Systematic Reduction of Sign Errors in Many-Body Calculations of Atoms and Molecules

The self-healing diffusion Monte Carlo algorithm (SHDMC) is shown to be an accurate and robust method for calculating the ground state of atoms and molecules. By direct comparison with accurate configuration interaction results for the oxygen atom, we show that SHDMC converges systematically towards the ground-state wave function. We present results for the challenging ${\mathrm{N}}_2$ molecule, where the binding energies obtained via both energy minimization and SHDMC are near chemical accuracy ($1\mathrm{kcal}/\mathrm{mol}$). Moreover, we demonstrate that SHDMC is robust enough to find the nodal surface for systems at least as large as ${\mathrm{C}}_{20}$ starting from random coefficients. SHDMC is a linear-scaling method, in the degrees of freedom of the nodes, that systematically reduces the fermion sign problem.

Figure 1

Comparison of the values of the coefficients ${\lambda }_n$ corresponding to the first 250 excitations of a converged SHDMC trial wave function (large black circles) with a large configuration interaction calculation with single, double, triple, and quadruple excitations (CISDTQ) [16] wave function (small red circles) for the oxygen atom. ${\lambda }_0=0.9769$ is not shown. Inset: Residual projection ($R_P=1-|⟨{\Psi }_{\mathrm{SHDMC}}|{\Psi }_{\mathrm{CI}}⟩/⟨{\Psi }_{\mathrm{CI}}|{\Psi }_{\mathrm{CI}}⟩|$) as a function of the number of CSFs included: circles, $R_P$ obtained with the full CISDTQ norm; squares, $R_P$ obtained with the truncated CISDTQ norm.

Figure 2

Total energies obtained for ${\mathrm{N}}_2$ with VMC and DMC methods for wave functions optimized via EMVMC [10] (squares) and SHDMC (circles) as a function of $\sum _n({\lambda }_n^{\mathrm{MCSCF}}{)}^2$. The lines are parabolic extrapolations to 1. The dot-dashed line represents the scalar relativistic core-corrected estimate of the exact energy (see Table ). The shaded area is the region of chemical accuracy.

Figure 3

Proof of principle of SHDMC for larger systems. Initial evolution of the average local energy for a SHDMC run with branching [8] generated for ${\mathrm{C}}_{20}^{+2}$, with random initial coefficients (see text). Inset: Calculated icosahedral cluster ${\mathrm{C}}_{20}^{+2}$.