Molecular Systems introduces the reader to the broad array of Fragmentation and embedding methods that are currently available or under development to facilitate accurate calculations on large, complex systems such as proteins, polymers, liquids and nanoparticles. These methods work by subdividing a system into subunits, called fragments or subsystems or domains. Calculations are performed on each fragment and then the results are combined to predict properties for the whole system. Topics covered include: * Fragmentation methods * Embedding methods * Explicitly correlated local electron correlation methods * Fragment molecular orbital method * Methods for treating large molecules This book is aimed at academic researchers who are interested in computational chemistry, computational biology, computational materials science and related fields, as well as graduate students in these fields.

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Fragmentation

Toward Accurate Calculations onComplex Molecular Systems

Edited by

Mark S. Gordon

Iowa State University, USA

This edition first published 2017 © 2017 by John Wiley & Sons Ltd

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Library of Congress Cataloging-in-Publication Data

Names: Gordon, M. S. (Mark S.), editor. Title: Fragmentation : toward accurate calculations on complex molecular systems / edited by Professor Mark S. Gordon, Iowa State University, USA. Description: Chichester, UK ; Hoboken, NJ : John Wiley & Sons, Inc., 2017. | Includes bibliographical references and index. Identifiers: LCCN 2016057161 (print) | LCCN 2016058050 (ebook) | ISBN 9781119129240 (cloth) | ISBN 9781119129257 (pdf) | ISBN 9781119129264 (epub) Subjects: LCSH: Fragmentation reactions. | Electron configuration. Classification: LCC QD281.F7 F738 2017 (print) | LCC QD281.F7 (ebook) | DDC 547/.128--dc23 LC record available at https://lccn.loc.gov/2016057161

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CONTENTS

List of Contributors

Preface

1 Explicitly Correlated Local Electron Correlation Methods

1.1 Introduction

1.2 Benchmark Systems

1.3 Orbital-Invariant MP2 Theory

1.4 Principles of Local Correlation

1.5 Orbital Localization

1.6 Local Virtual Orbitals

1.7 Choice of Domains

1.8 Approximations for Distant Pairs

1.9 Local Coupled-Cluster Methods (LCCSD)

1.10 Triple Excitations

1.11 Local Explicitly Correlated Methods

1.12 Technical Aspects

1.13 Comparison of Local Correlation and Fragment Methods

1.14 Summary

Appendix A: The LCCSD Equations

Appendix B: Derivation of the Interaction Matrices

Acknowledgments

References

2 Density and Potential Functional Embedding: Theory and Practice

2.1 Introduction

2.2 Theoretical Background

2.3 Density Functional Embedding Theory

2.4 Potential Functional Embedding Theory

2.5 Summary and Outlook

Acknowledgments

Note

References

3 Modeling and Visualization for the Fragment Molecular Orbital Method with the Graphical User Interface FU, and Analyses of Protein–Ligand Binding

3.1 Introduction

3.2 Overview of FMO

3.3 Methodology

3.4 GUI Development

3.5 Conclusions

Acknowledgments

References

4 Molecules-in-Molecules Fragment-Based Method for the Accurate Evaluation of Vibrational and Chiroptical Spectra for Large Molecules

