One-particle Green's function of interacting two electrons using analytic solutions for a three-body problem: comparison with exact Kohn--Sham system

TL;DR

This paper analytically solves a three-electron quantum system to derive exact Green's functions, compares them with Kohn-Sham approximations, and analyzes how interactions affect electronic properties and energy gaps.

Contribution

It provides the first explicit analytic solutions for three-electron eigenstates and constructs exact Green's functions, enabling systematic comparison with Kohn-Sham results.

Findings

Exact solutions for three-electron eigenstates obtained analytically.

Discrepancy in energy gaps increases with interaction strength.

Exact and Kohn-Sham Green's functions can differ significantly in shape.

Abstract

For a three-electron system with finite-strength interactions confined to a one-dimensional harmonic trap, we solve the Schroedinger equation analytically to obtain the exact solutions, from which we construct explicitly the simultaneous eigenstates of the energy and total spin for the first time. The solutions for the three-electron system allow us to derive analytic expressions for the exact one-particle Green's function (GF) for the corresponding two-electron system. We calculate the GF in frequency domain to examine systematically its behavior depending on the electronic interactions. We also compare the pole structure of non-interacting GF using the exact Kohn--Sham (KS) potential with that of the exact GF to find that the discrepancy of the energy gap between the KS system and the original system is larger for a stronger interaction. We perform numerical examination on the behavior of GFs in real space to demonstrate that the exact and KS GFs can have shapes quite different from each other. Our simple model will help to understand generic characteristics of interacting GFs.