Photoinduced Excited-State Energy-Transfer Dynamics of a Nitrogen

Article pubs.acs.org/JPCC

Photoinduced Excited-State Energy-Transfer Dynamics of a Nitrogen-Cored Symmetric Dendrimer: From the Perspective of the Jahn−Teller Effect Jing Huang,†,‡,§ Likai Du,†,‡,§ Jun Wang,†,‡,§ and Zhenggang Lan*,†,‡,§ †

Key Laboratory of Biobased Materials, Qingdao Institute of Bioenergy and Bioprocess Technology and §The Qingdao Key Lab of Solar Energy Utilization and Energy Storage Technology, Qingdao Institute of Bioenergy and Bioprocess Technology, Chinese Academy of Sciences, Qingdao, 266101 Shandong, People’s Republic of China ‡ University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China S Supporting Information *

ABSTRACT: We report an interesting view to understand the ultrafast excited-state energy-transfer (EET) process in the D3-symmetric dendrimer tris(4-ethynylphenyl)amine (TEPA) from the perspective of the well-known E⊗e Jahn−Teller (JT) effect. Upon excitation to two lowest excited states (S1 and S2) with doubly degenerate E symmetry, two sets of e vibrational modes, dihedral angle twist and strong pyramidalization near the nitrogen core, lead to the JT distortion and symmetry lowering. Through the excited-state dynamics simulation with the onthe-fly surface-hopping approach at the TDDFT level, we find that the system may either travel three equivalent minima of S1 state or undergo the nonadiabatic transitions between S1 and S2 states. These motions induce the ultrafast EET among different branches and the reorientation of the transition dipole moments, finally leading to the ultrafast fluorescence anisotropy decay. This energy-transfer mechanism can provide some interesting insights on the excited-state dynamics of large dendrimers with three equivalent branches and transition metal complexes with C3 symmetry.

■

T-NPTPA,25,29 A-DSB,30,31 Gn,32 and so on. With timeresolved spectroscopy, they reported that a group of tribranched molecules with a nitrogen core show a very fast energy transfer among different branches on the ∼30−100 fs time scale. Through the investigation of a series of branched molecules with similar building blocks (chromophores) and different core units (C, N, and P), they also tried to discuss the intramolecular EET within the framework of Förster theory and touched the coherent/incoherent EET.26 In their studies, the anisotropy decay time for N(DSB)3 is ∼57 fs, and similar faster components were also observed in C(DSB)4 and P(DSB)3, indicating the ultrafast intramolecular EET in such systems.30 However, for the latter two systems [C(DSB)4 and P(DSB)3] the second much slower decay time scale was also observed, which is attributed as the Förster-type energy transfer between the DSB chromophore segments.26 Later, Samuel and coworkers investigated much larger C3-symmetric systems with a truxene core,28 which also indicated that the EET between different branches are biexponential with a faster initial 500 fs and followed by a slower 3−8 ps. These interesting findings invoked other experimental studies, which also found the ultrafast EET among different arms in multibranched symmetric dendrimers.33,34

INTRODUCTION Organic dendrimers show attractive photovoltaic properties such as efficient photoharvesting,1,2 unidirectional energy transfer,3 enhanced two-photon absorption,4−6 and high photoluminescence quantum yield.7,8 These properties make them potentially useful in photovoltaic materials for solar energy conversion,1,3,9 fluorescence sensors,10,11 organic lightemitting diodes,12,13 and organic lasers.8,14 The fundamental photophysics and photochemistry of these novel dendritic macromolecular architectures involve the exciton formation and movement (localization or delocalization),15−18 especially the intramolecular excited-state energy transfer (EET) between different subunits (branches).3,19−22 A number of experimental studies have been carried out to study the intramolecular EET between different subunits for branched molecules.23−25 In particular, the organic dendrimers with rotational symmetry (such as the existence of C3 or C4 rotational axes) have attracted considerable attention since the efficient EET is expected among those equivalent branches. With the development of laser technology, time-dependent fluorescence depolarization may provide useful information on the energy-transfer processes of such high-symmetric systems.25−29 This special technology allows us to monitor the orientation change of the emission dipole with respect to the absorption dipole, which is caused by the variation of local excitation with time. Goodson III and collaborators studied the intramolecular EET mechanism of several dendrimers, such as © XXXX American Chemical Society

Received: December 15, 2014 Revised: March 16, 2015

A

DOI: 10.1021/jp512496z J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

Article

■

THEORETICAL METHODS Electronic-Structure Calculations. This work mainly focuses on the simplified TEPA dendrimer (Figure 1), which

