Monte Carlo

Algo for performing Monte Carlo:

  • Initialization: Fix the number of MC steps, Choose initial \(R\) and variational parameters \(\alpha\) and calculate \(|\psi_T^\alpha (R)|^2\).
  • Initialise the energy and the variance and start the MC calc.
    • Calc a trail pos \(R_p = R + r \cdot step\) wehere \(r\) is a random var $r \in [0,1].
    • Metropolis algo to accept or reject this move \(w = P(R_p)/P(R)\)
    • If the step is accepted, then we set \(R = R_p\)
    • Update averages
  • Finish and compute final averages

General MCMC

  • We have ProbDist (possibly high dimensionality). Want to sample from it or find expected value of it. Analytical solution is implausible.
  • MCMC:
    • Perform “random walk” through probdist favoring values with higher probs.
    • Starting point -> pick random nearby point -> evaluate its probability. If it has higher prob than start: move there, otherwise stay (or move to that point with low prob)
    • Then we are visiting every point in the probdist propotional to how probable that point is.

Markov chain

  • Can be said in two ways:
    • The next state is only dependent on the current state
    • The current state is only dependent on the previous state
  • Requirement:
    • The chain has to be ergodic:
      • It must visit every point in the domain, and will visit tem a proportionate amount to their probability.
      • To be ergodic, the chain must be:
        • Irreducible - for every state there is a positive probability of moving to any other state
        • Aperiodic - the chain must not get trapped in cycles

Variational Monte Carlo

  • Can include functions from different sub-files
  • Must have some initial variables: Seed, number of: particles, steps, dimensions
  • How to choose step size(for Harmonic Oscillator):
    • SE from the ??
    • \(k(r) = u(r)/r\)
    • \(r(0) = const\), \(r(\infty) = 0\)
    • \(u(0) = 0\), \(u(\infty) = 0\)
    • Harmonic Oscill: \(-\frac{\hbar^2}{2m} \frac{d^2}{dr^2}u(r) + 1/2 m \omega^2r^2u(r) = E u(r)\)
    • Scale equations, make them dimensionless (to atomic units)
      • \(\rho = \gamma r\) , \([r] = length\), \([\gamma] = length^{-1}\)
      • \(r = \rho/\gamma\)
      • \(-\frac{\hbar^2\gamma^2}{2m} \frac{d^2}{d\rho^2}u + 1/2 m \omega^2\rho^2/\gamma^2 u = E u\)
      • Can choose and tune \(\gamma\) as we want to remove all the constants. Done by setting all constants = 1 and solving for gamma.
      • Here: \(\gamma = \sqrt{\hbar/m\omega}\)
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