Variational Monte Carlo Flexibility

VMC technique merges Monte Carlo integration [55, 56]) with the energy Variational Principle [57]. It is basically a statistical method to compute quantum expectation values once an approximate wave function ^t is given. More specifically, VMC allows one to obtain a statistical estimate of the quantity where /(R) = ^r(R-)2 is a normalized probability distribution of the random variable R, and Oioc(Rfc) is the numerical value of the local operator Oioc at the point R* in configuration space. In the right-hand side of Eq. 2 the Na points Rfc are distributed in such a way to "sample" the distribution/: if an infinite number of points were drawn and collected, their density would reproduce the form of the distribution / [58],

Since we are restricted to sample only a finite number Ns of points, Eq. 2 represents only an estimate of the exact value of the integral. It can be shown [55] that the statistical error in estimating (O) using Eq. 2 decreases following (iVs)-1/2 for large Ns. This error is also proportional to the square root of the variance of OiOC(R) over the distribution /, i.e.

if Var(O) is bound. Differently, although the mean value in Eq. 2 might still exist, it is no longer possible to give an estimate of the error, and the result of the Monte Carlo integration might be difficult to interpret. It is interesting to note that the statistical error of the estimate does not depend explicitly on the dimensionality of the configurational space, but only on the properties of the local operator via Var(0), and on the number of sampled points. When the dimensionality Ds of the problem grows, this property makes Monte Carlo integration advantageous with respect to ordinary lattice methods, that, to achieve a chosen error, require a number of points that grows exponentially with Ds. As a rough estimate, Monte Carlo integration beats lattice methods already when Ds = 4 ~ 6, especially for non smooth functions.

If Ofa = Eioc = H^t/^t, Eq. 2 and Eq. 3 allow to compute the mean energy and its variance, and so to optimize a trial wave function in order to obtain the "best" possible description for a given system. Specifically, both (H) and Var(H) have their minimum value if and only if ifx is the exact wave function. Therefore, minimizing {H) and

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