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We review the Bethe-Salpeter equation (BSE) approach to the calculation ofelectronic excitation energies of molecular systems. We recall the generalGreen's function many-theory formalism and give the working equations of theBSE approach within the static GW approximation with and without spinadaptation in an orbital basis. We apply the method to the pedagogical exampleof the H2 molecule in a minimal basis, testing the effects of the choice of thestarting one-particle Green's function. Using the non-interacting one-particleGreen's function leads to incorrect energy curves for the first singlet andtriplet excited states in the dissociation limit. Starting from the exactone-particle Green's function leads to a qualitatively correct energy curve forthe first singlet excited state, but still an incorrect energy curve for thetriplet excited state. Using the exact one-particle Green's function in the BSEapproach within the static GW approximation also leads to a number ofadditional excitations, all of them being spurious except for one which can beidentified as a double excitation corresponding to the second singlet excitedstate.

Electronic excitation energies of molecular systems from the Bethe-Salpeter equation: Example of the H2 molecule