Phonon-Driven Exciton Dissociation at DonorâAcceptor Polymer

J. Phys. Chem. B 2008, 112, 495-506

495

Phonon-Driven Exciton Dissociation at Donor-Acceptor Polymer Heterojunctions: Direct versus Bridge-Mediated Vibronic Coupling Pathways† Hiroyuki Tamura,£ John G. S. Ramon,‡ Eric R. Bittner,‡ and Irene Burghardt*,£ De´ partement de Chimie, Ecole Normale Supe´ rieure, 24 rue Lhomond, F-75231 Paris cedex 05, France, and Department of Chemistry and Texas Center for SuperconductiVity, UniVersity of Houston, Houston, Texas 77204 ReceiVed: September 10, 2007; In Final Form: NoVember 14, 2007

We present a molecular-level, quantum dynamical analysis of phonon-driven exciton dissociation at polymer heterojunctions, using a linear vibronic coupling model parametrized for 3 electronic states and 24 vibrational modes. Quantum dynamical simulations were carried out using the multiconfiguration time-dependent Hartree method. In this study, which significantly extends the two-state model of Tamura et al. (Tamura, H.; Bittner, E. R.; Burghardt, I. J. Chem. Phys. 2007, 126, 021103), we focus on the role of bridge states, which can mediate the decay of the photogenerated exciton and possibly interfere with the direct transition toward an interfacial charge-separated state. Both the direct and bridge-mediated pathways are found to depend critically on the dynamical interplay of high-frequency CdC stretch modes and low-frequency ring-torsional modes. The dynamical mechanism is interpreted in terms of a hierarchical electron-phonon model, leading to the identification of generalized reaction coordinates for the nonadiabatic process. Variation of the vibronic coupling model parameters in a realistic range provides evidence that the direct exciton decay pathway is not dynamically robust, and bridge-mediated pathways can become dominant. The ultrafast, coherent dynamics is of pronounced nonequilibrium character and cannot be modeled by conventional kinetic equations. The predicted femtosecond to picosecond decay times are consistent with time-resolved spectroscopic observations.

I. Introduction The generation and decay of excitonic states plays a crucial role for the optoelectronic properties of polymer-based electronic devices such as organic light-emitting diodes (OLEDs) and solar cells.1-9 Excitons, that is, electron-hole quasi-particles held together by Coulombic interactions,6 are generated by polaron recombination, as in OLEDs, or else by the primary photoexcitation step, as in photovoltaic systems. At a molecular level, excitonic states correspond to localized or delocalized electronhole densities extending over the π-conjugated polymer strands. Excitons can feature electronic couplings, for example, to charge-transfer states, giving rise to exciton dissociation that possibly occurs on extremely short, ultrafast time scales.6,10-12 These processes are intrinsically quantum mechanical in nature and necessitate a molecular-level interpretation. In the present work, we address such a molecular-level treatment of exciton dissociation, focusing on the elementary events at polymer heterojunctions.13-17 Polymer heterojunctions, that is, interfaces between different phase-segregated polymer materials,13-17 are an important ingredient in device design and function since they act as highly efficient exciton dissociation sites. The band offset at the heterojunction is designed so as to overcome the exciton binding energy of about B ∼ 0.5 eV16,18 (i.e., considerably larger than that in inorganic semiconductors, with B typically in the 50 meV range). Here, we focus on so-called bulk polymer †

Part of the “James T. (Casey) Hynes Festschrift”. * To whom correspondence should be addressed. [email protected]. £ Ecole Normale Supe ´ rieure. ‡ University of Houston.

E-mail:

heterojunctions,13-16 which exhibit a highly folded interfacial area; this results in a high probability for the exciton to reach the interface within its diffusion length (on the order of ∼10 nm19-21). The exciton decay toward a charge-separated state (“exciplex”)10,16,17 is largely determined by molecular-level electronic interactions at the interface. Once the exciton has reached the interface, its dissociation can be extremely fast, on the order of femtoseconds to picoseconds. To a first approximation, the efficiency of charge generation depends on the ratio between B and the band offset ∆E between the two polymer species.16,18 If ∆E > B, the exciton is destabilized and tends to decay; this is desirable for photovoltaic applications. Conversely, if ∆E < B, the exciton is stable, and this is best suited for LED applications. However, intermediate cases where ∆E = B cannot straightforwardly be addressed in terms of donor-acceptor energetics. This applies, for example, to the TFB/F8BT heterojunction system under consideration in the present work, involving parallel chains of the poly[9,9dioctylfluorene-co-bis(N,N-(4-butylphenyl))bis(N,N-phenyl-1,4phenylenediamine)] (TFB) and poly[9,9-dioctylfluorene-cobenzothiadiazole] (F8BT) polymer species.16,17 For this system, recent time-resolved photoluminescence studies suggest that both exciton decay and regeneration phenomena are observed, with the earliest events falling into a (sub)picosecond regime.10,11 A detailed understanding of the exciton dissociation process necessitates an analysis of the electron-phonon coupling effects that mediate the exciton decay. In organic π-conjugated semiconductors, the coupling between the elementary electronic processes and the material’s phonon modes is very pronounced,6,22 much stronger than that in inorganic semiconductors. Direct evidence of electron-phonon coupling is provided by the experimental observation of subpicosecond scale coherent

10.1021/jp077270p CCC: $40.75 © 2008 American Chemical Society Published on Web 12/15/2007

496 J. Phys. Chem. B, Vol. 112, No. 2, 2008 nuclear motions following photoexcitation.23 Further, the ultrafast nature of the electronic decay processes10,12,24 suggests the presence of coherent vibronic coupling mechanisms, possibly determined by conical intersection topologies, very similar to the photophysics of polyatomic molecules and to Jahn-Teller effects in solids.25-27 These vibronic coupling mechanisms will be at the center of the present study. In continuation of our previous work on this system,28-32 we present here a microscopic, molecular-level picture of the processes at the TFB/F8BT heterojunction. Our study involves a realistic quantum dynamical model comprising three electronic states and an explicit 24 mode representation of the phonon distribution. A partial account of this work has been given in ref 33. The model is based on a semiempirical electronic structure characterization of the polymer interface18,31,34,35 in conjunction with a linear vibronic coupling (LVC) Hamiltonian25,26 suitable to describe electron-nuclear coupling in polyatomic systems, solids, and disordered phases. Efficient multiconfigurational techniques, that is, the multiconfiguration timedependent Hartree (MCTDH) method,36-39 are applied for the explicit propagation of the multidimensional quantum state. Typically, the dynamics associated with this model corresponds to fastspossibly ultrafastsprocesses at avoided crossings and conical intersection topologies.25-27,40-42 We further apply recently developed variants of the LVC model, involving the identification of effective modes suitable to describe the short-time dynamics43-45 along with an extension of this concept in terms of a hierarchical electron-phonon (HEP) formulation.28,29,46,47 This hierarchical representation is derived from the initial N-mode-parametrized LVC model (see eq 1 below) using orthogonal coordinate transformations by which effective modes are constructed that successively unravel the dynamics as a function of time. (Formally, these collective modes fulfill certain moment conservation properties which conform to the moment, or cumulant, expansion of the propagator.29,44) We have previously applied this construction to a twostate model of the TFB/F8BT heterojunction28,29 describing the decay of the photogenerated exciton toward an interfacial charge-transfer state. The present study generalizes our previous work so as to include the influence of an additional electronic state acting as a “bridge”. The picture emerging from our analysis of refs 28 and 29 is consistent with an ultrafast (femtosecond to picosecond scale) dynamical process at the heterojunction interface and strongly emphasizes the coherent, nonequilibrium nature of the dynamics. The dynamical role of the phonon bath is underscored by explicitly identifying generalized reaction coordinates based upon the effective mode description. This is in marked contrast to conventional kinetic descriptions which assume (multi)exponential decay characteristics. By contrast, our analysis is in general agreement with a non-Markovian master equation description, as proposed in ref 30. While the results of refs 28 and 29 and, similarly, the present analysis are restricted to zero temperature, dissipative effects due to an external environment are accounted for in terms of a “dissipative closure” of the mode hierarchy. An extension to finite temperatures will be addressed in future work. An important aspect of our analysis of refs 28 and 29 relates to the respective roles of the two phonon branches that characterize the polymer heterojunction system, that is, a highfrequency branch composed of CdC stretch modes and a lowfrequency branch composed of ring-torsional modes. Despite the fact that the high-frequency modes dominate the coupling to the electronic subsystem, the low-frequency modes are shown

