There are different calculations depending on what is the major cause of charge carriers recombining. You might have direct recombination across the band gap, releasing a photon of energy equal to the gap. This means finding matrix elements from integrating over wavefunctions in the semiconductor. (Getting the wavefunctions is another first principle calculation of course!)
An old example is Lasher and Stern, Phys. Rev. 133, A553 (1964).