Stochastic nodal surfaces in quantum Monte Carlo calculations

Treating the fermionic ground state-problem as a constrained stochastic optimization problem, a formalism for fermionic quantum Monte Carlo is developed that makes no reference to a trial wave function. Exchange symmetry is enforced by nonlocal terms appearing in the Green's function corresponding to an additional walker propagation channel. Complemented by a treatment of diffusion that encourages the formation of a stochastic nodal surface, we find that an approximate long-range extension of walker cancellations can be employed without introducing significant bias, reducing the number of walkers required for a stable calculation. A proof-of-concept implementation is shown to give a stable fermionic ground state for simple harmonic and atomic systems.

Figure 1

Schematic of wave function formation arising from competing walker propagation channels (shown for three fermions in a harmonic well as in Fig. 4).

Figure 2

The diffusive propagation of two nearby walkers of opposite sign located at $x_1$ and $x_2\Rightarrow {\psi }_D\left(x\right)=G_D(x,x_2,\delta \tau )-G_D(x,x_1,\delta \tau )\equiv G_2-G_1$ (black dashed line). The red (blue) shaded region shows the portion of $G_2$ ($G_1$) that can be canceled in the propagation.

Figure 3

Propagation schemes satisfying Eq. (13), applied to the walkers in Fig. 2. (a) Traditional DMC propagation [Eq. (14)]. (b) Our propagation scheme [Eq. (16)]. Note that in (a) there is overlap of the $+\text{ve}$ and $-\text{ve}$ walker distributions. The same is not true for (b).

Figure 4

The wave function of three noninteracting fermions with coordinates $x,y$, and $z$ in a one-dimensional harmonic well, integrated and viewed along the (1,1,1) projection. The analytic nodal surface is shown as a dotted black line. From this projection, the nodal pockets can be clearly seen. (a) Bosonic collapse from DMC with exchange moves but without a stochastic nodal surface. (b) From DMC with exchange moves and a stochastic nodal surface. (c) Analytic bosonic ground state. (d) Analytic fermionic ground state.

Figure 5

DMC calculations of the helium atom ground state, with electrons having opposite spin (parahelium), and the exited (triplet) state with electrons having parallel spin (orthohelium), calculated using a stochastic nodal surface. A time step of $\delta \tau ={10}^{-3}$ atomic units was used with $\delta {\tau }_{\text{eff}}$ = 0.5 atomic units (see Sec. 3d). It is clear to see that the exited (fermionic) state shows larger fluctuations than the (bosonic) ground state. This is due to cancellations between oppositely signed walkers contributing to fluctuations in the growth estimator of the energy. The reference energies are from VMC optimization of many-parameter trial wave functions [21], accurate to within a few $\mu \mathrm{Ha}$.

Figure 6

The denominator of Eq. (22) vs DMC time step for the simulation used to produce Fig. 4 (${10}^4$ walkers, $\delta \tau =\delta {\tau }_{\text{eff}}={10}^{-3}$ atomic units). Note the $y$-axis scale. For the purposes of this plot, the trial wave function was set to the analytic fermionic ground state [shown in Fig. 4]. We can see that the denominator remains large and roughly constant. Inset: the denominator as a fraction of its maximum possible value (obtained if the walkers are all of the same sign as the analytic wave function).

Figure 7

The effect of $\delta {\tau }_{\text{eff}}$ on the nodal surface of a two-fermion system in one dimension. Left: $\delta {\tau }_{\text{eff}}=0.1$ Right: $\delta {\tau }_{\text{eff}}=1.0$. Red circles (blue squares) represent the location of positive (negative) walkers. The background is shaded according to ${\psi }_{D,\text{eff}}\left(x\right)$, where the nodal surface can be seen as a bright line separating the positive (red) and negative (dark blue) nodal pockets. It is clear that increasing $\delta {\tau }_{\text{eff}}$ leads to a smoother nodal surface that is closer to the analytic nodal surface at $x=y$.

