Second order coalescence conditions of molecular wave functions

Kato's cusp condition gives the exact first order dependence of molecular wave functions on interparticle separation near the coalescence of two charged particles. We derive conditions correct to second order in interparticle separation, which concern second order derivatives of the wave function at the coalescence point. For identical particle coalescence, we give equations correct to third order. In addition to a universal, particle dependent term, a system and state dependent term arises in the higher order conditions, which we interpret as an effect of Coulombic screening. We apply our analysis to the standard orbital-based methods of quantum chemistry and discuss the implications for Jastrow- and R12-type correlation factors.