Compact and Flexible Basis Functions for Quantum Monte Carlo Calculations

Molecular calculations in quantum Monte Carlo frequently employ a mixed basis consisting of contracted and primitive Gaussian functions. While standard basis sets of varying size and accuracy are available in the literature, we demonstrate that reoptimizing the primitive function exponents within quantum Monte Carlo yields more compact basis sets for a given accuracy. Particularly large gains are achieved for highly excited states. For calculations requiring non-diverging pseudopotentials, we introduce Gauss-Slater basis functions that behave as Gaussians at short distances and Slaters at long distances. These basis functions further improve the energy and fluctuations of the local energy for a given basis size. Gains achieved by exponent optimization and Gauss-Slater basis use are exemplified by calculations for the ground state of carbon, the lowest lying excited states of carbon with $^5S^o$, $^3P^o$, $^1D^o$, $^3F^o$ symmetries, carbon dimer, and naphthalene. Basis size reduction enables quantum Monte Carlo treatment of larger molecules at high accuracy.

💡 Research Summary

This paper addresses a central challenge in quantum Monte Carlo (QMC) simulations: the need for compact yet accurate basis sets. Traditional quantum chemistry (QC) relies on a mixture of contracted Gaussian functions and primitive Gaussians. While contracted functions reproduce the shape of atomic orbitals over a reasonable range, they cannot capture the electron‑nucleus cusp because they have zero gradient at the origin. Primitive Gaussians, on the other hand, have the wrong long‑range asymptotic behavior. In QMC, however, matrix elements are evaluated by stochastic integration, freeing practitioners from the analytical integral constraints that dictate the exclusive use of Gaussians in QC. The authors exploit this flexibility in two ways.

First, they re‑optimize the exponents of the primitive Gaussian functions directly within the VMC (variational Monte Carlo) framework. Using the linear method, all wave‑function parameters—including Jastrow factors, CSF coefficients, orbital coefficients, and the primitive exponents—are simultaneously optimized to minimize a weighted sum of the energy and the variance of the local energy (weights 0.95 and 0.05, respectively). The contracted functions remain fixed and are spline‑represented for computational efficiency, while the primitive functions stay analytic, allowing straightforward gradient evaluation. This re‑optimization yields a substantial gain in both total energy and the RMS fluctuations of the local energy (σ) across all tested systems.

Second, the authors introduce a novel primitive form called the Gauss‑Slater (GS) function, designed for calculations that employ non‑divergent pseudopotentials. The GS function is defined as

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