Analytic solution of the Schrodinger equation for an electron in the field of a molecule with an electric dipole moment

TL;DR

This paper presents an exact analytic solution to the Schrödinger equation for an electron interacting with a molecule possessing a permanent electric dipole moment, expanding the class of solvable quantum potential problems.

Contribution

It introduces a novel tridiagonal matrix approach allowing exact solutions for non-central dipole potentials, including both bound and scattering states, and computes critical dipole moments for electron capture.

Findings

Exact solutions for all energy states including scattering states.

Critical dipole moments for electron capture in neutral molecules.

Series expansion using orthogonal polynomials with recursion relations.

Abstract

We relax the usual diagonal constraint on the matrix representation of the eigenvalue wave equation by allowing it to be tridiagonal. This results in a larger solution space that incorporates an exact analytic solution for the non-central electric dipole potential cos(theta)/r^2, which was known not to belong to the class of exactly solvable potentials. As a result, we were able to obtain an exact analytic solution of the three-dimensional time-independent Schrodinger equation for a charged particle in the field of a point electric dipole that could carry a nonzero net charge. This problem models the interaction of an electron with a molecule (neutral or ionized) that has a permanent electric dipole moment. The solution is written as a series of square integrable functions that support a tridiagonal matrix representation for the angular and radial components of the wave operator. Moreover, this solution is for all energies, the discrete (for bound states) as well as the continuous (for scattering states). The expansion coefficients of the radial and angular components of the wavefunction are written in terms of orthogonal polynomials satisfying three-term recursion relations. For the Coulomb-free case, where the molecule is neutral, we calculate critical values for its dipole moment below which no electron capture is allowed. These critical values are obtained not only for the ground state, where it agrees with already known results, but also for excited states as well.