Momentum eigenfunctions in the complex momentum plane. II. Relation of secondary singular points to one-electron, local, potential functions

Abstract

In a previous paper [J. Chem. Phys. 64, 4375 (1976)] it was shown that the points P=±i√−2E in the complex momentum plane (E is the one-electron energy eigenvalue) are singular points of the one-electron momentum eigenfunction obtained from a one-electron local potential function. In the present research a whole series of additional (secondary) singular points are located. These depend on the specific nature of the one-electron potential function in particular the types of exponentials contained in the potential. For bound states these singular points (like the primary singular points) lie on the imaginary axis but are further from the origin than the points ±i√−2E. Moreover, the positions in the P plane of the primary singularites are independent of direction of the momenturm vector and the same is also true of the secondary singular points for each of the (independent particle) potential functions considered in the present research, a major simplification. Since the primary singular points are closer to the origin than any others, they determine the radius of convergence of power series expansions and also the transformations which provide analytic continuation of the power series. The new singular points do not affect the convergence of series expansions used in past research. Collision amplitudes, in Born approximation, representing elastic scattering of an electron by a static, local, potential field also possess primary singular points on the imaginary axis of q (q being the magnitude of a momentum difference vector q) and secondary singular points further out on the imaginary axis. (The amplitudes for inelastic scattering also possess such singular points.) Again, the secondary singularities do not affect the convergence of series expansions used in past research [e.g., J. Chem. Phys. 57, 4357 (1972)]. Moreover, the positions of the singular points in the q-plane are independent of the direction of q. In momentum representation the Hartree–Fock equations are integral equations with kernel functions which are collision amplitudes, in Born approximation, for elastic scattering with exchange. The exchange amplitudes arise from nonlocal potentials, but approximate local potentials can be obtained by replacing the exchange amplitudes by functions only of a difference vector q and applying an inverse Fourier transform. The collision amplitudes corresponding to several local potential functions proposed by other investigators are examined for primary singular points. It is found that the primary singular point in the q-plane is related to one-electron eigenvalues in the same way for each of the potentials and hence a general expansion of the corresponding amplitude is suggested. From the inverse Fourier transform a series is obtained for the local potential. If the coefficients of the series are adjusted to give minimum energy for the system then the variable terms of the series constitute a basis set for the representation of local potential functions. For atoms, two different basis sets are discussed, one leading to a linear combination of Yukawa functions. The nonlinear Hartree–Fock equations can be arranged as integral equations in two different ways which lead, with the same approximations, to two different local potential functions for the same filled orbitals. If the same potential functions are applied to obtain excited (virtual) orbitals one set corresponds to a negative ion and the other to a neutral atom. The conditions on the atomic orbitals used in LCAO treatments is discussed.