How far do electrons delocalize?
Electron delocalization is central to chemical bonding, but it is also a fundamentally nonclassical and nonintuitive quantum mechanical phenomenon. Tools to quantify and visualize electron delocalization help to understand, teach, and predict chemical reactivity. We develop a new approach to quantify and visualize electron delocalization in real space. Our electron delocalization range function EDR (r⃗;u) quantifies the degree to which electrons at point r⃗ in a calculated wavefunction delocalize over length scale u. Its predictions are physically reasonable. For example, EDR (r⃗;u=0.25 bohr ) is close to one at points r⃗ in the cores of first-row atoms, consistent with the localization of core electrons to ~0.25 bohr. EDR (r⃗;u=1 bohr ) is close to one at points r⃗ in typical covalent bonds, consistent with electrons delocalizing over the length of the bond. Our approach provides a rich representation of atomic shell structure; covalent and ionic bonding; the delocalization of excited states, defects, and solvated electrons; metallic and insulating systems; and bond stretching and strong correlation.