Tamura et al. to play a key role in the dynamical process. Indeed, refs 28 and 29 suggest that the exciton decay does not take place in the absence of the low-frequency components of the phonon bath. As mentioned above, the present analysis is concerned with a generalization of this model to include additional states which play the role of bridge states, thus introducing new indirect dynamical pathways. Even though a two-state model involving the lowest-lying excitonic and charge-transfer states is a natural first choice as a reduced representation, it is clear that these relevant states are embedded in a continuous distribution of many states18,31,32 that could act as bridge states for the transfer process. We will thus address the possibility that the indirect, bridge pathways play a significant role or even a dominant role as compared with the direct pathway. Since our recent semiempirical31 and time-dependent density functional theory (TDDFT)32 electronic structure studies have shown that an additional charge-transfer-type state features a strong coupling to the photogenerated exciton state, we focus here on a three-state model including this additional state. The experimental study of ref 10 suggests that such an intermediate state could indeed play a crucial role in the photocurrent generation. In view of the fact that the actual material exhibits a statistical ensemble of interface configurations,48-50 the present study should be interpreted as a snapshot of a single realization, potentially corresponding to a single-molecule experiment.51 In order to investigate realistic variations of the dynamical properties over the ensemble, we will further consider the dependence of the dynamics on various system parameters (i.e., energetic separations, distribution of frequencies, etc.). We will thus be able to draw certain conclusions regarding the dynamical stability of the pathways under investigation. The present analysis will eventually have some bearing on the microscopic engineering of device function, beyond the conventional notion of band gap engineering (i.e., material design based on donoracceptor energetics).9 Indeed, the dynamical properties of the system as a function of the local interface structure can be a determining factor for the material function. The remainder of this paper is organized as follows. Section II provides an introduction to the linear vibronic coupling model used, along with several variants of this model, involving an effective mode representation and a hierarchical chain representation. Section III gives a brief account of the TFB/F8BT heterojunction system, and section IV addresses the dynamics of exciton dissociation at the heterojunction. Section V concludes, and an appendix addresses the construction of the effective mode vibronic coupling Hamiltonian used in section II. II. Theory: Vibronic Coupling Models A. Linear Vibronic Coupling Model. As in our previous work addressing a two-state model for the exciton decay toward a charge-transfer state,28,29 we will use a linear vibronic coupling (LVC) model25,27 parametrized for a representative set of 24 phonon modes comprising the high-frequency CdC stretch and low-frequency ring-torsional branches of the two polymer species constituting the heterojunction interface. Details of the parametrization via semiempirical calculations are given in section III and refs 18 and 31. The use of the LVC model is appropriate in view of the moderate nuclear displacements during the process. In the present context, a three-state LVC model is applied so as to capture the effect of an additional intermediate state, which can act as a bridge between the exciton and charge-transfer

Direct versus Bridge-Mediated VC Pathways

J. Phys. Chem. B, Vol. 112, No. 2, 2008 497

states.33 The general form of the vibronic coupling model considered here is as follows

(

)

(12) (13) κ(1) i xi λi xi λi xi N ωi (2) (23) H) (p2i + x2i )1 + λ(12) i xi κi xi λi xi + C (1) 2 i)1 i)1 (23) (3) λ(13) i xi λi xi κi xi N

∑

∑

Here, 1 and C denote the unit matrix and a coordinate independent constant matrix, respectively; the ωi, xi, and pi ) -i∂/∂xi are the frequencies, displacements, and momenta along the vibrational normal modes. Mass- and frequency-weighted coordinates are used, along with atomic units (p ) 1). The diagonal and off-diagonal potential terms correspond to the diabatic potentials and couplings. The model eq 1 allows for the presence of conical intersections between pairs of electronic states at nuclear configurations, where the diabatic coupling vanishes and the adiabatic states become degenerate.25-27,41,42 In section IV, quantum dynamical simulations using the MCTDH method will be presented for this model. Future extensions of the model will address the inclusion of secondand higher-order anharmonicities, especially with regard to the low-frequency modes. While the three-state model eq 1 is here employed so as to account for the phonon-driven nonadiabatic dynamics at the heterojunction interface, we do not explicitly take into account exciton migration processes5,48 prior to the decay at the interface.52 However, the LVC model as such can be straightforwardly extended to account for site-site transfer. B. Effective Mode Construction. Besides the LVC form eq 1, we further apply a recently developed effective mode LVC representation,43-45 which separates the Hamiltonian into a vibronic coupling part involving a subset of (few) effective modes, while the remaining (many) modes, which are not coupled to the electronic subsystem, are absorbed into a residual part

H ) Heff + Hres

(2)

This construction relies on the observation that the N modes appearing in the vibronic coupling potential eq 1 generate cumulative contributions, which give rise to collective tuning, coupling, and shift effects acting on the coupled surfaces. For an electronic nel state system, Neff ) 1/2nel(nel + 1) effective modes are generated, corresponding to the number of elements in the upper diagonal form of the LVC potential eq 1. An orthogonal coordinate transformation, X ) Tx, is introduced, which generates a new coordinate set X containing these collective modes, along with (N - Neff) residual modes. For the present case of three electronic states, that is, Neff ) 6 effective modes, Heff can be cast in the following form33

Ωi (P2i + X2i )1 + Heff ) i)1 2 (Ki + Di)Xi Λ(12) i Xi 6

∑

6

∑ i)1

(

Λ(13) i Xi

)

Λ(12) i Xi

(Ki - Di)Xi Λ(23) i Xi +

Λ(13) i Xi

Λ(23) i Xi 6

6

K(3) i Xi

∑ ∑ dij(PiPj + XiXj)1 + C i)1 j)i+1

(3)

where the parameters and modes X are defined by extension of

the two-state construction described in refs 29 and 45; see Appendix 1 for a detailed derivation. Two of the six effective modes (i.e., X1 and X2) are chosen as topology-adapted modes which span the branching plane41,42,45,53 for a given pair of electronic states (here, states 1 and 2), that is, the plane along which the degeneracy at the conical intersection is lifted; see Figure 3. Several of the coupling constants in eq 3, that is, the Λ(12) (i g 2), Di (i g 3), Ki (i ) 4, 5, 6), Λ(13) (i ) 5, 6), and i i (23) Λ6 , are zero by construction. The residual modes {Xi, i ) 7, ..., N} contained in Hres do not couple directly to the electronic subsystem but couple bilinearly to the effective modes N N Ωi dij(PiPj + XiXj)1 (4) (P2i + X2i )1 + i)7 2 i)1 j)7,j>i N

