Vibronic Resonance Along Effective Modes Mediates Selective Energy Transfer in Excitonically Coupled Aggregates

We recently proposed effective normal modes for excitonically coupled aggregates which exactly transform the energy transfer Hamiltonian into a sum of one-dimensional Hamiltonians along the effective normal modes. Identifying physically meaningful vibrational motions which maximally promote vibronic mixing suggested an interesting possibility of leveraging vibrational-electronic resonance for mediating selective energy transfer. Here we expand on the effective mode approach elucidating its iterative nature for successively larger aggregates, and extend the idea of mediated energy transfer to larger aggregates. We show that energy transfer between electronically uncoupled but vibronically resonant donor-acceptor sites does not depend on the intermediate site energy or the number of intermediate sites. The intermediate sites simply mediate electronic coupling such that vibronic coupling along specific promoter modes leads to direct donor-acceptor energy transfer bypassing any intermediate uphill energy transfer steps. We show that interplay between the electronic Hamiltonian and the effective mode transformation partitions the linear vibronic coupling along specific promoter modes to dictate the selectivity of mediated energy transfer, with a vital role of interference between vibronic couplings and multi-particle basis states. Our results suggest a general design principle for enhancing energy transfer through synergistic effects of vibronic resonance and weak mediated electronic coupling, where both effects individually do not promote efficient energy transfer. The effective mode approach proposed here paves a facile route towards four-wavemixing spectroscopy simulations of larger aggregates without severely approximating resonant vibronic coupling.


I. INTRODUCTION
Impulsively excited vibrational wavepackets often accompany ultrafast electronic energy or charge transfer dynamics as mere spectators and could be understood within the Born-Oppenheimer approximation. However, specific vibrational motions can sometimes strongly couple with electronic motions causing breakdown of the adiabatic framework to drive ultrafast internal conversion between electronic states. Such examples may include initial steps of photosynthesis 1 , photochemistry of vision 2 , and phase transitions in quantum materials 3 . Identifying vibrational motions which promote vibronic mixing opens an interesting avenue of driving state selective photochemistry such as inhibiting 'promoter' modes to extend excited state lifetime 4 , or driving promoter modes to modulating charge transfer in organic crystals 5,6 , transition metal dichalcogenides 7 and donor-bridge-acceptor molecules 8 .
Path integral 9 and effective modes schemes 10,11 can in principal treat all intramolecular Franck-Condon (FC) vibrational modes of the system on the same footing to provide numerically exact microscopic view of quantum decoherence even for molecular aggregates. While calculations of population transfer rates including the full multidimensional vibrational subspace are now feasible, calculations of spectroscopic signatures of four-wave mixing spectroscopies, which scale as fourth power of the number of basis states may be too expensive still.
Demonstrations of mode-selective photochemistry suggest that a differentiability between spectroscopically meaningful spectator and promoter vibrational motions is desirable for guiding synthetic design and optical spectroscopy. Promoter modes correspond to inter-or intramolecular vibrational modes with specific motions, and sometimes definite symmetries, which mix electronic degrees of freedom to dominate the short-time quantum dynamics 10 . Promoter modes are therefore spectroscopically interesting to identify. Effective modes constructed by combining several intramolecular FC vibrations through bilinear couplings 11 may not provide a spectroscopically meaningful distinction between promoter versus spectator vibrational modes.
Several theoretical approaches [12][13][14] for simulating quantum dynamics and four-wavemixing spectroscopies of molecular aggregates have relied on the above distinction to treat promoter modes explicitly in the system Hamiltonian, while quantum relaxation of singly-excited states due to rest of the bath is treated using quantum Master equations, or symmetric and asymmetric Brownian oscillators when a symmetry based distinction 15 between singly-excited electronic states is possible. A numerically exact treatment of the system Hamiltonian of a molecular aggregate with several singly-excited states and vibrational modes of spectroscopic interest requires an exponentially scaling basis set 16 , and becomes computationally impractical for four-wavemixing simulations which further scale to the 4 th power of basis set size. This often necessitates a description of vibronic excitons which relies on scaling down vibrational dimensionality of the basis by restricting vibrational excitations on ground electronic states. Such approximations, namely the one-particle approximation 17 and numerically similar coherent exciton scattering approximation 18 , are very successful in describing linear spectroscopic properties of large molecular aggregates 19 and organic thin films 20 . However, we have recently shown 16 that specific situations, such as vibronic resonances 21 cause multi-particle basis states to gain substantial oscillator strength. The resulting exciton delocalization, vibrational distortion field and quantum dynamics cannot be described under such approximations.
