Assessing Orbital Optimization in Variational and Diffusion Monte Carlo

TL;DR

The paper addresses whether orbital optimization (OO) at the variational Monte Carlo level can improve diffusion Monte Carlo results for a correlated 2D magnetic material CrSBr. It systematically compares several OO strategies (OS, SR, SR+LS) and Jastrow enhancements across growing active spaces, evaluating effects on $E_{\mathrm{VMC}}$, $E_{\mathrm{DMC}}$, fixed-node bias, locality error, and mixed-estimator biases. The main findings show that OO robustly lowers $E_{\mathrm{VMC}}$ and reduces mixed-estimator biases, and modestly improves fixed-node error, but increases locality error, resulting in $E_{\mathrm{DMC}}$ that does not beat the DFT+U orbital baseline under the studied pseudopotential framework. The work highlights the potential of OO for improving variational and observable biases while identifying locality as a key bottleneck, suggesting avenues such as all-electron calculations or alternative Hamiltonians to realize OO benefits in DMC for correlated materials.

Abstract

In this work, we investigate the fidelity of orbital optimization in variational Monte Carlo to improve diffusion Monte Carlo results on correlated magnetic systems, using CrSBr as a model system. We compare the performance of different optimization methods, showing that stochastic reconfiguration is a robust and reliable optimizer. We show that short range Jastrow factors are important for improving diffusion Monte Carlo, regardless of the quality of orbitals. Large active spaces are required to converge the variational energy, but ulitmately orbital optimization produces worse diffusion Monte Carlo energies when compared to standard orbitals from density functional theory. We show that this increased bias is due to larger locality errors from the use of pseudopotentials, while the fixed-node error is actually improved by using orbital optimization. Additionally, for observables other than energy, orbital optimization produces a systematically smaller mixed-estimator bias. Ultimately, we believe orbital optimization provides a reliable method to improve variational and pure fixed-node energies as well as lower mixed-estimator bias.

Figures (10)

• Figure 1: Structure of the quasi-2D CrSBr system in a ferromagnetic configuration, made with VESTAmomma_vesta_2011. The magnetic moments on the chromium atoms are also indicated.
• Figure 2: Orbital optimization using different optimization strategies with different active spaces, (\ref{['fig:oo_n37']} & \ref{['fig:oo_n37_var']}) with 37 orbitals and (\ref{['fig:oo_n60']} & \ref{['fig:oo_n60_var']}) with 60 orbitals.
• Figure 3: Comparison of different optimization strategies in VMC and DMC. In PBE+U orbitals, only the jastrow is optimized and is plotted using the top $x$-axis, whereas the rest optimize the orbitals with different active space sizes and are shown across the bottom $x$-axis. Note that in (\ref{['fig:vmc_compare']}), all error bars are smaller than the points.
• Figure 4: Comparison of OO using different Jastrow factors, with VMC shown in (\ref{['fig:jastrow_vmc']}) and DMC shown in (\ref{['fig:jastrow_dmc']}). Note that in (\ref{['fig:jastrow_vmc']}), there errorbars are smaller than the points on this scale.
• Figure 5: Comparison of optimizations with different active space sizes using the SR optimizer. Energy is shown in (\ref{['fig:act_energy']}) and variances are shown in (\ref{['fig:act_variance']}).