4.1 Introduction

4.2 Computational Methods and Theory

4.3 Results and Discussion

4.4 Summary

4.5 Conclusions

Acknowledgments

References

5 Effective Fragment Molecular Orbital Method

5.1 Introduction

5.2 Effective Fragment Molecular Orbital Method

5.3 Summary and Future Developments

References

6 Effective Fragment Potential Method: Past, Present, and Future

6.1 Overview of the EFP Method

6.2 Milestones in the Development of the EFP Method

6.3 Chemistry at Interfaces and Photobiology

6.4 Future Directions and Outlook

References

7 Nucleation Using the Effective Fragment Potential and Two-Level Parallelism

7.1 Introduction

7.2 Methods

7.3 Results

7.4 Conclusions

Acknowledgments

References

8 Five Years of Density Matrix Embedding Theory

8.1 Quantum Entanglement

8.2 Density Matrix Embedding Theory

8.3 Bath Orbitals from a Slater Determinant

8.4 The Embedding Hamiltonian

8.5 Self-Consistency

8.6 Green’s Functions

8.7 Overview of the Literature

8.8 The One-Band Hubbard Model on the Square Lattice

8.9 Dissociation of a Linear Hydrogen Chain

8.10 Summary

Acknowledgments

References

9

Ab initio

Ice, Dry Ice, and Liquid Water

9.1 Introduction

9.2 Computational Method

9.3 Case Studies

9.4 Concluding Remarks

9.5 Disclaimer

Acknowledgments

References

10 A Linear-Scaling Divide-and-Conquer Quantum Chemical Method for Open-Shell Systems and Excited States

10.1 Introduction

10.2 Theories for the Divide-and-Conquer Method

10.3 Assessment of the Divide-and-Conquer Method

10.4 Conclusion

References

11 MFCC-Based Fragmentation Methods for Biomolecules

11.1 Introduction

11.2 Theory and Applications

11.3 Conclusion

Acknowledgments

References

Index

EULA

List of Tables

Chapter 1

Table 1.1

Table 1.2

Table 1.3

Table 1.4

Table 1.5

Table 1.6

Table 1.7

Table 1.8

Table 1.9

Table 1.10

Table 1.11

Table 1.12

Chapter 2

Table 2.1

Table 2.2

Chapter 3

Table 3.1

Table 3.2

Chapter 5

Table 5.1

Table 5.2

Table 5.3

Table 5.4

Table 5.5

Table 5.6

Table 5.7

Chapter 6

Table 6.1

Chapter 8

Table 8.1

Chapter 10

Table 10.1

Table 10.2

Table 10.3

Table 10.4

Table 10.5

Table 10.6

Table 10.7

Table 10.8

Chapter 11

Table 11.1

Table 11.2

Guide

Cover

Table of Contents

Preface

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List of Contributors

Emily A. Carter

School of Engineering and Applied Science, Princeton University, USA

Garnet K.L. Chan

Frick Chemistry Laboratory, Department of Chemistry, Princeton University, USA

Ajitha Devarajan

Office of University Development, University of Michigan, USA

Johannes M. Dieterich

Department of Mechanical and Aerospace Engineering, Princeton University, USA

Dmitri G. Fedorov

Research Center for Computational Design of Advanced Functional Materials (CD-FMat), National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Japan

Alexander Gaenko

Advanced Research Computing, University of Michigan, USA

Kandis Gilliard

Department of Chemistry, University of Illinois at Urbana–Champaign, USA

Mark S. Gordon

Ames Laboratory of United States Department of Energy, USA

Department of Chemistry, Iowa State University, USA

Pradeep K. Gurunathan

Department of Chemistry, Purdue University, USA

Xiao He

School of Chemistry and Molecular Engineering, East China Normal University, China

NYU-ECNU Center for Computational Chemistry, NYU Shanghai, China

So Hirata

Department of Chemistry, University of Illinois at Urbana–Champaign, USA

Jan H. Jensen

Department of Chemistry, University of Copenhagen, Denmark

Carlos A. Jiménez-Hoyos

Frick Chemistry Laboratory, Department of Chemistry, Princeton University, USA

K. V. Jovan Jose

*

Department of Chemistry, Indiana University, USA

Murat Keçeli

Department of Chemistry, University of Illinois at Urbana–Champaign, USA

Argonne National Laboratory, USA

Kazuo Kitaura

Graduate School of System Informatics, Kobe University, Japan

Christoph Köppl

Institute for Theoretical Chemistry, University of Stuttgart, Germany

Caroline M. Krauter

Department of Mechanical and Aerospace Engineering, Princeton University, USA

Jinjin Li

Department of Chemistry, University of Illinois at Urbana–Champaign, USA

National Key Laboratory of Science and Technology on Micro/Nano Fabrication, Department of Micro/Nano Electronics, Shanghai Jiao Tong University, China

Jinfeng Liu

School of Chemistry and Molecular Engineering, East China Normal University, China

Qianli Ma

Institute for Theoretical Chemistry, University of Stuttgart, Germany

Hiromi Nakai

Department of Chemistry and Biochemistry, School of Advanced Science and Engineering, Waseda University, Japan

