Theory of optically forbidden d-d transitions in strongly correlated crystals.

A general multiband formulation of the linear and nonlinear optical response functions for realistic models of correlated crystals is presented. Dipole-forbidden d-d optical transitions originate from vertex functions, which we consider assuming the locality of an irreducible four-leg vertex. The unified formulation for second- and third-order response functions in terms of the three-leg vertex is suitable for practical calculations in solids. We illustrate the general approach by consideration of intra-atomic spin-flip contributions, with an energy of 2J, where J is a Hund exchange, in the simplest two-orbital model.