The Equations of Motion Method: An Approach to the Dynamical Properties of Atoms and Molecules

This chapter is concerned with the equations of motion method as a many-body approach to the dynamical properties of atoms and molecules. In a wide range of spectroscopic experiments one is primarily concerned with just dynamical properties. These dynamical properties include excitation energies and oscillator strengths in optical spectroscopy, the dynamic or frequency-dependent polarizability in light scattering studies, photoionization cross sections, and elastic and inelastic electron scattering cross sections. These experiments probe the response of an atom or molecule to some external perturbation. If one is concerned with these properties one should develop a formalism which aims directly at these properties. Of course this idea is not novel. For example, one might try to calculate the appropriate Green’s functions whose poles, and residues at these poles, are directly the excitation energies and transitions densities, respectively. One could also attempt to solve the time-dependent Schrödinger equation directly, e.g., in the time-dependent Hartree—Fock approximation. The approach to these dynamical properties of atoms and molecules which we will discuss is based on the equations of motion formalism as suggested by Rowe.(1) This is a very practical formalism based on the equations of motion for excitation operators defined as operators that convert one stationary state of a system into another state