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

In currently most popular explicitly correlated electronic structure theories, the dependence of the wave function on the interelectronic distance r_ij is built via the correlation factor f (r_ij). While the short-distance behavior of this factor is well understood, little is known about the form of f (r_ij) at large r_ij. 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 r_ij as