## 2 Materials and Methods: Model Excitonic Hamiltonian
The excitation to which they correspond and the labeling used in the paper are shown in Figure 2, see also Figure 1. There are four local (Frenkel) excitons, two on QD\({}_{\rm A}\) and two on QD\({}_{\rm B}\), labeled 1 to 4, and four charge transfer (CT) excitons for which the hole and the electron are not on the same QD, labeled 5 to 8. When the two dots are of different sizes, one obtains hetero donor-acceptor dimers. The exciton basis corresponds to eight zero order singly excited electronic singlet states of the dimer. In the basis of the exciton antisymmetrized singlet basis functions, the matrix of the Hamiltonian (Equation (1)) takes the following form [35; 77; 78]:  
\[H_{ij,i^{\prime}j^{\prime}}=\left(E_{i}^{e}-E_{j}^{h}\right)\delta_{ii^{\prime }}\delta_{jj^{\prime}}-J_{ij,i^{\prime}j^{\prime}}+K_{ij,i^{\prime}j^{\prime}} \tag{3}\]  
with the Coulomb  
\[J_{ij,i^{\prime}j^{\prime}}=\sum_{\sigma_{1}\sigma_{2}}\iint d\mathbf{r}_{1}d \mathbf{r}_{2}\phi_{i}^{h}(\mathbf{r}_{1},\sigma_{1})\phi_{j}^{*}(\mathbf{r}_ {2},\sigma_{2})\left(\frac{e^{2}}{4\pi\epsilon\epsilon_{0}|\mathbf{r}_{1}- \mathbf{r}_{2}|}\right)\phi_{j^{\prime}}^{h*}(\mathbf{r}_{1},\sigma_{1})\phi_ {j^{\prime}}^{e}(\mathbf{r}_{2},\sigma_{2}) \tag{4}\]  
Figure 1: (**A**) The two hole and one electron single particle states considered in the model to build the excitonic level structure of a heterodimer. The arrows represent the two different kinds of excitations leading to excitons, a local exciton (hole and electron on the same quantum dot (QD), exciton h2e\({}_{\rm B}\)) and a charge transfer one (CT) with the hole on QD\({}_{\rm A}\) and the electron on QD\({}_{\rm B}\) (h1\({}_{\rm B}\)-e\({}_{\rm A}\)). c.b. and v.b. stand for conduction band and valence band, respectively. (**B**) The resulting excitonic level structure; left scale for an isolated dimer, right scale for an ensemble of \(N=1000\) dimers, for which the size of QD\({}_{\rm A}\) and QD\({}_{\rm B}\) is drawn from a Gaussian distribution with a width, \(\sigma\), of 5% in diameter (D\({}_{\rm A}=2.25\) nm, D\({}_{\rm B}=3.85\) nm). Because of the inherent size dispersion, each level of the single dimer (left) corresponds to an exciton band (right). For the two sizes shown in panel A, the h2e\({}_{\rm B}\) exciton band overlaps with the h1\({}_{\rm B}\)-e\({}_{\rm A}\) one. The arrows show the Coulomb coupling between the exciton h1e\({}_{\rm A}\) and the other excitons in the band. For the details on the Coulomb couplings between different types of excitons, see SI Section 3 and Figure S3; for details about the size distribution effects, see SI Section 5.  
and exchange integrals  
\[K_{ij,j^{\prime}j^{\prime}}=\sum_{\sigma_{1}\sigma_{2}}\iint d\mathbf{r_{1}}d \mathbf{r_{2}}\phi_{i}^{h}(\mathbf{r_{1}},\sigma_{1})\phi_{j}^{e}(\mathbf{r_{2} },\sigma_{2})\left(\frac{e^{2}}{4\pi\varepsilon\varepsilon_{0}|r_{1}-r_{2}|} \right)\phi_{j}^{h_{e}}(\mathbf{r_{2}},\sigma_{2})\phi_{j}^{e}(\mathbf{r_{1}}, \sigma_{1}) \tag{5}\]  
In a dimer, the Coulomb interactions \(V_{Coulomb}(|\mathbf{r}-\mathbf{r}^{\prime}|)\) (Equation (1)) can either couple two single excitons localized on the same QD or couple the two QDs, see Figure S3, Section 3 of the SI. The two particle, six dimensional integrals (Equations (4) and (5)) are computed numerically using a double Monte Carlo integration scheme (see details in the SI Section 3 and Figures S4 and S5). The Coulomb integrals are screened by the dielectric constant of the material, see Section 3 and Figures S6 and S7 of the SI for details. Our numerical implementation for the computation of the Coulomb couplings was benchmarked with respect to known analytical expressions for particular Coulomb matrix elements [79]. We computed exciton coupling surfaces as a function of the dot radius (1 to 3 nm) for two QDs, QD\({}_{\text{A}}\) and QD\({}_{\text{B}}\), in a dimer at fixed surface to surface distances (0.1 to 0.6 nm) by numerical integration.