Normally, theoretical considerations on the ultrafast EET in ordinary organic dendrimers generally involve the exciton theory,17,21 which considers the model Hamiltonian describing the energies of the individual units and their couplings. Then the quantum evolution of the electronic density matrix provides physical insight on the EET processes. Alternatively, the electronic-structure calculations on the molecular excited states may also propose the EET mechanism.35−37 For example, Martı ́nez and co-workers suggested that different branches of the phenylacetylene dendrimers (C3 axis and the benzene ring core) are weakly coupled at the ground-state equilibrium geometry, while their couplings strongly increase in the excitedstate minimum. However, the understanding of EET dynamics of realistic dendrimers at the full-atomic level requires a comprehensive nonadiabatic dynamics simulation, since only this approach properly understands excited-state dynamics, photoactive nuclear degrees of freedom, coupled electron− nucleus motion, etc.38−40 However, until now these nonadiabatic dynamics studies mainly focused on the tree-type molecules with nonequivalent branches,38−41 which do not display any intramolecular rotational symmetry. As noticed by Goodson III, Samuel, and De Schryver, the high-symmetric organic dendrimers with at least 3-fold rotational symmetry should in principle possess two degenerate electronic states.24,28,32,42 When these molecules are excited to the electronic states carrying the degenerate irreducible representations, the symmetry of the molecular system will be broken, leading to the finite energy splitting of these two states. Such symmetry-broken phenomenon should be the result of the famous Jahn−Teller (JT) effect.43 Since the vibronic couplings are essential in this situation, the nonadiabatic dynamics simulation is required to understand the EET mechanisms more comprehensively from the JT perspective. To the best of our knowledge, no theoretical attempt has been made to explore the excited-state nonadiabatic dynamics in such high-symmetrical organic dendrimers at the all-atomic level, especially from the JT perspective. Most experimental works have investigated the threebranched dendrimers with rather large molecular size (>60 atoms), which are beyond the treatable limit of current electronic-structure calculations and nonadiabatic dynamics simulation. As our first efforts to treat the excited-state dynamics of such dendrimers, this work considered a simplified model system, nitrogen-cored D3 symmetry dendrimer tris(4ethynylphenyl)amine (TEPA), which should in principle capture the main feature of EET dynamics of similar systems. We examined the potential energy surface (PES) of the two lowest excited states and explain the possible JT effects. Then the on-the-fly surface-hopping dynamics at the TDDFT level was used to examine the excited-state processes and to determine the relevant nuclear motions responsible for the symmetry breaking in the JT distortion. The energy-transfer process was also analyzed by the time-dependent transition density matrix. We believe that this work provides useful information on understanding the EET mechanism of highsymmetric organic dendrimers with several equivalent branches as well as the excited-state JT processes of high-symmetric transition metal complexes, such as aluminum(III)−tris(8hydroxyquinoline) (Alq3),44−46 Ru(bpy)32+,47 Os(bpy)32+,48 and so on.

Figure 1. Molecular structure of TEPA.

has a nitrogen core and three identical arms. Each arm (block) includes one benzene ring and one ethynylene bond; thus, we named these blocks B1−B3 in sequence. Several electronicstructure methods, such as TDDFT with a few functionals (TDDFT/PBE0,49 TDDFT/BH&HLYP,50,51 TDDFT/CAMB3LY52), CIS, and SCS-ADC(2),53−56 were employed to examine the excited-state surfaces. The quality of results obtained at the TDDFT levels were checked by comparing with the values with the SCS-ADC(2) level, which is commonly thought to be a rather accurate method. The SCS-ADC(2), TDDFT/BH&HLYP, TDDFT/PBE0, and CIS calculations with the def2-SVP basis set57,58 were carried out with the Turbomole 6.359 program package, while the TDDFT/CAM-B3LYP/6-31G* calculation was performed with the Gaussian 0960 program. Transition Density Analysis. The electronically excited states may involve local excitation (Bi → Bi LE) or charge transfer (Bi → Bj CT) excitation between different branches. Here, the recently proposed analysis method based on the oneelectron transition density matrix61−64 was employed to gain useful insight into the EET between different arms in the TEPA dendrimer. The one-electron transition density matrix is defined as TrsEG = ⟨E|a +r a s|G⟩

(1)

TEG rs

where defines the transition density between molecular orbitals (MOs) s and r. |E⟩ (|G⟩) refers to the excited (ground) state many-electron wave function. a+r (as) is the creation (annihilation) operator for an electron at the rth (sth) MO. The transformation of the transition density matrix from the MO basis to the atomic orbital (AO) basis is performed as follows T[AO] = C·T[MO]·Ct