Hres )

∑

∑ ∑

Given an initial ground-state equilibrium position {Xi ) 0} for all phonon modes, the new representation of eqs 2-4 entails that a vertical photoexcitation process from the ground state to one of the excited states will result in a displacement of only the effective modes {Xi, i ) 1, ..., 6} from their equilibrium positions. The residual modes will experience a time-delayed displacement as a result of the coupling to the effective modes. This leads to a sequential picture of the dynamical process, by which the primary excitation in the effective mode subspace propagates toward the residual mode subspace. This picture is borne out by a formal analysis as detailed in refs 29, 43, 44, and 47, showing that the short-time dynamics is entirely determined by the effective modes. In particular, it can be shown that the truncation of the Hamiltonian eqs 2-4 at the level of Heff leaves the first three Hamiltonian moments Mn ) 〈ψ0|Hn|ψ0〉, n ) 1, ..., 3 (and the associated cumulants) unaffected.43,44 C. Hierarchical Electron-Phonon (HEP) Model. As shown in refs 28, 29, 46, and 47, further transformations can be introduced in the residual-mode subspace, in such a way that a chain-like structure is obtained. This translates the dynamics to a sequential propagation of the initial excitation through the chain. By analogy with our recent two-state analysis,28,29 Hres can be transformed to a band-diagonal structure, yielding the nthorder Hamiltonian n

H(n) ) Heff +

H(l) ∑ res l)1

(5)

where each lth-order residual term now comprises six modes 6l+6 i-1 Ωi dij(PiPj + XiXj)1 (6) (P2i + X2i )1 + i)6l+1 2 i)6l+1 j)i-6 6l+6

H(l) res )

∑

∑ ∑

For 6 + 6n ) N, the nth-order Hamiltonian H(n) is equivalent to the original Hamiltonian eq 1. We refer to eqs 5 and 6 as a hierarchical electron-phonon (HEP) model.28,29 Successive orders of the HEP Hamiltonian can be shown to conserve the moments Mn (and associated cumulants) of the original Hamiltonian up to the (2n+3)rd order. (This follows from the development described in the Appendix of ref 29, which carries over directly to an arbitrary number of electronic states.) Truncation of the HEP Hamiltonian creates a series of reduced dimensionality models, associated with approximate nth-order propagators which accurately reproduce

498 J. Phys. Chem. B, Vol. 112, No. 2, 2008

Tamura et al.

the dynamics of the overall system up to a certain time. While the lowest-order descriptionsHeff onlyscan be appropriate, for example, for a rapid passage through a conical intersection region,43 the present system will be shown to necessitate a higher-order analysis. D. Reduced Dimensionality LVC Model. The nth-order truncated Hamiltonian eq 5 can be back-transformed to a reduced dimensionality LVC form for M ) 6n + 6 modes M)6n+6

H(n) )

∑ i)1

Ω ˜i ˜ 2i )1 + (P ˜ 2i + X 2 M)6n+6

∑ i)1

(

κ˜ (1) ˜ i λ˜ (12) ˜ i λ˜ (13) ˜i i X i X i X

)

λ˜ (12) ˜ i κ˜ (2) ˜ i λ˜ (23) ˜ i + C (7) i X i X i X λ˜ (13) ˜ i λ˜ (23) ˜ i κ˜ (3) ˜i i X i X i X

The transformation X ˜ )T ˜ X diagonalizes the matrix of bilinear couplings {dij} of eq 6. By this construction, a hierarchy of (M ) 6n + 6)-dimensional LVC models can be obtained, which are formally equivalent to the HEP model of eq 5. The two representations, eqs 5 and 6 versus eq 7, are complementary as far as the interpretation of the dynamics is concerned. The HEP Hamiltonian, eqs 5 and 6, provides a sequential unravelling of the dynamics as a function of time and is particularly appropriate in view of introducing systembath-type approximations. By contrast, the form of eq 7 has the advantage that it is identical to the original LVC model, and the dynamics of all modes can be considered in terms of the coupling to the electronic subsystem. This interpretation will be useful, for example, in the analysis of the high-frequency versus low-frequency motions characterizing the heterojunction system under consideration (see section IV). E. Vibronic Coupling Model Including Dissipation. Dissipation due to the coupling to an external bath can be accounted for by different mechanisms which capture vibrational relaxation, vibrational dephasing, electronic dephasing, and so forth. These effects can be added to the finite-dimensional system Hamiltonian in terms of master equations or else by using explicit representations of an external bath. Here, we are particularly interested in the case where the finite-dimensional vibronic coupling model is taken to represent an infinitedimensional (N f ∞) system, and one can assume that all short and intermediate time scale effects are indeed contained in the system Hamiltonian and its HEP representation. The infinitedimensional nature of the overall system is then most naturally accounted for in the framework of a truncated HEP model by introducing a “dissipative closure” of the hierarchy. By this technique, one prevents the occurrence of artificial recurrences that propagate back along the chain, and an effectively irreversible dynamics results. This construction is closely related to Mori theory54-56 as well as generalized Langevin formulations.56,57 Following our discussion of ref 29, we introduce dissipation by coupling a collection of external bath modes (typically 1050 additional modes) to the highest member of the truncated HEP hierarchy n

H(n) diss ) Heff +

(n) H(l) ∑ res + Hbath l)1

(8)

Here, the bath Hamiltonian is given as follows NB

H(n) bath )

6n+6 NB ωB,i 2 2 + xB,i )1 + dBij (Pi pB,j + XixB,j)1 (pB,i 2 i)1 i)6n+1 j)1 (9)

