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 non-local 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.
Deep Dive into Zero-variance zero-bias quantum Monte Carlo estimators of the spherically and system-averaged pair density.
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 non-local 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 calculatin
Two-electron distribution functions occupy an important place in electronic structure theory between the simplicity of one-electron densities and the complexity of the many-electron wave function. In particular, the systemaveraged electron pair density, the probability density of finding two electrons separated by the relative position vector u, also known as the electron position intracule density I(u), plays an important role in qualitative and quantitative descriptions of electronic systems. Position intracule densities have been extensively used to analyze shell structure, electron correlation, Hund's rules and chemical bonding (see, e.g., Refs. . Density functional theory-like approaches have been proposed based on the position intracule density [32][33][34][35][36] or on the closelyrelated Wigner intracule density [37][38][39].
Position intracule densities have been extracted from experimental X-ray scattering intensities for small atoms and molecules [40][41][42]. They have been calculated in the Hartree-Fock (HF) approximation for systems ranging from small atoms to large molecules [7,12,16,19,21,27,[43][44][45][46]. Calculations using common quantum chemistry correlated methods such as second-order Møller-Plesset perturbation theory, multi-configurational self-consistent-field (MCSCF) and configuration interac-tion approaches have been limited to atoms and small molecules [6,11,14,18,24,28,47,48]. Very accurate calculations using Hylleraas-type explicitly-correlated wave functions have been done only for the helium and lithium isoelectronic series [1,5,8,15,17,[49][50][51][52][53]. Variational Monte Carlo (VMC) calculations using Jastrow-Slater wave functions have been used to compute correlated position intracule densities for atoms from helium to neon and some of their isoelectronic series [22, 23, 25, 26, 29-31, 52, 54, 55]. In this paper, we show that the calculation of position intracule densities using quantum Monte Carlo (QMC) methods can be made much more accurate and efficient, opening new possibilities of investigation.
The position intracule density associated with an N -electron (real) wave function Ψ(R), where R = (r 1 , r 2 , …, r N ) is the 3N -dimensional vector of electron coordinates (ignoring spin for now), is defined as the quantum-mechanical average of the delta-function operator δ(r iju)
where r ij = r jr i and the sum is over all electron pairs, and its spherical average is
Among the most important properties of I(u) are the normalization sum rule (giving the total number of elec-tron pairs)
the electron-electron cusp condition [56] dI(u) du
and, for finite systems, the exponential decay at large u determined by the spherically averaged one-electron density n(u) evaluated at the distance u from the chosen origin
where I is the vertical ionization energy (see Refs. [57][58][59].
The moments of the radial intracule density M k = ∞ 0 du 4πu 2+k I(u) are related to physical observables [60], in particular M -1 is just the electron-electron Coulomb interaction energy
In standard correlated methods based on an expansion of the wave function in Slater determinants, the important short-range part of the position intracule density converges very slowly with the one-electron and manyelectron basis. The advantage of employing QMC methods [61] lies in the possibility of using compact, explicitlycorrelated wave functions which are able to describe properly the short-range part of I(u). However, the problem in this approach is that one calculates the average of a delta-function operator which has an infinite variance. As for other probability densities, the standard procedure in QMC approaches simply consists of counting the number of electron pairs separated by the distance u within ∆u encountered in the Monte Carlo run. More precisely, e.g. in VMC calculations, I(u) is estimated as the statistical average
over the trial wave function density Ψ(R) 2 , of the local “histogram” estimator I histo L (u, R)
where
Here and in the following, f (R) 2 . For u = 0, this histogram estimator has a finite but large variance for small u or small ∆u. Although the use of importance sampling can help decrease the variance [22,62,63], the calculation of position intracule densities in QMC remains very inefficient: very long runs have to be performed to reach an acceptably small statistical uncertainty. Moreover, the histogram estimator has a discretization error : even in the limit of an infinite sample M → ∞, I histo (u) remains only an approximation of first order in ∆u to the position intracule density I(u) of the wave function Ψ(R). Note however that it is possible to greatly reduce the discretization error by choosing a flexible analytic form for I(u) that obeys known conditions (such as the cusp condition of Eq. ( 4)) and fitting in each interval ∆u the integral of I(u) (instead of I(u)) to the computed data. This approach has been used to calculate radial electron densities for atoms [
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