## 3 Results  
### Effect of the QD Size Dispersion on Excitonic Level Structure  
In Figure 4, we show examples of the effect of the size dispersion on the energies of the zero order exciton levels (left) and of the eigen excitons (right) for ensembles of 1000 dimers of three different types: In panel A, a heterodimer built with QD\({}_{\text{A}}\) and QD\({}_{\text{B}}\) of mean diameters equal to D\({}_{\text{A}}\) = 2.35 nm and D\({}_{\text{B}}\) = 3.85 nm with 3% size dispersion and a surface to surface linker distance L = 0.2 nm. The two mean sizes were tuned so that the local exciton h2e\({}_{\text{B}}\) is isoenergetic with the CT exciton h1\({}_{\text{B}}\)-e\({}_{\text{A}}\). In panel B, a heterodimer built with QD\({}_{\text{A}}\) and QD\({}_{\text{B}}\) of mean diameters of D\({}_{\text{A}}\) = 2.7 nm and D\({}_{\text{B}}\) = 3.5 nm with 5% size dispersion for L = 0.2 nm. In this case, there is no exact resonance between excitonic bands on each dot, but the bands h1e\({}_{\text{A}}\) and h2e\({}_{\text{B}}\) overlap, which leads to efficient interdot coupling between these two local excitons. In panel C, we show the level structure of the excitonic bands for a homodimer with a mean diameter of 2.5 nm, a size dispersion of 5%, and L = 0.55 nm. In the case of a homodimer, all the excitonic bands are resonant. At the single particle level, the size distribution induces wider energy distributions for the small (QD\({}_{\text{A}}\)) dots than for the large ones (QD\({}_{\text{B}}\)), as expected because confinement effects are larger for smaller sizes. For the same reason, the energy distributions of the single particle levels are also wider for electrons than holes and for hole 2 than for hole 1, see Figure S2 of Section 1 of the SI. As a consequence, in a heterodimer where QD\({}_{\text{A}}\) is smaller than QD\({}_{\text{B}}\), the confinement effects lead to wider energy distributions for the local h2e\({}_{\text{A}}\) and h1e\({}_{\text{A}}\) and for the CT h2\({}_{\text{A}}\)-e\({}_{\text{B}}\) excitons. From the random size distributions of the two QDs (see Figure S10, Section 5 of the SI), we obtain energy distributions of the zero order exciton that are also fitted to Gaussian distributions (Figure 4 left), with widths ranging from 10 to 80 meV depending on the type of excitation involved in each exciton. The values of the parameters of the Gaussian fits to the zero order energy distributions of the exciton for the QD parameters used in Figure 4A are given in Table S1. In addition to this intrinsic size dispersion effect, the exciton energy distribution will also be affected by environment effects, which we do not explicitly take into account here. The distributions of the coupling strengths also depend on the distribution of the one particle wave function. They are plotted for selected exciton pairs in Figure S11, Section 5 of the SI. Those of the transition dipole matrix elements are plotted in Figure S12 for the dimer with QD sizes of panel A. They also exhibit Gaussian distributions, and the parameters are given in Table S1. Figure 4: Dispersion of the energy levels of the zero order excitons (left) and of the eigen excitons (right) induced by size dispersion. The fits to Gaussian distributions are shown in black lines, see Table S1 of Section 4 of the SI for the Gaussian parameters. Computed for 1000 dimers of three different types. **(A)** A heterodimer with QD\({}_{\text{A}}\) (D\({}_{\text{A}}\) = 2.35 nm) and QD\({}_{\text{B}}\) (D\({}_{\text{B}}\) = 3.85 nm) with 3% size dispersion for the short L = 0.20 nm ligand length. Note how the Coulomb interactions between the zero order exciton bands that overlap (exciton eh\({}_{\text{B}}\) and hh\({}_{\text{B}}\)-e\({}_{\text{A}}\)) lead to a splitting of the corresponding eigen exciton ones. **(B)** A heterodimer with D\({}_{\text{A}}\) = 2.7 nm and D\({}_{\text{B}}\) = 3.5 nm and L = 0.2 nm and 5% dispersion in diameter. In this case, the exciton bands of the two local excitons h1e\({}_{\text{A}}\) and h2e\({}_{\text{B}}\) overlap.