Orbital forces and chemical bonding in density-functional theory: Application to first-row dimers

The local-density-functional description of chemical bonding in first-row homonuclear dimers is analyzed in terms of both static and dynamic orbital forces. The dynamic orbital force of the ith molecular orbital is equal to the negative of the derivative of the one-electron energy ${\varepsilon }_i$ with respect to the pth nuclear coordinate, i.e., -∂${\varepsilon }_i$/∂$X_p$. The static orbital force is also equal to a derivative of the one-electron energy, but the differentiation is carried out with the orbital held fixed, i.e., (-∂${\varepsilon }_i$/∂$X_p$)ψ. It is shown that the static force is the orbital’s contribution to the total Hellmann-Feynman force, whereas the dynamic force describes the change in the total force due to change in the orbital’s occupation number. The chemical bond force in the first-row dimers is observed to be a delicate balance between bonding and antibonding orbital forces. Most of the bond force comes from the 2${\sigma }_g$ orbital and to a lesser extent from the 1${\pi }_u$ state. The polarization of the core orbitals in ${\mathrm{N}}_2$, ${\mathrm{O}}_2$, and ${\mathrm{F}}_2$ is found to originate indirectly through their interaction with the 3${\sigma }_g$ orbital. The dynamic orbital force gives accurate predictions about the change of equilibrium bond distances accompanying electronic ionization and excitation. The formalism and results are related to earlier Hartree-Fock studies.