Figure 8

The DMC energy of a lithium atom as a function of the effective time step $\delta {\tau }_{\text{eff}}$ used to define the stochastic nodal surface. For each value of $\delta {\tau }_{\text{eff}}$, the energy was obtained from a simulation of ${10}^4$ walkers for ${10}^5$ iterations with a time step of ${10}^{-3}$ atomic units. The DMC energy is shaded to $\pm$ the reblocked error. The blue dotted line is at the nonrelativistic fermionic energy obtained from a Hylleraas-type expansion, accurate to within a basis set error of $<{10}^{-9}$ Ha [25]. The inset shows the effect of increasing $\delta {\tau }_{\text{eff}}$ beyond sensible values.

Figure 9

The evolution of a DMC calculation of a beryllium atom. The calculation was carried out using ${10}^4$ walkers for ${10}^4$ time steps, each of $\delta \tau ={10}^{-3}$ atomic units. An effective time step of $\delta {\tau }_{\text{eff}}=1.35$ atomic units, derived in a preliminary calculation from Eq. (25), was used to describe the stochastic nodal surface. In the upper panel, the dashed line is at the target population. In the lower panel, the dashed line is at $-14.66654$ Ha, the energy obtained from a Hylleraas-type expansion, accurate to within $2\times {10}^{-4}$ Ha of the exact value [24]. The DMC estimate of the energy is $-14.665\pm 0.07$ Ha.

Figure 10

Schematic of cancellations via explicit pairing (upper two panels) and a stochastic nodal surface (lower two panels). Red circles represent positive walkers, blue squares represent negative walkers, and empty shapes represent canceled walkers. When using explicit pairing, walkers are first paired according to some criterion and then canceled. This cancellation is often only partial and may take place over several iterations [13, 14, 15]. When using a stochastic nodal surface, the diffused wave function is evaluated for the configuration of each walker, and any walker with the wrong sign is immediately removed from the simulation.

Figure 11

A Voronoi wave function for two noninteracting fermions in a 1D harmonic oscillator. Red circles (blue squares) represent positive (negative) walkers. The wave function is positive (negative) in red (dark blue) shaded regions. The emerging stochastic nodal surface at $x_1=x_2$ can be clearly seen. Any walker crossing this surface in the next iteration will be removed from the simulation.

Figure 12

The DMC energy as a function of target population for three noninteracting fermions in a 1D harmonic well. The result for each population was calculated using $5\times {10}^4$ iterations with a time step of $\delta \tau ={10}^{-3}$ atomic units. Shown are calculations using two different values of $\delta {\tau }_{\text{eff}}$ (0.1 is shown in blue and converges faster than 0.05, shown in orange). The inset shows the same data plotted against the inverse population. For large populations, we found a deviation from power-law behavior where the convergence is instead exponential. For $\delta {\tau }_{\text{eff}}=0.1$ the best fit converges as $N^{-0.79}exp(-N/3542)$ and gives an energy of $4.497\pm 0.003$ hartree in the infinite population limit. The analytic energy is 4.5 hartree.

Figure 13

The DMC energy as a function of the target walker population for a boron atom. The result for each population was calculated using $5\times {10}^4$ iterations with a time step of $\delta \tau ={10}^{-3}$ atomic units. $\delta {\tau }_{\text{eff}}$ was set to 1.35 atomic units, to facilitate comparison with our calculations of the beryllium atom. The inset shows the same data, plotted against the inverse population. The exact energy shown is at $-24.65386608\pm 2\times {10}^{-9}$ Ha, which is the result obtained in the infinite-basis limit of an explicitly correlated Gaussian basis set expansion [28]. The best-fit power law converges as $N^{-0.6}$ and gives an energy of $-24.67\pm 0.1$ hartree in the infinite population limit.

Figure 14

The energy of a system on noninteracting fermions in a 1D harmonic well, as a function of the number of fermions. The DMC calculations were carried out using 5000 walkers for ${10}^4$ iterations with a time step of $\delta \tau ={10}^{-3}$ atomic units. The statistical errors in the energy are smaller than the widths of the lines.