Self-healing diffusion quantum Monte Carlo algorithms: Direct reduction of the fermion sign error in electronic structure calculations

We develop a formalism and present an algorithm for optimization of the trial wave function used in fixed-node diffusion quantum Monte Carlo (DMC) methods. The formalism is based on the DMC mixed estimator of the ground-state probability density. We take advantage of a basic property of the walker configuration distribution generated in a DMC calculation, to (i) project out a multideterminant expansion of the fixed-node ground-state wave function and (ii) to define a cost function that relates the fixed-node ground-state and the noninteracting trial wave functions. We show that (a) locally smoothing out the kink of the fixed-node ground-state wave function at the node generates a new trial wave function with better nodal structure and (b) we argue that the noise in the fixed-node wave function resulting from finite sampling plays a beneficial role, allowing the nodes to adjust toward the ones of the exact many-body ground state in a simulated annealing-like process. Based on these principles, we propose a method to improve both single determinant and multideterminant expansions of the trial wave function. The method can be generalized to other wave-function forms such as pfaffians. We test the method in a model system where benchmark configuration-interaction calculations can be performed and most components of the Hamiltonian are evaluated analytically. Comparing the DMC calculations with the exact solutions, we find that the trial wave function is systematically improved. The overlap of the optimized trial wave function and the exact ground state converges to 100% even starting from wave functions orthogonal to the exact ground state. Similarly, the DMC total energy and density converges to the exact solutions for the model. In the optimization process we find an optimal noninteracting nodal potential of density-functional-like form whose existence was predicted in a previous publication [Phys. Rev. B 77, 245110 (2008)]. Tests of the method are extended to a model system with a conventional Coulomb interaction where we show we can obtain the exact Kohn-Sham effective potential from the DMC data.

Figure 1

(Color online) (a) Schematic representation of trial wave function (${\Psi }_T$, blue dots), fixed-node ground state (${\Psi }_{FN}$, purple continuous), ground state ($\Psi$, black dash and dots), and new trial wave function (${\tilde{\Psi }}_T$, red dashed line) in the direction perpendicular to the nodal surface $\left(x\right)$. We show that smoothing the kink in the fixed-node wave function ${\Psi }_{FN}$ moves the nodes of ${\tilde{\Psi }}_T$ toward the nodes of the ground state $\Psi$. (b) Schematic representation of how the nodal surface evolves, shown with increasing purple line thickness, after each iteration in the algorithm. The noise introduced in the nodes by random fluctuations of the walkers is assumed to correct itself if the statistics is increased from one iteration to the next.

Figure 2

(Color online) Self-healed DMC run obtained using the method described in Sec. 3. Black points denote the average value of the local energy for each DMC step. Green points mark the reference energy used for population control. Orange lines mark the average energy of the trial wave function. The horizontal blue line marks the energy of the ground state in the full CI calculation. Vertical lines mark the steps when the coefficients of wave function are updated. Inset: detail of the DMC run for the first $10000$ steps (same conventions as in the main figure).

Figure 3

(Color online) Values of the coefficients of the multideterminant expansion (small green circles) as compared with a full CI calculation (large black circles). The DMC statistical errors of the coefficients is equal to the radius of the green circles.

Figure 4

(Color online) Change in the values of the multideterminant expansion as the DMC self-healing algorithm progresses. Light gray colors denote older coefficients while darker ones denote more converged results. The initial nonzero coefficients are highlighted in red squares.

Figure 5

(Color online) Logarithm of the residual projection $R_P$ [see Eq. (43)] as a function of the total weighted number of configurations along the complete run $N_w$. The lines are a guide to the eyes. Inset: projection of the DMC self-healed wave function onto the full CI ground state as function of the logarithm of $N_w$.

Figure 6

(Color online) Energy of the DMC run as a function of the number of DMC steps used to gather statistical data of the wave function in the previous block. The statistical error bars for the first three points on the left were not calculated. The statistical error bars of the points on the right were smaller than the size of the symbols. Blue squares denote calculations starting from a bad trial wave function, while the red circles mark the results obtained from an initial trial wave function corresponding to the best blue square on the right (see text). Green diamonds were generated starting from the best red circle.

Figure 7

(Color online) Values of the coefficients of the multideterminant expansion (small green circles) obtained from the DMC run for two electrons in a square box with a Coulomb interaction in the highly correlated limit. The statistical errors in the values of the coefficients are equal to the size of the red bar.

Figure 8

Density of the ground state of two spinless electrons with Coulomb interaction in a square box. We choose one of the two degenerate ground states, reducing the symmetry of the density to $D_2$. (a) Left side of the density of the many-body ground state constructed with the converged coefficients shown in Fig. 7. (b) Kohn-Sham noninteracting density constructed as explained in Ref. 17.

Figure 9

Kohn-Sham potential for two spinless electrons in a square box corresponding to the ground state of Figs. 7, 8. The potential was constructed using the methods explained in Ref. 17.

Figure 10

(a) Effective nodal potential, (b) one-body Jastrow, and (c) two-body Jastrow factors obtained by minimizing Eq. (29), in which the multideterminant expansion of Fig. 7 has been replaced by a single determinant function.