Krishnan Raghavachari

Department of Chemistry, Indiana University, USA

Michael A. Salim

Department of Chemistry, University of Illinois at Urbana–Champaign, USA

Max Schwilk

Institute for Theoretical Chemistry, University of Stuttgart, Germany

Lyudmila V. Slipchenko

Department of Chemistry, Purdue University, USA

Olaseni Sode

Department of Chemistry, University of Illinois at Urbana–Champaign, USA

Department of Chemistry, Biochemistry, and Physics, The University of Tampa, USA

Casper Steinmann

Department of Physics, Chemistry and Pharmacy, University of Southern Denmark, Denmark

Hans-Joachim Werner

Institute for Theoretical Chemistry, University of Stuttgart, Germany

Theresa L. Windus

Ames Laboratory of United States Department of Energy, USA

Department of Chemistry, Iowa State University, USA

Sebastian Wouters

Center for Molecular Modelling, Ghent University, Belgium

Frick Chemistry Laboratory, Department of Chemistry, Princeton University, USA

Kiyoshi Yagi

Department of Chemistry, University of Illinois at Urbana–Champaign, USA

Theoretical Molecular Science Laboratory, RIKEN, Japan

Takeshi Yoshikawa

Department of Chemistry and Biochemistry, School of Advanced Science and Engineering, Waseda University, Japan

Kuang Yu

Department of Mechanical and Aerospace Engineering, Princeton University, USA

John Z. H. Zhang

School of Chemistry and Molecular Engineering, East China Normal University, China

NYU-ECNU Center for Computational Chemistry, NYU Shanghai, China

Department of Chemistry, New York University, USA

Tong Zhu

School of Chemistry and Molecular Engineering, East China Normal University, China

NYU-ECNU Center for Computational Chemistry, NYU Shanghai, China

Note

*

Current address: School of Chemistry, University of Hyderabad, India

Preface

Electronic structure theory, also referred to as ab initio quantum chemistry (QC), has attained a high level of maturity and reliability for gas-phase molecules of modest size. Unfortunately, the formal scaling of these methods such as Hartree–Fock (HF), density functional theory (DFT), second-order perturbation theory (MP2), coupled cluster theory (CC), and multi-reference (MR) methods hinder their application to large molecules, to condensed phase systems or to excited electronic state potential energy surfaces. These limitations are especially severe for methods that account for electron correlation, such as MP2, CC, and MR methods, since their scaling with system size is steeper than for the simpler HF and DFT methods. There is therefore a need for computational strategies that nearly retain the accuracy of the most reliable methods while greatly reducing the scaling of these methods as a function of system size. While researchers who are interested in simulations of large molecular systems have often turned to classical molecular mechanics (MM) force fields, MM methods are limited in their applicability. While there are a few exceptions, classical MM cannot realistically treat bond making/bond breaking (the essence of chemistry) or excited state phenomena.

One effective QC approach that has become increasingly popular is referred to as fragmentation (broadly defined) or embedding theory. Fragmentation commonly refers to the physical subdivision of a large molecule into fragments, each of whose energy can be computed on a different compute node, thereby making the overall computation highly parallel. Fragmentation methods of this type scale nearly linearly with system size and can take advantage of massively parallel computers. Fragmentation methods of this type are discussed in Chapters 3, 5, 6, 7, 10, and 11. An alternative approach to physical fragmentation of a molecule is to fragment the wave function, by employing localized molecular orbitals to separate the wave function into domains that can be separately correlated. This approach is based on the fact that electron correlation is short-range. Chapter 1 provides an excellent discussion of local electron correlation methods by one of the leaders in the field.

Embedding methods are similar to fragmentation methods in that a total system is partitioned into multiple subsystems, in a manner that allows the incorporation of interactions among the subsystems. Like fragmentation and local orbital approaches, embedding methods reduce the steep scaling of traditional electronic structure methods. Embedding methods frequently involve multiple levels of theory. Approaches to embedding methods are discussed in Chapters 2, 4, 8, and 9.

The methods that are discussed in this book provide an exciting path forward to the accurate study of large molecules and condensed phase phenomena.

