# Coupled Self-Consistent RPA Equations for Even and Odd Particle Numbers. Tests with Solvable Models

## TL;DR

This paper develops coupled self-consistent RPA equations for even and odd particle numbers, enabling self-consistent calculation of occupation probabilities, and demonstrates their effectiveness with solvable models like Lipkin and Hubbard.

## Contribution

Introduces a novel coupled equation framework for even and odd particle numbers based on the same correlated vacuum, extending SCRPA methods.

## Key findings

- Very good results in Lipkin model tests
- Accurate occupation probabilities for odd particle numbers
- Unified approach improves consistency between even and odd cases

## Abstract

Coupled equations for even and odd particle number correlation functions are set up via the equation of motion method. For the even particle number case this leads to self-consistent RPA (SCRPA) equations already known from the literature. From the equations of the odd particle number case the single particle occupation probabilities are obtained in a self-consistent way. This is the essential new procedure of this work. Both, even and odd particle number cases are based on the same correlated vacuum and, thus, are coupled equations. Applications to the Lipkin model and the 1D Hubbard model give very good results.