Coupling relations between molecular electronic and geometrical degrees of freedom in density functional theory and charge sensitivity analysis

The mapping concepts, reflecting the equilibrium, ground-state coupling between the electronic and geometrical degrees of freedom of both closed and open molecular systems are explored within density functional theory (DFT) and charge sensitivity analysis (CSA). After a brief overview of the principal derivatives and relations, including alternative Legendre transformed representations of molecular states, the explicit transformations are derived for the mapping between the nuclear coordinates Q, specifying the molecular geometry, and the 'electronic' coordinates represented by the electron density rho(r), in the local resolution, or the vector N = [N(alpha)] of the atomic electron populations, in the atoms-in-molecule (AIM) resolution. They fall into two categories: the electron following, transforming a given shift of the nuclear coordinates into the conjugate relaxation of the electronic 'coordinates' rho or N, and the electron preceding, 'translating' a specified displacement of these electronic degrees of freedom into the conjugate geometry relaxation. Algorithms for determining such transformations are discussed with a special emphasis placed upon the semiempirical CSA modeling in atomic resolution. Several additional mapping quantities are identified, which reflect the coupling between molecular electronic descriptors, e.g. the number of electrons or the chemical potential, and the relevant geometry-related quantities, e.g. Q or the forces acting on nuclei. Implications of the mapping concepts for chemistry are briefly examined and possible areas for their application are identified.