Alchemical insights into approximately quadratic energies of iso-electronic atoms

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.