  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 appearing the atomic states 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 differing 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.