(2)

[AO]

where T is the one-electron density matrix in the AO basis and C is the MO coefficient matrix. Furthermore, the transition density matrix TEG,[Lo] is formed in the orthogonal Löwdin orbital basis by37,65 TEG,[Lo] = (S[AO])1/2 ·TEG,[AO]·(S[AO])1/2

(3)

where S is the AO overlap matrix. For an excited state, the contributions of the transition between two atoms (a and b) in principle should be given by simply adding all relevant AO B


Article

The Journal of Physical Chemistry C transitions. Thus, the contributions of the a → b transition can be simply expressed as E Bab =

σji(t ) = v ·Fji =

∑ (TEG,[Lo])2rs r∈a

(4)

s∈b

≈

The contribution of the transition from unit A to unit B of the whole molecule can be obtained by summing BEab as shown in eq 5

ΩEG AB =

(5)

b∈B

In this work, we define each arm of the dendrimer (Bi) as one unit to investigate the excitation characters. If A ≠ B, eq 5 represents the contribution of the CT transition from arm Bi to Bj, whereas it refers to the LE transition Bi → Bi if A = B. In this way, we can easily view all intraunit LE and interunit CT contributions for an excited state. Nonadiabatic Dynamics Method. The nonadiabatic dynamics was simulated with the on-the-fly trajectory surfacehopping (TSH) method implemented in our in-house program named “JADE”.66 Since all technical details can be found therein, we only outline the necessary theoretical issues here. In the TSH method, the nuclear motion on the single surface is described by a single classical trajectory R(t), computed by numerical integration of Newton’s equations. The velocity− Verlet algorithm67 is used for nuclear propagation. The electronic wave function satisfies the time-dependent Schrödinger equation, and the propagation of the quantum amplitudes (ci(t)) of each state along the nuclear trajectory R(t) can be given by the following set of coupled differential equations iℏ

dcj(t ) dt

=

∑ ci(t )[Hji − iℏvFji] i

|ΨK (r , R(t ))⟩ =

ciK, a =

Fji ≡ ⟨ϕj|∇R |ϕi⟩

(8)

(9)

Then the transition probability to jump from one potential surface to another is evaluated on the basis of Tully’s fewest switches algorithm Pij =

2Δt Re(ci*cj) ·v ·Fji |ci|2

εa − εi K Fi , a ωK

(13)

ΦCSF i,a

In the above equation, is the CI coefficient and is a singlet spin adapted configuration state functions. εa and εi are the energies of virtual and occupied molecular orbitals, respectively. Also, ωK corresponds to the excited energy. FKi,a represents the solution of the TDDFT pseudoeigenvalue problem.78,79 As is well known, TDDFT does not give a very precise description of PES crossings between the excited and the ground states,80,81 while it can be used to treat the degeneracy of electronically excited states. In the current system, the energy-transfer process is relevant to the nonadiabatic dynamics between the two lowest excited states (S1 and S2); thus, the TDDFT method should be a reasonable approach to treat this system under the suitable functional. Hence, the photoinduced nonadiabatic dynamics for TEPA dendrimer was studied by the on-the-fly TSH simulations at the TDDFT level. Certainly, selection of the functional is very essential in the reasonable treatment of excited states of TEPA dendrimer. After the careful benchmark calculations, we select the TDDFT/ BH&HLYP/def2-SVP in the direct dynamics simulation, since this level gives rather consistent results with SCSADC(2) and TDDFT/CAM-B3LYP (see below). The initial conditions (such as geometries and velocities) in TSH dynamics were prepared based on the Wigner distribution82 of normal modes of the ground state. Then they were created by putting these snapshots into the S1 and S2 state vertically. All of the relevant energies, gradients, and nonadiabatic couplings were calculated analytically in the manner of “on-the-fly”. The step times were 0.4 fs for the nuclear motion and 0.004 fs for the electronic propagation. In principle, the nonadiabatic coupling vector should be used to rescale the atomic velocities at hops for energy conservation, because this approach can be derived from more rigorous semiclassical approaches.83,84 In the current simulation, the nonadiabatic coupling vector is not available and we rescale the atomic velocities uniformly.75 The current work involves the very huge amount of calculations because a large number of trajectories should be computed to achieve the statistical limit. To save the total

In our implementation, the solution of eq 6 is implemented by performing a unitary propagator for the quantum amplitudes.68 In the adiabatic representation, the electronic wave functions are eigenfunctions of the electronic Hamiltonian and thus

