Optimization of configuration interaction coefficients in multideterminant Jastrow–Slater wave functions

Abstract

A quantum Monte Carlo method for obtaining multideterminant Jastrow-Slater wave functions for which the energy is stationary with respect to variations of CI coefficients is presented. It is a generalization of a recently developed approach to the optimization of single particle functions ͓C. Filippi and S. Fahy, J. Chem. Phys. 112, 3523 ͑2000͔͒. Using ground state calculations of the atoms Be, C, and Ne and the dimer Si 2 as illustrative examples, the method is shown to converge rapidly and to significantly lower the energy in most cases.

Energy and Variance Optimization of Many-Body Wave Functions

“…The reason is that, for a sufficiently flexible variational wave function, it is possible to lower the energy on the finite set of Monte Carlo (MC) configurations on which the optimization is performed, while in fact raising the true expectation value of the energy. On the other hand, if the variance of the local energy is minimized, each term in the sum over MC configurations is bounded from below by zero and the problem is far less severe [5].Nevertheless, in recent years several clever methods have been invented that optimize the energy rather than the variance [6,7,8,9,10,11,12,13,14,15]. The motivations for this are four fold.…”

“…Nevertheless, in recent years several clever methods have been invented that optimize the energy rather than the variance [6,7,8,9,10,11,12,13,14,15]. The motivations for this are four fold.…”

“…The generalized eigenvalue method of Nightingale and Melik-Alaverdian [9] is the most efficient choice for optimizing linear parameters, but for nonlinear parameters they use variance minimization. The effective fluctuation potential method [10,11,12,13,14] is the most successful method for nonlinear parameters and has been applied to optimizing the orbitals [11,14] and the linear coefficients in a multideterminantal wave function [12,14], and, has been extended to excited states [14]. It is not straightforward to use this method to optimize Jastrow factors, but Prendergast, Bevan and Fahy [13] have formulated a version for periodic systems and have optimized an impressively large number of parameters.…”

Simple and efficient approach to the optimization of correlated wave functions

“…However, the determinantal part of the wave function solely determines the DMC energy [3] and often needs to be reoptimized to obtain accurate results [4]. A practical and simple approach to calculate the optimal determinantal component is therefore particularly important to a wide and successful application of QMC methods.In recent years, several methods to optimize the wave function through energy minimization have been proposed [4,5,6,7,8,9,10,11,12,13,14,15,16]. A direct approach to energy minimization entails to compute the gradient and the Hessian of the energy with respect to the desired parameters.…”

“…However, the Hessian with respect to the orbital parameters in the determinant is affected by higher statistical noise so that devising a stable energy-minimization scheme is more difficult [10]: the resulting approach is for instance less stable that the simple stochastic reconfiguration (SR) method [11,12] and, during the optimization, may have to reduce to the inefficient SR to retain stability. To date, the most successful method remains the energy fluctuation potential (EFP) method [4,13,14,15,16] which determines the optimal determinantal component as the solution of an effective Hamiltonian iteratively constructed via Monte Carlo sampling. The approach has been used to optimize the orbitals [4,14], the linear coefficients in front of the determinants [4,15], and has been extended to excited states [4].…”