Bias cancellation in one-determinant fixed-node diffusion Monte Carlo: Insights from fermionic occupation numbers

The accuracy of the fixed-node diffusion Monte Carlo (FNDMC) depends on the node location of the supplied trial state ${\mathrm{\Psi }}_T$. The practical FNDMC approaches available for large systems rely on compact yet effective ${\mathrm{\Psi }}_T$, most often containing an explicitly correlated single Slater determinant (SD). However, SD nodes may be better suited to one system than to another, which may possibly lead to inaccurate FNDMC energy differences. It remains a challenge how to estimate nonequivalence or appropriateness of SDs. Here we use the differences of a measure based on the Euclidean distance between the natural orbital occupation number (NOON) vector of the SD and the exact solution in the NOON vector space, which can be viewed as a measure of SD nonequivalence and as a qualitative measure of the expected degree of nondynamic-correlation-related bias in FNDMC energy differences. This is explored on a set of small noncovalent complexes and covalent bond breaking of ${\mathrm{Si}}_2$ vs ${\mathrm{N}}_2$. It turns out that NOON-based measures well reflect the magnitude and sign of the bias present in the data available, thus providing insights into the nature of bias cancellation in SD FNDMC energy differences.

Figure 1

Comparison of FNDMC and CCSD/aug-VDZ results for a set of noncovalent complexes: deviations from the reference $\epsilon$ (kcal/mol), $T_1$ diagnostics for dimolecular complexes $S$ (supersystem, yellow) and noninteracting constituents $C$ (cyan) and their differences $\mathrm{\Delta }{T}_1$, rescaled NOON-space distances of the correlated post-HF NOONs from the Slater point $d$, their differences $\delta$, and the corresponding bond-multiplicity-rescaled differences $\kappa$. For definitions, see Sec. 2.

Figure 2

Comparison of the FNDMC and CCSD/aug-VDZ results for ${\mathrm{Si}}_2$ and ${\mathrm{N}}_2$. For description, see the caption of Fig. 1.