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Origin of Aggregation-Induced Enhanced Emission: A Role of Pseudo-Degenerate Electronic States of Excimers Formed in Aggregation Phases

Origin of aggregation-induced enhanced emission (AIEE) is investigated considering cyano-substituted 1,2-bis(pyridylphenyl)ethene (CNPPE) as an example. On the basis of ONIOM calculations using the time-dependent density functional theory (TD-DFT), we found that pseudo-degeneracy of excimers formed in solid phase plays an important role in the appearance of AIEE. The electron density difference delocalized over molecules gives rise to small diagonal vibronic coupling constants (VCCs), which suggests that the internal conversion is more suppressed in solid phase than in solution phase. The reduction of the off-diagonal VCCs owing to the packing effect is elucidated by vibronic coupling density (VCD) analysis. The pseudo-degeneracy enables fluorescence from the high singlet excited states against Kasha's rule because the electron density difference and the overlap density between the excited states vanish. A Hubbard model of a pseudo-degenerate electronic system is constructed to explain the vanishing mechanism. We propose the following design principle for AIEE: a candidate molecule for AIEE should have pseudo-degenerate adiabatic electronic states in the aggregation phases originating from the excimer formation.


Introduction
In contrast to concentration quenching, aggregation-induced enhanced emission (AIEE) yields strong luminescence in the aggregation phases. 1,2 The restriction of intramolecular motions is generally accepted as the reason behind AIEE, that is, the restriction of intramolecular rotations or vibrations arising from the physical constraint in the aggregation phases blocks non-radiative transition pathways. 3 Hexaphenylsilole (HPS) is one of the AIEE dyes, and the restricted rotation of the side phenyl ring is found to be a key factor for AIEE. [4][5][6] This is corroborated by quantum mechanical and molecular mechanical (QM/MM) calcula-tions where the Huang-Rhys factors and reorganization energies at low-frequency modes are reduced in the solid phase compared to the gas phase owing to the packing effect. 7 In addition, HPS having a bulky shape does not form a cofacial configuration in the solid phase, 1,2 which is considered to prevent concentration quenching.
Cofacial aggregation is regarded as one of the reasons behind concentration quenching. This is because a destructive alignment of the transition dipole moment renders the lowest excited state that is symmetry forbidden. 8 Nevertheless, some dyes have been reported to be emissive in spite of the formation of cofacial aggregations. [9][10][11][12][13][14][15][16] In general, according to Kasha's rule, emissions have been attributed to the lowest excited states. 17 However, if all the radiative and non-radiative transitions from a higher excited state to all the lower excited states are suppressed, fluorescence from the higher excited state can be expected to occur against Kasha's rule. Sato et al. have already reported that, in fluorescent dopants employed in organic lightemitting diodes (OLEDs), radiative and non-radiative transitions from a triplet excited state T n (n > 1) to all the lower triplet excited states can be suppressed due to the pseudodegenerate electronic states. [18][19][20] The pseudo-degeneracy leads to cancellation of the overlap density between the excited states, which generates T n excitons with long lifetimes. This enables the fluorescence via higher triplets (FvHT) mechanism for OLEDs, that is fluorescence utilizing the reverse intersystem crossing (RISC) from T n to singlets. FvHT is different from thermally activated delayed fluorescence (TADF) in that TADF undergoes the thermally activated RISC from T 1 to S 1 by decreasing the energy difference, ∆E S 1 −T 1 . 21,22 The FvHT mechanism is proposed to explain the mechanism of electroluminescence in some OLED dopants with large ∆E S 1 −T 1 , where T 1 excitons cannot overcome the energy difference thermally. 19,23 A cyano-substituted 1,2-bis(pyridylphenyl)ethene (CNPPE) (Fig. 1) has been reported to exhibit the AIEE behavior in solid phase. 24 The rate constants of the non-radiative transitions are decreased from > 1.0 × 10 10 s −1 in CH 2 Cl 2 solution to 5.0 × 10 7 s −1 in solid 3 phase. The rate constants of the radiative transitions are comparable between CH 2 Cl 2 solution (> 2.0 × 10 7 s −1 ) and solid phase (1.3 × 10 8 s −1 ). Accordingly, the fluorescence quantum yield is increased from 0.002 to 0.72 by aggregation. Since CNPPE forms cofacial configurations in solid phase, the occurrence of concentration quenching is predicted. Some cofacial CNPPE molecules have C i symmetry in the crystal structure. This suggests the possibility of pseudo-degenerate electronic states delocalized over the cofacial molecules. In this case, fluorescence from higher singlets than S 1 is expected to occur against Kasha's rule because the transitions between the excited states can be suppressed as in the case of the FvHT mechanism. The electronic states delocalized over molecules may decrease the vibronic coupling constants (VCCs). 25 These indicate that the internal conversion can be more suppressed in solid phase than in solution phase as long as excimer formation occurs in solid phase.
