Fig. 1: Exciton energies in the hydrogen dimer.

BSE excitation energies (i.e., energy required by the system to excite a given optical mode) are reported as a function of the interatomic separation R (MBD excitation energies are shown as inset). The first two BSE excitation modes (cyan) correspond to 1s → 2s transitions and undergo no dipolar splitting due to their spherical symmetry. BSE excitations 3–8 instead correspond to 1s → 2p transitions (degenerate only in the limit of vanishing interatomic coupling), and exhibit growing splitting at short interatomic distance. Optical vdW forces for a given excitation mode correspond to the inverse of the exciton-energy gradient. Gray balls schematically indicate atoms, while arrows (whose colors replicate the corresponding energy curves) give intuitive description of the correlation arising between dipolar displacements in the two H atoms. In excitons 3 and 8 (blue and black, respectively), atomic dipole modes are oriented along the direction joining the two atoms. Parallel dipoles (exc. 3) imply attractive coupling, while repulsion is found when the two dipoles are antiparallel, in analogy to the electrostatic case. Similar mechanisms arise in excitons 4–7, where dipole moments are instead oriented orthogonally to the H–H direction.