Basic Mathematical Properties of Electronic Wave Functions in Momentum Space

Abstract

The analytic behavior of momentum eigenfunctions in the complex momentum plane is examined. One-electron models are considered including those based on local potentials as well as the Hartree-Fock equations. Methods for the location and characterization of singular points are described. Conformal mapping is employed to obtain eigenfunction expansions which are convergent for both real and complex values of the independent variable. Except by accident the singular points of momentum eigenfunctions are branch points except for atomic hydrogen where poles are encountered. The existence of branch points can eliminate certain series which might be employed to represent momentum eigenfunctions. Examples are discussed. An example is given in which the order of summation of basis sets affects convergence in the complex momentum plane. The connection between first order collision amplitudes and the Schrodinger equation in momentum space is discussed and singular points for the amplitudes are located. An exact local potential is described and the associated amplitude is examined. Other examples illustrating the importance of complex function theory in momentum space are described.