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Schödinger's Equation
Phys 5870: Modern Computational Methods in Solids
Adrian E. Feiguin
Schödinger's Equation
Variational Methods
Examples of linear variational calculations
The infinite potential well
Exercise 2.1: Infinite potential well
Hydrogen atom
Exercise 2.2: Hydrogen atom
The Hartree-Fock method
The Born-Oppenheimer approximation
The helium atom
A program for the helium ground-state
Exercise 3.1: Helium atom
Many electron systems and the Slater determinant
Two-particle case
General case: the many body wave-function
Hartree-Fock theory
The Hartree-Fock equations
Koopman's theorem
Matrix form of the Hartree-Fock equations
Density Functional Theory
Introduction
Some considerations about exchange
Functionals and functional derivatives
The Coulomb (Thomas-Fermi) functional
Hohenberg-Kohn theorems
H-K theorem I
H-K theorem II
DFT formalism and derivation of the Kohn-Sham equations
Non-interacting case
Interacting system
The local density approximation - LDA
Limitations
More about exchange
Solution to the Kohn-Sham equations
Pros and Cons of the DFT
Methods for band-structure calculations
The tight-binding approximation
General case: Linear Combination of Atomic Orbitals
Example 5.1: Single s band
Some remarks on the tight-binding method
Limitations of the tight-binding model
Plane Waves
Matrix elements
Orthogonalized plane waves
The Pseudopotential Method
Construction of pseudopotentials
Empirical pseudo-potentials
The cellular (Wigner-Seitz) method
Remarks about the cellular method
The Muffin-tin potential
The Augmented plane-wave method (APW)
Matching the boundary conditions
Matrix elements
Some remarks about the APW method
The LAPW method
Adding electron-electron interactions
Random sequences
Pseudo-random number generators
Testing for randomness and uniformity
Moments
Autocorrelation
Visual test
Statistical errors
Non-uniform random distributions
Exponential distribution
von Neumann rejection
Random walk methods: the Metropolis algorithm
Exercise 9.1: The Gaussian distribution
Monte Carlo integration
Simple Monte Carlo integration
Monte Carlo error analysis
Exercise 10.1: One dimensional integration
Exercise 10.2: Importance of randomness
Variance reduction
Importance Sampling
Exercise 10.3: Importance sampling
Exercise 10.4: The Metropolis algorithm
Monte Carlo Simulation
The Canonical Ensemble
The Metropolis algorithm
Exercise 13.1: Classical gas in 1D
The Ising model
Boundary conditions
Physical quantities
Exercise: One-dimensional Ising model
Simulation of the 2D Ising model
The heat capacity
The magnetic susceptibility
Metropolis algorithm
Boundary conditions
Initial conditions and equilibration
Tricks
Exercise 13.3: Equilibration of the 2D Ising model
Measuring observables
Exercise 13.4: The correlation time
Exercise 13.5: Comparison with exact results
The Ising phase transition
Exercise 13.6: Qualitative behavior of the 2D Ising model
Exercise 13.7: Critical slowing down
Quantum Monte Carlo
Variational Monte Carlo
World Line Monte Carlo
Suzuki-Trotter transformation and the equivalent classical system
Monte Carlo simulation with worldlines
Measurement and averaging
Determinantal (or Auxiliary Field) Monte Carlo
Projector Monte Carlo
Sign problem revisited
Bibliography
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Adrian E. Feiguin 2009-11-04