∑

∑ ∑

The coupling parameters {dBij } are sampled according to a specified spectral density, which is here taken to be Gaussian, as detailed in section IV.B.3. This type of spectral density represents a non-Markovian realization of the external bath. (By contrast, the Ohmic spectral density we employed in ref 29 implies a Markovian approximation; see also refs 58 and 59 for related applications.) While the calculations detailed in section IV.B.3 are restricted to a zero temperature setting, that is, a wave function representation in the combined system-bath space, the construction of eqs 8 and 9, naturally extends to a bath at finite temperature. Reference 60 briefly comments on the implications of the zero temperature case. III. The TFB/F8BT Heterojunction The TFB/F8BT bulk heterojunction can be characterized as a type-II system (see, e.g., ref 16), that is, both HOMO and LUMO energies are higher in one component polymer (here, TFB) than in the other (here, F8BT). Type-II heterojunctions generally exhibit an efficient charge separation at the polymer interface, in particular, if the band offset ∆E is substantially larger than the exciton binding energy B ∼ 0.5 eV. In the case of TFB/F8BT, the band offset is however small, that is, ∆E ∼ 0.1 eV for the HOMO offset (but ∆E ∼ 0.5 eV for the LUMO offset) according to the semiempirical results employed here; hence, the exciton is only marginally destabilized. Indeed, the blend material exhibits a considerable fluorescence intensity,16 and TFB/F8BT has therefore been used for LED-type applications rather than for photovoltaics. In ref 10, it has been suggested that the exciton initially decays but is regenerated on a picosecond time scale. Our recent dynamical calculations28,29 for a two-state model indicate that a rapid decay of the exciton is observed, followed by a picosecond-scale coherent dynamics which partially repopulates the exciton state. As described in our previous publications,30,31 the TFB/F8BT interface has been characterized by semiempirical calculations for a polymer lattice dimer model consisting of two parallel cofacial chains with the F8 subunits of each in juxtaposition (see Figure 1). The vibronic coupling model eq 1 was parametrized using a localized Wannier function basis, within a single-configuration interaction (SCI) approximation. In particular, the model parameters correspond to the vertical SCI energies, gradients, and interstate couplings computed at the ground-state equilibrium configuration of the polymer dimer for a set of 24 normal modes. These modes cover the highfrequency (CdC stretch) and low-frequency (ring-torsional) branches of the phonon distribution. More specifically, we consider 12 high-frequency modes in the range of 0.18-0.20 eV (1452-1613 cm-1) and 12 low-frequency modes in the range of 0.011-0.012 eV (89-97 cm-1). Details of the model and parameters may be found in refs 18, 31, 34, and 35 and in the Supporting Information. The exciton (XT) state, which is the lowest excited state with significant oscillator strength, is mainly localized on the F8BT moiety, while the lowest-lying interfacial charge-transfer state



Figure 1. Schematic illustration of (a) the molecular composition of the TFB/F8BT donor-acceptor heterojunction and (b) the photoexcitation process and potential crossings. The blue, red, and purple lines indicate the XT, CT, and IS diabatic potentials, respectively. (Dotted lines indicate the corresponding adiabatic potentials.) The red and purple arrows indicate the direct XT f CT versus indirect XT f IS f CT exciton dissociation pathways.

(CT) corresponds to a TFB+-F8BT- charge-separated state. (However, the CT state is not purely polaronic since it does carry oscillator strength through mixing with excitonic configurations on the F8BT moiety.31,32) Among the manifold of remaining states, an additional charge-transfer-type state is found to exhibit strong coupling to the XT state; this state will be considered as an intermediate state (IS) in the following; see Figure 1. This state likely corresponds to an experimentally observed intermediate state10 which is conjectured to play a crucial role in the photocurrent generation; this state is thought to represent an interfacial geminate polaron pair. The threestate model eq 1 is thus constructed for the XT, CT, and IS states. IV. Exciton Dissociation Pathways In this section, we address the dynamical pathways that result from the three-state vibronic coupling model of eq 1, parametrized for the TFB/F8BT heterojunction as described in the preceding section. We use both the LVC model eq 1 in its original form as well as its HEP variant, that is, eqs 2 and 3 in conjunction with eqs 5 and 6, along with the reduced dimensionality LVC analogue eq 7. Since the present analysis is an extension of the two-state XT-CT model of the TFB/F8BT heterojunction which we proposed in our previous work,28,29 we begin with a brief summary of this study (section IV.A). Against this background, the central purpose of the present analysis is to examine the possible role of intermediate “bridge” states in the exciton dissociation process. As mentioned above, we focus on an additional charge-transfer-type state identified in the electronic structure studies of refs 31 and 32, which is here included in a three-state model including this bridge state (section IV.B). By carrying out the explicit quantum dynamics according to eq 1, the interference between direct and indirect (i.e., bridge-induced) dynamical pathways is fully captured. Besides leading to a more realistic scenario of the exciton dissociation process, this analysis will further allow us to assess the validity of the reduced twostate model. In view of the statistical distribution of interface structures in the actual material,48-50 we also investigate the effect of varying the molecular parameters, for example, energy gaps, couplings, and frequencies, within a realistic range (section IV.C). In particular, changes in the energetic position of the intermediate state can result from variations in the interface orientation and distance, as shown in ref 32. (In addition,

uncertainties in the electronic structure characterization of the material must, of course, be accounted for.) We find that the direct XT-CT decay, which was at the center of our previous analysis,28,29 is, in general, not dynamically robust, while the indirect pathway(s) can become dominant. This emphasizes that the dynamical properties of the actual material will depend, to a significant extent, on the multistate nature of the system. While the two-state picture of refs 28 and 29 is a natural first approximation, a more realistic description necessitates the inclusion of bridge states. All of the calculations reported on below have been carried out using the MCTDH method36-38 (Heidelberg MCTDH package39) for multidimensional wave functions representing up to 24 phonon modes. In addition, 12 modes representing a Gaussian bath were added in selected calculations (see sections II.E and IV.B.3). The MCTDH wave function corresponds to a multiconfigurational form

Ψ(x,t) )

∑J AJ(t)ΦJ(x,t)

where the configurations ΦJ are Hartree products p

ΦJ(x1, ..., xp,t) )

φj (xκ,t) ∏ κ)1 k

(Here, the composite index J ) {j1, ..., jp} has been used.) The high dimensionality of the present calculations necessitates extensive mode combination,38 that is, several (typically four) phonon modes are combined in a given particle φjk(xκ,t). In particular, the 24 mode calculation described below comprised 6 four-dimensional particles with 6 single-particle functions each; see also the details of the calculations as specified in ref 29. A. Two-State XT-CT Model. Here, we briefly summarize the key results of the analysis of refs 28 and 29 for a reduced XT-CT model of the heterojunction. Starting from the threestate vibronic coupling matrix eq 1, this reduced two-state model is obtained by restricting the matrix form to the (12) ) (XT,CT) subspace. Our previous analysis has shown that the two-state XT-CT model results in an ultrafast (∼200 fs) XT state decay followed by coherent oscillations. Further analysis in terms of an effective mode model and the associated HEP decomposition, that is, the two-state version of the models discussed in sections II.B


Figure 2. For the two-state XT/CT model, the adiabatic and diabatic potentials are shown as a function of one of the high-frequency coordinates (XCdC) and one of the low-frequency coordinates (Xtorsion). This representation refers to the original LVC model eq 1 or else to the reduced dimensionality LVC model eq 7. The Franck-Condon geometry is indicated by a black circle. While the motion remains markedly diabatic along the high-frequency mode XCdC, the lowfrequency dynamics along Xtorsion facilitates the XT f CT transition. This explains the strong influence of the low-frequency modes on the nonadiabatic process. (Alternative explanations can be given in terms of the HEP model, as detailed in refs 28 and 29).

