  We here present a fresh approach to Frenkel excitons in cubic semiconductor
crystals, with a special focus on the spin and spatial degeneracies of the
electronic states. This approach uses a second quantization formulation of the
problem in terms of creation operators for electronic states on all lattice
sites -- their creation operators being true fermion operators in the
tight-binding limit valid for semiconductors hosting Frenkel excitons. This
operator formalism avoids using cumbersome ($6N_s$ x $6N_s$) Slater
determinants -- 2 for spin, 3 for spatial degeneracy and $N_s$ for the number
of lattice sites -- to represent state wave functions out of which the Frenkel
exciton eigenstates are derived. A deep understanding of the tricky Coulomb
physics that takes place in the Frenkel exciton problem, is a prerequisite for
possibly diagonalizing this very large matrix analytically. This is done in
three steps: (i) the first diagonalization, with respect to lattice sites,
follows from transforming excitations on the $N_s$ lattice sites
$\mathbf{R}_\ell$ into $N_s$ exciton waves $\mathbf{K}_n$, by using appropriate
phase prefactors; (ii) the second diagonalization, with respect to spin,
follows from the introduction of spin-singlet and spin-triplet electron-hole
pair states, through the commonly missed sign change when transforming
electron-absence operators into hole operators; (iii) the third
diagonalization, with respect to threefold spatial degeneracy, leads to the
splitting of the exciton level into one longitudinal and two transverse modes,
that result from the singular interlevel Coulomb scattering in the small
$\mathbf{K}_n$ limit.
