Quadratic response functions within the time-dependent Hartree-Fock approximation

The response of an atomic or molecular system to an external perturbation is studied within the framework of time-dependent Hartree-Fock theory. Quadratic response functions are defined and analyzed with regard to their poles and residues, from which an expression for matrix elements of an arbitrary operator between excited states is deduced. For one-particle operators the matrix elements $〈m\left|A\right|n〉$ are correct through first order in correlation and the hypervirial relation $(E_n-E_m) 〈m\left|A\right|n〉=〈m\left|\mathrm{}\right[A,H\left]\mathrm{}\right|n〉$ is satisfied. Spurious poles, which have no counterpart in an exact theory, are encountered in the quadratic response functions. Their appearance is attributed to well-known deficiencies in the time-dependent Hartree-Fock approximation.