Gamma Distribution Model To Provide a Direct Assessment of the Overall Quality of Quantum Monte Carlo-Generated Electron Distributions † Braden Coles, Paul Vrbik, Robert D. Giacometti, and Stuart M. Rothstein* Department of Chemistry, Brock UniVersity, St. Catharines, Ontario L2S 3A1 Canada ReceiVed: July 23, 2007; In Final Form: NoVember 19, 2007 Our objective is to assess the accuracy of simulated quantum Monte Carlo electron distributions of atoms and molecules. Our approach is first to model the exact electron distribution by a linear combination of gamma distribution functions, with parameters chosen to exactly reproduce highly accurate literature values for a number of selected moments for the system of interest. In application to the ground-state electron distributions of helium and dihydrogen, a high level of accuracy of the model was confirmed upon comparing its predicted moments, not used in the model's parametrization, to those calculated from high-level theory. Next, we generated electron-electron and electron-nucleus distributions for dihydrogen from electron positions outputted from a variety of quantum Monte Carlo algorithms. Upon juxtaposition of the simulated distributions with the putatively exact one that we derived from the model, we quantified the error in simulated distributions. The most accurate distributions were obtained from no-compromise reptation quantum Monte Carlo, a recently developed algorithm designed to ameliorate the distributions’ time-step bias. Marginally less accurate distributions were generated from fixed-node diffusion Monte Carlo with descendant counting and detailed balance. Introduction Decades of research devoted to developing quantum Monte Carlo methods have firmly established the advantages of this approach to electronic structure calculations: rapid convergence with basis-set size, favorable scaling with the number of electrons, no calculation and storage of large numbers of integrals, and codes that are naturally suited for parallel computation. Monographs 1,2 and recent reviews, e.g., ref 3, have been devoted to these issues. Benchmark calculations suggest that the most commonly employed variant, fixed-node diffusion Monte Carlo 4 (FNDMC), estimates the energy with accuracy on par with CCSD(T), although not yet to within chemical accuracy. 5 FNDMC samples the mixed distribution, ΨΦ 0 , where Φ 0 is “exact”, except for the incorrect nodes 6 imposed by the importance sampling function, Ψ. This so-called “nodal error” introduces a positive bias in the simulated energy. 7 In addition to this, there is yet another significant source of error that arises in both FNDMC and the more-recently developed reptation quantum Monte Carlo 8,9 (RQMC) ap- proach: “time-step bias”. This error is a result of mathematical moves being made in accordance with the so-called “short-time Green's function”, G 0 G b . This quantity is exact only in the limit of zero time step: a small interval of imaginary simulation time. 10-12 Although in practice there are elegant ways to improve the sampling, e.g., refs 13 and 14, and thus reduce the bias, in principle the time step, and consequently the bias associated with its use, cannot be reduced to zero. Alternatively, of course, one may simply reduce the value used for the time step, but this adversely affects the efficiency of the simulation. 15 Normally, expectation values of operators that do no commute with the Hamiltonian are biased by the inputted importance sampling function. Nevertheless, both FNDMC and variational Monte Carlo (VMC) recover expectation values as if they had been drawn from the “exact” distribution, Φ 0 2 , by employing methods of “descendant counting”, e.g., refs 16 and 17, and by averaging variationally distributed quantities with accumulated past and future branching factors, e.g., refs 18 and 19, respectively. On the other hand, RQMC algorithms directly sample the “exact” distribution, at the middle of the reptile. Here and above we qualified the word “exact”, as all of these distributions suffer from time-step and nodal error. We are presently concerned with assessing the accuracy of simulated quantum Monte Carlo electron-electron and electron- nucleus distributions. The normal practice, comparing the simulated energy and a selection of other properties with accurate determinations in the literature, is indirect and monitors only isolated regions of the distributions. Instead, our objective is to provide a direct assessment of the overall quality of the simulated distributions. Our approach is first to model the unknown, truly exact electron distributions by linear combina- tions of gamma distributions. The model is parametrized to reproduce literature results for a broad range of moments, and its efficacy verified by its accurately reproducing literature values for moments not used in its parametrization. Next, curves for quantum Monte Carlo-generated distributions are constructed by spline-fitting histograms of simulated electron positions. Finally, errors in the simulated distributions are revealed upon juxtaposition of the simulated distributions with the putatively exact ones that are derived from the model. This paper is organized as follows: In the following section we introduce the gamma distribution model and apply it to electron-nucleus and electron-electron distributions of ground- † Part of the “William A. Lester, Jr., Festschrift”. * Corresponding author. Department of Physics. Phone: 905-688-5550 x3401. Fax: 905-682-9020. E-mail: srothste@brocku.ca. 2012 J. Phys. Chem. A 2008, 112, 2012-2017 10.1021/jp075790e CCC: $40.75 © 2008 American Chemical Society Published on Web 02/06/2008