List of Contributors xi
Preface xv
1 Explicitly Correlated Local Electron Correlation Methods 1
Hans-Joachim Werner, Christoph K¶ppl, Qianli Ma, and Max Schwilk
1.1 Introduction 1
1.2 Benchmark Systems 3
1.3 Orbital-Invariant MP2 Theory 6
1.4 Principles of Local Correlation 9
1.5 Orbital Localization 10
1.6 Local Virtual Orbitals 12
1.6.1 Pseudo-Canonical Pair-Specific Orbitals 12
1.6.2 Projected Atomic Orbitals 16
1.6.3 Pair Natural Orbitals 18
1.6.4 Linear Scaling PNO Generation 22
1.6.5 Orbital-Specific Virtuals (OSVs) 23
1.7 Choice of Domains 24
1.8 Approximations for Distant Pairs 26
1.8.1 Bipolar Multipole Approximations of Electron Repulsion Integrals 26
1.8.2 Approximations of Distant Pair Energies 29
1.9 Local Coupled-Cluster Methods (LCCSD) 33
1.9.1 Weak Pair Approximations 35
1.9.2 Long-Range Cancellations of Terms in the LCCSD Equations 36
1.9.3 Projection Approximations 39
1.10 Triple Excitations 41
1.11 Local Explicitly Correlated Methods 41
1.11.1 Pno-lmp2-f 12 42
1.11.2 Pno-lccsd-f 12 49
1.12 Technical Aspects 53
1.12.1 Local Density Fitting 53
1.12.2 Parallelization 56
1.13 Comparison of Local Correlation and Fragment Methods 57
1.14 Summary 60
Appendix A: The LCCSD Equations 63
Appendix B: Derivation of the Interaction Matrices 65
References 67
2 Density and Potential Functional Embedding: Theory and Practice 81
Kuang Yu, Caroline M. Krauter, Johannes M. Dieterich, and Emily A. Carter
2.1 Introduction 81
2.2 Theoretical Background 82
2.3 Density Functional Embedding Theory 84
2.3.1 Basic Theory 84
2.3.1.1 Definition of the Embedding Potential 85
2.3.1.2 Optimization Procedure 85
2.3.2 Embedding Potential Constructionâ"Implementations in Planewave Codes 86
2.3.2.1 Implementation with Pseudopotentials in ABINIT 87
2.3.2.2 Implementation with PAW in VASP 87
2.3.2.3 Penalty Functionsâ"Damping the Unphysical Oscillations 91
2.3.2.4 Illustrative Example 93
2.3.3 Embedded Cluster Calculations 94
2.3.3.1 Calculation of Embedding Integrals 94
2.3.3.2 Evaluation of the Total Energy 96
2.3.3.3 Examples 97
2.4 Potential Functional Embedding Theory 101
2.4.1 Basic Theories and Technical Details 102
2.4.1.1 Definition of Energies 102
2.4.1.2 Optimized Effective Potential (OEP) Scheme for Exact Kinetic Energy 103
2.4.1.3 Energy Gradient 104
2.4.1.4 Summary of the Code Structure 105
2.4.2 Illustrative Examples 106
2.4.2.1 AlP Diatomic 107
2.4.2.2 H 2 O on MgO (001) 108
2.5 Summary and Outlook 109
Acknowledgments 111
References 111
3 Modeling and Visualization for the Fragment Molecular Orbital Method with the Graphical User Interface FU, and Analyses of Proteinâ"Ligand Binding 119
Dmitri G. Fedorov and Kazuo Kitaura
3.1 Introduction 119
3.2 Overview of FMO 120
3.3 Methodology 120
3.3.1 FMO/PCM Formulation in the Presence of Dummy Atoms 120
3.3.2 New Analyses Defining the Desolvation Penalty in the Proteinâ"Ligand Binding 122
3.3.2.1 Asymmetric Binding Analysis (ABA) 122
3.3.2.2 Symmetric Binding Analysis (SBA) 123