Hji = εiδji

(12)

cKi,a

where v is a vector of nuclear velocities and Hji is the electronic Hamiltonian. The nonadiabatic coupling vector (Fji) between the states j and i can be expanded in a set of known electronic functions (ϕi) (7)

∑ ciK,a|ΦCSF i , a (r , R(t ))⟩ i,a

(6)

Hji ≡ ⟨ϕj|He|ϕi⟩

(11)

2Δt

where Δt is the integration time step. The detailed evaluation of nonadiabatic coupling elements can be found in refs 66 and 69−77 and the Supporting Information. The electronic ground state can be represented by a single Slater determinant built from the occupied Kohn−Sham orbitals, while the wave function for the excited electronic states can be approximated by the CIS-type expansion. It is worth noting that the expansion for the excited state K can be, in principle, obtained according to the Casida’s assignment ansatz.78 Following previous work,69,70,72−77 the wave function ΨK can be written as

∑ BabE a∈A

∂ ϕ ∂t i ⟨ϕi(t )|ϕj(t + Δt )⟩ − ⟨ϕi(t + Δt )|ϕj(t )⟩ ϕj

(10)

The nonadiabatic coupling terms can be reformulated as the scalar product of the velocity vector v and the nonadiabatic coupling vector Fji69−75 C


Article

The Journal of Physical Chemistry C computational time, most trajectory propagations stop at 250 fs, since the nonadiabatic population decay dynamics is essentially over within this time scale. However, a few typical trajectories were propagated up to 900 fs for further detailed analysis of molecular motions. Additionally, it is well known that the standard trajectory surface-hopping method suffers from the overcoherent problem.85−89 To avoid it, we introduce the decoherence correction proposed by Granucci and Persico90 during the dynamics process with the relevant constant C = 0.1. Jahn−Teller (JT) Effect. The JT theorem states that the nonlinear molecules with high symmetry belonging to nonAbelian point group can distort to low-symmetry configurations if the molecular system stays on the degenerated electronic states carrying degenerate irreducible representations. At the ground-state equilibrium geometry, the two lowest singlet excited states (S1 and S2) are degenerate with the E irreducible representation. Thus, the electronic degeneracy can be easily destroyed by the symmetry-breaking vibrations of the TEPA molecule, due to the vibronic coupling according to the JT theorem.43,91−93 For this D3-symmetric molecule, a doubly degenerate vibrational e mode can couple two degenerated E states, leading to the so-called E⊗e JT effect.43,91,92 Due to the presence of the C3 rotational axis, some intramolecular coordinates or physical observables localized on one branch can be easily transformed to their counterparts on the next branch, namely 2 Ĉ3α1 = α2 , Ĉ3 α1 = α3

Figure 2. Important internal coordinates of TEPA: (a) bond distances (r1, r2, r3), (b) bond angles (θ1, θ2, θ3), (c) twisting dihedral angles (τd1, τd2, τd3), and (d) pyramidalization angles (τp1, τp2, τp3).

lowest adiabatic singlet excited states (S1 and S2) become doubly degenerate (E symmetry). The relevant transitions are HOMO → LUMO and HOMO → LUMO+1, and the involved orbitals are plotted in Figure S1, Supporting Information. We subsequently optimized the geometry of the S1 state under the D3 symmetry constraint at the TDDFT/PBE0 and TDDFT/BH&HLYP levels (Table S2, Supporting Information). In contrast to the (S0)min geometry, all three N−C bond distances (ri) are slightly shortened, while the C−C−N−C dihedral angles (τd) become smaller by about 3−5° at different levels. Furthermore, the optimized structure with D3 symmetry at different theoretical levels seems to be very close to each other (Table S2, Supporting Information). On the excited states, the vibronic coupling may remove the S1/S2 degeneracy, and thus, it is important to clarify which nuclear motions play the dominant roles here. To figure out the JT effective modes, the geometries of the S1 minimum [(S1)min] without any symmetry constraint (C1) was performed. Interestingly, the optimization results were highly dependent on the theoretical levels. Although S1 optimization at the TDDFT/BH&HLYP level was performed without any symmetry constraint, the resulting geometry of (S1)min shows C2 symmetry. From D3 symmetric (S1)min to C2 symmetric (S1)min, τd1 changes slightly while τd2 and τd3 decreases and increases, respectively, by ∼7°. Meanwhile, τp1 and τp2 increase by +4.4° and −4.4°, respectively. To double check, we try to perform S 1 optimization with the C2-symmetry constraint, and the same geometry is obtained. Therefore, we suspect that the τd twist motions as well as the pyramidalizations of τp are the effective vibrational modes for the D3 → C2 symmetry-lowing processes. However, the different (S1)min with C2v symmetry is found at the TDDFT/PBE0 level, which displays the large twisting motions on τd1, τd2, and τd3, as shown in Table S2 and Figure S2, Supporting Information. However, when the (S1)min geometry at the TDDFT/PBE0 level was taken as the initial guess to reperform the S1 optimization at the TDDFT/ BH&HLYP level, we obtain the same structure as the previous TDDFT/BH&HLYP results. The reverse situation is also

