Fractional Charge by Fixed-Node Diffusion Monte Carlo Method

Fixed-node diffusion Monte Carlo (FNDMC) method is a stochastic quantum many-body approach that has a great potential in electronic structure theory. We examine how FNDMC total energy $E(N)$ satisfies exact constraints, linearity and derivative discontinuity, versus fractional electron number $N$, if combined with mean-field trial wave functions that miss such features. H and Cl atoms with fractional charge reveal that FNDMC method is well able to restore the piecewise linearity of $E(N)$. The method uses ensemble and projector ingredients to achieve the correct charge localization. A water-solvated ${\mathrm{Cl}}^{-}$ complex illustrates superior performance of FNDMC method for charged noncovalent systems.

Figure 1

Hydrogen cube: total energy $E(N)$ per atom vs electron occupation number $N$ per atom obtained by mean-field (HF, PBE) and FNDMC (DMC) methods. FNDMC error bars (not shown) are smaller than the symbol size.

Figure 2

Chlorine cube: total energy $E(N)$ per atom vs electron occupation number $N$ per atom obtained by mean-field (HF, PBE, HSE06) and FNDMC (DMC) methods. FNDMC error bars (not shown) are smaller than the symbol size.

Figure 3

Hydrogen cube: total energy $E(N)$ per atom vs electron number per atom from VMC with ${\mathrm{\Psi }}_S$ (S) and Slater-Jastrow ${\mathrm{\Psi }}_T$ (SJ) using PBE orbitals, and, FNDMC (DMC) simulation.

Figure 4

Relatative errors (RE) of the interaction energies for the solvated anion ${\mathrm{Cl}}^{-}({\mathrm{H}}_2\mathrm{O}{)}_6$ complex (depicted) computed by RHF, DFT, and FNDMC simulations using ${\mathrm{\Psi }}_T$ based on the same orbitals, compared vs the CCSD(T)/CBS reference. Gray region indicates the benchmark level (1%). FNDMC error bars (not shown) are smaller than the symbol size.