# Analytic model of a multi-electron atom

## TL;DR

This paper introduces a fully analytical approximation for multi-electron atoms using a hydrogen-like basis with a single-effective charge, enabling accurate, convergent calculations of atomic properties including correlation effects.

## Contribution

It develops a novel analytical perturbation method employing a hydrogen-like basis that captures correlation effects and allows closed-form summations over states.

## Key findings

- Accurately describes the entire spectrum of multi-electron atoms.
- Provides results comparable to multi-configuration Hartree-Fock at second order.
- Achieves convergence and accuracy independent of the number of electrons.

## Abstract

A fully analytical approximation for the observable characteristics of many-electron atoms is developed via a complete and orthonormal hydrogen-like basis with a single-effective charge parameter for all electrons of a given atom. The basis completeness allows us to employ the secondary-quantized representation for the construction of regular perturbation theory, which includes in a natural way correlation effects, converges fast and enables an effective calculation of the subsequent corrections. The hydrogen-like basis set provides a possibility to perform all summations over intermediate states in closed form, including both the discrete and continuous spectra. This is achieved with the help of the decomposition of the multi-particle Green function in a convolution of single-electronic Coulomb Green functions. We demonstrate that our fully analytical zeroth-order approximation describes the whole spectrum of the system, provides accuracy, which is independent of the number of electrons and is important for applications where the Thomas-Fermi model is still utilized. In addition already in second-order perturbation theory our results become comparable with those via a multi-configuration Hartree-Fock approach.