Microscopic derivation of Frenkel excitons in second quantization

Starting from the microscopic Hamiltonian describing free electrons in a periodic lattice, we derive the Hamiltonian appropriate to Frenkel excitons. This is done through a grouping of terms different from the one leading to Wannier excitons. This grouping makes the atomic states appear as a relevant basis to describe Frenkel excitons in the second quantization. Using them, we derive the Frenkel exciton creation operators as well as the commutators which rule these operators and which make the Frenkel excitons differ from elementary bosons. The main goal of the present paper is to provide the necessary grounds for future works on Frenkel exciton many-body effects, with the composite nature of these particles treated exactly through a procedure similar to the one we have recently developed for Wannier excitons.