## 3 Results
### Effect of the QD Size Dispersion on Excitonic Level Structure
**(C)** Energies of the eigen exciton for a homodimer of QD with a diameter of 2.5 nm and L = 0.55 nm. The size dispersion is 5% in diameter. All the bands overlap, but the splitting is small because L is larger, except for the higher energy exciton bands that involve the excitation of h2 hole. We can clearly see in Figure 4A left that the bands (centered at 2.44 eV) of excitons h2e\({}_{\text{B}}\) and h1B-e\({}_{\text{A}}\) overlap and are well separated from the other pairs of excitons. There is also an overlap between the two exciton bands that involve the local exciton h2e\({}_{\text{A}}\) and the CT one h2\({}_{\text{A}}\)-e\({}_{\text{B}}\). For each of these bands, the effect of the Coulomb electronic coupling is enhanced because of accidental resonances. The sizes of panel B were engineered to be able to probe interdot coherences between local excitons and intradot ones. There is efficient coupling between the local exciton bands of h2e\({}_{\text{B}}\) and h1e\({}_{\text{A}}\) (panel B left) that partially overlap, as well as between the higher exciton bands h2e\({}_{\text{A}}\) and the CT one h2\({}_{\text{A}}\)-e\({}_{\text{B}}\). For the homodimer (panel C), all the exciton bands overlap. The zero order excitons are coupled by the Coulomb interactions, which also exhibit a dispersion in their strengths owing to the size dispersion effect on the wave functions (see Figure 3). The spectrum of the energies of eigenstates of the \(8\times 8\) excitonic Hamiltonians of the 1000 dimers obtained by diagonalizing the Coulomb interactions are shown on the right hand side of panels A, B, and C of Figure 4. Overall, the distributions of eigen exciton energies remain Gaussian, which can be understood because the values of the Coulomb couplings are small. Compared with zero order, one observes a splitting of the two bands of excitons that were engineered to overlap, h2e\({}_{\text{B}}\) and h1B-e\({}_{\text{A}}\) in panel A and of all the bands in panel C. The splitting is larger when the Coulomb interactions are large and the surface-to-surface distance between the QDs is small. In panel B, there is no significant splitting of the bands h1e\({}_{\text{A}}\) and h2e\({}_{\text{B}}\) because they are not in exact resonance. However, because of the bands overlap, there are accidental degeneracies that increase the efficiency of the Coulomb coupling. This stronger effective coupling is responsible for a larger distortion of the zero order distribution, as can be seen in the values of the fits reported in Table S1. Typically, the bands of the two excitons that overlap at higher energy, h2e\({}_{\text{A}}\) and h2\({}_{\text{A}}\)-e\({}_{\text{B}}\) in panels A and B, are less effectively coupled because the Coulomb interaction between these two excitons is smaller and there are less accidental degeneracies because these two bands are more spread in energy. On the other hand, in the homodimer (panel C), one observes a large splitting of the four higher energy bands that corresponds to the four excitons (h2e\({}_{\text{B}}\), h2e\({}_{\text{A}}\), h2\({}_{\text{A}}\)-e\({}_{\text{B}}\), h2\({}_{\text{B}}\)-e\({}_{\text{A}}\)). Note that, in the case of the homodimer, the Coulomb interactions lead to a redistribution of the oscillator strength and break the rule of a purely bright and a purely dark eigen exciton, as would be the case for homodimers of molecules.