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 rij as

$r_{12}^\rho \,e^{B r_{12}}$
r12ρeBr12 and obtained simple analytic expressions for the system dependent values of ρ 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 rne−βr, n = 0,…,N, 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.

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