The above challenges suggest that theoretical approaches which can identify spectroscopically meaningful promoter vibrational motions, reduce vibrational dimensionality of the multidimensional energy transfer problem, and treat non-adiabatic vibronic coupling exactly can serve as a useful tool to guide molecular design, and spectroscopic experiments and simulations. Early theoretical treatments [22][23][24] of energy transfer in a dimer have analyzed the problem in terms of physically motivated tuning and correlation effective modes, akin to longitudinal and totally symmetric deformations of a quantum particle in a 2-dimensional (2D) box potential. For a given FC active intramolecular vibrational mode,q A andq B on molecules A and B respectively, the dimer energy transfer Hamiltonian is transformed as a sum of separable 1D Hamiltonians, that is, H(q A ,q B ) =Ĥ(q + ) +Ĥ(q − ). Jonas and co-workers have shown 21,25,26 that anti-correlated motions alongq − tune the singly-excited state energy gaps, and are solely responsible for driving non-adiabatic energy transfer between vibronically resonant states. Theq − mode is akin to the tuning vector in conical intersections that defines the direction of Hellmann-Feynman forces 27 in non-adiabatic transitions. Correlated motions alongq + do not tune singly-excited state energy gaps, play no role in vibronic mixing, and can be treated under the adiabatic framework. As far as the role of vibronic coupling in influencing the dynamics and spectroscopic signatures is concerned, the dimensionality of the problem is reduced toĤ(q − ) while still treating vibronic coupling exactly.
The above analysis of dimer energy transfer in terms of physically intuitive tuning and correla-tion vibrational modes motivated our earlier work 28 where an extension of these effective modes to larger molecular aggregates was proposed. Taking the example of a 3-mer, we showed that linear combination of correlation and tuning modes akin to dimer and subsequent Gram-Schmidt orthogonalization yields physically meaningful effective normal modes of the aggregate -a global correlation mode which does not tune any energy gap, a global tuning mode which tunes all nearestindex energy gaps, and a second-nearest index tuning mode. The above transformation preserves the vibrational frequencies of the system and does not yield bilinearly coupled effective modes.
Interestingly, expressing the Hamiltonian in terms of the effective modes leads to a sum of 1D Hamiltonians, one along each such effective mode, such that the role of individual effective modes in promoting vibronic mixing can be individually analyzed. The new physical insights gained from this approach suggested an interesting design principle of leveraging vibronic resonances to mediate selective energy transfer to the acceptor in the presence of an intermediate site.
Here we extend the effective-mode approach and its applications. We elucidate the iterative structure of effective normal modes for successively larger aggregates. Starting from specific 3mer examples, it is analytically shown that the effective mode transformation partitions the linear vibronic coupling along specific modes with a crucial role played by the electronic Hamiltonian.
The design of the electronic Hamiltonian ultimately selects the promoter mode by rearranging vibronic couplings to constructively interfere only along specific effective modes. The physical intuition so gained is utilized to generalize idea of selectively mediating energy transfer to Λ type systems where multiple intermediate sites and uphill energy transfer steps may be involved. We consider the special case of vibronic resonance between the donor and acceptor excitons, now actively investigated in several photosynthetic proteins 1 , singlet exciton fission 29 candidates, and organic polymers 30 . We show that interference between resonant vibronic couplings along different effective modes is phase-independent and always leads to larger overall couplings. This interference plays a vital role in determining the selectivity of mediated energy transfer between the donor and acceptor by suppressing weaker vibronic couplings with intermediate sites. The manuscript is organized as follows. Section II formalizes the effective mode approach highlighting its iterative nature. The section concludes with a physically intuitive picture for the nature of derived effective modes. Section III applies the effective mode formalism to identify promoter modes in general Λ-or V-type systems along which vibronic coupling strengths are maximized. The section illustrates interesting effects such as role ofĤ elec in determining the promoter mode, interference between vibronic couplings along different effective modes, and its role in determining the selectivity of mediate energy transfer. Section IV presents the conclusions.

II. THEORY
Our recent work generalized the pairwise tuning and correlation modes of an excitonic dimer to a N-mer with total V intramolecular FC active modes on each molecule. The dimer tuning and correlation modes,q − andq + respectively, could be combined to give global tuning and correlation modes,Q − andQ + respectively. A Gram-Schmidt orthogonalization process leads to the residual modes such that the NV -dimensional Hamiltonian can be written as a sum of 1D Hamiltonians along the effective modesQ + ,Q − and NV − 2 residual modesR. Below we will sketch the derivation for a N-mer, and in the process elucidate the iterative nature of the scheme for successively larger aggregates.

A. Hamiltonian
Each molecule of the aggregate is assumed to be a two electronic level system with site basis states |G and |E . The molecules are electronically coupled through Coulomb interactions between their ground to excited state transition dipoles. The resulting electronic basis for the aggregate is constructed from a tensor product of the site basis of each molecule. This results in a set of N singly-excited electronic basis states, where N = 5 for the case of 5-mer considered here. A singly-excited state |I denotes the state |G A G B . . . E I . . . G E , where only the I th molecule is electronically excited. The purely electronic part of the singly-excited Hamiltonian of the aggregate is given byĤ elec = ∑ I ε I |I I| + ∑ L<I J LI (|L I| + |I L|) , where the Coulomb coupling matrix element J LI between any two singly-excited electronic states |L and |I is a real quantity. ε I is the electronic site energy for state |I . It is assumed that only the Coulomb integrals contribute to electronic coupling with negligible electron exchange, although the approach described below can be extended to include charge-transfer couplings as well 28 . Note that no specific spatial arrangement and mutual electronic couplings has been assumed.