(14)

where Ĉ 3 is the 3-fold rotational operator and α can be the bond distance, bond angle, dihedral angle, and other quantities localized on one branch. It is not easy to directly use the original coordinates/ quantities to present the JT effect generally; thus, in the current analysis we utilize the symmetry-adapted ones constructed by the linear combinations of α1, α2, and α3 1 (α1 + α2 + α3) St = (15) 3 Sa =

1 (2α1 − α2 − α3) 6

(16)

Sb =

1 (α2 − α3) 2

(17)

where St is the totally symmetric combination of all quantities. Both Sa and Sb correspond to two equivalent JT effective quantities with e symmetry. For instance, when they represent the JT active coordinates, the PESs within the two-dimensional space spanned by Sa and Sb display a well-defined double-cone topology.

■

RESULTS Ground and Excited States. The critical coordinates of the TEPA molecule for photoinduced processes are labeled in Figure 2. The ground-state geometry was obtained from optimization with D3 symmetry. The geometry optimization without symmetry constraint (C1) leads to the same structure, confirming that the D3 structure is a real minimum. The optimized structures of the ground state (S0) are very similar at different theoretical levels, implying the weak method dependence for the ground-state calculations (Table S1, Supporting Information). At the S0 minimum [(S0)min] geometry, the two D


Article


Figure 3. PES scan of (a) S1 and (b) S2 excited state between the two (S1)min located by the BH&HLYP and PBE0 theories. Point 1 in the x axis represents C2-(S1)min at the TDDFT/BH&HLYP level, and Point 8 in the x axis donates C2v-(S1)min at the TDDFT/PBE0 level. Then all intermediate points (Points 2−7) are generated by the linear interpolation from Point 1 to Point 8. Different levels (SCS-ADC(2), red; BH&HLYP, green; CAM-B3LYP, blue; CIS, magenta; PBE0, black) are considered.

Scheme 1. (a) Classification of Electronic Transition within Three Blocks (B1, B2, B3);a (b) Illustration of the JT Effectb

a Each square stands for the transition from the Bx (horizontal) to By (longitudinal) blocks. The color codes in this figure indicate the amplitude of transition intensity. The deeper color represents the larger transition contribution. bThree (S1)min and TS with C2 symmetry are shown, associated with their related transition density of S1 and S2.

underestimation of the CT state.79,80 It is well known that the CIS method, on the other hand, could strongly overestimate the CT state;80,94 thus, the strong rising of the PES at the CIS level is observed. Overall, we believe that only C2-symmetric (S1)min exists on the S1 state, and the TDDFT/BH&HLYP/ def2-SVP level gives the reasonable description here. Three branches (Figure 1) are used to build intraunit and interunit electronic transitions. Transition density analyses were performed for the lowest six excited states (Figures S4 and S5, Supporting Information). At both D3 (S0)min and C2 (S1)min geometries, the lower E excited states (S1/S2) are mainly composed of LE components with minor and partial CT components. The S3 and S6 states (total-symmetric A1 state) are also assigned as the LE state that is mainly composed by the symmetric combination of all LE transitions at three branches. Both S3 and S6 states also display the rather minor contribution of the CT characters. The situations of the higher E excited states (S4/S5 states) are similar to that of S1/S2 states. Thus, the state with obvious/pure CT character does not play an essential role in the current EET dynamics. To further confirm the feasibility of TDDFT/BH&HLYP and TDDFT/CAM-B3LYP, the PESs for the three lowest excited states as a function of linear interpolated coordinate between two (S1)min (D3 and C2 symmetries at the TDDFT/ BH&HLYP level) were constructed at the SCS-ADC(2), BH&HLYP, CAM-B3LYP, and CIS levels. The details are presented in Figure S6, Supporting Information. Results indicate that the PES profiles obtained at the BH&HLYP level are very close to its counterpart at the TDDFT/CAM-

