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

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Roland Assaraf, Jul 02, 2015 Available from:- [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 · 2.33 Impact Factor - [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 · 3.12 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this work, an improved approach for computing cluster dissociation rates using Monte Carlo (MC) simulations is proposed and a discussion is provided on its applicability as a function of environmental variables (e.g., temperature). With an analytical transformation of the integrals required to compute variational transition state theory (vTST) dissociation rates, MC estimates of the expectation value for the Dirac delta delta(q(rc)-q(c)) have been made free of the discretization error that is present when a prelimit form for delta is used. As a by-product of this transformation, the statistical error associated with delta(q(rc)-q(c)) is reduced making this step in the calculation of vTST rates substantially more efficient (by a factor of 4-2500, roughly). The improved MC procedure is subsequently employed to compute the dissociation rate for Lennard-Jones clusters X(13-n)Y(n) (n=0-3) as a function of temperature (T), composition, and X-Y interaction strength. The X(13-n)Y(n) family has been previously studied as prototypical set of systems for which it may be possible to select and stabilize structures different from the icosahedral global minimum of X(13). It was found that both the dissociation rate and the dissociation mechanism, as suggested by the statistical simulations, present a marked dependence on n, T, and the nature of Y. In particular, it was found that a vacancy is preferentially formed close to a surface impurity when the X-Y interaction is weaker than the X-X one whatever the temperature. Differently, the mechanism was found to depend on T for stronger X-Y interactions, with vacancies being formed opposite to surface impurities at higher temperature. These behaviors are a reflex of the important role played by the surface fluctuations in defining the properties of clusters.The Journal of Chemical Physics 07/2008; 128(24):244515. DOI:10.1063/1.2937914 · 3.12 Impact Factor