Total forces in the diffusion Monte Carlo method with nonlocal pseudopotentials

We report exact expressions for atomic forces in the diffusion Monte Carlo (DMC) method when using nonlocal pseudopotentials. We present approximate schemes for estimating these expressions in both mixed and pure DMC calculations, including the pseudopotential Pulay term and the Pulay nodal term. Harmonic vibrational frequencies and equilibrium bond lengths are derived from the DMC forces and compared with those obtained from DMC potential-energy curves. Results for four small molecules show that the equilibrium bond lengths obtained from our best force and energy calculations differ by less than $0.002\text{Å}$.

Figure 1

(Color online) Investigation of the time step behavior of DMC forces (a.u.) and DMC energies (Ha) for the SiH molecule when using the small basis set. Upper graph: the pure DMC forces $F_{\text{pure}}^{\text{tot}}$ and $F_{\text{pure}}^{\text{HFT}}$ as a function of the DMC time step: circles indicate forces on the Si atom and crosses indicate forces on the H atom. A quadratic form is fitted to these forces. The standard errors of these forces are indicated unless they are smaller than the line width of the fitted functions. Middle graph: the same information as the upper graph for mixed DMC calculations. Lower graph: the DMC energy as a function of time step for comparison, a cubic form is fitted to the energies.

Figure 2

Dependence of the future-walking pure DMC HFT force estimates (a.u.) on the future-walking projection time (a.u.). Upper right graph: the HFT force on the Si atom for the SiH molecule as calculated within the mixed DMC and future-walking pure DMC method for different bond lengths using the small basis set. A Morse potential is fitted to these forces. Upper left graph: future-walking pure DMC forces calculated for different bond lengths plotted against the future-walking projection time. The mixed DMC force corresponds to zero future-walking projection time. Lower graphs: the same information as the upper graphs for the large basis set. When using the small or large basis set, we see that the future-walking projection time of 10 a.u. is long enough to obtain accurate pure DMC estimates.

Figure 3

(Color online) Upper graph: three different estimates of the pure DMC forces, $F_{\text{pure}}^{\text{HFT}}$, $F_{\text{pure}}^{\text{HFT}+\text{P}}$, and $F_{\text{pure}}^{\text{tot}}$, for the Si and H atoms of the SiH molecule evaluated at different bond lengths. These estimates are defined in Eqs. (12, 13, 14, 15, 16, 17) and are calculated with the small basis set. The Morse potential is fitted to the forces. Middle graph: the same information as for the upper graph with the three estimates, $F_{\text{mix}}^{\text{HFT}}$, $F_{\text{mix}}^{\text{HFT}+\text{P}}$, and $F_{\text{mix}}^{\text{tot}}$, calculated within the mixed DMC method. These estimates are defined in Eqs. (6, 7, 8, 9, 10, 11). Lower graph: the DMC energy at different bond lengths, a Morse potential fitted to the DMC energies and the derivative of the Morse potential corresponding to the forces on the Si and H atoms. To guide the reader, the dotted vertical line corresponds to the zero-force geometry obtained from the fitted DMC energy curve.