Simple and efficient approach to the optimization of correlated wave functions

Abstract

We present a simple and efficient method to optimize within energy minimization the determinantal component of the many-body wave functions commonly used in quantum Monte Carlo calculations. The approach obtains the optimal wave function as an approximate perturbative solution of an effective Hamiltonian iteratively constructed via Monte Carlo sampling. The effectiveness of the method as well as its ability to substantially improve the accuracy of quantum Monte Carlo calculations is demonstrated by optimizing a large number of parameters for the ground state of acetone and the difficult case of the 1 1 B1u state of hexatriene.Over the last decade, quantum Monte Carlo (QMC) methods have been employed to accurately compute electronic properties of large molecular and solid systems where conventional quantum chemistry approaches are extremely difficult to apply [1]. A crucial step in both the variational (VMC) and the diffusion Monte Carlo (DMC) approach is the construction of the trial wave function Ψ T which is usually chosen of the Jastrow-Slater form, that is Ψ T = J Φ, where Φ is a small expansion in Slater determinants and J the positive Jastrow correlation factor.Although considerable progress has been made (principally using the variance minimization approach [2]) in the construction of optimal Jastrow factors, relatively little attention has been given to the optimization of the determinantal part of the wave function. Methods such as Hartree-Fock (HF) or a small scale configuration interaction (CI) are used as a practical way of constructing the determinantal component, which is generally not reoptimized when the Jastrow factor is added. 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. The use of an estimate of the Hessian characterized by reduced statistical fluctuations [9, 10] yields a simple and robust optimization algorithm for the Jastrow 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 i...

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