Optimization of Ground- and Excited-State Wave Functions and van der Waals Clusters

Abstract

A quantum Monte Carlo method is introduced to optimize excited-state trial wave functions. The method is applied in a correlation function Monte Carlo calculation to compute ground- and excited-state energies of bosonic van der Waals clusters of up to seven particles. The calculations are performed using trial wave functions with general three-body correlations.

Energy and Variance Optimization of Many-Body Wave Functions

“…In fact, any method that attempts to minimize the energy, by minimizing the energy evaluated on a finite sample of Monte Carlo points, is bound to require a very large sample and therefore be highly inefficient for the reason discussed in the introduction. Our modifications of the straightforward expressions for the gradient and Hessian are similar in spirit to the work of Nightingale and Melik-Alaverdian [9]. A straightforward minimization of the energy on a Monte Carlo sample results in a symmetric Hamiltonian matrix, but they derive a nonsymmetric Hamiltonian matrix that yields exact parameters from a finite sample in the limit that the basis functions span an invariant subspace.…”

“…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].…”

Amplitude Determinant Coupled Cluster with Pairwise Doubles

“…We variationally optimize the energy of the pADCCD wave function using the linear method (LM) [26,27] where |Ψ x and |Ψ y are shorthand for derivatives of |Ψ with respect to the xth and yth wave function parameters µ x and µ y , respectively, and Ψ 0 ≡ |Ψ . After solving this eigenvalue problem for c, one updates the parameters by…”