This mechanism is different from what is proposed in styrylbenzene, i.e. di, ? tri, ? ? and tetra(styryl)benzene, ? which asserts that the origin of the AIEE behavior is attributed to blocking the trans-cis photoisomerization of stilbene unit in solid phase.
In this study, we investigated the role of pseudo-degeneracy in the appearance of AIEE considering CNPPE as an example. Vibronic coupling density (VCD) analyses were performed to elucidate the local picture of VCC. [26][27][28] To explain the results obtained by the TD-DFT calculations, we discuss the electron density difference and overlap density in the pseudo-degenerate electronic system based on the Hubbard model.

Theory
We consider the internal conversion from an initial vibronic state |Φ mi (r, Q) associated with electronic m and vibrational mi states to a final vibronic state |Φ nj (r, Q) . Here, r = (r 1 , · · · , r i , · · · r N ) is a set of N electronic coordinates, and Q = (Q 1 , · · · , Q α , · · · Q M ) is a set of M mass-weighted normal coordinates. Within the crude adiabatic approximation, 29,30 the vibronic states are represented as the product of vibrational and electronic states fixed at the nuclear configuration R 0 : |Φ mi (r, Q) = |χ mi (Q) |Ψ m (r; R 0 ) . R 0 is chosen as the equilibrium nuclear configuration of the ground or excited optimized structures. The rate constant of the internal conversion from electronic state m to n is expressed as 31 where P mi (T ) is the Boltzmann distribution function of the initial vibronic state at temperature T , and E mi and E nj are the eigenenergies of |Φ mi (r, Q) and |Φ nj (r, Q) , respectively.
E mi is represented as the sum of electronic E m and vibrational energies. V mn,α is the offdiagonal VCC given by whereĤ(r, R) is the molecular Hamiltonian, and R is a set of nuclear configuration. V n := V nn is called the diagonal VCC. Ignoring the Duschinsky effect, which means vibrational modes do not change during an excitation, the matrix element of the vibrational states is written as where |n mi,α is a vibrational state of a single mode. The Franck-Condon (FC) overlap integral is expressed as 32 n mi,α |n nj,α = n mi,α !n nj,β ! 2 n mi,α +n nj,α e − 1 Here, g n,α is the dimensionless diagonal VCC (the Huang-Rhys factor): where ω n,α is the angular frequency of vibrational mode α. In general, a rate constant of an internal conversion is small when diagonal and off-diagonal VCCs are small. 31 Particularly, the dependence of the rate constant on the diagonal VCC is strong, and the reduction of the diagonal VCCs significantly contributes to the suppression of the internal conversion. 31 Within a single mode approximation at 0 K, the rate constant of the internal conversion is reduced to 31 where the delta function is replaced with the Gaussian function which represents the density of states with the linewidth σ.
In addition to the internal conversion, vibrational relaxation from FC to adiabatic (AD) states is a non-radiative process that should suppress for emission. 31 Within the crude adiabatic representation assuming the harmonic approximation, the reorganization energy due to vibrational relaxation is evaluated by 6 Thus, the reduction of the diagonal VCCs leads to the small reorganization energy that depends on the square of the diagonal VCCs.
