Trial wavefunctions Monte Carlo sampling

In variational Monte Carlo (VMC), one samples, using the Metropolis rejection method, the square of an assumed trial wavefunction, j where I = r, are the coordinates of all the particles... 

To obtain the importance sampling version of diffusion quantum Monte Carlo, we first multiply the basic equation, Eq. [9] by a trial wavefunction /q and define a new function f - x A /o, which is the product of the true wavefunction and the trial wavefunction. After several pages of rearrangement, one may obtain the basic equation for DQMC with importance sampling, ... 

Importance sampling, difference schemes, and their combinations all have the desirable characteristic of giving small errors for good trial wavefunctions and no errors in the limit of exact trial wavefunctions. Difference calculations have the additional desirable characteristic of correcting good trial wavefunctions to yield better ones. Rather than calculate a complete wavefunction, one may calculate the much smaller correction to a trial wavefunction. The statistical error normally associated with Monte Carlo calculations may then be limited to the correction term and thus reduced in size. 

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