Sum rules for nonlinear optical properties of molecules

It is explicitly shown, that calculated sum over all states of the nonlinear optical properties (NLO) of molecular system (polarizability, hyperpolarizability, etc.) is exactly zero when wave functions which satisfy the Hellmann–Feynman theorem is used. These sum rules are correct for sum over energy levels (i.e., translation, rotation, oscillation and electronic) in case of any one-electron properties which can be described by the Hamiltonian H0 + λH(1), where H0 is the Hamiltonian of the unperturbed molecule and λ and H(1) represent external perturbation and the one-electron operator of any physical property, respectively.