  By using the composite many-body theory for Frenkel excitons we have recently
developed, we here derive the ground state energy of $N$ Frenkel excitons in
the Born approximation through the Hamiltonian mean value in a state made of
$N$ identical $\mathbf{Q=0}$ excitons. While this quantity reads as a density
expansion in the case of Wannier excitons, due to many-body effects induced by
fermion exchanges between $N$ composite particles, we show that the Hamiltonian
mean value for $N$ Frenkel excitons only contains a first order term in
density, just as for elementary bosons. Such a simple result comes from a
subtle balance, difficult to guess a priori, between fermion exchanges for two
or more Frenkel excitons appearing in Coulomb term and the ones appearing in
the $N$ exciton normalization factor - the cancellation being exact within
terms in $1/N_{s}$ where $N_{s}$ is the number of atomic sites in the sample.
This result could make us naively believe that, due to the tight binding
approximation on which Frenkel excitons are based, these excitons are just bare
elementary bosons while their composite nature definitely appears at various
stages in the precise calculation of the Hamiltonian mean value.