and II.C, highlights several aspects:28,29 (i) From the branching plane representation of the potential in the {X1, X2} coordinates, one can infer that the system exhibits a conical intersection, but the Franck-Condon geometry lies significantly below this intersection. The dynamics evolves in an extended avoidedcrossing region in the vicinity of the intersection. (ii) The highfrequency CdC stretch modes are strongly coupled to the electronic subsystem and therefore constitute the first level of the effective mode hierarchy (Heff), that is, the modes {X1,X2,X3} are all of high-frequency character. (iii) Despite the predominance of the high-frequency modes in the electronic coupling, the low-frequency ring-torsional modes which appear at the next order (H(1) res) of the mode hierarchy play a key role in the XTCT transfer. Indeed, exciton dissociation is not observed in the absence of the low-frequency modes. The dynamical interplay between the high-frequency and lowfrequency modes is apparently a central feature in the process. The analysis of refs 28 and 29 shows that the dynamics is markedly diabatic if confined to the high-frequency subspace (Heff), exhibiting repeated coherent crossings. These are strongly perturbed by the coupling to the low-frequency modes, entailing a net transfer to the CT state along with dephasing and energy transfer between the phonon branches (as observed at the H(1) level and higher orders of the HEP hierarchy). These observations are further illustrated by the reduced potential of Figure 2, showing a cut of the LVC potential along one of the high-frequency modes (here denoted XCdC) and one of the low-frequency modes (here denoted Xtorsion). The cut refers to the original LVC form eq 1shere in a two-state subspaces but could equally be taken with respect to the reduced LVC model eq 7 for n g1. (By contrast, n ) 0 would correspond to the effective Hamiltonian Heff of eq 3, which does not feature any low-frequency modes.) This representation suggests that XT f CT crossings can be induced by the low-frequency

Tamura et al. motions, while the high-frequency motion remains of pronounced diabatic character. Indeed, the Landau-Zener estimate (or an analogous estimate using Zhu-Nakamura theory)40 of transition rates for the low-frequency direction yields probabilities of about PXTfCT = 0.5, while the crossing probability in the high-frequency direction is always small, PXTfCT = 0.05.29 That is, the low-frequency dynamics is, by far, more favorable to an XT-CT transition. The representation of Figure 2 also suggests that if the potential along the Xtorsion direction was changed in such a way that the avoided crossing region would become inaccessible, this could have a strong impact on the overall dynamics. In section IV.C, we show that this is indeed observed for certain parameter variations. B. Three-State XT-CT-IS Model. We now turn to an analysis of the full three-state system, for which we again apply both the LVC and HEP models of section II. The goal of the present analysis is to examine to what extent the two-state picture summarized in the preceding section has to be modified in view of including perturber states. 1. LVC Potentials and Dynamics. Figure 3 shows the diabatic XT, CT, and IS potential energy surfaces (PES) for a cut through the XT-CT branching plane (panel a) and the XT-IS branching plane (panel b), respectively. Recall from the discussion of section II.B that a topology-adapted representation is employed, according to which the first two effective modes are chosen along the principal directions which lift the degeneracy at the conical intersection of a given pair of electronic states, that is, (X1,X2)XT,CT or (X′1,X′2)XT,IS. (In the notation of eq 1, the (12) electronic subspace would refer either to the (XT,CT) pair of states or else to the (XT,IS) pair.) Since these two planes are, in general, not orthogonal to each other, it is not possible to choose an effective mode representation which simultaneously includes the two branching planes. In Figure 3, the corresponding adiabatic potentials (S1, S2, S3) are shown as well (panels c and d). As can be inferred from the figure, the exciton dissociation process is determined by a landscape of multiple intersecting surfaces, and the branching plane topology is a key factor in the nonadiabatic dynamics. Of particular interest for the present analysis is the role of the intermediate state (IS) that lies above the exciton state but could be accessible due to its strong diabatic coupling to the XT state and due to the significant extension of the wavepacket (see Figure 3b). Figure 4a shows the population evolution obtained from MCTDH simulations for the overall 3-state, 24-mode system according to eq 1. An ultrafast initial XT f CT decay takes place (∼50% at 150 fs), followed by an oscillatory behavior of the XT and CT populations, which is found to persist over the 3 ps observation interval. These observations are in qualitative agreement with the experimentally observed exciton decay and regeneration,10,16 even though a steady overall increase of the CT population is eventually observed in our simulation. The IS population experiences an initial increase but does not rise above an average of ∼10% beyond 500 fs. Yet, one cannot exclude the possibility that this state acts as a “bridge” facilitating the XT-CT transfer. To verify whether the direct XT f CT pathway remains dominant or whether the indirect pathway via the IS state, XT f IS f CT, plays a substantial role in the dynamics, we carried out two additional simulations. First, we compared with a twostate calculation, which was restricted to the XT-CT subspace (Figure 4b), and second, a modified three-state calculation was carried out, where the XT-CT diabatic coupling was artificially



Figure 3. Branching plane projections of the coupled three-state PESs: (a) XT, CT, and IS PESs projected onto the XT-CT branching plane and (b) XT and IS PESs projected onto the XT-IS branching plane; (c) and (d) are the corresponding adiabatic representations. In (b), the wavepacket width and trajectory of the wavepacket center (blue arrow) are also indicated. The two branching space projections are associated with two different sets of effective modes X corresponding to the choice (12) ) (XT,CT) versus (12) ) (XT,IS) in the matrix potential of eq 1. The white, blue, red, and purple circles indicate conical intersection points and minima on the XT, CT, and IS states, respectively. The dotted lines indicate (diabatic) seam lines.

set to zero, such that only the indirect pathway was permitted (Figure 4c). The first XT/CT simulation is in agreement with the two-state scenario of refs 28 and 29; see section IV.A (Note, though, that the results are slightly different from those of refs 28 and 29 due to differences in the parametrization). In the second case, that is, in the absence of a direct XT-CT coupling, a subpicosecond exciton decay and concomitant rise of the CT state population is also observed (Figure 4c), even though the process is somewhat slower than in the fully coupled threestate dynamics. As in the full simulation of Figure 4a, the IS population tends toward an average of ∼10%, that is, the IS state facilitates the XT-CT transition but does not act as an additional acceptor state. Overall, the dynamics of Figure 4a is therefore the result of a superposition of the direct (XT f CT) and indirect (XT f IS f CT) pathways. Given that the XT f IS transfer is endothermic (with a 0.2 eV barrier)10,16,61 in a conventional kinetic picture, the efficiency of the indirect pathway is entirely a consequence of the coherent, quantum dynamical character of the process. 2. HEP Analysis. We now address the HEP construction, which is expected to shed further light on the dynamical process, similar to the two-state analysis of refs 28 and 29; see section IV.A. In particular, we aim to verify whether the conclusions obtained for the two-state XT/CT model regarding the respective roles of the high-frequency versus low-frequency modes carry

over to the three-state case. Again, the HEP analysis should lead to the identification of a “minimal” reduced dimensionality model. The effective mode construction described in section II.B (and detailed in the Appendix), along with the HEP model of section II.C, leads to a structure of the mode hierarchy which features the high-frequency versus low-frequency phonon branches in alternationsvery similar to the two-state model of refs 28 and 29. The zeroth-order, that is, H(0) ) Heff of eq 3, is entirely composed of high-frequency modes, that is, the modes {X1, ..., X6}, are all of CdC stretch type. By contrast, the next order of the hierarchy, that is, the modes {X7, ..., X12} constituting H(1) res, are of the low-frequency, ring-torsional type, whereas the next order H(2) res, with modes {X13, ..., X18}, is again of highfrequency type. On the basis of the observations summarized in section IV.A, one would expect that the mode hierarchy has to be carried at least to the first order, H(1), to obtain a qualitatively correct picture of the dynamics. As shown in Figure 5, the high-frequency modes by themselves, at the level of the six-mode H(0) approximation, indeed cannot reproduce the exciton decay and CT rise (even though they correctly reproduce the shortest-time dynamics, on the order of 50 fs). At this level of approximation, quantum phase coherence effects are substantially overestimated, leading to a weak, periodic population transfer. By contrast, inclusion of the low-frequency modes, at the level of the 12-mode H(1) approximation, results in a qualitatively correct description of the


Tamura et al.

