Excitation operators and the extended random-phase approximation. I. Theory

An analysis of extended random-phase approximations is made with emphasis on the justification of the simplifying assumptions used in applications. It is demonstrated that in a complete operator space, the equations of motion in expectation value form, supplemented by the ground-state annihilation condition, yield the exact result for excitation operators and are equivalent to the operator equation of motion. Certain operators that are normally neglected have to be included to obtain the exact solution; some of these additional operators are shown to be important while others are expected to have small effects. The random-phase approximation and its self-consistent variants are analyzed, and a number of improvements that can be introduced without increasing the difficulty of the calculations are suggested.