A common set of V intramolecular vibrational modes are present on the ground and excited electronic state of each molecule, such that d I j denotes the FC displacement on the singly-excited electronic state of the I th molecule along the j th mode. Note that there is no loss of generality because the set of FC displacements on each molecule can be different. The dimensionless vibrational coordinate for the j th mode on molecule I is denoted by the unit vector operatorq I j . The corresponding nuclear momentum unit vector operator is denoted byp I j . The ground electronic state Hamiltonian of the aggregate is given byĤ G = ∑ I ∑ V j=1 1 2 ω j (p 2 I j +q 2 I j ). We can then write the singly-excited electronic state Hamiltonian,Ĥ N for the case of a N-mer, as - The energy is defined in frequency units. Indices I and L run over the molecules A to E of the 5-mer.

B. Effective Normal Modes
The iterative nature of effective normal modes can be understood through mathematical induction by deriving effective modes for successively larger aggregates starting from a dimer. In our earlier work 28 , we derived effective modes for a 3-mer. Below we build up from this work to derive a set of effective normal modes for a 5-mer, corresponding to each set of the total V intramolecular vibrational modes per molecule. In doing so, we will highlight the general structure as well as the FC displacements associated with these delocalized normal modes.
The singly-excited 5-mer HamiltonianĤ 5 is 5V-dimensional in the starting intramolecular vibrational basis. For any given set j of intramolecular modes with frequency ω j on each molecule, we will assume equal FC displacements for simplicity. That is, d I j = d j for any molecule I. Since only linear transformations are involved in the derivation, such a simplification does not limit the generality of the method. For example, see the derivation of effective normal modes with unequal FC displacements for the case of a 3-mer in Section S1.1. Minor relative differences in vibrational frequencies and FC displacements in disordered aggregates such as a photosynthetic protein will only manifest on longer timescales.
Using the definitions in Eqn. 2 and Eqn. 3 of ref. 28 for the case of a N-mer, the j th set of global tuning and the correlation vectors, G j and C j respectively, are linear combinations of all the pairwise tuning (g p n ) and correlation vectors (c p ), and written compactly as - Here index p n denotes the N − 1 nearest-neighbor combinations, and index p denotes all the possible N(N − 1)/2 pairs of sites. Note that the usage of term nearest-neighbor should be interpreted in terms of nearest indices and not in terms of spatial proximity. For the case of global tuning vector for a 5-mer, m p n j = (m AB j , m BC j , m CD j , m DE j ). For the case of global correlation vector, n p j = (n AB j , n AC j , n AD j . . . n DE j ). Both denote a vector of unknown coefficients that dictates the weightage of pairwise tuning (correlation) modes towards the global tuning (correlation) mode. Eqn. 2 represents a dot product of this 'weighting' vector with another vector formed by the contributions of pairwise tuning(correlation) motions towards adjusting the singly-excited state energies of the two sites that form the pair, through linear vibronic coupling. Patra et al. 28 show that the unknown coefficients can be obtained by imposing intuitive geometric constraints. For example, the global tuning vector G j is orthogonal to all second-nearest neighbor energy gap tuning vectors. Similarly, the global correlation vector C j is orthogonal to all the pairwise tuning vectors in the system, as well as the global tuning vector G j . Imposing such constraints to determine the unknown coefficients, and normalizing the resulting vibrational modes yields global tuning and correlation modes for the 5-mer,Q + j andQ − j respectively.
As explained by Patra et al. 28 , the remaining N − 2 residual modes are determined by imposing the constraint in each of the N electronic domains that the total linear vibronic coupling arising from the j th set of intramolecular vibrational modes remains the same in the delocalized vibrational basis. Note that this results in an overdetermined system of N − 2 unknowns with N constraints, yielding linearly dependent residual modes. This leads to a certain flexibility in designing the linearly independent residual modes using Gram-Schmidt orthogonalization (Section II C). After orthonormalization, the resulting set of effective normal modes for the 5-mer can be expressed as is a column vector of the j th set of delocalized effective modes for the 5-mer. The superscripts on the three residual modes indicate the diabatic electronic sites on which the residual effective mode has FC displacements. The physical meaning and choice of these residual modes will be discussed in Section II C. q j = (q A j ,q B j ,q C j ,q D j ,q E j ), is a column vector of the j th set of intramolecular vibrational modes. The orthogonal transformation matrix U 5×5 , shown in Eqn. S12, transforms the j th set of intramolecular vibrational modes into delocalized effective normal modes of the aggregate. A similar transformation U 3×3 is shown in Section S1.1. Two crucial points that highlight the generality of the approach should be noted here -1. No j subscript on the transformation U is intentional because the same orthogonal transformation is valid for any set j in the total V sets of intramolecular modes. This is highlighted schematically in Figure 1 and discussed in Section II C. 2. The coefficients of orthogonal transformation U also determine the set of FC displacements along the corresponding j th set of delocalized effective modes in Q j . The set FC displacements for the 3and 5-mer are tabulated in Tables S1 and S2 respectively. The derivation of 5-mer effective modes is detailed in Section S1.2.
