Spatial shape of electron delocalization: structure of the laplacian of the negative exchange-correlation density

The Laplacian of the negative exchange-correlation density (with respect to coordinate r(2)), nabla<(r)2>(2)[-Gamma(sigma1)(sigma2)(XC) (r(1),r(2))] = nabla(r)2(2)X(sigma1)(sigma2)(r(1),r(2)), is proposed as an instrument for the analysis of electron delocalization in real space. It determines local concentrations in the amount of electrons that are delocalized from a reference point r(1) over space. Integration of the reference coordinate r(1) over an atomic basin Omega(n) gives the function nabla(2)X(sigma1)(sigma2)(Omega(n);r), which contains detailed information about the spatial shape of the delocalization that originates from an atom in a molecule. Its isosurface representations are richly structured and resemble molecular orbitals in their complexity and partly also in their shape. The sum over all nabla(2)X(sigma1)(sigma2)(Omega(n);r) functions of a molecule equals the Laplacian of the electron density nabla(2)rho(r), for which it provides a meaningful partitioning into atomic contributions.