Energies of atoms are approximately quadratic in nuclear charge

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 find energy differences between iso-electronic atoms to be approximately quadratic in their nuclear charges, $\Delta E \approx -(1 + 2\gamma \sqrt{Ne-1})\,\Delta Z\, \bar{Z}$. $\gamma \approx 0.3766 \pm 0.0020$ Ha corresponds to a universal constant, and $Ne$, $\Delta Z$, and $\bar{Z}$ respectively to electron number, and nuclear charge difference and average. We compare the formula's predictive accuracy using experimental numbers and numerical results obtained via DFT for the entire periodic table up to Radon. A detailed discussion of the atomic Helium-series is included. Finally, we show the applicability of AIT by predicting trends between electron affinities from ionization energies with linear corrections, resulting in a Spearman's rank correlation coefficient of 0.931.