Phys 5870: Modern Computational Methods in Solids

Adrian E. Feiguin

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• 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
• About this document ...

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Adrian E. Feiguin 2009-11-04

Reference: https://web.northeastern.edu/afeiguin/phys5870/phys5870/phys5870.html