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.

Download full-text


Available from: Roland Assaraf, Oct 06, 2015
24 Reads
  • Source
    • "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] "
    [Show abstract] [Hide abstract]
    ABSTRACT: We provide a pedagogical introduction to the two main variants of real-space quantum Monte Carlo methods for electronic-structure calculations: variational Monte Carlo (VMC) and diffusion Monte Carlo (DMC). Assuming no prior knowledge on the subject, we review in depth the Metropolis-Hastings algorithm used in VMC for sampling the square of an approximate wave function, discussing details important for applications to electronic systems. We also review in detail the more sophisticated DMC algorithm within the fixed-node approximation, introduced to avoid the infamous Fermionic sign problem, which allows one to sample a more accurate approximation to the ground-state wave function. Throughout this review, we discuss the statistical methods used for evaluating expectation values and statistical uncertainties. In particular, we show how to estimate nonlinear functions of expectation values and their statistical uncertainties.
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: The combination of continuum many-body quantum physics and Monte Carlo methods provide a powerful and well established approach to first principles calculations for large systems. Replacing the exact solution of the problem with a statistical estimate requires a measure of the random error in the estimate for it to be useful. Such a measure of confidence is usually provided by assuming the central limit theorem to hold true. In what follows it is demonstrated that, for the most popular implementation of the variational Monte Carlo method, the central limit theorem has limited validity, or is invalid and must be replaced by a generalized central limit theorem. Estimates of the total energy and the variance of the local energy are examined in detail, and shown to exhibit uncontrolled statistical errors through an explicit derivation of the distribution of the random error. Several examples are given of estimated quantities for which the central limit theorem is not valid. The approach used is generally applicable to characterizing the random error of estimates, and to quantum Monte Carlo methods beyond variational Monte Carlo.
    Physical Review E 02/2008; 77(1 Pt 2):016703. DOI:10.1103/PhysRevE.77.016703 · 2.29 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We pursue the development and application of the recently introduced linear optimization method for determining the optimal linear and nonlinear parameters of Jastrow-Slater wave functions in a variational Monte Carlo framework. In this approach, the optimal parameters are found iteratively by diagonalizing the Hamiltonian matrix in the space spanned by the wave function and its first-order derivatives, making use of a strong zero-variance principle. We extend the method to optimize the exponents of the basis functions, simultaneously with all the other parameters, namely, the Jastrow, configuration state function, and orbital parameters. We show that the linear optimization method can be thought of as a so-called augmented Hessian approach, which helps explain the robustness of the method and permits us to extend it to minimize a linear combination of the energy and the energy variance. We apply the linear optimization method to obtain the complete ground-state potential energy curve of the C(2) molecule up to the dissociation limit and discuss size consistency and broken spin-symmetry issues in quantum Monte Carlo calculations. We perform calculations for the first-row atoms and homonuclear diatomic molecules with fully optimized Jastrow-Slater wave functions, and we demonstrate that molecular well depths can be obtained with near chemical accuracy quite systematically at the diffusion Monte Carlo level for these systems.
    The Journal of Chemical Physics 06/2008; 128(17):174101. DOI:10.1063/1.2908237 · 2.95 Impact Factor
Show more