Figure 6. XT and CT state populations for the H(1) diss, 3-state 12-mode dynamics including dissipation (via an explicit 12-mode oscillator bath conforming to a Gaussian spectral density; see section IV.B.3). The full 24-mode calculation in the absence of dissipation (black lines) is shown for comparison; see Figure 4a. (IS populations are not shown.)

In Figure 6, the HEP hierarchy according to eq 8 has been carried to the first order (n ) 1, i.e., 12 modes), that is, we (1) consider the Hdiss approximation, such that the external bath modes couple to the low-frequency torsional modes constituting the H(1) res order of the hierarchy. The bath is composed of NB ) 12 external modes, which are also chosen to be of low-frequency type (within the range of 0.004-0.029 eV) so as to be resonant with the ring-torsional modes. The bath modes are sampled in an equidistant fashion in accordance with a discretized Gaussian spectral density, that is Figure 4. Time-evolving state populations for (a) the full 3-state 24mode wavepacket propagation, (b) a 2-state 24-mode calculation that is restricted to the (XT,CT) subspace, and (c) a complementary 3-state 24-mode calculation where the indirect (XT f IS f CT) pathway was selected by setting the XT-CT diabatic coupling to zero.

Figure 5. CT populations for the truncated HEP Hamiltonian H(n) of eq 5 for n ) 0 (6 modes), n ) 1 (12 modes), and n ) 2 (18 modes), as compared with the exact 24-mode result.

dynamics. Indeed, the low-frequency modes induce vibrational energy redistribution and dephasing effects that eventually lead to an irreversible exciton decay. Finally, the H(2), 18-mode approximation is essentially identical to the exact, 24-mode result on a ∼1 ps time scale. A similar interpretation holds for the indirect pathway in the absence of XT-CT coupling. The main elements of this analysis are similar to the twostate case described in refs 28 and 29. This suggests that the dynamical coupling pattern of high-frequency versus low-frequency modes apparently shows no marked dependence on the electronic state character. Indeed, it is largely determined by the fact that the high-frequency CdC stretch modes always tend to provide a dominant contribution to the electronic couplings. 3. Exciton Decay Including Dissipation. In view of the fact that the finite-dimensional model under consideration tends to overemphasize coherent effects on intermediate and long time scales, we address here the inclusion of dissipation as discussed in section II.E. In particular, an explicit NB-mode bath conforming to a Gaussian spectral density is added in accordance with the Hamiltonian eqs 8 and 9.

NB

Ji(ω) )

∑ dBij δ(ω-ωj) 98 J0 exp[(ω - Ω)2]/σ2 NBf∞

j)1

with J0 ) 3 × 10-5 au, Ω ) 0.014 eV, and σ ) 0.003 au. An identical set of bath coupling constants {dij} is chosen for all torsional modes i. The Gaussian bath is efficient in inducing energy relaxation and dephasing in the low-frequency mode subspace. This results in a pronounced attenuation of the coherent oscillatory behavior as demonstrated in Figure 6.62 A realistic modeling of the process is presumably intermediate between the strongly coherent, oscillatory evolution of Figure 4a and the partially damped dynamics of Figure 6. C. Sensitivity to Molecular Parameters. Given that the parametrization of the vibronic coupling model eq 1 is a function of the particular interface geometry and that the actual material exhibits a statistical distribution of such geometries, the sensitivity of the dynamics to parameter variations is an important issue to be addressed. In this section, we explore selected parameter variations, which allow us to draw conclusions on the robustness (or lack thereof) of the exciton dissociation pathways addressed in the preceding section. In the following, we address three types of changes of the model parameters: (i) a shift of the IS state energetic position, in such a way that the IS state lies between the CT and XT states at the Franck-Condon geometry; see Figure 7 (case I); (ii) a shift of both the IS and CT state positions (see case II in Figure 7); (iii) a change of the low-frequency ring-torsional mode frequencies by a factor of 2, that is, frequencies are chosen so as to cover the interval of 0.03-0.04 eV (240-320 cm-1) instead of 0.015-0.02 eV (120-160 cm-1) in the original model of section IV.B (case III). Cases I and II are motivated by the recent TD-DFT calculations carried out by two of us,32 indicating that the intermediate IS state is located between the CT and XT states. Case III is representative of frequency



Figure 7. Schematic showing the Franck-Condon energies of the modified systems discussed in section IV.C. On the left-hand side, the FC energies of the CT, XT, and IS states of the original system of section IV.B are shown. Case I corresponds to an energy lowering of the IS state, while case II corresponds to a change of both the IS and CT energies. Case III of section IV.C corresponds to the original FC energies, but the frequencies of the low-frequency phonon band are modified.

Figure 8. Time-evolving state populations for case I, with the IS state shifted to lower energies. Here, the IS state acts as a second reservoir state, competing with the CT state.

variations in a realistic range, as induced, for example, by modifications of the monomer side chains. For case I, Figure 8 shows that the modified IS state, shifted to lower energy, acts as a second reservoir state competing with the CT state. This is in contrast to the situation discussed in the preceding section, where the IS state exclusively acts so as to catalyze the XT-CT transfer. Both the direct pathway (unchanged as compared with the situation of section IV.B) and the indirect pathway are efficient in this case. For case II (energetic lowering of both the CT and IS states), Figure 9 illustrates separately the combined XT/CT/IS dynamics (panel a) and the direct two-state XT-CT pathway (panel b). Clearly, the direct XT-CT exciton dissociation pathway is substantially modified and becomes inefficient in this case. The fast, oscillatory decay that is observed in the overall dynamics (panel a) is entirely due to the indirect XT f IS f CT pathway. This result may first seem surprising, given that the CT state has been energetically stabilized (see Figure 7), and one might expect an increased transfer efficiency via the direct pathway. Likewise, case III, with slightly increased frequencies of the ring-torsional modes, exhibits an “inactive” direct XT f CT pathway, as illustrated in Figure 10. Again, the indirect pathway remains efficient, thus leading to an overall fast decay. What is the reason for the inefficiency of the direct XT-CT pathway in cases II and III, in contrast to case I? In both situations, the topology illustrated in Figure 2 is modified, either by changing the CT energetics or by changing the torsional frequencies. In both cases, this change can make the avoided crossing along the Xtorsion direction essentially inaccessible, as illustrated in Figures 9 and 10 (panel c). Hence, the XT-CT topology tends to be “fragile” with respect to small changes in the molecular parameters. By contrast, the indirect pathway turns out to be robust with regard to the selected parameter variations.

Figure 9. Time-evolving state populations for case II, with both the IS state and the CT state shifted to lower energies. Panel a shows the overall three-state dynamics, which exhibits a fast XT state decay. Panel b shows the XT-CT dynamics, in the absence of the IS state; here, essentially no population transfer takes place. The rapid XT decay of panel a must therefore be exclusively due to the indirect XT f IS f CT pathway. Panel c shows a cut of the XT/CT potential along a representative torsional coordinate, illustrating that the absence of the XT decay in panel b is due to the fact that the avoided crossing is inaccessible along the torsional coordinate. This 1D representation corresponds to a cut of the potential representation of Figure 2 along the low-frequency coordinate Xtorsion.