1Explicitly Correlated Local Electron Correlation Methods

Hans-JoachimWerner, Christoph Köppl, Qianli Ma, and Max Schwilk

Institute for Theoretical Chemistry, University of Stuttgart, Germany

1.1 Introduction

Accurate wave function methods for treating the electron correlation problem are indispensable in quantum chemistry. A well-defined hierarchy of such methods exists, and in principle, these methods allow to approach the exact solution of the non-relativistic electronic Schrödinger equation to any desired accuracy. A much simpler alternative is density functional theory (DFT), which is probably most often used in computational chemistry. However, its failures and uncertainties are well known, and there is no way for systematically improving or checking the results other than comparing with experiment or with the results of accurate wave function methods.

Due to the steep scaling of the computational resources (CPU-time, memory, disk space) with the molecular size, conventional wave function methods such as CCSD(T) (coupled-cluster with single and double excitations and a perturbative treatment of triple excitations) can only be applied to rather small molecular systems. For example, the CPU-time of CCSD(T) scales as , where is a measure of the molecular size (e.g., the number of correlated electrons) and even the simplest electron correlation method, MP2 (second-order Møller-Plesset perturbation theory) scales as . This causes a “scaling wall” that cannot be overcome. Even with massive parallelization and using the largest supercomputers, this wall can only be slightly shifted to larger systems. However, it is well known that electron correlation in insulators is a short-range effect. The pair correlation energies decay at long-range with R− 6, where R is the distance between two localized spin orbitals. Therefore, the steep scaling is unphysical. It results mainly from the use of canonical molecular orbitals, which are usually delocalized over larger parts of the molecule.

The scaling problem can be much alleviated by exploiting the short-range character of electron correlation using local orbitals and by introducing local approximations. This was first proposed in the pioneering work of Pulay et al. [1–6], and in the last 20 years enormous progress has been made in developing accurate local correlation methods. There are two different approaches, both of which are based on the use of local orbitals. The traditional one is to treat the whole molecule in one calculation and to apply various approximations that are based on the fast decay of the correlation energy. We will denote such methods “local correlation methods.” A large variety of such approaches has been published in the past [7–59].

The second approach is the so-called “fragmentation methods” [60–87], in which the system is split into smaller pieces. These pieces are treated independently, mostly using conventional methods (although the use of local correlation methods is also possible). The total correlation energy of the system is then assembled from the results of the fragment calculations. Various methods differ in the way in which the fragments are chosen and the energy is assembled. A special way of assembling the energy using a many-body expansion is used in the so-called incremental methods [88–96], but these also belong to the group of fragmentation methods. Fragmentation methods will be described in other chapters of this volume and are therefore not the subject of this chapter. However, in Section 1.13, we will comment on the relation of local correlation and fragmentation methods.

Another problem of the CCSD(T) method is the slow convergence of the correlation energy with the basis set size. Very large basis sets are needed to obtain converged results, and this makes conventional high-accuracy electron correlation calculations extremely expensive. This problem is due to the fact that the wave function has a cusp for r12 → 0, where r12 is the distance between two electrons. The cusp is due to the singularity of the Coulomb operator , and cannot be represented by expanding the wave function in antisymmetrized products of molecular orbitals (Slater determinants). This leads to the very slow convergence of the correlation energy with the size of the basis set, and in particular with the highest angular momentum of the basis functions. This problem can be solved by including terms into the wave function that depend explicitly on the distance r12, and these methods are known as “explicit correlation methods” [97–155].

The combination of explicit correlation methods with local approximations has been particularly successful [140–153]. As will be explained and demonstrated later in this chapter, this does not only drastically reduce the basis set incompleteness errors, but also strongly reduces the errors caused by local approximations. Local correlation methods employ two basic approximations. The first is based on writing the total correlation energy as a sum of pair energies. Each pair describes the correlation of an electron pair (in a spin-orbital formulation), or, more generally, the correlation of the electrons in a pair of occupied local molecular orbitals (LMOs). Depending on the magnitude of the pair energies, it is possible to introduce a hierarchy of “strong,” “close,” “weak,” or “distant” pairs [7,18,31,32]. Different approximations can be introduced for each class, ranging from a full local coupled-cluster (LCCSD) treatment for strong pairs to a non-iterative perturbation correction for distant pairs, which can be evaluated very efficiently using multipole approximations [12, 13]. We will denote such approximations as “pair approximations.” The second type of local approximations is the “domain approximation,” which is applied to each individual pair. A domain is a subset of local virtual orbitals which is spatially close to the LMO pair under consideration. Asymptotically, the number of orbitals in each pair domain (the “domain sizes”) become independent of the molecular size. Also the number of pairs in each class (except for the distant pairs) becomes independent of the molecular size. This leads to linear scaling of the computational effort as a function of molecular size, as has already been demonstrated for LMP2 and up to the LCCSD(T) level of theory more than 25 years ago [12–18].

