2 with ρ$_{ab}$ ( t ) ≡ 〈 e$_{a}$ | ˆ ρ ( t ) | e$_{b}$ 〉 and ω$_{ab}$ = ( ϵ$_{a}$ − ϵ$_{b}$ ) / ℏ . The first term on the right hand side of Eq. 5 describes the fully coherent dynamics in the absence of ˆ H$_{sb}$ and the second term describes the irreversible dynamics from the interaction with the phonon bath. Correlations in bath-fluctuations on different sites [20] are taken into account by using the relation g n $_{q}$g m ∗ q = g m q g n ∗ q = g 2 $_{q}$e − R mn /R corr , where R$_{mn}$ ≡ | R$_{mn}$ | is the dis- tance between the sites and R$_{corr}$ is the phonon correla- tion length [21]. Using this relation for the electron-phonon couplings, the cross-correlation C$_{mn}$ ( ω ) can now be written as, where the frequency correlation function C ( ω ) = 2 π [ n ( ω )+1]( J ( ω ) − J ( − ω )), with n ( ω ) the Bose-Einstein distribution and the spectral density of the bath J ( ω ) = ∑ $_{q}$| g$_{q}$ | $^{2}$ω2 $_{q}$δ ( ω − ω$_{q}$ ), where δ ( ω ) is the Dirac delta func- tion. For the Hamiltonian specified in Eq. 1, the Redfield tensor elements are given by where δ$_{ij}$ is the Kronecker delta and with P denoting the Cauchy principal value of the in- tegral and U$_{na}$ = 〈 s$_{n}$ | e$_{a}$ 〉 the transformation matrix el- ements relating the site basis {| s$_{n}$ 〉} and the excitonic basis {| e$_{a}$ 〉} . Since | e$_{a}$ 〉 = ∑ n U$_{na}$ | s$_{n}$ 〉 , then | U$_{na}$ | 2 can be interpreted as the contribution of the n -th site to the a -th eigenstate of ˆ H$_{s}$ . Hereafter, we will focus on QDs at low temperature (10 K) as our prototypical experimental realization. We neglect possible inversion asymmetry of the crystal, and therefore the contribution of the piezoelectric coupling, and focus instead on the deformation potential coupling. As shown by Calarco et al. [22], the spectral density describing this coupling (in the absence of an external electric field) is given by where, Θ( ω ) is the Heaviside step function. We use typical values for GaAs QDs for our numerical simula- tions [23, 24], giving η = ( D e − D h ) 2 4 π $^{2}$ρu$^{5}$ℏ = 0 . 035 ps 2 and ω$_{c}$ = √ 2 u $^{2}$/l2 = 1 . 41 ps − $^{1}$, where D$_{e}$ ( D$_{h}$ ) is the de- formation coupling potential for electron (hole), u the speed of sound within the quantum dot, ρ its mass den- sity and l the ground state localization length, assumed to be the same for electron and holes. We also assume a correlation length, R$_{corr}$ , of 3 nm. Using Eqns. 7-9, the population transfer rates, k$_{ab}$ , be- tween the eigenstates a → b is given by, This equation is central to our insight into designing exci- tonic transfer. Though derived for a different regime, the form of Eq. 11 is similar to the widely used rate equation in F¨ orster theory (incoherent limit), k Forster ab ∝ J 2 $_{ab}$I$_{ab}$ , where J is the electronic coupling, and I is the spectral overlap integral [25]. Both equations are a product of two terms: one dealing primarily with the description of the system ( ζ in this paper or J 2 for F¨ orster theory), and an- other largely depending on and arising from the system- bath interaction ( C ( ω ) in this paper or I for F¨ orster the- ory). F¨ orster theory has been applied in fluorescence res- onance energy transfer (FRET) to design chromophores for biosensing assays [2, 26, 27] and was recently verified experimentally for semiconductor QDs [28]. We use the simple structure of Eq. 11 to gain microscopic and exper- imentally relevant insight into engineering directed and optimized EET. While C ( ω$_{ab}$ ) depends on the overlap of system eigenenergies with the spectral properties of the phonon bath (e.g., lattice vibrations, solvent, protein environ- ment, etc.), ζ depends on the transformation matrix U in Eq. 9, determined by the relative magnitude of electronic couplings with respect to site energies, site connectivity, and spatial correlation between sites. The aim is then to maximize (minimize) the product of these two factors in Eq. 11 to favor (suppress) the desired rates. To illustrate the applicability of Eq. 11, we choose two three-site examples to highlight the importance of the phonon bath interaction to achieve directed EET. Since multiplying a system Hamiltonian by a scalar does not change the maximum exciton transfer probability in the coherent limit, the distances and site energies for the two cases are chosen so that the Hamiltonians are related by a multiplicative factor (3 . 5 ˆ H 1 s = ˆ H 2 $_{s}$). As a conse- quence, the ζ values are roughly the same for cases 1 and 2 (Table I); slight differences are introduced by the bath correlation term in Eq. 9. In the cases chosen, the fully coherent evolution gives a maximum probability of finding an excitation of 5% (1%) for site 1(2). Therefore, any difference between the two examples is a result of interaction with the environment. In contrast to the ζ factors, the C ( ω$_{ab}$ ) values for cases 1 and 2 at 10 K differ by at least one order of magnitude due the position of the transition frequencies with respect to the spectral density (Table I). Any changes to the system Hamiltonian can affect C ( ω$_{ab}$ ). In our examples, the scalar factor mentioned above changes the C ( ω$_{ab}$ ) such that the largest population transfer rate is switched J ( ω ) = Θ( ω ) ηω $^{3}$e − ω $^{2}$/ω2 c , (10) Γ$_{ab,cd}$( ω$_{dc}$ ) = 1 2 ζ$_{ab,cd}$C ( ω$_{dc}$ ) + i 2 π ζ$_{ab,cd}$ P (∫ ∞ −∞ C ( ω ) ω$_{dc}$ − ω d ω ) , (8) ζ$_{ab,cd}$ = ∑ n,m ( U − $^{1}$)$_{an}$ U$_{nb}$ ( U − $^{1}$)$_{cm}$ U$_{md}$e − R mn /R corr , (9) R$_{ab,cd}$ = Γ$_{db,ac}$( ω$_{ca}$ ) + Γ ∗ $_{ca,bd}$( ω$_{db}$ ) − δ$_{bd}$ ∑ e Γ$_{ae,ec}$( ω$_{ce}$ ) − δ$_{ac}$ ∑ e Γ ∗ $_{be,ed}$( ω$_{de}$ ) , (7) C$_{mn}$ ( ω ) = e − R mn /R corr C ( ω ) , (6) k$_{ab}$ = R$_{bb,aa}$ = ζ$_{ab,ba}$C ( ω$_{ab}$ ) . (11)