Energy and Variance Optimization of Many-Body Wave Functions

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 ${\mathrm{NO}}_2$ and decapentaene (${\mathrm{C}}_{10}{\mathrm{H}}_{12}$) 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.

Figure 1

Energy of ${\mathrm{NO}}_2$ versus iteration number for energy minimization. Inset: the later iterations on an expanded scale and also the energies from minimizing the variance and minimizing the linear combination. The linear combination yields almost as good an energy as energy minimization.

Figure 2

Same as Fig. 1 but for the rms fluctuations of the local energy $\sigma$ rather than the energy. The linear combination $\sigma$ is half way between those from energy minimization and variance minimization.

Figure 3

The autocorrelation time, $T_{\mathrm{corr}}$, of ${\mathrm{NO}}_2$ versus iteration number. Energy minimization gives the smallest $T_{\mathrm{corr}}$, variance minimization the largest, and, the mixed minimization a value that is close to that from energy minimization.

Figure 4

Same as Fig. 1 but for decapentaene (${\mathrm{C}}_{10}{\mathrm{H}}_{12}$).