From the present discussion, it is clear that a reduced XTCT model is not generally sufficient to describe the exciton dissociation dynamics. The influence of intermediate, bridgetype states should be considered as an essential ingredient in the description of the polymer interface. Another important aspect relates to the role of the PES topology in the ultrafast dynamics. The robustness or fragility of the dynamics with regard to parameter variations essentially hinges on the underlying topology (e.g., avoided crossings in several dimensions or “funnel”-type topologies at a conical intersection). The identification of these relevant topological structures necessitates the molecular-level analysis proposed here. V. Concluding Remarks We have demonstrated the extension of our vibronic coupling analysis of refs 28 and 29 for a polymer heterojunction system so as to include the possible influence of a bridge state. On the basis of a molecular-level semiempirical electronic structure treatment,18,31,34,35 all states and diabatic couplings are parametrized for a representative set of 24 phonon modes of highfrequency and low-frequency type. While the present analysis singles out a particular additional state (IS), which has been shown to exhibit significant electronic coupling to the XT and CT states, the basic features of this analysis would carry over


Figure 10. Time-evolving state populations for case III, with the original FC energies but the frequencies of the torsional (low-frequency) phonon branch increased by a factor of 2. Panel a again shows the overall three-state dynamics; panel b shows the XT-CT dynamics in the absence of the IS state; and panel (c) shows a 1D cut along a representative torsional coordinate. Similar to case II, the direct XTCT pathway becomes inefficient, while the indirect XT-IS-CT pathway mediates the pronounced population transfer shown in panel a.

to several or many perturber states, providing indirect XT f IS f CT pathways. The main result of the present analysis is that additional pathways via bridge states can have a pronounced influence on the dynamics and could even dominate the dynamics in cases where the direct XT-CT transfer pathway is inefficient (see section IV.C). The high-dimensional avoided-crossing and conical intersection topologies which characterize the electronphonon system, are not necessarily robust, and the availability of multiple pathways is presumably a key feature that makes the exciton decay processes efficient. While the two-state XTCT model addressed in refs 28-30 is a natural first choice as a reduced model for the exciton decay, it is not generally sufficient as a minimal model for the process. The ultrafast dynamical events described by the vibronic coupling model eq 1 and its HEP variant eq 5 necessitate a detailed quantum dynamical analysis and are not compatible with a simple kinetic (exponential or multiexponential) picture of the process. The explicit dynamics considered here could be recast in terms of a non-Markovian generalized kinetic description, for example, at the level of the generalized master equation discussed in ref 30. The coherent, oscillatory behaviors observed in our simulations indicate that electronic and nuclear coherence are preserved on the short time scales under consideration. This is in accordance with experimental observations, for example, of coherent motions of the nuclear backbone in conjugated polymer species.24 With regard to the TFB/F8BT heterojunction

Tamura et al. system, our observations should relate to the picosecond scale decay and exciton regeneration phenomenon reported in the time-resolved photoluminescence experiments of refs 10 and 16. The HEP analysis reported in sections II.C and IV.B.2 emphasizes that the short and intermediate time scales are determined by few, collective phonon modes. These modes can be interpreted as generalized reaction coordinates for the chargetransfer process. Importantly, both the lowest order of the HEP hierarchy (involving the high-frequency CdC stretch modes) and the first order (involving the low-frequency ring-torsional modes) are essential to give a qualitatively correct picture of the process. This applies both to the direct XT f CT pathway and to the indirect, bridge-mediated XT f IS f CT pathway. The crucial role of the ring-torsional modes in the dynamical events confirms the observations of refs 22 and 63, which address the spectroscopic signature of both types of phonon modes in phenylene-based polymers. All of these results would carry over to an initial parametrization involving a very large number N of phonon modes, for example, thousands of modes. While dynamical calculations according to eq 1 would not be feasible for such a “macrosystem”, the HEP model gives a correct reduced dimensionality description of the ultrafast dynamics. Finally, while the molecular-level analysis proposed here refers to a particular realization of the polymer interface, among a statistical distribution of such realizations in the actual material, we have attempted to address realistic parameter variations which could characterize different local configurations and chemically modified polymer species. An important finding is that certain variations, for example, of energy gaps and of the low-frequency mode spectrum can “deactivate” the direct XTCT pathway while the XT-IS-CT pathway becomes dominant. As mentioned above, this provides a strong indication that the presence of multiple pathways plays a key role in the actual material system. The microscopic information obtained by the present analysis should thus contribute to identifying the key parameters determining the dynamical properties observed both in single-molecule experiments and in bulk experiments monitoring ensembles of interface structures. Acknowledgment. We thank Andrey Pereverzev, Clara Christ, Etienne Gindensperger, and Lorenz Cederbaum for constructive discussions. This work was supported by the ANR05-NANO-051-02 and ANR-NT05-3-42315 Projects, by NSF Grant CHE-0345324, and by the Robert Welch Foundation. Appendix A. Effective Mode Construction The construction of the effective six-mode Hamiltonian eq 3 is analogous to the two-state/three-mode construction described in refs 28 and 29. Alternative schemes are possible,43 all of which can be interrelated by orthogonal transformations in the effective mode subspace. We proceed by constructing a topology-adapted LVC model as described in ref 29, focusing on a conical intersection between a chosen pair of states (here, 1 and 2 with respect to the vibronic coupling matrix of eq 1). Two orthogonal branching plane coordinates are identified, that is, the {X1,X2} modes, which represent the principal directions along which the degeneracy at the (12) conical intersection is lifted. The remaining coordinates {X3, ..., X6} are then successively obtained by an orthogonalization procedure. This procedure, as detailed below, can be equally applied to another combination of branching plane coordinates (see the XT/CT vs XT/IS branching plane



representations of Figure 3). However, it is not possible to simultaneously create topology-adapted representations for several branching spaces since these are not generally orthogonal. We first outline the construction of the (12) branching plane coordinates, which subsume the collective effects of the {λ(12) i } 1/ (κ(1) - κ(2))} vibronic coupling contributions. and {κ-(12) ) 2 i i i Following ref 29, we first construct an effective coupling mode X′1 N

X′1 )

t1ixi ∑ i)1

t1i )

λ(12) i

N

2 1/2 [λ(12) ∑ i ] ) i)1

λh(12) ) (

λh(12)

(A.1)

To determine X′2, an initial guess is made using the {κ-(12) } i tuning parameters N

X′′2 )

t′2ixi ∑ i)1

t′2i )

κ-(12) i

N

[κi∑ i)1

κ-(12) ) (

-(12)

κ

(12) 2 1/2

])

(A.2)

Then, X′′2 is orthogonalized against X′1 by a rotation in the X′1X′′2 plane, X′2 ) (X′′2 - θX′1)/N2, where θ is the X′1-X′′2 inner product and N2 is a normalization constant (i.e., using the Gram-Schmidt orthogonalization scheme). In the next step, the remaining coordinates are orthogonalized against the branching space modes {X′1,X′2}, and the coupled equations N

∑ i)1

N

K′iX′i )

κ+(12) xi ∑ i i)1

are solved to obtain the {K′i} parameters; here, κ+(12) ) 1/2(κ(1) i i + κ(2) i )}. Following this, a third effective coordinate is introduced so as to subsume all shift terms by the condition N

K3X3 )

K′iX′i ∑ i)3

(A.3)

Up to this point, the construction is identical to the two-state case described in ref 29. The additional steps leading to the identification of the modes {X4, ..., X6} are analogous to the construction of X3. For example, the mode X4 is obtained by identifying N

Λ(13) 4 X4 )

Λ′i(13) X ′′i ∑ i)4

(A.4)

from an intermediate set of modes {X′′4, ..., X′′N}, which were orthogonalized against the modes {X′1,X′2,X′3}, obtained as described above, with the parameters {Λ′i(13) } obtained from the coupled equations N

N

Λ′i(13) X′′i ) ∑ λ(13) ∑ i xi i)1 i)1 and K(3) A similar procedure is followed to obtain Λ(23) i i . From this prescription, one can deduce the conditions stated in section IV.B above, that is, Λ(13) (i ) 5, 6) and Λ(23) are zero by i 6 construction.

