Self-consistent electron–nucleus cusp correction for molecular orbitals

Abstract

We describe a method for imposing the correct electron-nucleus (e-n) cusp in molecular orbitals expanded as a linear combination of (cuspless) Gaussian basis functions. Enforcing the e-n cusp in trial wave functions is an important asset in quantum Monte Carlo calculations as it significantly reduces the variance of the local energy during the Monte Carlo sampling. In the method presented here, the Gaussian basis set is augmented with a small number of Slater basis functions. Note that, unlike other e-n cusp correction schemes, the presence of the Slater function is not limited to the vicinity of the nuclei. Both the coefficients of these cuspless Gaussian and cusp-correcting Slater basis functions may be self-consistently optimized by diagonalization of an orbital-dependent effective Fock operator. Illustrative examples are reported for atoms (H, He and Ne) as well as for a small molecular system (BeH 2 ). For the simple case of the He atom, we observe that, with respect to the cuspless version, the variance is reduced by one order of magnitude by applying our cusp-corrected scheme.

Discontinuities of Kinetic Energy Densities within Finite and Complete Basis Sets

“…In other situations, however, discontinuities of kinetic energy densities may have tangible consequences. For instance, when imposing nuclear cusp conditions on molecular orbitals − it might be beneficial to go beyond the standard equations for spherically averaged quantities and try to impose cusp conditions with explicit directional dependence. The fact that, in H 2 , discontinuities of τ­( r ) can be as large as 75% of the value of τ­( r ) at the nucleus (Figure ) may explain why the spherically averaged nuclear cusp condition is harder to impose for hydrogens than for heavier nuclei …”

Vmc Optimization of Ultra-Compact, Explicitly-Correlated Wave

“…Satisfying the cusp conditions is a relevant asset in quantum Monte Carlo calculations since it significantly reduces the variance of the local energy during random sampling. 39 It is also known that fulfilling cusp conditions is relevant to obtain an adequate description of the electron energy distributions in double photoionization. 40…”

Toward quantum-chemical method development for arbitrary basis functions

“…However, even contracted GTOs (cGTO) fail to describe both the nuclear cusp and the exponential decay of the electron density 6 . In addition, no matter the number of GTO basis functions used, the derivatives are always wrong at the nucleus, which causes singularities and computational failure for example in quantum Monte Carlo calculations [7][8][9] . Furthermore the description of some properties such as the nuclear-magnetic resonance (NMR) shielding tensors or a description of Rydberg-like or autoionising states [10][11][12][13] directly involves the nuclear cusp or the asymptotic tail, making physically accurate basis functions desirable 14,15 .…”