Time-Dependent Gutzwiller Approximation: Interplay with Phonons

We discuss the interplay of phonons with electronic correlations in the small amplitude-limit of the time-dependent Gutzwiller approximation. In this approach the time-dependent Lagrangian for the Hubbard model can be computed with a Gutzwiller wave-function in infinite dimensions and subsequently expanded up to quadratic order in density and double occupancy fluctuations. The dynamics of the latter show a strong softening around the metal-insulator transition giving rise to an interesting interplay with phonons which couple to the hopping.