next up previous contents
Next: Performing VMC calculations of Up: Variational Quantum Monte Carlo Previous: Electrostatic Energies

Accumulating Averages


In the flow chart of the VMC algorithm, (figure gif), it is simply stated that the energy of the walker is accumulated after moving each of the electrons. In fact we choose a slightly more complicated formula for updating each of the quantities being calculated[26]. After each proposed move, whether it is rejected or not, each quantity <Q> is updated such that


where tex2html_wrap_inline6265 is any quantity of interest such as the kinetic energy, potential energy or total energy associated with particle i, and p is the probability of accepting the move from tex2html_wrap_inline6271 to tex2html_wrap_inline6273 . It is possible simply to accumulate tex2html_wrap_inline6275 at just the new points on the walk, but the combination in Eq.(gif), of values at the old and new points allows information about points which are rejected to be included and reduces the contribution from ``unlikely'' moves which are accepted. By this means, the variance of the expectation value of Q is reduced. It has been shown in Ref.[26] that the accumulation of tex2html_wrap_inline6279 gives a correct sampling of the probability density, tex2html_wrap_inline6281 . This can be demonstrated by considering each term, tex2html_wrap_inline6283 , and tex2html_wrap_inline6285 , separately and calculating the probability distribution which each term samples. By adding together these two probability distributions, it is shown that the combination of the two terms does indeed sample from the correct total probability distribution.

tex2html_wrap_inline6275 is the energy of particle i when the configuration is at tex2html_wrap_inline6291 The probability of being in configuration tex2html_wrap_inline6125 and evaluating the energy of particle i, i.e. the probability of arriving at configuration tex2html_wrap_inline6125 as a result of making a move of particle i to tex2html_wrap_inline6301 , should obviously be tex2html_wrap_inline6303 if the sampling is being done correctly.

At the old position of electron i, tex2html_wrap_inline6307 , electron i can move anywhere within the range of the maximum step size, i.e. within a volume V. The probability of arriving at tex2html_wrap_inline5837 from tex2html_wrap_inline6315 is tex2html_wrap_inline6317 . The probability of evaluating tex2html_wrap_inline6319 after accepting a move tex2html_wrap_inline6321 is therefore


The probability of being at tex2html_wrap_inline6125 and rejecting any move tex2html_wrap_inline6325 is


The sum of these two probabilities, i.e. accumulating tex2html_wrap_inline6327 as in Eq.(gif), gives the probability density tex2html_wrap_inline6303 , as required.

next up previous contents
Next: Performing VMC calculations of Up: Variational Quantum Monte Carlo Previous: Electrostatic Energies

Andrew Williamson
Tue Nov 19 17:11:34 GMT 1996