Accurate forces in quantum Monte Carlo calculations with nonlocal pseudopotentials

Calculating accurate forces within variational and diffusion Monte Carlo (VMC and DMC) methods is a very challenging problem. We derive expressions for the contribution to the Hellmann-Feynman force from nonlocal pseudopotentials for use within the VMC and DMC methods. Equilibrium bond lengths and harmonic vibrational frequencies are calculated from the Hellmann-Feynman forces and compared with those obtained from the energies at the Hartree-Fock, VMC, and pure DMC levels. Results for five small molecules show that the equilibrium bond lengths obtained from the force and energy calculations differ by less than $0.007\mathrm{\AA }$ at the DMC level.

Figure 1

Illustration of the coordinate transformation of the nonlocal pseudopotential integration grid when displacing an atom centered at the origin (center of the small circle) by the vector $\mathbf{\Delta }$ (center of the large circle). The electron is at ${\mathbf{r}}_i$. The integration grid at zero displacement consists of four points $r_i{\mathbf{u}}_1$, etc., which are transformed by the atomic displacement to $\mathbf{\Delta }+\mid {\mathbf{r}}_i-\mathbf{\Delta }\mid {\mathbf{u}}_1$, etc.

Figure 2

(Color online) Upper right graph: HFT forces in a.u. on the Si atom in the SiH molecule as calculated within the VMC, mixed DMC, and future walking pure DMC methods for five different bond lengths. Upper left graph: Future walking pure DMC forces plotted against the future walking projection time between 0.5 and $10\mathrm{a.u.}$ The mixed DMC forces correspond to the zero future walking projection time and the VMC forces are plotted at $-0.5\mathrm{a.u.}$ to guide the reader. Lower graphs: same as the upper ones with the forces now evaluated on the H atom.