Alchemical insights into approximately quadratic energies of iso-electronic atoms

TL;DR

This paper introduces an analytical estimate for energy differences between iso-electronic atoms based on the Alchemical Integral Transform, validated with experimental and DFT data across the periodic table, revealing an approximately quadratic energy behavior.

Contribution

It presents a novel simple analytical formula for predicting energy differences between iso-electronic atoms, grounded in the Alchemical Integral Transform approach.

Findings

The formula accurately predicts energy differences across the periodic table.

Energy differences exhibit approximately quadratic behavior with respect to nuclear charge.

Validation against experimental and DFT data confirms the formula's effectiveness.

Abstract

Accurate quantum mechanics based predictions of property trends are so important for materials design and discovery that even inexpensive approximate methods are valuable. We use the Alchemical Integral Transform (AIT) to study multi-electron atoms, and to gain a better understanding of the approximately quadratic behavior of energy differences between iso-electronic atoms in their nuclear charges. Based on this, we arrive at the following simple analytical estimate of energy differences between any two iso-electronic atoms, $\Delta E \approx -(1 + 2\gamma \sqrt{N_e-1})\,\Delta Z\, \bar{Z}$. Here, $\gamma \approx 0.3766 \pm 0.0020$ Ha corresponds to an empirical constant, and $N_e$, $\Delta Z$, and $\bar{Z}$ respectively to electron number, and nuclear charge difference and average. We compare the formula's predictive accuracy using experimental numbers and non-relativistic, numerical results obtained via DFT (pbe0) for the entire periodic table up to Radon. A detailed discussion of the atomic Helium-series is included.