Quantum Monte Carlo for Atoms, Molecules and Solids
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VMCThe VMC method exploits the QMC formalism above to evaluate the expectation values of a known trial function, ΨT, rather than the ground state Φ0 [Reynolds3]. This is useful when the function ΨT cannot be integrated analytically or by other numerical methods. The method is especially valuable for investigating novel wave function forms. VMC is also used as a precursor to DMC as a method to optimize  and to generate initial walker distributions.
The importance sampled Schrödinger equation without the local-energy dependent term is a Fokker-Plank equation with solution
One can generate an ensemble of pseudo particles {Ri} via the related Langevin Equation, which can be integrated over a small time step  , to give a prescription for moving a pseudo particle with coordinates R to a new position R′,
Here D is the diffusion constant (1/2) and χ is a Gaussian random variable with zero mean and variance of 2Dδτ. Once the ensemble has converged to the distribution  properties such as the local energy can be sampled. In practice Metropolis sampling is introduced to further speed convergence and eliminate any dependence on the time step.
Trial functions for VMC can be constructed in many ways and specifics are treated separately below. Download 242 Kb. Share with your friends:
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