FIG. 1. Effective Mode Transformation. Effective normal modes for an aggregate of N molecules each having V intramolecular FC modes. Each set of total V set of modes corresponds to a particular set of N effective modes. The effective modesQ + j do not tune any energy gaps and are mere spectators in vibronically enhanced energy transfer. Other effective modes can either tune energy gaps, nearest neighbor energy gaps in case ofQ − and non-nearest neighbor in case of N − 2 residual modesQ R , to affect short-term non-adiabatic dynamics depending on how strongly they couple different electronic degrees of freedom. Using the effective mode transformation, the NV dimensional energy transfer Hamiltonian can be written as a sum of N ×V 1D Hamiltonians, one along each effective mode.
Overall, the orthogonal transformation U N×N between intramolecular and delocalized effective modes allows one to express the linear vibronic coupling terms of the aggregate HamiltonianĤ N in Eqn. 1 in terms of the delocalized effective normal modes. As explained schematically in Figure   1, this amounts to writing the NV -dimensional Hamiltonian in Eqn.1 as a summation of N ×V 1D Hamiltonians, one along each effective mode. The 1D Hamiltonians for the 5-mer are shown in Eqns. S13. From this simplified structure, the set of global correlation modes {Q + j } V j=1 , akin to totally symmetric deformation of a N-dimensional box, do not tune any singly-excited state energy gaps in the aggregate, and play no role in mixing the electronic domains inĤ elec . As shown in our previous work 28 for the case of a 3-mer, in order to ascertain the role of remaining N(V − 1) modes in promoting vibronically enhanced energy transfer, significantly faster calculations along the reduced 1D Hamiltonians can be carried out, such that effective modes which promote vibronic mixing versus those which are mere spectators could be differentiated. While extensive effectivemode schemes 10 which can treat all the intramolecular vibrational modes on the same footing have been demonstrated for excitonically coupled aggregates 11 , the unique aspects of the current approach, although may be only limited to a few spectroscopically observed vibrations of interest, is that the motions along these effective vibrational modes are physically interpretable, and their frequencies directly relate to spectroscopic observables. The effective mode structure proposed here is not bilinearly coupled (orthogonal) and preserves the vibrational frequencies of the system. Thus a reduction in the dimensionality of the vibrational sub-space is made possible by identifying promoter versus spectator vibrational motions. We have demonstrated this aspect in our original communication 28 , and will further extend that in the current manuscript. Further, as opposed to approximations 16 relying on multi-particle basis set truncation, the non-adiabatic vibronic mixing and the resulting enhancement of energy or charge transfer is treated numerically exactly in the effective mode transformation.

C. Iterative Nature of Physically Meaningful Effective Modes
Because the design of the effective normal modes preserves the vibrational frequencies of the system with no bilinear couplings between modes, an iterative structure for the effective modes of successively larger aggregates becomes possible. As mentioned above, the N electronic domains for N − 2 (to be determined) residual modes lead to a certain flexibility in constructing the residual effective modes through Gram-Schmidt orthogonalization. We start with the case of a 3-mer, where the global correlation modeQ + j does not tune any singly-excited energy gap, and the global tuning modeQ − j tunes all the nearest-neighbor energy gaps. The only choice of residual mode turns out to be the modeQ AC j which tunes the remaining A −C energy gap, or the second nearest-neighbor energy gap in the system. The superscript indicates the electronic domains with non-zero FC displacements along the effective mode.
A general schematic for choosing the residual modes for a N-mer is shown in Figure 2A. Applying this schematic to a 5-mer, the residual effective modes becomeQ ACE j ,Q AE j ,Q BD j . The choice of residual effective modes fixes the electronic domains in which the residual mode has zero and non-zero FC displacements. Other equivalent choices of residual modes are also possible and simply lead to residual modes with the same set of FC displacements but on different electronic domains. This is shown in Section S1.3. The knowledge of effective modes and corresponding FC displacements for smaller aggregates, such as a 3-mer, implies that the unknown FC displacements along the residual effective modes for a larger aggregate, such as a 5-mer, are already known. This is depicted in Figure 2B in the form of color-coding, where modes of similar colors have the same set of FC displacements with the electronic domains denoted by the superscripts. The FC displacement along the global correlation modeQ + j is also known -d j / √ N in each of the N electronic domains. An important point to also recall here is that for a given N-mer, the transformation U N×N holds for any set j of the total V sets of intramolecular FC modes. Thus for any larger aggregate the only unknowns are the FC displacements along the global tuning modeQ − j and along the new residual modes, such asQ ACEG j for the case of a 7-mer. on the residual modes indicate the electronic domains with non-zero FC displacements. For any given effective mode, each curved line connecting two electronic domains corresponds to the energy gap tuning vector between the