The critical question is, of course, how quickly the correlation energy as well as relative energies (e.g., reaction energies, activation energies, intermolecular interaction energies, and electronic excitation energies) converge with the domain sizes and how they depend on the pair approximations. The domain sizes which are necessary to reach a certain accuracy (e.g., 99.9% of the canonical correlation energy) depends sensitively on the choice of the virtual orbitals. As is known since the 1960s, fastest convergence is obtained with pair natural orbitals (PNOs) [156], and this has first been fully exploited in the seminal PNO-CI and PNO-CEPA methods of Meyer [157, 158], and somewhat later also by others [159–163]. The problem with this approach is that the PNOs are different for each pair and non-orthogonal between different pairs. This leads to complicated integral transformations and prevented the application of PNO methods to large molecules for a long time. The method was revived by Neese and coworkers in 2009 and taken up also by others (including us) later on [32,33,48–57,146–150]. The problem of evaluating the integrals was overcome by using local density-fitting approximations [22]. Furthermore, the integrals are first computed in a basis of projected atomic orbitals (PAOs), which are common to all pairs, and subsequently transformed to the pair-specific PNO domains [54,146,147,150]. Also, hybrid methods, in which so-called orbital-specific virtuals (OSVs) [164–167] are used at an intermediate stage, have been proposed [53, 146, 147]. Later sections of this chapter will explain these approaches in some detail.

Local approximations have also been developed for multi-reference wave functions [168–176]. The description of these methods is beyond the scope of the current article, but we mention that recently very efficient and accurate PNO-NEVPT2 [175] (N-electron valence state perturbation-theory) and PNO-CASPT2 [176] (complete active space second-order perturbation theory) methods have been described.

In the current article, we will focus on new developments of well-parallelized PNO-LMP2-F12 and PNO-LCCSD-F12 methods recently developed in our laboratory. These methods also have a close relation to the methods of Neese et al. [54–57, 153]. After introducing some benchmark systems, which will be used later on, we will first outline the principles of local correlation and describe the choice of the local occupied and virtual orbitals as well as of the domains. The convergence of the correlation energy as a function of the domain sizes will be demonstrated for various types of virtual orbitals for LMP2. Subsequently, based on these foundations, we will discuss more advanced approximations for distant pairs and close/weak pair approximations used in local coupled cluster methods. Next, we will present an introduction to local explicit correlation methods, and demonstrate the improvements achieved by the F12 approach both for LMP2-F12 and LCCSD-F12. Finally, we will describe some important technical details, such as local density fitting and parallelization. A summary concludes the chapter.

1.2 Benchmark Systems

Some large molecules and reactions, which we have used extensively to benchmark our methods [32, 145, 147, 148], are shown in Figures 1.1 and 1.2. For easy reference, we have given short names to some of the molecules, which are shown in the figure and will be used throughout this article. Reaction I is the last step in the synthesis of androstendione. In reaction II, testosterone is esterified to make it more lipophilic for a longer retention time in body tissues. Reaction III is the dissociation of a gold(I)-aminonitrene complex (AuC41H45N4P, for simplicity denoted Auamin, see Figure 1.1). This reaction is taken from Ref. [177] and plays an important role in catalytic aziridination and insertion reactions. The Auamin molecule has three phenyl and three mesityl groups and therefore strong long-range dispersion interactions are expected.

Figure 1.1 Benchmark molecules and reactions.

Figure 1.2 Visualization of a selection of large molecules mentioned in Tables 1.3 and 1.4.

In Figure 1.2