Beyond the locality approximation in the standard diffusion Monte Carlo method

We present a way to include nonlocal potentials in the standard diffusion Monte Carlo method without using the locality approximation. We define a stochastic projection based on a fixed node effective Hamiltonian, whose lowest energy is an upper bound of the true ground-state energy, even in the presence of nonlocal operators in the Hamiltonian. The variational property of the resulting algorithm provides a stable diffusion process, even in the case of divergent nonlocal potentials, like the hard-core pseudopotentials. It turns out that the modification required to improve the standard diffusion Monte Carlo algorithm is simple.

Figure 1

Energies for the carbon pseudoatom with $\tau =0.08H^{-1}$ at the given DMC generation. ${\Psi }_T$ is an antisymmetrized geminal power wave function with a three-body Jastrow factor (Refs. 12, 15). We report results for the locality approximation ($H^{\alpha ,\gamma }$ with $\alpha =1$ and $\gamma =0$) and the algorithm with $T$ moves ($\alpha =0$, $\gamma =0$).

Figure 2

(Color online) $E_{\mathrm{MA}}(\alpha ,0)$ (green, dashed line) and $E_{\mathrm{FN}}(\alpha ,0)$ (blue, dotted line) energies for the silicon pseudoatom with different values of the effective Hamiltonian parameter $\alpha$. The DMC energies with locality approximation (red, solid line), corresponding to $E_{\mathrm{MA}}(1,0)$, are reported in all panels for reference. We used three different ${\Psi }_T$’s, which have the same determinantal part. A more accurate ${\Psi }_T$ corresponds to a smaller difference between its variational energy $\left(E_{\mathrm{VMC}}\right)$ and the $E_{\mathrm{MA}}(0,0)$ energy, reported on the abscissa.

Figure 3

(Color online) The same as Fig. 2, but for the carbon pseudoatom.