two domains, such that pairwise combinations of all such vectors gives rise to the overall effective mode. This set of effective normal modes correspond to each of the total V sets of intramolecular vibrations. The subscript j to indicate the j th set is therefore suppressed here for clarity. (b) Iterative structure of the effective normal modes for successively larger aggregates. The form of residual modes is based on the schematic shown in panel a. The global correlation mode,Q + has a FC displacement of 1 √ N in each of the N electronic domains. The global tuning modeQ − and the residual modes are color coded to map effective modes which have identical sets of FC displacements. The electronic domains with non-zero FC displacements along residual modes are shown in superscripts. The map indicates that several effective normal modes and FC displacements for a general N-mer are equivalent to effective modes for smaller aggregates. For example, for a 7-mer, the only unknowns are the new colors, that is, the FC displacements alongQ − , and along the new residual modeQ ACEG . Table S3 lists the FC displacements in the diabatic site basis along all the delocalized effective modes shown in Fig. 2B. The derivation of FC displacements in Table S3 follows the above formalism and discussed in more detail in Section S1. To emphasize the iterative nature of effective modes, Figure 3 illustrates the pattern of FC displacements and the diabatic site potentials along the j th set of effective modes for a 3-mer and 5-mer with a generalĤ elec . Panel A plots the pattern of FC displacements along each effective mode in the diabatic site basis. The direction and size of arrow on top of each site map to the signs and magnitudes of FC displacements on the respective sites. Panel B plots the diabatic site potentials with the FC displacements along the effective modes shown on scale with the classical turning points. All diabatic sites have equal FC displacements alongQ + modes, such that motions alongQ + cause no change in relative energy gaps. In contrast, motions alongQ − tune relative energy gaps between nearest neighbor excitons, but do not tune the second-nearest neighbor energy gaps. In the same fashion, motions along effective modes such asQ ACE do not tune the relative B − E energy gap. The pattern of FC displacements, including the sign and magnitude, for the 5-mer effective modesQ AC andQ BD are identical to those along the 3-mer effective modesQ AC . Similarly the 5-mer effective modeQ ACE has FC displacements identical to the 3-mer global tuning modeQ − . Table S3, it is seen that the transformation to effective normal modes essentially partitions the linear vibronic couplings arising from intramolecular FC displacements unequally along the delocalized effective modes. The effective modes that are delocalized over more number of sites contribute to smaller vibronic couplings. For example, compare FC displacements for a 2-mer versus a 7-mer, corresponding to the global tuning modeQ − in Table S3. It may be expected that more delocalized effective modes influence the vibronic dynamics on longer timescales on account of reduced vibronic coupling. However, Section III A and Section III B 1 illustrate the crucial role of the purely electronic HamiltonianĤ elec in ultimately determining the dominant promoter mode by rotating 26,28 the linear vibronic couplings to strongly couple only specific electronic domains. Section III B 2 discusses that constructive or destructive interference between vibronic coupling matrix elements 16,28,41 arising from different electronic sites, effective modes or vibrational frequencies are secondary effects that also dictate the total vibronic mixing along the promoter effective modes.

III. APPLICATIONS
In the following sections, the above formalism will be applied to generalize the concept of mediated energy transfer proposed 28 in our previous work. Using examples of Λ-or V-type aggregates, we will apply the effective mode approach to identify the dominant promoter modes which

A. Promoter Modes in a 3-mer
We will use the formalism described in Section II to identify the promoter modes in a 3-mer system along which vibronic mixing between excitons is maximized. As we will see, this understanding is necessary in order to generalize the idea of mediated energy transfer to larger aggregates, and is made possible through the effective-mode approach presented here. Section S1 derives the U 3×3 transformation for a 3-mer with unequal intramolecular FC displacements d A , d B , d C along any given set of intramolecular vibrational modes on sites A, B and C, respectively. The transformation U 3×3 connects the set of intramolecular vibrational modes to the corresponding set of effective normal modesQ + ,Q − andQ AC as - Here . Note that the subscript corresponding to j th set of FC vibrational modes has been dropped in Eqn. 3 for brevity. As shown in Section S1.1, the FC displacements along the effective modes on each electronic domain can be derived from U 3×3 in a straightforward manner.
For any given set of modes, the linear vibronic coupling part of the 3-mer Hamiltonian,Ĥ LV 3 , can then be expressed as a sum of vibronic coupling Hamiltonians along individual 3-mer effective d AC A and d AC C are FC displacements along theQ AC effective mode. These are defined as d AC As seen from Eqn. 5, onlyQ − andQ AC dependent Hamiltonians tune the diabatic site energy gaps and responsible for mixing electronic domains inĤ elec . The extent to whichQ − andQ AC promote vibronic mixing depends crucially on the structure ofĤ elec . This point is illustrated below.
