Derivatives of the polarization propagator including orbital relaxation effects

In this article, we relate derivatives of the polarization propagator used in many-body theory to the nonlinear (quadratic) polarization propagator, and we relate derivatives of the quadratic polarization propagator to the nonlinear propagator of the next higher order, the cubic polarization propagator. We restrict the analysis to differentiation with respect to parameters eta for which the derivative of the Hamiltonian can be written as a sum of one-electron operators. Geometrical derivatives are obtained by specializing to the parameter eta to the alpha coordinate of nucleus I. We treat orbital relaxation explicitly by allowing for the eta dependence of creation and annihilation operators in the propagators. This treatment entails an extension of the geometrical derivative relations among response functions proven by Olsen and Jorgensen [J. Chem. Phys. 82, 3235 (1985)], because the propagator derivatives may involve changes in the one-electron orbitals that do not appear in the susceptibility derivatives. These results underlie the relations between Raman intensities and electric-field shielding tensors, which have been explained in terms of nonlocal polarizability and hyperpolarizability densities. The results suggest an alternative computational route to geometrical or other derivatives of both linear- and nonlinear-response functions: these derivatives can be evaluated without numerical differentiation, directly from the propagator of the next higher order.