分子及凝聚态系统物性的计算模拟：从电子机构到分子动力学=Computer sim-ulations of molecules and condensed matters:from electronicPDF电子书下载

1 Introduction to Computer Simulations of Molecules and Condensed Matter 1

1.1 Born-Oppenheimer Approximation and the Born-Oppenheimer Potential Energy Surface 2

1.2 Categorization of the Tasks in Computer Simulations of Molecules and Condensed Matters 6

1.2.1 Electronic Structure Calculations 6

1.2.2 Geometry Optimization,Stationary Points on PES,Local Minimum,and Transition State 7

1.2.3 Meta-Stable State and Transition State Searching 10

1.2.4 Molecular Dynamics for the Thermal Effects 12

1.2.5 Extensions of MD:Enhanced Sampling and Free-Energy Calculations 13

1.2.6 Path Integral Simulations for the Quantum Nuclear Effects 15

1.3 Layout of the Book 16

2 Quantum Chemistry Methods and Density-Functional Theory 18

2.1 Wave-Function Based Method 18

2.1.1 The Hartree and Hartree-Fock Approximations 19

2.1.2 Beyond the Hartree-Fock Approximation 22

2.2 Density-Functional Theory 23

2.2.1 Thomas-Fermi Theory 23

2.2.2 Density-Functional Theory 25

2.2.3 Exchange-Correlation Energy 27

2.2.4 Interpretation of the Kohn-Sham Energies 29

3 Pseudopotentials,Full Potential,and Basis Sets 31

3.1 Pseudopotential Method 32

3.1.1 Generation of the Pseudopotential 33

3.1.2 Implicit Approximations 38

3.1.2.1 Frozen Core 38

3.1.2.2 Core-Valence Linearization 39

3.1.2.3 Pseudoization 39

3.2 FP-(L)APW+lo Method 40

3.2.1 LAPW Basis Functions 43

3.2.2 APW+lo Basis Functions 44

3.2.3 Core States 45

3.2.4 Potential and Density 46

4 Many-Body Green Function Theory and the GW Approximation 47

4.1 Green Function Method 49

4.1.1 The Green Function 49

4.1.2 The Dyson Equation 52

4.1.3 Self-Energy:Hedin Equations 54

4.1.4 The Quasiparticle Concept 57

4.2 GW Approximation 58

4.3 G0W0 Approximation 61

4.4 Numerical Implementation of an All-Elctron G0W0 Code:FHI-gap 67

4.4.1 Summary of the G0W0 Equations 68

4.4.2 The Mixed Basis 70

4.4.3 Matrix Form of the G0W0 Equations 72

4.4.4 Brillouin-Zone Integration of the Polarization 74

4.4.5 The Frequency Integration 78

4.4.6 Flowchart 80

5 Molecular Dynamics 82

5.1 Introduction to Molecular Dynamics 84

5.1.1 The Verlet Algorithm 85

5.1.2 The Velocity Verlet Algorithm 87

5.1.3 The Leap Frog Algorithm 89

5.2 Other Ensembles 90

5.2.1 Andersen Thermostat 92

5.2.2 Nosé-Hoover Thermostat 93

5.2.3 Nosé-Hoover Chain 102

5.2.4 Langevin Thermostat 104

5.2.5 Andersen and Parrinello-Rahman Barostats 106

5.3 Examples for Practical Simulations in Real Poly-Atomic Systems 109

6 Extension of Molecular Dynamics,Enhanced Sampling and the Free-Energy Calculations 116

6.1 Umbrella Sampling and Adaptive Umbrella Sampling Methods 118

6.2 Metadynamics 128

6.3 Integrated Tempering Sampling 131

6.4 Thermodynamic Integration 134

7 Quantum Nuclear Effects 141

7.1 Path-Integral Molecular Simulations 145

7.1.1 Path-Integral Representation of the Propagator 145

7.1.2 Path-Integral Representation of the Density Matrix 149

7.1.3 Statistical Mechanics:Path-Integral Molecular Simulations 153

7.1.4 Staging and Normal-Mode Transformations 161

7.1.5 Evaluation of the Zero-Point Energy 171

7.2 Extensions Beyond the Statistical Studies 180

7.2.1 Different Semiclassical Dynamical Methods 181

7.2.2 Centroid Molecular Dynamics and Ring-Polymer Molecular Dynamics 184

7.3 Free-Energy with Anharmonic QNEs 191

7.4 Examples 194

7.4.1 Impact of QNEs on Structures of the Water-Hydroxyl Overlayers on Transition Metal Surfaces 194

7.4.2 Impact of Quantum Nuclear Effects on the Strength of Hydrogen Bonds 203

7.4.3 Quantum Simulation of the Low-Temperature Metallic Liquid Hydrogen 212

7.5 Summary 226

Appendix A Useful Mathematical Relations 228

A.1 Spherical Harmonics 228

A.2 Plane Waves 229

A.3 Fourier Transform 229

A.4 Spherical Coordinates 229

A.5 The Step(Heaviside) Function 230

Appendix B Expansion ofa Non-Local Function 231

Appendix C The Brillouin-Zone Integration 234

C.1 The Linear Tetrahedron Method 234

C.1.1 The Isoparametric Transfromation 236

C.1.2 Integrals in One Tetrahedron 238

C.1.3 The Integration Weights 239

C.2 Tetrahedron Method for q-Dependent Brillouin-Zone Integration 240

C.2.1 Isoparametric Transformation 241

C.2.2 The Integration Region 242

C.2.3 Polarizability 244

C.2.3.1 Polarisability on the Real Frequency Axis 244

C.2.3.2 Polarisability on the Imaginary Frequency Axis 246

Appendix D The Frequency Integration 248

References 251

Acknowledgements 271