Supporting Information Available: List of parameters used for the 3-state 24-mode model eq 1. This information is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Friend, R. H.; Gymer, R. W.; Holmes, A. B.; Burroughes, J. H.; Marks, R. N.; Taliani, C.; Bradley, D. D. C.; Santos, D. A. D.; Bre´das, J. L.; Logdlund, M.; Salaneck, W. R. Nature 1999, 397, 121. (2) Granstro¨m, M.; Petritsch, K.; Arias, A. C.; Lux, A.; Andersson, M. R.; Friend, R. H. Nature 1998, 395, 257. (3) Halls, J.; Cornil, J.; dos Santos, D. A.; Silbey, R.; Hwang, D. H.; Holmes, A. B.; Bre´das, J. L.; Friend, R. H. Phys. ReV. B 1999, 60, 5721. (4) Schmidt-Mende, L.; Fechtenko¨tter, A.; Mu¨llen, K.; Moons, E.; Friend, R. H.; MacKenzie, J. D. Science 2001, 293, 1119. (5) Bre´das, J. L.; Beljonne, D.; Cropceanu, V.; Cornil, J. Chem. ReV. 2004, 104, 4971. (6) Scholes, G. D.; Rumbles, G. Nat. Mater. 2006, 5, 683. (7) Cropceanu, V.; Cornil, J.; da Silva Filho, D. A.; Olivier, Y.; Silbey, R.; Bre´das, J.-L. Chem. ReV. 2007, 107, 926. (8) Gu¨nes, S.; Neugebauer, H.; Sariciftci, N. S. Chem. ReV. 2007, 107, 1324. (9) Organic Light-Emitting DeVices; Mu¨llen, K., Scherf, U., Eds.; Wiley-VCH: New York, 2006. (10) Morteani, A. C.; Sreearunothai, P.; Herz, L. M.; Friend, R. H.; Silva, C. Phys. ReV. Lett. 2004, 92, 247402. (11) Morteani, A. C.; Sreearunothai, P.; Herz, L. M.; Friend, R. H.; Silva, C. Ultrafast Phenomena XIV; Springer Series in Chemical Physics, Volume 79; Springer: Heidelberg, Germany, 2005. (12) Ai, X.; Beard, M. C.; Knutsen, K. P.; Shaheen, S. E.; Rumbles, G.; Ellingson, R. E. J. Phys. Chem. B 2006, 110, 25462. (13) Yu, G.; Gao, J.; Hummelen, J. C.; Wudl, F.; Heeger, A. J. Science 1995, 270, 1789. (14) Halls, J. J. M.; Walsh, C. A.; Greenham, N. C.; Marseglia, E. A.; Friend, R. H.; Moratti, S. C.; Holmes, A. B. Nature 1995, 376, 498. (15) Peumans, P.; Uchida, S.; Forrest, S. R. Nature 2003, 425, 158. (16) Morteani, A. C.; Friend, R. H.; Silva, C. In Organic Light-Emitting DeVices; Mu¨llen, K., Scherf, U., Eds.; Wiley-VCH: New York 2006; p 35. (17) Morteani, A. C.; Dhoot, A. S.; Kim, J. S.; Silva, C.; Greenham, N. C.; Murphy, C.; Moons, E.; Cina, S.; Burroughes, J. H.; Friend, R. H. AdV. Mater. 2003, 15, 1708. (18) Bittner, E. R.; Ramon, J. G. S. In Quantum Dynamics of Complex Molecular Systems; Burghardt, I., Micha, D. A., Eds.; Springer: Heidelberg, Germany, 2006. (19) Xue, J.; Rand, B. P.; Uchida, S.; Forrest, S. R. AdV. Mater. 2005, 17, 66. (20) Stevens, M. A.; Silva, C.; Russell, D. M.; Friend, R. H. Phys. ReV. B 2001, 63, 165213. (21) Halls, J.; Pichler, K.; Friend, R. H.; Moratti, S.; Holmes, A. Appl. Phys. Lett. 1996, 68, 3120. (22) Tretiak, S.; Saxena, A.; Martin, R. L.; Bishop, A. R. Phys. ReV. Lett. 2002, 89, 097402. (23) Lanzani, G.; Cerullo, G.; Brabec, C.; Sacriftci, N. S. Phys. ReV. Lett. 2003, 90, 047402. (24) Kersting, R.; Lemmer, U.; Mahrt, R. F.; Leo, K.; Kurz, H.; Ba¨ssler, H.; Go¨bel, E. O. Phys. ReV. Lett. 1993, 70, 3820. (25) Ko¨ppel, H.; Domcke, W.; Cederbaum, L. S. AdV. Chem. Phys. 1984, 57, 59. (26) Bersuker, I. B.; Polinger, V. Z. Vibronic Interactions in Molecules and Crystals; Springer Series in Chemical Physics; Springer: New York, 1989. (27) Ko¨ppel, H.; Domcke, W.; Cederbaum, L. S. In Conical Intersections; Domcke, W., Yarkony, D. R., Ko¨ppel, H., Eds.; World Scientific: New Jersey, 2004; Vol. 15, p 323. (28) Tamura, H.; Bittner, E. R.; Burghardt, I. J. Chem. Phys. 2007, 126, 021103. (29) Tamura, H.; Bittner, E. R.; Burghardt, I. J. Chem. Phys. 2007, 127, 034706. (30) Pereverzev, A.; Bittner, E. R. J. Chem. Phys. 2006, 125, 104906. (31) Ramon, J. G. S.; Bittner, E. R. J. Phys. Chem. B 2006, 110, 21001. (32) Ramon, J. G. S.; Bittner, E. R. J. Chem. Phys. 2007, 126, 181101. (33) Tamura, H.; Ramon, J. G. S.; Bittner, E. R.; Burghardt, I. Phys. ReV. Lett. Submitted; arXiv:0707.2163 [cond-mat.soft]. (34) Bittner, E. R.; Ramon, J. G. S.; Karabunarliev, S. J. Chem. Phys. 2005, 122, 214719. (35) Karabunarliev, S.; Bittner, E. R. J. Chem. Phys. 2003, 118, 4291. (36) Meyer, H.-D.; Manthe, U.; Cederbaum, L. S. Chem. Phys. Lett. 1990, 165, 73. (37) Manthe, U.; Meyer, H.-D.; Cederbaum, L. S. J. Chem. Phys. 1992, 97, 3199.

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