Zero-variance zero-bias quantum Monte Carlo estimators of the spherically and system-averaged pair density

Cornell Theory Center, Cornell University, Ithaca, New York 14853, USA.
The Journal of Chemical Physics (Impact Factor: 2.95). 07/2007; 126(24):244112. DOI: 10.1063/1.2746029
Source: PubMed


We construct improved quantum Monte Carlo estimators for the spherically and system-averaged electron pair density (i.e., the probability density of finding two electrons separated by a relative distance u), also known as the spherically averaged electron position intracule density I(u), using the general zero-variance zero-bias principle for observables, introduced by Assaraf and Caffarel. The calculation of I(u) is made vastly more efficient by replacing the average of the local delta-function operator by the average of a smooth nonlocal operator that has several orders of magnitude smaller variance. These new estimators also reduce the systematic error (or bias) of the intracule density due to the approximate trial wave function. Used in combination with the optimization of an increasing number of parameters in trial Jastrow-Slater wave functions, they allow one to obtain well converged correlated intracule densities for atoms and molecules. These ideas can be applied to calculating any pair-correlation function in classical or quantum Monte Carlo calculations.

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    • "Even for a fixed probability distribution, it is possible to use various estimators for X, some of which have smaller variance than others, since one has the freedom to add any quantity with zero expectation value. This has been exploited to construct improved estimators for diverse observables [16] [17] [18] [19] [20]. There is often a compromise to be found between a low computation time per iteration t s and a low variance V[X] "
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