A quantity of central importance in a quantum Monte Carlo calculation is the many-electron trial wave function. In the commonly used Slater-Jastrow form, this takes the form of a linear combination of one or more Slater determinants of one-electron orbitals taken from a simple DFT or Hartree-Fock calculation, multiplied by a Jastrow factor which depends on the relative positions of pairs of particles.

Evidently it is important to expand the HF/DFT orbitals in a basis set that is both practical and physical. Unfortunately, these two qualities are somewhat mutually exclusive. The cusp behavior of s-like molecular or crystalline orbitals corresponds to the r-dependence of Slater orbitals which go as exp(-Zr). However, the multidimensional integrals required by Hartree-Fock are extremely time-consuming in a Slater function basis. It is this computational problem to which Gaussian bases are perfectly suited. The integrals can be done analytically, and the savings in time allows for large basis sets at minimal cost. A problem with Gaussian bases lies in their behavior near r=0. All Gaussian functions of the form f(r)exp(-Zr^2) have zero gradient at the origin, thereby rendering it impossible for the trial wave function to satisfy the nuclear cusp conditions.

Because the determinant(s) is such a large factor in the energy and quality of the trial wavefunction, it seems beneficial to try to solve the cusp problem within the Gaussian bases, rather than apply a patch within the Jastrow correlation function. Our improved sampling methods concentrate on modifying the Gaussian basis functions near the core, while preserving continuity (to avoid bias and fluctuations) and minimizing additional computational effort.