Using local operator fluctuations to identify wave function improvements

A method is developed that allows analysis of quantum Monte Carlo simulations to identify errors in trial wave functions. The purpose of this method is to allow for the systematic improvement of variational wave functions by identifying degrees of freedom that are not well described by an initial trial state. We provide proof of concept implementations of this method by identifying the need for a Jastrow correlation factor and implementing a selected multideterminant wave function algorithm for small dimers that systematically decreases the variational energy. Selection of the two-particle excitations is done using the quantum Monte Carlo method within the presence of a Jastrow correlation factor and without the need to explicitly construct the determinants. We also show how this technique can be used to design compact wave functions for transition metal systems. This method may provide a route to analyze and systematically improve descriptions of complex quantum systems in a scalable way.

Figure 1

Visual representation of the path through Hilbert space from the initial to the exact wave function taken by the exact (black) and mimicked (red) projection operators, respectively.

Figure 2

Amplitude of two-body $b\to a$ bonding-to-antibonding excitation ${\rho }_{a_{\uparrow }{a}_{\downarrow }{b}_{\uparrow }{b}_{\downarrow }}\left(\mathbf{R}\right)$ versus local energy $E_L\left(\mathbf{R}\right)$ for two different trial wave functions, with corresponding principal components of the distribution indicated. The Slater-Jastrow wave function used to generate (a) did not include the CSF corresponding to this two-body excitation, while the wave function used to generate (b) does. The principal components rotate upon the addition of this CSF.

Figure 3

Comparison of normalized signal strength for different estimators of relative CSF importance for stretched dimers of ${\mathrm{H}}_2,{\mathrm{N}}_2,{\mathrm{O}}_2$, and ${\mathrm{F}}_2$, respectively. The indicated bars are the determinant coefficients taken from a CISD calculation, the signal drawn from the $\langle (E_L-\langle H\rangle )A_k\rangle$ estimator, and the determinant coefficients taken from an optimized multi-Slater-Jastrow wave function, respectively. The CSFs are arranged such that the optimized final CSF weight declines monotonically from top to bottom. Each indicated excitation is a one- or two-particle excitation that includes both itself and any symmetry-related partners. For example, $({\pi }_i,{\pi }_i)\to ({\pi }_i,{\pi }_i)$ is a two-particle excitation that excites a bonding $\pi$-orbital electron to an antibonding ${\pi }^{*}$ orbital of the same angular momentum $(x$ or $y)$ in each spin channel. On the other hand, $({\pi }_i,{\pi }_j)\to ({\pi }_j,{\pi }_i)$ involves a two-body exchange.

Figure 4

Added spin-up or spin-down CSF excitations vs associated variational Monte Carlo energy in a multi-Slater-Jastrow wave function for the CSF ordering suggested by conventional CISD for each considered model system.

Figure 5

Normalized covariance signal of CSFs versus the decrease in energy obtained from adding a CSF to the trial state. Significant negative correlations exist between the two values. The shading is provided as a visual guide.

Figure 6

Correlator signal $\langle \mathrm{\Psi }\left|(\widehat{H}-\langle H\rangle )c_1^{†}{c}_1\right|\mathrm{\Psi }\rangle$ as a function of the parameter $\theta$ appearing in the trial state $cos\theta |D_1\rangle +sin\theta |D_2\rangle$. In this example, we have chosen ${\epsilon }_1=0$ and ${\epsilon }_2=1$ and allowed $\mathrm{\Delta }$ to assume several values between 0 and 1.

Figure 7

(a) Covariance of the 1RDM with the local energy $E_L$ for TiO in the J12 wave function. (b) Covariance of the pair distribution $g\left(\mathbf{r}\right)$ with the local energy $E_L$ for the J12, J30, and JS30 wave functions in both spin channels (right): J12, Slater-Jastrow state with 12 parameters per atom in the three-body part of the Jastrow factor; J30, J12, but with 30 three-body terms instead of 12; JS30, J30, but with a spin-dependent two-body portion of the Jastrow factor.

Figure 8

Total TiO VMC energy vs the number of included CSFs using different Jastrow factors. Note that the decline in energy is quite modest with respect to the number of included CSFs, but falls dramatically when spin dependence is incorporated in the two-body portion of the Jastrow factor.