observed when we start from the TDDFT/BH&HLYP (S1)min geometry and perform the TDDFT/PBE0 optimization. To confirm which functional gives correct description of the excited states, further ab initio calculations were performed by scanning the PESs of S1 and S2 states between the two (S1)min obtained at the BH&HLYP and PBE0 levels. Various theoretical methods, including SCS-ADC(2)/def2-SVP, TDDFT/PBE0/def2-SVP, TDDFT/BH&HLYP/def2-SVP, TDDFT/CAM-B3LYP/6-31G*, and CIS/def2-SVP, were employed. As shown in Figure 3, the PES profile of S1 increases from the first (S1)min (at the BH&HLYP level) to the second (S1)min (at the PBE0 level) at all used theoretical levels except PBE0. In addition, the results at the BH&HLYP and CAM-B3LYP levels are very similar to those at the more accurate SCS-ADC(2) level. This strongly indicates that only C2-symmetric (S1)min appears on the S1 state and PBE0 gives an artifact here. To understand possible reasons, we subsequently examined the electronic character of the S1 state. At the (S1)min obtained by the TDDFT/PBE0 optimization, the HOMO is located on one arm while the LUMO distributes throughout another two arms (Figure S2, Supporting Information). Thus, the S1 state, corresponding to HOMO → LUMO, at such C2vsymmetric geometry displays the completely CT character. Conversely, at the C2-symmetric (S1)min obtained at the TDDFT/BH&HLYP level, the HOMO is delocalized over three arms and the LUMO distributed to two of them (Figure S3, Supporting Information). Therefore, the S1 state may only display the rather weak and partial CT character. Thus, the results at the PBE0 level should be an artifact due to its strong E


Article


starting from S1 or S2 individually (Figure S7a and S7b, Supporting Information). Due to the inclusion of the decoherent corrections, internal consistency (time-dependent trajectory occupations and electronic populations) is achieved (Figure 4) during the propagation. Although the ground electronic state (S0) was also included in the current surface-hopping calculations, we do not see any trajectory jumping back to the S0 within the simulation time scale. This indicates that no further effective nonadiabatic decay pathway exists for the internal conversion to the electronic ground state. Thus, the system, in principle, should stay in the excited state and may return to the ground state by means of fluorescence emission or slow nonradiative transitions with much longer time scales. This is consistent with the experimental studies.31,95 To address the potential influence of the total-symmetric A1 (S3) state on the EET dynamics, we examine the timedependent energies of three lowest excited states along the evolution of the sample trajectory (Figure S8a, Supporting Information). Although the S2 and S3 states may come closer in the propagation, in most cases, this event happens when the trajectory moves on the S1 state. To get a more precise view on the role of the S3 state in the excited-state dynamics, we did the preliminary trajectory-surface-hopping calculations by taking all three lowest excited states into account, and 48 trajectories starting from the S2 state are propagated. Throughout the dynamics, the contributions of the S3 state are really minor (Figure S8b, Supporting Information), indicating that the S3 state does not play an essential role in the current energytransfer process. Thus, only S1 and S2 states are important for the EET process of TEPA. Moreover, the current simulation indicates that after 20 fs the S2 state occupation is always very low (∼20%) during the dynamics propagation. Concerning the similar S1/S2 PES profiles at the TDDFT/BH&HLYP and SCSADC(2) levels, it is reasonable to assume that the S2 population should also remain at low values if the SCS-ADC(2) level was used in the nonadiabatic dynamics simulation. In this case, we still expect that the contribution of the even higher state (S3) should also be very minor. The EET mechanism is analyzed by the time-dependent transition density for a typical trajectory (Figure 5a). The dynamics simulation for this trajectory lasts to 900 fs (Figure 5b). The ultrafast decays and increases of each LE transition component (TB1, TB2, TB3) indicate the efficient EET among these branches with the time evolution. Interestingly, sudden changes of LE contributions are observed at the hopping events (Figure 5a, labeled as the red dotted lines), which indicates that the nonadiabtic transition is a key issue responsible for the intramolecular EET process. In addition, the fluctuation of LE components is also observed beyond the hopping events, which implies that the EET between different branches is also possible when the system moves on a single adiabatic electronic state. The time evolutions of critical geometrical parameters (i.e., bond distance, bond angle, dihedral angle, and pyramidalization angle) were examined. The small fluctuations are observed for three N−C bond distances (r1, r2, r3) as well as three C−C−N bond angles (θ1, θ2, θ3) (Figure S9, Supporting Information). In contrast, the dihedral and pyramidalization angles (Figure 5c and 5d) display the obvious large-amplitude motions. Thus, the dihedral angle and pyramidalization should be the major reaction coordinates in this dynamics process, which is consistent with the results in electronic-structure calculations.

