Asymptotic forms for the nonrelativistic coulomb propagator

The nonrelativistic Coulomb Green's function G(r 1 , r 2 , k) = G(x,y,k) (where x ≡ r 1 + r 2 + r 12 , y ≡ r 1 + r 2 − r 12 ) was worked out in closed form by Hostler in 1963. There is as yet no closed form for the corresponding propagator K(x,y,t) . The unavailability of this function is a serious defect of Feynman's path‐integral formulation of quantum mechanics. It has moreover created a bottleneck in our formalism for generating atomic and molecular eigenvalue spectra by time‐dependent Green's function techniques. As, perhaps, a first step toward the ultimate realization of a closed form, we have worked out asymptotic forms for K in the limiting domains as Z → 0, t → 0, and r → ∞. We have exploited the structure of the propagator, whereby K = F exp (iS) , in terms of the classical action integral S . Our main result is a first‐order approximation to the Coulomb propagator, which can be connected to a corresponding approximation for G . We also propose a simpler function K̃ for possible application in the aufbau of many‐electron Green's functions. As such, this function might play a role analogous to that of Slater‐type orbitals in conventional quantum chemistry.