Variational formulation of dynamical electronic response functions in the presence of nonlocal exchange interactions

We consider the dynamical electronic response function in theoretical frameworks that include nonlocal exchange interactions, such as the Bethe-Salpeter equation with the frequency independent approximation of the screened interaction, Hartree-Fock, and range-separated Hybrid DFT approaches. Within these pictures, we demonstrate that any time-dependent electronic linear response function allows for a formulation which is variational in the electronic density matrix. To achieve our goal, we consider the usual form of a response function, written in terms of a screened and a bare electronic vertices (``bare-screen''), and perform an exact rewriting in terms of purely screened electronic vertices (``screen-screen''). Within the ``screen-screen'' formulation, the response function can be written as a stationary point of a functional of the exact density matrix. Further, we show that the imaginary part of any electronic response can be written in the form of a generalized Fermi golden rule, by introducing an exact complementary rewriting in terms of vertices related by complex conjugation (``screen*-screen''). The screen-screen formulation can be further extended partitioning the electronic interaction in separate contributions, expressing the response in terms of partially screened electronic vertices (``partial screen-partial screen''), preserving the stationary properties. We numerically validate the effectiveness of our formalism by calculating the optical conductivity of graphene, which exhibits strong excitonic effects. To do so, we solve the Bethe-Salpeter Equation on a tight-binding model, including exchange effects in the response of graphene. Our findings show the advantages of the variationality of the screen-screen formulation over the others both in convergence properties and robustness with density-matrix approximations.