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

A general multiband formulation of linear and non-linear optical response functions for realistic models of correlated crystals is presented. Dipole forbidden d-d optical transitions originate from the vertex functions, which we consider assuming locality of 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 intraatomic spin-flip contributions, with the energy of 2J, where J is a Hund exchange, in the simplest two-orbital model.