Quantum Monte Carlo with variable spins: Fixed-phase and fixed-node approximations

We study several aspects of the recently introduced fixed-phase spinor diffusion Monte Carlo method, in particular, its relation to the fixed-node method and its potential use as a general approach for electronic structure calculations. We illustrate constructions of spinor-based wave functions with the full space-spin symmetry without assigning up or down spin labels to particular electrons, effectively “complexifying” even ordinary real-valued wave functions for Hamiltonians without spin terms. Interestingly, with proper choice of the simulation parameters and spin variables, such fixed-phase calculations enable one to reach also the fixed-node limit. The fixed-phase approximation has several desirable properties when compared to the fixed-node approximation. The fixed-phase solution provides a straightforward interpretation as the lowest bosonic state in a given effective potential generated by the many-body approximate phase, whereas nodal boundary conditions are defined through less intuitive and complicated hypersurfaces with one dimension less than the original configuration space. In addition, the divergences of the local energy and drift at real wave function nodes are smoothed out to lower dimensionality when the wave function is complexified, thus decreasing the variation of sampled quantities and eliminating artificial nodal domain issues that can occur in the fixed-node formalism. We illustrate some of these properties on calculations of selected first-row systems that recover the fixed-node results with quantitatively similar levels of the corresponding biases. At the same time, the fixed-phase approach opens new possibilities for more general trial wave functions with further opportunities for increasing accuracy in practical calculations.

Figure 1

FPDMC energy of the C atom with varying spin masses ${\mu }_s$ (or equivalently the spin time step ${\tau }_s=\tau /{\mu }_s$). The initial spin configurations were chosen so that the wave function decomposes into a product of determinants.

Figure 2

Percentage error of the correlation energies in the fixed-node (FN) and fixed-phase (FP) extrapolated calculations.

Figure 3

Total energies for the Be atom with a two-configuration wave function. The top $x$ axis shows the actual spin time step for the FN calculation, which is linearly extrapolated to zero time step with an energy of $-14.66071$(5) Ha. The bottom $x$ axis indicates the spin mass for the FP calculation, where with each value configurations are initialized such that $s_1=s_3=s$ and $s_2=s_4=s^{\prime }$. We hold the spatial time step fixed at $\tau =0.001{\mathrm{Ha}}^{-1}$ throughout.

Figure 4

${\mathrm{N}}_2$ binding curve for the ${}^1\mathrm{\Sigma }_g$ molecular state using a HF nodal surface or phase. The horizontal line indicates the experimental dissociation energy. The experimental error bar is too small to be visible on this scale. The small increase in FP underbinding comes from a slightly smaller fixed-phase bias in the N atom and a slightly larger bias in the dimer with the HF phase. We note that the results need not to be necessarily identical for Hamiltonians with ECPs due to marginal differences in localization errors from two close, but nonidentical trial wave functions.