The VCD is expressed as the density form of the VCC: [26][27][28] V mn,α = dx η mn,α (x ), where x = (x, y, z) is a three dimensional coordinate. The diagonal VCD η n,α (x ) := η nn,α (x ) is defined by Here, ∆ρ nm (x ) is the electron density difference between |Ψ n (r; R 0 ) and the reference state |Ψ m (r; R 0 ) is taken as the electronic state in the equilibrium nuclear configuration.ρ(x ) is the electron density operator defined bŷ where ψ * iσ (x ) and ψ jτ (x ) are spatial orbitals, andĉ † iσ andĉ jτ are creation and annihilation operators, respectively. i and j refer to the orbital indices, and σ and τ refer to the spin where u(x ) is the electron-nucleus potential acting on a single electron. 7 The off-diagonal VCD is defined by where ρ mn (x ) is the overlap density between |Ψ m (r; R 0 ) and |Ψ n (r; R 0 ) : The VCD enables us to understand the vibronic couplings (VCs) arising from the electronic factor ∆ρ nm (x ) or ρ mn (x ) and the vibrational factor v α (x ). It should be noted that the disappearance of ∆ρ nm (x ) and ρ mn (x ) gives rise to the suppression of the internal conversions via the reduction of V n,α and V mn,α , respectively. Since the transition dipole moment also depends on ρ mn (x ), 31 the disappearance of ρ mn (x ) suppresses both the radiative and nonradiative transitions.  were fixed during the geometry optimizations and vibrational analyses for the ground and excited states.

Methods of calculations
The CNPPE in CH 2 Cl 2 solution was modeled as a single molecule with C 1 symmetry. The ground and excited states of a single molecule in solution, Monomer Model, were computed at the M06-2X/6-31G(d,p) and TD-M06-2X/6-31G(d,p) levels of theory, respectively, including the solvent effect through the polarizable continuum model (PCM). 35 The above calculations were carried out using the Gaussian 09. 36,37 The VCCs and VCD were calculated using our own code.

Results and discussion
Vibronic coupling constants (VCCs) and vibronic coupling density There are a few possibilities for selecting dimers from the crystal structure. The total energies of excited states of the three types of dimer models were compared (Fig. S2). The S 1 and S 2 states of Dimer Model 1 are energetically more stable than those of the other dimer models, and the fluorescence from Dimer Model 1 is expected. Therefore, we concentrate ourselves on Dimer Model 1. Table 1 lists the excited states of Dimer Model 1 at the S 0 and S 2 optimized structures. From the selection rule of the electric dipole transition, S 1 (A g ) is symmetry-forbidden and S 2 (A u ) is symmetry-allowed (Laporté rule). Although, according to Kasha's rule, an emission does not occur from the second excited states, the fluorescence from S 2 is possible if all the transitions from S 2 to S 1 are suppressed.  : Diagonal VCCs of (a) S 0 @S 2 and (b) S 1 @S 2 as well as off-diagonal VCCs of (c) S 0 @S 2 ← S 2 @S 2 and (d) S 1 @S 2 ← S 2 @S 2 for Dimer Model 1.
between S 2 and S 0 is enabled with the large oscillator strength (see Table 1). It should be noted, because of the extremely small diagonal VCCs of S 1 @S 2 , that the internal conversion from S 2 to S 1 is suppressed, thereby enabling the fluorescence from S 2 . expressed as where φ HO/LU (X 1 /X 2 ) denotes the HOMO/LUMO of molecule X 1 /X 2 comprising the model.  (Table 1). Since the NHOMO/HOMO and LUMO/NLUMO are pseudo-degenerate, S 1 and S 2 are pseudo-degenerate. Figure 5 (a) and (b) show the electron density differences of S 2 @S 2 -S 0 @S 2 , ∆ρ 20 , and S 1 @S 2 -S 0 @S 2 , ∆ρ 10 , respectively. ∆ρ 20 and ∆ρ 10 exhibit similar distributions. On the contrary, the electron density difference of S 2 @S 2 -S 1 @S 2 , ∆ρ 21 VCCs of S 0 @S 2 ← S 2 @S 2 could be much larger than those of S 1 @S 2 ← S 2 @S 2 . However, the values of these off-diagonal VCCs are comparable (Figs. 3 (c) and (d)) because ρ 20 is symmetrically localized on the atoms while ρ 21 is localized on the bonds. In general, an overlap density that is symmetrically localized on atoms weakly couples to a potential derivative, resulting in small off-diagonal VCCs. 31 Therefore, the off-diagonal VCCs of S 0 @S 2 ← S 2 @S 2 are not large.
