Grid-based diffusion Monte Carlo for fermions without the fixed-node approximation

A diffusion Monte Carlo algorithm is introduced that can determine the correct nodal structure of the wave function of a few-fermion system and its ground-state energy without an uncontrolled bias. This is achieved by confining signed random walkers to the points of a uniform infinite spatial grid, allowing them to meet and annihilate one another to establish the nodal structure without the fixed-node approximation. An imaginary-time propagator is derived rigorously from a discretized Hamiltonian, governing a non-Gaussian, sign-flipping, branching, and mutually annihilating random walk of particles. The accuracy of the resulting stochastic representations of a fermion wave function is limited only by the grid and imaginary-time resolutions and can be improved in a controlled manner. The method is tested for a series of model problems including fermions in a harmonic trap as well as the He atom in its singlet or triplet ground state. For the latter case, the energies approach from above with increasing grid resolution and converge within $0.015E_h$ of the exact basis-set-limit value for the grid spacing of 0.08 a.u. with a statistical uncertainty of ${10}^{-5}{E}_h$ without an importance sampling or Jastrow factor.

Figure 1

Examples of sign-preserving (left) and sign-flipping (right) moves for two fermions (labeled “1” and “2”) with the same spin on a 2D grid. The grid points are ordered in the increasing $x$ coordinates first and then in the increasing $y$ coordinates so that ${\mathit{r}}_1<{\mathit{r}}_2$ in both cases before the walker moves. In the move shown on the left, the coordinates of particles 1 and 2 are in a canonical order after the move, preserving the sign of $c$. The move shown on the right brings the coordinates of particles 1 and 2 into a noncanonical order (i.e., ${\mathit{r}}_1<{\mathit{r}}_2$ no longer holds), which needs to be permuted once to be canonical ordered, causing a sign flip in $c$. In the case of N electrons the new particle coordinates are sorted and the phase is determined from the parity of associated permutation.

Figure 2

Left: The energy evaluated by the growth estimator, $E$, as a function of the average number of walkers, $\langle N_w\rangle$, for the $S=0$, 1, and 2 ground states of the four noninteracting fermions in a 1D harmonic trap. Right: The number of walkers, $N_w$, as a function of the Monte Carlo steps, $N_{\text{MC}}$. The energy onset, $\omega$, was held fixed at 4.25, 5.25, and 8.25 $E_h$ for the states with $S=0$, 1, and 2, respectively.

Figure 3

The number of walkers, $N_w$, as a function of the Monte Carlo steps, $N_{\text{MC}}$, in grid DMC for two spin-1/2 fermions in a 3D harmonic trap in the triplet ground state for several grid spacings $\delta$. The energy onset was held fixed at $\omega =4.5E_h$ and imaginary time step $\tau =0.1$ a.u.

Figure 4

The energies of the He atom in the ${}^{3}S$ (left) and ${}^{1}S$ (right) states obtained by grid DMC with the exact nodal constraint as a function of the grid spacing ($\delta$). Statistical errors are shown with vertical bars. Horizontal dashed lines indicate the exact nonrelativistic energies [33]. Quadratic fit of the grid-DMC energies are drawn with solid curves.

Figure 5

Comparison of $p_n$ [Eq. (11) or (A3)] versus Gaussian function [Eq. (A6)] with $\tau =0.015$ and $\delta =0.1$.