3.3.2.3 Symmetric Binding Analysis with Separated Cavitation (SBAC) 123
3.3.2.4 Fragment-Wise Elaboration of SBA in FMO 124
3.3.2.5 Fragment-Wise Elaboration of SBAC 127
3.3.3 Application of Analyses to Proteinâ"Ligand Binding 127
3.4 GUI Development 128
3.4.1 Outline of FU 128
3.4.2 Modeling and Result Visualization 129
3.4.2.1 Modeling of an FKBP Protein Complex 129
3.4.2.2 Creating FMO Input 129
3.4.2.3 Running FMO in GAMESS 131
3.4.2.4 Visualizing FMO Results 131
3.4.3 An Overview of Using FU for a Complex System 133
3.4.4 Examples of Scripting in FU 133
3.4.4.1 Converting Multiple PDB Files into Z-matrix Files 133
3.4.4.2 Drawing Dipole Moments with Arrows 135
3.5 Conclusions 136
Acknowledgments 137
References 137
4 Molecules-in-Molecules Fragment-Based Method for the Accurate Evaluation of Vibrational and Chiroptical Spectra for Large Molecules 141
K. V. Jovan Jose and Krishnan Raghavachari
4.1 Introduction 141
4.2 Computational Methods and Theory 142
4.3 Results and Discussion 146
4.3.1 MIM Method for Geometry Optimization 146
4.3.2 MIM Method for Evaluating IR Spectra (MIM-IR) 146
4.3.3 MIM Method for Evaluating Raman Spectra (MIM-Raman) 149
4.3.4 MIM Method for Evaluating VCD Spectra (MIM-VCD) 151
4.3.5 MIM Method for Evaluating ROA Spectra (MIM-ROA) 154
4.3.6 Two-Step-MIM Scheme for Evaluating Raman and ROA Spectra 156
4.4 Summary 157
4.5 Conclusions 158
Acknowledgments 159
References 159
5 Effective Fragment Molecular Orbital Method 165
Casper Steinmann and Jan H. Jensen
5.1 Introduction 165
5.1.1 Effective Fragment Potentials 166
5.1.2 Fragment Molecular Orbital Method 167
5.2 Effective Fragment Molecular Orbital Method 168
5.2.1 Correlation Energies in the EFMO Method 170
5.2.2 The EFMO Gradient 172
5.2.3 Timings and Computational Efficiency 173
5.2.4 Biochemistry with EFMO 174
5.2.5 Fully Integrated EFMO 178
5.2.6 Remarks, Notes, and Comments 179
5.3 Summary and Future Developments 180
References 180
6 Effective Fragment Potential Method: Past, Present, and Future 183
Lyudmila V. Slipchenko and Pradeep K. Gurunathan
6.1 Overview of the EFP Method 183
6.2 Milestones in the Development of the EFP Method 185
6.2.1 EFP1 Water Model 185
6.2.2 EFP (EFP2) General Model 187
6.3 Present: Chemistry at Interfaces and Photobiology 192
6.3.1 OH Radical Solvated in Water 192
6.3.2 EFP for Macromolecules and Polymers 198
6.4 Future Directions and Outlook 202
References 203
7 Nucleation Using the Effective Fragment Potential and Two-Level Parallelism 209
Ajitha Devarajan, Alexander Gaenko, Mark S. Gordon, and Theresa L. Windus
7.1 Introduction 209
7.2 Methods 211
7.2.1 Brief Overview of DNTMC 211
7.2.2 Brief Overview of EFP 213
7.2.3 Overview of the Two-Level Parallelism Approach 215
7.3 Results 217
7.3.1 Evaporation Rate of Water Hexamer Cluster at 243K 217
7.3.2 Ion Mediated Nucleation 218
7.3.3 Evaporation Rate of Sulfuric Acid from Neutral Sulfuric Acid Dimer Clusters 219
7.3.4 Two-Level Parallel DNTEFP Performance Analysis 221