B3LYP levels and the more accurate SCS-ADC(2) level. Since the above analyses show the validation of TDDFT/ BH&HLYP/def2-SVP in the description of TEPA excited states, we select this approach for qualitative understanding of the EET dynamics of the model TEPA dendrimer system. The E⊗e Jahn−Teller Effect. For the D3-symmetric TEPA molecule, the symmetry breaking motion on the excited state is well described by the E⊗e JT effects. Since both S1 and S2 belong to the E symmetry with double degeneracy, the molecular vibration motions with the e symmetry may lift the degeneracy. When only the first-order vibronic couplings are involved, the PESs of two electronic states within the twodimensional coordinate space spanned by degenerate e modes display a circular symmetry with the center at the highsymmetry geometry.68 However, when the second-order vibronic couplings are taken into account, the S1 state should display three coexisting minima, and the JT mechanism for this case is depicted in Scheme 1b. Moreover, different minima show different electronic characters (Scheme 1). From the transition density analysis of each (S1)min with C2 symmetry (Scheme 1b), we find that the electronic transition of S1 involves two of three arms and S2 involves another one. Meanwhile, different S1 minima may display rather distinctive LE electronic characters. For example, near one (S1)min region (top configuration in Scheme 1b), the LE within B1 and B2 may be important for the S1 electronic character, while the B3 → B3 LE are dominant for S2. However, the B1 → B1 and B3 → B3 LE transitions play essential roles in S1 at the second (S1)min (down right of Scheme 1b). A similar case holds for the transition state (TS) connecting different S1 minima. Moreover, since different LE transitions are involved at each (S1)min, we expect that the orientation of transition dipole moments varies dramatically on the S1 surface. Nonadiabatic Dynamics. To gain further physical insights about the energy-transfer process, the nonadiabatic surfacehopping dynamics simulation was performed. Since S1 and S2 states are doubly degenerate in the Franck−Condon region, the same number of trajectories starts from both of them. As shown in Figure 4, the fractional trajectory occupation of the S2 state decays very quickly within 20 fs, and afterward the number of trajectories propagating on the S1 and S2 states nearly retains 4:1 within the current simulated time scale. This similar dynamics behavior is also observed in the trajectory simulation

Figure 4. Time-dependent fractional trajectory occupations and electronic populations of TEPA electronic states obtained from the nonadiabatic dynamics simulation. F


Article


Figure 5. Time-dependent quantities for a typical trajectory of TEPA: (a) transition density within 250 fs, (b) transition density within 900 fs, (c) dihedral angle C−C−N−C, and (d) pyramidalization C−N−C−C.

■

DISCUSSION In order to provide a more intuitive picture on the interbranch EET process and the relevant vibrational motions responsible for lowering symmetry (JT effects), we employ the symmetryadapted analysis for the reactive coordinates and transition densities (eqs 16 and 17). Figure 6a and 6b shows the symmetrized twist dihedral angles (the combination of τd1, τd2, and τd3) and relevant pyramidalization angles (the combination of τp1, τp2, and τp3), respectively, which should be the esymmetric vibrational motions responsible for the JT distortion. The three green chars in each figure correspond to three equivalent (S1)min. Obviously, the system travels around the high-symmetry geometry, most importantly going back and forth through three (S1)min. Very interestingly, the evolution of the symmetric LE transitions (Ta and Tb) forms a perfect equilateral triangle as shown in Figure 6c, also reflecting the typical JT feature. In fact, the peripheral and central components in Figure 6c correspond to the symmetrized transition densities of S1 and S2, respectively (Figure S10, Supporting Information). Overall, when the trajectory goes back and forth among three (S1)min on the single excited state or performs the hops, the effective intramolecular EET takes place among different branches of dendrimers. The most striking step toward characterization of the JT effect is to investigate the topology of the adiabatic PES within the space spanned by the two JT active coordinates.43,91−93 Thus, we plotted the adiabatic PES as a function of two orthogonal e modes (Qx and Qy, Figure 7). For illustration purposes, the Qx and Qy coordinates here do not correspond to any particular realistic coordinates but represent a pair of typical JT-effective e modes. For the E⊗e JT effect with a C3 rotational axis, the excited-state PES resembles a so-called “Mexican hat” when only the first-order vibronic couplings are included.43 However, three minima appear on the lower adiabatic surface when the second-order vibronic couplings are involved91−93 (Figure 7). For the current system, the dihedral angle twist and the strong pyramidalization near the nitrogen core are the JT active coordinates. Thus, we also selected a pair of dimensionless

Figure 6. Time-dependent two equivalent symmetrized quantities with the e symmetry responsible for the JT distortion: (a) Da and Db (combined by τd1, τd2, and τd3), (b) Pa and Pb (combined by τp1, τp2, and τp3), and (c) Ta and Tb (combined by TB1, TB2, and TB3). (Three green chars indicate three equivalent minima.)

