R-matrix theory of charge transfer

The $R$-matrix theory of Wigner is applied to the description of charge exchange. By making use of the correct coordinates for each arrangement, the boundary conditions are properly satisfied. Unphysical long-range couplings do not appear, and the use of "electronic traveling factors" is avoided. The wave function for different arrangements are nonorthogonal, and are essentially equivalent to the "intersecting waves" of Delos [Phys. Rev. A 23, 2301 (1981)]. It is shown under appropriate simplifying assumptions that the $R$-matrix formalism reduces to the high-energy two-state approximation of Kramers and Brinkman. The main task of the $R$-matrix theory is the determination of the matrix elements of the Hamiltonian between basis functions belonging to different arrangements. This is achieved by methods of functional transfer combined with appropriate simplifications arising from the smallness of $\frac{m}{M}$. This leads to the central result that the transformation of a highly oscillating nuclear radial function from one arrangement to another brings with it a highly oscillating function of the electronic coordinates of the new arrangement. This causes off-diagonal elements of the Hamiltonian matrix to decrease rapidly as the energy increases. Consequently the cross section for charge transfer also decreases with increasing energy.