The singly-excited 3-mer electronic Hamiltonian is shown in Figure 4, and setup such that SinceĤ LV 3 is separable along 1D Hamiltonians, the contributions ofQ − andQ AC towards vibronic mixing can be individually analyzed. From Eqns. S18-S19, it is seen that the vibronic coupling matrix element alongQ − which mixes the donor-acceptor electronic domains is given . For symmetric V-and Λ-system site energies ∆ 1 = ∆ 2 = ∆. Eqn. S15 shows that an electronic transformation modifies the picture such that only one of the resulting ± linear combination states couples strongly to the intermediate state. The stronger coupling is denoted by bold arrows. The ± linear combinations of the donor-acceptor sites are denoted as mixed colors. In case of a more general asymmetric V-or Λ-system, the ± linear combinations are still weakly coupled. This weak coupling, zero for the symmetric case, is denoted as thin arrow between the ± states. The energy levels resulting after this intermediate transformation are shown for the case of equal electronic couplings, that is, J 1 = J 2 = J, although Eqn. S15 describes this more generally.
between the sites. This is so because the effective mode transformation andĤ elec rearrange the intramolecular FC displacements interfere constructively along the global tuning modeQ − . In contrast, as shown in Eqn. S21, alongQ AC , the corresponding vibronic coupling matrix element is . In this case, the vibronic coupling is diminished because it depends only on the difference of intramolecular FC displacements, which becomes negligible for case of identical molecules in an aggregate. Thus, for 'downhill' 3-mer (case 1), motions along the global tuning modeQ − maximally promote vibronic mixing, while vibrational motions alongQ AC are spectators in the process. This point was also shown in our earlier work 28 .
For case 2, we will start with a 'symmetric'

B. Mediated Energy Transfer via Vibronic Resonance
In the general Λ 3-mer described above, direct donor-acceptor electronic mixing is weak. Therefore any enhancement of electronic energy transfer will require mixing between electronic and vibrational degrees of freedom to either overcome or bypass any intermediate uphill steps. The following discussion considers a special case of vibronic mixing which exploits a resonance 21 between excitonic energy gap and a quanta of vibration on the acceptor exciton to enhance energy transfer. This vibronically enhanced transfer is enabled by strong mixing of the donor exciton with the resonantly 'selected' vibration on the acceptor.
We consider an excitonically coupled 3-mer with a specific vibrational mode which participates in vibronic resonance. Even for one specific FC vibrational mode per molecule, the dimensionality scales as 3 × n 3 vib . However, the above understanding of symmetric 'downhill' and Λ cases suggests a dominant role for only theQ AC effective mode reducing it to a 1D problem to a good approximation. The above expression follows from the analysis described in earlier refs. 16,28,41 . The vibrational ket v AC X corresponds to the vibrational base ket on exciton X along the delocalized effective modeQ AC with v quanta of vibrational excitation. The analysis neglects the second order energetic perturbations caused by coupling between neighboring resonant manifolds. Refs. 16,28 shows that this assumption holds well for small FC displacements and is sufficient to analytically describe the absorption spectra and population dynamics resulting from exact non-adiabatic eigenvectors In contrast to the A − C energy gap tuning motions which play the dominant role in a general Λ-system discussed above, for the 'downhill' system, global energy gap tuning motions alongQ − play the major role in mediating energy transfer. This was demonstrated in our earlier study 28 .
The important point is that the switching roles of effective modes is dictated by how electronic HamiltonianĤ elec rearranges linear vibronic mixing to constructively interfere only along specific effective modes. The above examples of mediated energy transfer in a general Λ-or V-system suggests that this could be a more general design principle made possible by selectively exploiting vibronic resonances in energetically disordered weakly coupled aggregates with dense FC-active vibrational spectral density, such as photosynthetic proteins. In the following subsections, we further comment on the general aspects of this idea.

Selectivity of Mediated Energy Transfer
The vibronic eigenvectors in Eqn. 6 suggest that the mediated energy transfer discussed above is independent of whether the intermediate site were optically dark. This is so because the eigenvec- We will utilize the analytic understanding of symmetric Λ or V system (Section III A) in order to think about selectivity in the more general Λ system. Figure 6 schematically shows the vibronic basis states which dominantly mix together in a general Λ versus V system. The relative strengths of vibronic coupling elements for a symmetric Λ system (Eqn. S29) suggests that even in a general system, the ratio of donor-acceptor to donor-intermediate site mixing, that is, j AC αγ / j AC β γ is strongly enhanced due to constructive and destructive interference between intramolecular vibronic coupling elements along theQ AC effective mode. Thus, vibronic basis states |β v AC β and |γ v AC γ couple very weakly compared to the strongly coupled resonant γ − α states, determining the selectivity as well as the timescale of population transfer to the acceptor α. Similarly, the ratio of donor-acceptor to acceptor-intermediate site mixing is j AC αγ / j AC αβ ∼ cot(θ ), where θ is the electronic mixing angle for the symmetric Λ-or V-system defined in Eqn. S25. Even for moderate-strong electronic mixing angle of 30 o , γ − α population transfer will be ∼3x faster than α − β exchange timescale. It is interesting to note how the interplay ofĤ elec and the effective modeQ AC rearranges vibronic couplings to couple only specific exciton domains. Note that the vibronic couplings j are multiplied by an additional factor coming from the matrix elements of the coordinate vector operator. Thus, difference in vibrational quantum numbers of the vibronically coupled states along with FC displacements determine the total strength of the matrix elements. For weak FC displacements along the delocalized effective modes, this factor is largest when the difference in vibrational quantum numbers is unity.