G


Article


However, in realistic situations, the initial highly coherent preparation and the large florescence anisotropy (∼0.7) are generally not feasible. For instance, the initial florescence anisotropy of the nitrogen-cored high-symmetric systems with three branches is always close to 0.4 in experimental observations. Thus, the wave function superposition of the double-degenerated states either does not exist at all or can be easily destroyed in the very early stage of dynamics.28,31 The initial fluorescence anisotropy of ∼0.4 indicates that in fact the dynamics from the S1 and S2 states evolve individually. Thus, it is still reasonable to employ the surface-hopping dynamics for qualitative understanding of the time-dependent fluorescence anisotropy decay. The decay of florescence anisotropy should be the result of the reorientation dynamics of the transition dipole moment. Previous experimental works clarified that many C3-symmetric systems display similar florescence anisotropy decay; we thus believe that the current simplified TEPA model can capture the main feature of the dipole-moment motion. Subsequently, the reorientation of the transition dipole was investigated by the time-dependent quantity of cos2 β,103 where β is the angle representing the evolution of the transition-dipole moment direction. In the surface-hopping dynamics, the population transfer in the nonadiabatic dynamics is completed after 20 fs and afterward the populations become stable. Thus, we take both 0 and 20 fs as the starting point to estimate the evolution of the transition dipole moment with time. Both give us very similar behaviors. As shown in Figure 8, the cos2 β decay is

Figure 7. (a) Illustrative adiabatic PES of S1 and (b) its contour projection. Here the Qx and Qy coordinates do not correspond to any particular realistic coordinates but represent a pair of the typical JTeffective e modes.

normal coordinates (Q1 and Q2) representing the symmetrized internal rotation coordinates (Da, Db) along the dihedral angle C−C−N−C. The PESs of the S1 and S2 states as a function of Q1 and Q2 were constructed (Figure S11, Supporting Information) for a better view of the double-cone topology in the vicinity of conical intersections. It is very necessary to point out that the nonadiabatic coupling elements are rather approximated in our approach, because the excited-state CIS-type wave functions were constructed from the linear-response TDDFT according to eqs 12 and 13. The more rigorous theoretical approach requires consideration of the higher order response terms within the TDDFT framework.96,97 Hence, we try to compute the nonadiabatic coupling vectors (derivative with respect to Q1 and Q2) by our numerical approaches and a more rigorous approach derived from the diabatic model Hamiltonian of the JT system. As shown in Figure S12, Supporting Information, both results are consistent with each other. Thus, our current approach can reproduce both the PES double-cone topology and the nonadiabatic couplings in the vicinity of the S1−S2 conical intersection. Upon excitation, the system keeps the high-symmetric geometry. Then starting from this unstable initial situation, the system should move along the potential energy hill and travel around three (S1)min with low symmetry. At these (S1)min the spatial orientation of the transition dipole should be very different since the LE transitions involve different arms. In addition, the nonadiabatic transition between S1 and S2 states also alters the LE transition and further modifies the orientation of the transition dipole moment. In short, the excited-state dynamics leads to the ultrafast EET among three branches of the high-symmetric TEPA molecule, which arises from the typical E⊗e JT mechanism. In experiments, the interactions between different branches were investigated using time-dependent fluorescence anisotropy, which is a powerful tool charactering the energy migration in multichromophore systems.15,24,98 Since a laser is the coherence light, the ultrashort pulse laser may excite two excited states simultaneously if they are double degenerate. This creates a wave function that is the superposition of two degenerate states. In other words, the strong electronic coherence exists in the initial preparation. Such strong quantum coherence may result in a very large florescence anisotropy up to 0.7 in the ideal cases.98−102 It is not feasible to use a trajectory surface-hopping approach to describe such strong initial coherence and extremely high fluorescence anisotropy in the ideal cases, because the initial samplings always consider that each trajectory starts from its own independent condition.

Figure 8. Time-dependent cos2 β decay.

ultrafast (

Photoinduced Excited-State Energy-Transfer Dynamics of a Nitrogen

Recommend Documents