Benchmarking the pseudopotential and fixed-node approximations in diffusion Monte Carlo calculations of molecules and solids

We performed diffusion Monte Carlo (DMC) calculations of the spectroscopic properties of a large set of molecules, assessing the effect of different approximations. In systems containing elements with large atomic numbers, we show that the errors associated with the use of nonlocal mean-field-based pseudopotentials in DMC calculations can be significant and may surpass the fixed-node error. We suggest practical guidelines for reducing these pseudopotential errors, which allow us to obtain DMC-computed spectroscopic parameters of molecules and equation of state properties of solids in excellent agreement with experiment.

Figure 1

Effect of Jastrow and localization error on VMC and DMC energy vs bond length curves for the $\mathrm{A}{\mathrm{g}}_2$ dimer using the $\mathrm{A}{\mathrm{g}}^{11}$ pseudopotential. The HF and VMC correspond to the expectation values of the Hamiltonian with and without the Jastrow term included, $\langle D{e}^J\left|\mathcal{H}\right|D{e}^J\rangle$ and $\langle D\left|\mathcal{H}\right|D\rangle$, respectively, where $\mathrm{D}{\mathrm{e}}^{\mathrm{J}}$ denotes the many-body Slater-Jastrow trial wave function.

Figure 2

(a) Total energies from VMC (upper values) and DMC (lower values) calculations for the $\mathrm{S}{\mathrm{n}}^{\mathrm{TN}}$ atom in its ground state using different values of the parameters in the Jastrow term. (b) Histogram of VMC total energies. (c) Histogram of DMC total energies. (d) Correlation between VMC and DMC total energies.

Figure 3

(a) Energy of ${\mathrm{Sn}}_2^{\mathrm{TN}}$ dimer as a function of its length (DMC energies with statistical error bars are given by black symbols, fit to Morse potential by red curve). (b) Deviations of DFT and DMC computed parameters from experiment for optimal bond length $R_{\mathrm{e}}$, (c) spectroscopic constant $w_e$, and (d) dissociation energy $D_e$. The molecules in (b), (c), and (d) are grouped by bonding mechanism—$s\text{−}s$ (green background), $s\text{−}p$ (red), $p\text{−}p$ (magenta), $d\text{−}d$ (blue), van der Waals (brown) — and by total mass in ascending order.

Figure 4

Core and valence electronic density as a function of distance from the nucleus in (a) $\mathrm{S}{\mathrm{n}}^{14}$ and (b) $\mathrm{S}{\mathrm{n}}^{22}$ pseudopotentials.

Figure 5

(a) A projection of the trial wave function and its nodes for the $\mathrm{M}{\mathrm{o}}_2$ dimer moving one electron in the basal plane for the Slater only (S), Slater-Jastrow (SJ), and Slater-Jastrow + backflow (SJB) trial wave functions. The nodes of wave function are modified by the backflow transformation. Deviations from experiment of DMC computed spectroscopic parameters of dimers employing different nodal structures of the trial wave function: (b) the optimal bond length $R_{\mathrm{e}}$, (c) the spectroscopic constant $w_{\mathrm{e}}$, and (d) the dissociation energy $D_e$. The molecules in (b), (c), and (d) are grouped by bonding mechanism $p\text{−}p$ (magenta) and $d\text{−}d$ (blue) and by total mass in ascending order.

Figure 6

Correlation between (a) geometric, (b) vibrational, and (c) energetic properties of solids and molecules from DMC calculations using the same pseudopotentials. Pearson correlation coefficient is shown above regression dashed lines.