The covariance term

Using the definition of the total sample variance we have

$$\begin{align*} \sigma^2_{m}& = \frac{\sigma^2}{m}+\frac{2}{mn^2}\sum_{i=1}^{m} \sum_{j < k}^{n}\tilde{x}_{ij}\tilde{x}_{ik}. \end{align*}$$

The first term is what we have used till now in order to estimate the standard deviation. However, the second term which gives us a measure of the correlations between different stochastic events, can result in contributions which give rise to a larger standard deviation and variance $\sigma_m^2$. Note also the evaluation of the second term leads to a double sum over all events. If we run a VMC calculation with say $10^9$ Monte carlo samples, the latter term would lead to $10^{18}$ function evaluations. We don't want to, by obvious reasons, to venture into that many evaluations.

Note also that if our stochastic events are iid then the covariance terms is zero.

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