Distributions and averages of electron density parameters: Explaining the effects of gradient corrections

We analyze the electron densities n(r) of atoms, molecules, solids, and surfaces. The distributions of values of the Seitz radius rs=(3/4πn)1/3 and the reduced density gradient s=|∇n|/(2(3π2)1/3n4/3) in an electron density indicate which ranges of these variables are significant for physical processes. We also define energy-weighted averages of these variables, 〈rs〉 and 〈s〉, from which local spin density (LSD) and generalized gradient approximation (GGA) exchange-correlation energies may be estimated. The changes in these averages upon rearrangement of the nuclei (atomization of molecules or solids, stretching of bond lengths or lattice parameters, change of crystal structure, etc.) are used to explain why GGA corrects LSD in the way it does. A thermodynamic-like inequality (essentially d〈s〉/〈s〉>d〈rs〉/2〈rs〉) determines whether the gradient corrections drive a process forward. We use this analysis to explain why gradient corrections usually stretch bonds (but not for example H–H bonds), reduce atomization and surface energies, and raise energy barriers to formation at transition states.