As depicted in Fig. 6, a crucial difference can arise between Λ-and V-systems. For any given set of inter-exciton vibronic couplings, the α − β vibronic mixing in the case of V-system imparts a larger α exciton (acceptor) character to β (intermediate) exciton. This is because lower FC progressions on α (with larger optical intensity) mix with β . Larger acceptor intensity donated to the intermediate site implies overall reduced selectivity compared to a Λ system. Note that in the above analysis it is assumed that exciton energy gap ∆ α,γ ∆ β ,γ , that is, the β − γ exciton energy gap is larger by several vibrational quanta along the resonant mode. For instance, for calculations in Fig. 5 these exciton energy gaps differ by ∼3x vibrational quanta. Similar selectivity arguments as above can be made in the other limit of exciton energy gaps as well. The interesting case of the two exciton energy gaps being comparable is discussed in the Section III B 2.

Interference between Vibronic Couplings
In Section III B we showed that for a general Λ model, both effective modesQ AC andQ − can contribute to vibronic mixing with the former being the dominant mode. In the presence of vibronic couplings along both effective modes, the resonant manifold in Eqn. 6 is modified by the addition of a third vibronic basis state alongQ − . The resulting resonant basis states |α 1 AC α |0 − α , |α 0 AC α |1 − α and |γ 0 AC γ 0 − γ will be denoted as α 10 , α 01 and γ 00 . Figure 7A schematically shows the resonant manifolds possible in a general Λ system with exciton energy gaps ∆ α,γ = ∆ β ,γ .
Analyzing the corresponding vibronic couplings in a symmetric Λ system (Eqns. S27 and S29) can again be useful. These are shown for the first resonant manifold in Fig. 7B. The interference between vibronic couplings is seen by a basis set rotation about the 3 rd electronic domain (γ) with a rotation angle η = arctan j − αγ / j AC α−γ . The relative strengths of vibronic couplings j AC αγ and j − αγ in a symmetric Λ system imply that mixing angle η is expected to be small even in a general system. Such a transformation decouples one state, while coupling the state with dominant α 10 character to γ 00 with larger vibronic coupling given by ( j AC ) 2 + ( j − ) 2 . This is formally similar to the transformation that decouples 16,41 correlated vibrations from the vibronic coupling Hamiltonian.
Interestingly, the interference between vibronic couplings described here is not phase dependent and always leads to larger overall couplings. This is distinctly different, for example, from the case of null excitons 44 where charge-transfer and Coulomb couplings can interfere destructively. The phase-independent interference between vibronic couplings manifests in more general contexts. In the context of multiple intramolecular vibrational modes in an excitonic dimer, ref. 41 has shown that depending on the width of vibronic resonance, vibronic coupling along multiple nearresonant (not exactly resonant) tuning modes can interfere constructively to result in larger overall vibronic coupling. For example, see Fig. 3 and Fig. 5 of ref. 41 . Extending this reasoning to the case of Λ 3-mer, in the presence of an additional near-resonant intramolecular mode, the constructive interference in panel A will be further enhanced with overall coupling ∑ 2 n=1 ( j AC n ) 2 + ( j − n ) 2 . This can be easily shown using transformations similar to panel A (see Eqn. S6 of Sahu et al. 16 for a related transformation). Enhancing the vibronic couplings arising from individual effective modes and near-resonant modes is one of the ways mutual interference of vibronic couplings leads to faster population transfer rates. The interference of vibronic couplings along multiple modes implies that vibrational motions along orthogonal effective modes can simultaneously drive the vibronic probability density evolution between electronic states. This is similar to the case of indirect coupling between independent vibrational modes discussed by Makri and co-workers 9 . This will be recalled in Section III C. However, interference between vibronic couplings can still channel energy transfer selectively to the acceptor. Fig. 7C shows the isoenergetic basis states in the second resonant manifold which are coupled by dominant vibronic couplings j AC XY and j − XY . Taking clues from the symmetric Λ model, for equal intramolecular FC displacements j − αγ , j − αβ and j AC β γ are exactly zero. The resulting dominant couplings are shown in Fig. 7C. The highlighted coupling element j AC αβ involves change in vibrational quanta of 2 and will be negligible even for moderately large intramolecular FC displacements considered here. From the discussion in Section III B 1, j AC αγ / j AC αβ ∼ cot(θ ), such that α − β coupling is weak even for moderate electronic mixing. This ratio is further strongly enhanced due to a change of 2 vibrational quanta in j AC αβ . Eqns. S27 and S29 show that j − β γ is weaker than the dominant j AC αβ coupling by a factor of cos(θ )/ resonances between different pairs of excitons arising from multiple vibrational modes. This will be a subject of our forthcoming publication. A related example is analyzed in our recent work 28 on a downhill 3-mer, where a common vibrational frequency couples multiple vibronic resonances (see Section S12 of Patra et al. 28 ).

Bypassing Uphill Energy Transfer
Donor-acceptor energy transfer in a Λ system involves an uphill energy transfer step, and there- The mechanism for mediated energy transfer proposed above could serve as an interesting design principle for efficiently mediating energy transfer by skipping uphill energy transfer steps all together. Energetically disordered aggregates with dense vibrational spectral density may be promising candidates to explore such mechanisms. Mediated vibronic coupling may be possible in singlet exciton fission 45 where charge-transfer (CT) states can be strongly coupled with (mutually weakly coupled) locally excited (LE) and correlated triplet (TT) states. However, owing to substantially higher lying CT states, the overall electronic mixing between LE and TT states may be too weak to promote efficient ultrafast formation of TT state.
