Too big, too small or just right? A benchmark assessment of density functional theory for predicting the spatial extent of the electron density of small chemical systems

Multipole moments are the first order responses of the energy to spatial derivatives of the electric field strength. The quality of density functional theory (DFT) prediction of molecular multipole moments thus characterizes errors in modeling the electron density itself, as well as the performance in describing molecules interacting with external electric fields. However, only the lowest non-zero moment is translationally invariant, making the higher order moments origin-dependent. Therefore, instead of using the 3 times 3 quadrupole moment matrix, we utilize the translationally invariant 3 times 3 matrix of second cumulants (or spatial variances) of the electron density as the quantity of interest denoted by $mathcal{K}$. The principal components of {mathcal{K}} are the square of the spatial extent of the electron density along each axis. A benchmark dataset of the prinicpal components of {mathcal{K}} for 100 small molecules at the coupled cluster singles and doubles with perturbative triples (CCSD(T)) at the complete basis set (CBS) limit is developed, resulting in 213 independent {mathcal{K}} components. The performance of 47 popular and recent density functionals is assessed against this Var213 dataset. Several functionals, especially double hybrids, and also SCAN and SCAN0 yield reliable second cumulants, although some modern, empirically parameterized functionals yield more disappointing performance. The H and Be atoms in particular are challenging for nearly all methods, indicating that future functional development could benefit from inclusion of their density information in training or testing protocols.