Kohn variational principle for a sphere of arbitrary radius

In this article, I consider the use of the Kohn variational principle in obtaining the reaction matrix, which determines the positive energy solution to the partial wave Schrödinger equation in coordinate space. The formulation considered is appropriate for electron scattering from an atom or molecule. In its usual form, the Kohn variational principle provides the relationship between the exact reaction matrix, independent of first-order errors in the trial wavefunction, and the trial reaction matrix. This relationship is usually called the Kato identity. It is obtained using the second form of Green's theorem for a sphere of infinite radius. I show how a Kohn-like variational principle, using Green's theorem for a sphere of arbitrary radius, provides an expression for the exact, coordinate space, integral equation amplitudes in terms of the trial amplitudes. The results are again exact to second order in the trial wavefunction. Applications of the present results to electron-molecule frame transformations and algebraic scattering theory are then suggested.