Direct iteration-variation method for scattering problems

We propose a direct iterative-variational scheme to solve the large sets of coupled integro-differential equations that arise in a variety of atomic and molecular scattering problems. The method, which is similar to direct configuration-interaction schemes of quantum chemistry, is applied within the linear algebraic (LA) prescription and involves the construction of an orthonormal basis from successive applications of the general LA matrix, labeled by channels and mesh points, to an initial guess for the solution vector. The solution vector is expanded in this basis, and the linear coefficients determined by a variational scheme. Since the basis is orthonormal, the procedure is guaranteed to converge within n iterations, where n is the order of the matrix. For all cases treated, the convergence is much more rapid. In addition, since a direct method is employed, only the potential, Green’s function, and solution vector need be stored. This formulation drastically reduces the central storage requirements of the LA method as well as improves the solution times since the integrals involved can be constructed from recursion relations. We apply the method to elastic scattering of electrons by ${\mathrm{N}}_2$ and LiH as well as inelastic collisions from ${\mathrm{H}}_2$ ${\mathrm{}}^{+}$.