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arXiv:cond-mat/0412634 · Physical Review Letters

Energy and Variance Optimization of Many-Body Wave Functions

Cyrus J. Umrigar, Claudia Filippi

Published Apr 21, 2005• 196 citations
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Abstract

We present a simple, robust, and efficient method for varying the parameters in a many-body wave function to optimize the expectation value of the energy. The effectiveness of the method is demonstrated by optimizing the parameters in flexible Jastrow factors that include 3-body electron-electron-nucleus correlation terms for the NO2 and decapentaene (C10H12) molecules. The basic idea is to add terms to the straightforward expression for the Hessian of the energy that have zero expectation value, but that cancel much of the statistical fluctuations for a finite Monte Carlo sample. The method is compared to what is currently the most popular method for optimizing many-body wave functions, namely, minimization of the variance of the local energy. The most efficient wave function is obtained by optimizing a linear combination of the energy and the variance.

DOI
10.1103/physrevlett.94.150201

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We study three wave function optimization methods based on energy minimization in a variational Monte Carlo framework: the Newton, linear, and perturbative methods. In the Newton method, the parameter variations are calculated from the energy gradient and Hessian, using a reduced variance statistical estimator for the latter. In the linear method, the parameter variations are found by diagonalizing a nonsymmetric estimator of the Hamiltonian matrix in the space spanned by the wave function and its derivatives with respect to the parameters, making use of a strong zero-variance principle. In the less computationally expensive perturbative method, the parameter variations are calculated by approximately solving the generalized eigenvalue equation of the linear method by a nonorthogonal perturbation theory. These general methods are illustrated here by the optimization of wave functions consisting of a Jastrow factor multiplied by an expansion in configuration state functions (CSFs) for the C2 molecule, including both valence and core electrons in the calculation. The Newton and linear methods are very efficient for the optimization of the Jastrow, CSF, and orbital parameters. The perturbative method is a good alternative for the optimization of just the CSF and orbital parameters. Although the optimization is performed at the variational Monte Carlo level, we observe for the C2 molecule studied here, and for other systems we have studied, that as more parameters in the trial wave functions are optimized, the diffusion Monte Carlo total energy improves monotonically, implying that the nodal hypersurface also improves monotonically.
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Gutzwiller functions are popular variational wave functions for correlated electrons in Hubbard models. Following the variational principle, we are interested in the Gutzwiller parameters that minimize, e.g., the expectation value of the energy. Rewriting the expectation value as a rational function in the Gutzwiller parameters, we find a very efficient way for performing that minimization. The method can be used to optimize general Gutzwiller-type wave functions both in variational and in fixed-node diffusion Monte Carlo.