  We review the Bethe-Salpeter equation (BSE) approach to the calculation of
electronic excitation energies of molecular systems. We recall the general
Green's function many-theory formalism and give the working equations of the
BSE approach within the static GW approximation with and without spin
adaptation in an orbital basis. We apply the method to the pedagogical example
of the H2 molecule in a minimal basis, testing the effects of the choice of the
starting one-particle Green's function. Using the non-interacting one-particle
Green's function leads to incorrect energy curves for the first singlet and
triplet excited states in the dissociation limit. Starting from the exact
one-particle Green's function leads to a qualitatively correct energy curve for
the first singlet excited state, but still an incorrect energy curve for the
triplet excited state. Using the exact one-particle Green's function in the BSE
approach within the static GW approximation also leads to a number of
additional excitations, all of them being spurious except for one which can be
identified as a double excitation corresponding to the second singlet excited
state.
