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## Accumulating Averages

In the flow chart of the VMC algorithm, (figure ), 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 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 to . It is possible simply to accumulate at just the new points on the walk, but the combination in Eq.(), 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 gives a correct sampling of the probability density, . This can be demonstrated by considering each term, , and , 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.

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

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

The probability of being at and rejecting any move is

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

Next: Performing VMC calculations of Up: Variational Quantum Monte Carlo Previous: Electrostatic Energies

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