  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.
