Atomic Radii Derived From the Expectation Value r 4 $$ \left\langle {r}^4\right\rangle $$

Abstract

The atomic radius as a fundamental chemical descriptor for the size of a chemical element is often used in physical chemistry. Many reference sets are available, based either on experiment or calculations. For example, Alvarez compiled a set of consistent van der Waals radii (Dalton Trans. 2013, 42, 8617) based on millions of measured interatomic non-bonded distances. In quantum mechanics, there are many ways in which the atom size can be defined and obtained because the atomic radius is not an observable. Here, we show that a theoretical measure can be based on expectation values such as $\left(r^2\right)$ and $\left(r^4\right)$. These are easily obtained from atomic electric moments, routinely generated by popular quantum chemistry codes, with full control over electronic structure, charge, spin state, etc. As such we obtain a measure for the size of free atoms H to Xe and demonstrate linear scaling of atomic size in the series as outermost s, p or d subshells are filled according to the Madelung rule. Radii derived from $\left(r^4\right)$ compare best to Alvarez's empirical reference set of van der Waals radii, and atomic radii from theoretical sources. Known periodic trends of atomic radii are well reproduced by our data. Furthermore, we demonstrate the dependence of atomic size on the electronic structure and spin state for d-block elements.