On the large interelectronic distance behavior of the correlation factor for explicitly correlated wave functions

TL;DR

This paper investigates the behavior of the correlation factor at large interelectronic distances in explicitly correlated wave functions, deriving new asymptotic forms and proposing a range-separated model that improves existing theories.

Contribution

It derives the large-distance asymptotic form of the correlation factor and introduces a novel range-separated model that enhances the accuracy of explicitly correlated wave functions.

Findings

f(r12) behaves as r12^{ ho} exp(B r12) at large r12

For helium-like ions, B is positive, leading to divergence of f(r12)

The new range-separated form improves the performance of R12/F12 methods

Abstract

In currently most popular explicitly correlated electronic structure theories the dependence of the wave function on the interelectronic distance rij is built via the correlation factor f(rij). While the short-distance behavior of this factor is well understood, little is known about the form of f(rij) at large rij. In this work we investigate the optimal form of f(r12) on the example of the helium atom and helium-like ions and several well-motivated models of the wave function. Using the Rayleigh-Ritz variational principle we derive a differential equation for f(r12) and solve it using numerical propagation or analytic asymptotic expansion techniques. We found that for every model under consideration, f(r12) behaves at large r12 as r12^{\rho} exp(B r12) and obtained simple analytic expressions for the system dependent values of {\rho} and B. For the ground state of the helium-like ions the value of B is positive, so that f(r12) diverges as r12 tends to infinity. The numerical propagation confirms this result. When the Hartree-Fock orbitals, multiplied by the correlation factor, are expanded in terms of Slater functions r^n exp(-{\beta}r), the numerical propagation reveals a minimum in f(r12) with depth increasing with N. For the lowest triplet state B is negative. Employing our analytical findings, we propose a new 'range-separated' form of the correlation factor with the short- and long-range r12 regimes approximated by appropriate asymptotic formulas connected by a switching function. Exemplary calculations show that this new form of f(r12) performs somewhat better than the correlation factors used thus far in the standard R12 or F12 theories.