Variance-minimization scheme for optimizing Jastrow factors

We describe a scheme for optimizing many-electron trial wave functions by minimizing the unreweighted variance of the energy using stochastic integration and correlated-sampling techniques. The scheme is restricted to parameters that are linear in the exponent of a Jastrow correlation factor, which are the most important parameters in the wave functions we use. The scheme is highly efficient and allows us to investigate the parameter space more closely than has been possible before. We search for multiple minima of the variance in the parameter space and compare the wave functions obtained using reweighted and unreweighted variance minimization.

Figure 1

(Color online) The unreweighted variance ${\sigma }_u^2$ for an all-electron neon atom plotted against one of the linear Jastrow parameters. All of the other linear parameters are set to zero. Different numbers of configurations were used to calculate the quartic coefficients of the unreweighted variance. In each case the configurations were distributed according to the square of the Hartree-Fock wave function. The Jastrow factor contained a total of 24 linear parameters. The curves for ${10}^5$ and ${10}^6$ configurations are indistinguishable in the figure.

Figure 2

(Color online) The unreweighted variance ${\sigma }_u^2$ for an all-electron neon atom plotted against the change in one of the linear Jastrow parameters. All the parameters in the Jastrow factor are set to large, random values. Different numbers of VMC-generated configurations were used to calculate the quartic coefficients of the unreweighted variance. In each case the configurations were distributed according to the square of a Slater-Jastrow wave function, where the Jastrow factor contained the random parameters, so that the resulting distribution was very unlike the ground-state distribution. The Jastrow factor contained a total of 72 linear parameters. The Slater wave function contained Hartree-Fock orbitals.

Figure 3

(Color online) The unreweighted variance ${\sigma }_u^2$ for a ${\mathrm{SiH}}_4$ molecule [with a Hartree-Fock silicon pseudopotential (Ref. 7)] plotted against the cutoff length for the electron-electron terms in the Jastrow factor $L_u$. The Jastrow factor is such that the local energy is continuous when an electron-electron separation passes through the cut-off length (Ref. 5). Different numbers of VMC-generated configurations were used to calculate the unreweighted variance. All of the linear Jastrow parameters are set to zero. In each case the configurations were distributed according to the square of the Hartree-Fock wave function. The Jastrow factor contained a total of 56 linear parameters, plus three cut-off lengths.

Figure 4

(Color online) The same as Fig. 3, except that all of the parameters in the Jastrow factor (including the cut-off lengths) have been optimized.

Figure 5

(Color online) The reweighted and unreweighted variance for an all-electron neon atom plotted against the change in the value of a linear Jastrow parameter ${\alpha }_1$. Plots are shown for the case in which all the parameters are set to zero and the case in which all the parameters have been optimized. The set of 100 configurations used to calculate the variance were distributed according to the square of the Hartree-Fock wave function. The Jastrow factor contained a total of 27 linear parameters.

Figure 6

(Color online) The same as Fig. 5 except that ${10}^4$ configurations were used to calculate the variance.

Figure 7

The VMC energy of an all-electron neon atom against the change in the value of a linear Jastrow parameter ${\alpha }_1$ from the value determined by self-consistent unreweighted variance minimization. The Jastrow factor was chosen to be poor, with no electron-nucleus or electron-electron-nucleus terms, and the same electron-electron terms were used for both parallel and antiparallel spins. There is only one optimizable parameter: ${\alpha }_1$.

Figure 8

(Color online) VMC energy of an all-electron neon atom against the change in the value of a linear Jastrow parameter ${\alpha }_1$ from the value determined by self-consistent unreweighted variance minimization. Specifically, the Jastrow factor was the best all-electron neon Jastrow factor described in Ref. 5. The reweighted and unreweighted energies calculated using $8\times {10}^5$ configurations distributed according to the square of the optimized wave function are also plotted. The statistical error bars in the VMC data are smaller than the symbols.

Figure 9

(Color online) Same as Fig. 8 except that the VMC variance is plotted against the change in the parameter value.

Figure 10

(Color online) VMC energy of an all-electron neon atom against the number of configurations used to optimize the linear Jastrow parameters in an unreweighted variance minimization. Eight optimization cycles were performed for each number of configurations in order to ensure that self-consistency was achieved. The Slater wave function contained Hartree-Fock orbitals. The error bars in the VMC data are smaller than the symbols.

Figure 11

(Color online) Unreweighted variance for an all-electron neon atom plotted against the value of the linear Jastrow parameter ${\alpha }_1$ for three different configuration distributions: the square of the Hartree-Fock wave function, the square of an optimized Slater-Jastrow wave function, and the square of a poor Slater-Jastrow wave function. ${10}^6$ configurations were used to calculate the unreweighted variance. The Jastrow factor contained a total of 36 linear parameters. The Slater wave function contained Hartree-Fock orbitals.

Figure 12

(Color online) VMC energy of an all-electron neon atom against the number of parameters in the Jastrow factor. Different numbers of configurations were used to carry out the unreweighted variance-minimization calculations. Six optimization cycles were performed in order to guarantee self-consistency. Very long VMC simulations were carried out using the optimized Jastrow factors in order to obtain the energies plotted in the graph. The VMC error bars are smaller than the symbols. The Slater wave function contained Hartree-Fock orbitals.