Note that the mechanism of bypassing uphill energy transfer discussed here is distinctly different from leveraging vibronic resonances as a quantum ratchet 46  where A implies site A excited while all other sites in their ground electronic state. The derivation of 5-mer effective modes is detailed in Section S1.2, and the diabatic site FC displacements along these modes are listed in Table S2. The 5-mer electronic Hamiltonian is shown in Fig. S2, and chosen such that the donor and acceptor are only indirectly coupled through weak electronic coupling mediated by the intermediate sites. Fig. 8A shows the relative electronic site energies on scale.  Fig. S2. Each molecule has one identical FC active vibrational mode, such that the Hamiltonian for each electronic state of the system, including the ground electronic state, is 5D. As in Section III B, the frequency of the vibrational mode is chosen so as to have a vibronic resonance between the donor-acceptor excitons. As we will see numerically, the dominant effective modes in this problem is only the A − E energy gap tuning modeQ AE with a minor role for global tuning modeQ − , such that the overall donor-acceptor vibronic mixing can be very well approximated with a 2D problem along the promoter modes.  Table S2, are shown on scale Q AE andQ − will interfere (Section III B 2) to simultaneously dictate the short-time evolution of the vibronic probability density, which could to a good approximation be visualized using the 2D diabatic potentials (Fig. 8B)   1. Role of n-particle basis states As discussed in Section II, resonant vibronic mixing requires explicit quantum treatment of vibrations in the system Hamiltonian. In case of extended aggregates, this enforces truncation of multi-particle basis sets for a feasible computation time. Often truncation up to 2-particle basis states is adequate to describe linear spectral lineshapes 20,47 . However, from a quantum dynamical perspective, resonant vibronic mixing enhances multi-particle basis states contributions through intensity borrowing. Description of fundamental properties of vibronic excitons such as delocalization, vibrational distortion radius, and energy transfer rates crucially depends 16 on multi-particle states.
Although multiple exciton energy gaps are present in the Λ system treated in Sections III A and III C, only one vibronic resonance between the donor-acceptor excitons exists. This may make Λ Hamiltonians seem similar to a dimer where 2-particle basis states can completely describe the system. However, this is not the case. Consider the Λ system depicted in Fig. 7, where exciton energy gaps are comparable to vibrational quanta. Recalling the discussion in Section III B 2, the dominant vibronic coupling j AC αγ dictates the selectivity of mediated energy transfer. As shown in Fig. 7C, vibronic excitons α 11 and α 20 couple with the donor exciton γ through two mixing channels. Using transformations from intramolecular |v A |v B |v C to delocalized |v + |v − v AC vibrational basis, it can be shown that vibrational basis states |0 + |1 − 1 AC and |0 + |0 − 2 AC have substantial 3-particle character. For example, following the analysis similar to Section S1 of ref. 41 , where the 2 nd term in both equalities indicates a substantial 3-particle character with vibrational excitations on two electronically unexcited states. Thus, it is fairly counter-intuitive that even a 3-mer with only one vibronic resonance requires 3-particle basis states to correctly describe interference between vibronic couplings and resulting selectivity of mediated transfer. Analogous reasoning for the 5-mer in Fig. 8 also suggests a vital role of multi-particle basis states, the essential idea stemming form the fact that ground state vibrational excitations on all coupled molecules contribute to any given singly-excited electronic states.

IV. CONCLUSIONS
We have extended the effective mode approach to highlight the physically intuitive and iterative It should be emphasized that these physical insights are made possible by the ability to analyze vibronic mixing contributions along specific effective modes.
Using the effective mode formalism, we have generalized the concept of mediated energy transfer to Λ or V type molecular aggregates. We have shown that synergy between weak mediated coupling and vibronic resonance allows for skipping any intermediate uphill energy transfer steps and enhancing vibronic mixing to near-perfect exciton delocalization. Interestingly, phase-independent interference between vibronic couplings along different effective modes enhances the dominant vibronic coupling along the promoter mode while suppressing weaker contributions from other effective modes. Such interference effects determine the selectivity of mediated energy transfer.
The approach presented here is a promising route for four-wavemixing spectroscopic simu-lations of excitonically coupled aggregates where rich spectroscopic signatures and interference effects 48,49 arising from dominant vibronic couplings are expected. This will be subject of future applications of this approach. Typically one-particle approximation to resonant vibronic coupling, although a severe approximation 16 , becomes unavoidable for simulating such systems with practical computation times. Further, the Λ-and V-type systems analyzed here can be fairly common in energetically disordered multi-chromophoric aggregates such as photosynthetic proteins and organic thin films where multiple near-resonant vibrational modes and exciton energy gaps are expected. Our analysis suggests that inclusion of multi-particle basis states (beyond two-particle approximation) becomes vital for calculations describing a functional role of vibronic mixing and coherences in enhancing energy and charge delocalization.

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request