7.4 Conclusions 223
Acknowledgments 223
References 224
8 Five Years of Density Matrix Embedding Theory 227
Sebastian Wouters, Carlos A. Jim©nez-Hoyos, and Garnet K.L. Chan
8.1 Quantum Entanglement 227
8.2 Density Matrix Embedding Theory 228
8.3 Bath Orbitals from a Slater Determinant 230
8.4 The Embedding Hamiltonian 232
8.5 Self-Consistency 234
8.6 Greenâs Functions 236
8.7 Overview of the Literature 237
8.8 The One-Band Hubbard Model on the Square Lattice 237
8.9 Dissociation of a Linear Hydrogen Chain 240
8.10 Summary 240
Acknowledgments 241
References 241
9 Ab initio Ice, Dry Ice, and Liquid Water 245
So Hirata, Kandis Gilliard, Xiao He, Murat Ke§eli, Jinjin Li, Michael A. Salim, Olaseni Sode, and Kiyoshi Yagi
9.1 Introduction 245
9.2 Computational Method 247
9.2.1 Internal Energy 248
9.2.2 Structure and Phonons 250
9.2.3 Spectra 251
9.2.4 Pressure Effects 252
9.2.5 Temperature Effects 253
9.2.6 Bornâ"Oppenheimer Molecular Dynamics 255
9.3 Case Studies 256
9.3.1 Ice-Ih 256
9.3.2 Ice-HDA 259
9.3.3 Ice-VIII 262
9.3.4 Liquid Water 266
9.3.5 CO 2 -I: Pressure Tuning of Fermi Resonance 272
9.3.6 CO 2 -I and III: Solidâ"Solid Phase Transition 277
9.3.7 CO 2 -I: Thermal Expansion 280
9.4 Concluding Remarks 284
9.5 Disclaimer 284
Acknowledgments 284
References 285
10 A Linear-Scaling Divide-and-Conquer Quantum Chemical Method for Open-Shell Systems and Excited States 297
Takeshi Yoshikawa and Hiromi Nakai
10.1 Introduction 297
10.2 Theories for the Divide-and-Conquer Method 298
10.2.1 Review of DC-SCF and DC-Based Correlation Theories 298
10.2.1.1 Dc-hf/dft 298
10.2.1.2 DC-Based Correlation Theory 300
10.2.1.3 Dual-Buffer DC-Based Correlation Method 301
10.2.2 Linear-Scaling Divide-and-Conquer Method for Open-Shell Systems 302
10.2.2.1 DC-USCF and DC-UMP 2 302
10.2.2.2 Expected Value of the Squared Spin Operator Å 2 304
10.2.3 Linear-Scaling Divide-and-Conquer Method for Excited-State Calculations 304
10.2.3.1 Dc-cis/tddft 304
10.2.3.2 Dc-sac/sacci 305
10.3 Assessment of the Divide-and-Conquer Method 307
10.3.1 Divide-and-Conquer Calculations for Open-Shell Systems 307
10.3.1.1 DC-USCF and DC-UMP 2 307
10.3.2 Excited-State Calculations based on the Divide-and-Conquer Method 313
10.3.2.1 Conjugated Aldehyde 313
10.3.2.2 Photoactive Yellow Protein 315
10.4 Conclusion 318
References 319
11 MFCC-Based Fragmentation Methods for Biomolecules 323
Jinfeng Liu, Tong Zhu, Xiao He, and John Z. H. Zhang
11.1 Introduction 323
11.2 Theory and Applications 324
11.2.1 The MFCC Approach 324
11.2.2 Electron Density and Total Energy 326
11.2.3 The EE-GMFCC Method for Energy Calculation 328
11.2.4 The EE-GMFCC-CPCM Method for Protein Solvation Energy 331
11.2.5 The EE-GMFCC-CPCM Method for Proteinâ"Ligand Binding Energy 337
11.2.6 The EE-GMFCC Method for Geometry Optimization and Vibrational Spectrum of Proteins 338
11.2.7 The EE-GMFCC-Based Ab Initio Molecular Dynamics for Proteins 340
11.3 Conclusion 345
Acknowledgments 346
References 346
Index 349
