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E-grāmata: Computer Simulations Of Molecules And Condensed Matter: From Electronic Structures To Molecular Dynamics

(Peking Univ, China), (Peking Univ, China)
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Computer Simulations Of Molecules And Condensed Matter: From Electronic Structures To Molecular DynamicsPalielināt
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This book provides a relatively complete introduction to the methods used in computational condensed matter. A wide range of electronic structure theories are introduced, including traditional quantum chemistry methods, density functional theory, many-body perturbation theory, and more. Molecular dynamics simulations are also discussed, with extensions to enhanced sampling and free-energy calculation techniques including umbrella sampling, meta-dynamics, integrated tempering sampling, etc. As a further extension beyond the standard Born-Oppenheimer molecular dynamics, some simulation techniques for the description of quantum nuclear effects are also covered, based on Feynman's path-integral representation of quantum mechanics. The book aims to help beginning graduate students to set up a framework of the concepts they should know before tackling the physical/chemical problems they will face in their research.
Preface v
1 Introduction to Computer Simulations of Molecules and Condensed Matter
1(16)
1.1 Born-Oppenheimer Approximation and the Born-Oppenheimer Potential Energy Surface
2(4)
1.2 Categorization of the Tasks in Computer Simulations of Molecules and Condensed Matters
6(9)
1.2.1 Electronic Structure Calculations
6(1)
1.2.2 Geometry Optimization, Stationary Points on PES, Local Minimum, and Transition State
7(1)
1.2.3 Metastable State and Transition State Searching
8(3)
1.2.4 Molecular Dynamics for the Thermal Effects
11(1)
1.2.5 Extensions of MD: Enhanced Sampling and Free Energy Calculations
12(1)
1.2.6 Path Integral Simulations for the Quantum Nuclear Effects
13(2)
1.3 Layout of the Book
15(2)
2 Quantum Chemistry Methods and Density-Functional Theory
17(14)
2.1 Wave Function-Based Method
17(5)
2.1.1 The Hartree and Hartree-Fock Approximations
18(3)
2.1.2 Beyond the Hartree-Fock Approximation
21(1)
2.2 Density-Functional Theory
22(9)
2.2.1 Thomas-Fermi Theory
22(2)
2.2.2 Density-Functional Theory
24(2)
2.2.3 Exchange-Correlation Energy
26(2)
2.2.4 Interpretation of the Kohn-Sham Energies
28(3)
3 Pseudopotentials, Full Potential, and Basis Sets
31(16)
3.1 Pseudopotential Method
32(7)
3.1.1 Generation of the Pseudopotential
33(5)
3.1.2 Implicit Approximations
38(1)
3.1.2.1 Frozen Core
38(1)
3.1.2.2 Core-Valence Linearization
38(1)
3.1.2.3 Pseudoization
39(1)
3.2 FP-(L)APW+lo Method
39(8)
3.2.1 LAPW Basis Functions
42(1)
3.2.2 APW+lo Basis Functions
43(1)
3.2.3 Core States
44(1)
3.2.4 Potential and Density
45(2)
4 Many-Body Green's Function Theory and the GW Approximation
47(34)
4.1 Green's Function Method
49(9)
4.1.1 The Green's Function
49(3)
4.1.2 The Dyson Equation
52(2)
4.1.3 Self-Energy: Hedin Equations
54(3)
4.1.4 The Quasi-Particle Concept
57(1)
4.2 GW Approximation
58(3)
4.3 G0W0 Approximation
61(5)
4.4 Numerical Implementation of an All-Electron G0W0 Code: FHI-Gap
66(15)
4.4.1 Summary of the G0W0 Equations
68(1)
4.4.2 The Mixed Basis
69(2)
4.4.3 Matrix Form of the G0W0 Equations
71(2)
4.4.4 Brillouin-Zone Integration of the Polarization
73(3)
4.4.5 The Frequency Integration
76(3)
4.4.6 Flowchart
79(2)
5 Molecular Dynamics
81(32)
5.1 Introduction to Molecular Dynamics
83(6)
5.1.1 The Verlet Algorithm
84(2)
5.1.2 The Velocity Verlet Algorithm
86(2)
5.1.3 The Leap Frog Algorithm
88(1)
5.2 Other Ensembles
89(18)
5.2.1 Andersen Thermostat
90(2)
5.2.2 Nose-Hoover Thermostat
92(8)
5.2.3 Nose-Hoover Chain
100(2)
5.2.4 Langevin Thermostat
102(2)
5.2.5 Andersen and Parrinello-Rahman Barostats
104(3)
5.3 Examples for Practical Simulations in Real Poly-Atomic Systems
107(6)
6 Extension of Molecular Dynamics, Enhanced Sampling and the Free-Energy Calculations
113(24)
6.1 Umbrella Sampling and Adaptive Umbrella Sampling Methods
115(9)
6.2 Metadynamics
124(2)
6.3 Integrated Tempering Sampling
126(3)
6.4 Thermodynamic Integration
129(8)
7 Quantum Nuclear Effects
137(82)
7.1 Path Integral Molecular Simulations
140(35)
7.1.1 Path Integral Representation of the Propagator
140(3)
7.1.2 Path Integral Representation of the Density Matrix
143(5)
7.1.3 Statistical Mechanics: Path Integral Molecular Simulations
148(8)
7.1.4 Staging and Normal-Mode Transformations
156(10)
7.1.5 Evaluation of the Zero-Point Energy
166(9)
7.2 Extensions Beyond the Statistical Studies
175(9)
7.2.1 Different Semiclassical Dynamical Methods
176(2)
7.2.2 Centroid Molecular Dynamics and Ring-Polymer Molecular Dynamics
178(6)
7.3 Free Energy with Anharmonic QNEs
184(4)
7.4 Examples
188(29)
7.4.1 Impact of QNEs on Structures of the Water-Hydroxyl Overlayers on Transition Metal Surfaces
188(8)
7.4.2 Impact of Quantum Nuclear Effects on the Strength of Hydrogen Bonds
196(9)
7.4.3 Quantum Simulation of the Low-Temperature Metallic Liquid Hydrogen
205(12)
7.5 Summary
217(2)
Appendix A Useful Mathematical Relations
219(4)
A.1 Spherical Harmonics
219(1)
A.2 Plane Waves
220(1)
A.3 Fourier Transform
220(1)
A.4 Spherical Coordinates
221(1)
A.5 The Step(Heaviside) Function
221(2)
Appendix B Expansion of a Non-Local Function
223(4)
Appendix C The Brillouin-Zone Integration
227(16)
C.1 The Linear Tetrahedron Method
227(6)
C.1.1 The Isoparametric Transfromation
229(3)
C.1.2 Integrals in One Tetrahedron
232(1)
C.1.3 The Integration Weights
232(1)
C.2 Tetrahedron Method for q-Dependent Brillouin-Zone Integration
233(10)
C.2.1 Isoparametric Transformation
235(1)
C.2.2 The Integration Region
236(1)
C.2.3 Polarizability
237(1)
C.2.3.1 Polarizability on the Real Frequency Axis
238(2)
C.2.3.2 Polarizability on the Imaginary Frequency Axis
240(3)
Appendix D The Frequency Integration
243(2)
References 245(16)
Acknowledgements 261(2)
Index 263
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