Sturmian basis functions in the coupled state impact parameter method for H++H scattering

When Sturmian basis functions are used in the standard coupled state impact parameter method for proton-hydrogen atom scattering, the correct boundary conditions cannot, in general, be satisfied. This means that for most choices of the initial and final states the approximate transition amplitude, as usually defined, does not exist. This difficulty is resolved and an approximate transition amplitude is defined which is a variational estimate of the exact transition amplitude for arbitrary initial and final states. In the Sturmian representation the exchange matrix elements can be calculated more directly by solving a set of coupled differential equations which is somewhat different from the set of equations given by Cheshire (1967). A procedure that does not require storage and interpolation of the matrix elements in order to exploit their time-symmetry is also described.