ISBN: 3-540-63961-6
TITLE: Computational Materials Science
AUTHOR: Ohno, Kaoru; Esfarjani, Keivan; Kawazoe, Yoshiyuki
TOC:

1. Introduction 1 
1.1 Computer Simulation as a Tool for Materials Science 1 
1.2 Modeling of Natural Phenomena 2 
2. Ab Initio Methods 7 
2.1 Introduction 7 
2.2 Electronic States of Many-Particle Systems 8 
2.2.1 Quantum Mechanics of Identical Particles 8 
2.2.2 The HartreeFock Approximation 11 
2.2.3 Density Functional Theory 18 
2.2.4 Periodic Systems 27 
2.2.5 Group Theory 32 
2.2.6 LCAO, OPW and Mixed-Basis Approaches 39 
2.2.7 Pseudopotential Approach 46 
2.2.8 APW Method 49 
2.2.9 KKR, LMTO and ASW Methods 52 
2.2.10 Some General Remarks 55 
2.2.11 Ab Initio O(N) and Related Methods 58 
2.3 Perturbation and Linear Response 64 
2.3.1 Effective-Mass Tensor 64 
2.3.2 Dielectric Response 66 
2.3.3 Magnetic Susceptibility 69 
2.3.4 Chemical Shift 71 
2.3.5 Phonon Spectrum 73 
2.3.6 Electrical Conductivity 75 
2.4 Ab Initio Molecular Dynamics 77 
2.4.1 CarParrinello Method 78 
2.4.2 Steepest Descent and Conjugate Gradient Methods 80 
2.4.3 Formulation with Plane Wave Basis 81 
2.4.4 Formulation with Other Bases 83 
2.5 Applications 85 
2.5.1 Application to Fullerene Systems 85 
2.5.2 Application to Point Defects in Crystals 93 
2.5.3 Application to Other Systems 95 
2.5.4 Coherent Potential Approximation 98 
2.6 Beyond the BornOppenheimer Approximation 100 
2.7 Electron Correlations Beyond the LDA 105 
2.7.1 Generalized Gradient Approximation 106 
2.7.2 Self-Interaction Correction 107 
2.7.3 GW Approximation 109 
2.7.4 Exchange and Coulomb Holes 115 
2.7.5 Optimized Effective Potential Method 116 
2.7.6 Time-Dependent Density Functional Theory 118 
2.7.7 Inclusion of Ladder Diagrams 119 
2.7.8 Further Remarks: Cusp Condition, etc. 122 
References 123 
3. Tight-Binding Methods 139 
3.1 Introduction 139 
3.2 Tight-Binding Formalism 140 
3.2.1 Tight-Binding Parametrization 141 
3.2.2 Calculation of the Matrix Elements 143 
3.2.3 Total Energy 144 
3.2.4 Forces 145 
3.3 Methods to Solve the Schrdinger Equation 
for Large Systems 146 
3.3.1 The Density Matrix O(N) Method 147 
3.3.2 The Recursion Method 151 
3.4 Self-Consistent Tight-Binding Formalism 153 
3.4.1 Parametrization of the Coulomb Integral U 157 
3.5 Applications to Fullerenes, Silicon 
and Transition-Metal Clusters 158 
3.5.1 Fullerene Collisions 159 
3.5.2 C_240 Doughnuts and Their Vibrational Properties 159 
3.5.3 IR Spectra of C_60 and C_60 Dimers 160 
3.5.4 Simulated Annealing of Small Silicon Clusters 164 
3.5.5 Titanium and Copper Clusters 165 
3.6 Conclusions 167 
References 168 
4. Empirical Methods and Coarse-Graining 171 
4.1 Introduction 171 
4.2 Reduction to Classical Potentials 172 
4.2.1 Polar Systems 173 
4.2.2 Van der Waals Potential 175 
4.2.3 Potential for Covalent Bonds 177 
4.2.4 Embedded-Atom Potential 179 
4.3 The ConnollyWilliams Approximation 181 
4.3.1 Lattice Gas Model 181 
4.3.2 The ConnollyWilliams Approximation 182 
4.4 Potential Renormalization 184 
4.4.1 Basic Idea: Two-Step Renormalization Scheme 184 
4.4.2 The First Step 186 
4.4.3 The Second Step 188 
4.4.4 Application to Si 190 
References 193 
5. Monte Carlo Methods 195 
5.1 Introduction 195 
5.2 Basis of the Monte Carlo Method 196 
5.2.1 Stochastic Processes 197 
5.2.2 Markov Process 199 
5.2.3 Ergodicity 202 
5.3 Algorithms for Monte Carlo Simulation 206 
5.3.1 Random Numbers 206 
5.3.2 Simple Sampling Technique 208 
5.3.3 Importance Sampling Technique 212 
5.3.4 General Comments on Dynamic Models 214 
5.4 Applications 216 
5.4.1 Systems of Classical Particles 216 
5.4.2 Modified Monte Carlo Techniques 219 
5.4.3 Percolation 223 
5.4.4 Polymer Systems 225 
5.4.5 Classical Spin Systems 230 
5.4.6 Nucleation 243 
5.4.7 Crystal Growth 247 
5.4.8 Fractal Systems 251 
References 263 
6. Quantum Monte Carlo (QMC) Methods 271 
6.1 Introduction 271 
6.2 Variational Monte Carlo (VMC) Method 271 
6.3 Diffusion Monte Carlo (DMC) Method 274 
6.4 Path-Integral Monte Carlo (PIMC) Method 278 
6.5 Quantum Spin Models 281 
6.6 Other Quantum Monte Carlo Methods 282 
References 282 
Appendix 
A. Molecular Dynamics and Mechanical Properties 285 
A.1 Time Evolution of Atomic Positions 287 
A.2 Acceleration of Force Calculations 291 
A.2.1 ParticleMesh Method 291 
A.2.2 The GreengardRockhlin Method 292 
References 296 
B. Vibrational Properties 297 
References 300 
C. Calculation of the Ewald Sum 301 
References 304 
D. Optimization Methods Used in Materials Science 305 
D.1 Conjugate-Gradient Minimization 305 
D.2 Broyden's Method 307 
D.3 SA and GA as Global Optimization Methods 309 
D.3.1 Simulated Annealing (SA) 309 
D.3.2 Genetic Algorithm (GA) 310 
References 312 
Index 315 
END