Consequently, the small electron density difference and overlap density between the pseudo-degenerate S 1 and S 2 excited states leads to the extremely small diagonal VCCs of S 1 @S 2 and the small off-diagonal VCCs of S 1 @S 2 ←S 2 @S 2 . We discuss the mechanism of the vanished electron density difference and overlap density in the pseudo-degenerate where all vibrational modes are vibronic active modes that provide the non-zero diagonal and off-diagonal VCCs. On the other hand, the reducible representation for the dimer belonging to C i symmetry is decomposed as where A g and A u are the totally and non-totally symmetric modes, respectively. The vibronic active mode for the diagonal VC is A g and that for the off-diagonal VC is A u . Therefore, the number of the vibronic active modes is almost the same in the monomer and dimer although the total number of vibrational modes is higher in the dimer. The numbers of irreducible representations of vibrational modes in excimers with C i , C 2 , and C s symmetry are summarized in Section S2.  (Table S4). The driving force of the excimer formation is the diagonal VCCs.
which is the steepest descent direction of the vibrational relaxation. 38 The effective mode is the intramolecular vibration rather than the intermolecular one. This indicates that the  4.98 × 10 −4 a.u. (Fig. 6 (c)), which is comparable to that of mode 113 in Dimer Model 1, 5.34 × 10 −4 a.u., (Fig. 3 (c)). Figure S5 shows Model. This is because vibrational mode 113 is the intramolecular vibration that is not easily affected by the surrounding molecules (Fig. S5). Therefore, the off-diagonal VCCs corresponding to these modes are not reduced by the packing effect. The packing effect in solid phase is visualized by the potential derivative.
The small diagonal VCCs contribute to the suppression not only of the internal conversion but also of the vibrational relaxation. 31 The reorganization energies due to vibrational relaxation were calculated to be ∆E = 0.582 eV for Monomer Model and ∆E = 0.275 eV for Dimer Model 1. For Monomer Model, since the potential energy surfaces of low-frequency torsional modes were not approximated to harmonic potentials, the reorganization energies for these vibrational modes were evaluated from the potential energy surfaces 31 instead of using Eq. (7) (Fig. S6). The reorganization energy of Dimer Model 1 is smaller than that of Monomer Model, and suppressed vibrational relaxation is expected in Dimer Model 1.
Furthermore, we calculated the electronic states of Decamer Model for the CNPPE solid (Fig. S7). Decamer Model has a large oscillator strength in the FC S 8 state (Table S7), and the S 8 exciton is expected to be generated by absorption. The electron density difference between the FC S 8 and S 0 states is delocalized as in the case for the combination of Dimer Model 1 and 2 (Fig. S8). The electronic states of Dimer Model 2 are localized on a single molecule after vibrational relaxation. Therefore, Decamer Model can be reduced to Dimer Model 1 in the adiabatic excited states. Thus, the modeling of the CNPPE solid using Dimer Model 1 is considered to be reasonable.
We also performed the calculations of Dimer Model 1 using the long-range corrected ωB97X-D functional ? in order to check the robustness of the results. S 2 (A u ) consisting of the HOMO-1-LUMO and HOMO-LUMO+1 excited configurations is symmetry-allowed from the Laporté rule (Table S8). In addition, the frontier orbitals in the AD S 2 state are delocalized over the molecules (Fig. S9), and the excited states do not have a charge transfer character. Therefore, the same conclusions as using the M06-2X functional can be deduced.

Hubbard model in a pseudo-degenerate electronic system
In this section, we discuss the disappearance mechanism of the electron density difference and overlap density based on the Hubbard model. We consider an excited electronic structure of a system consisting of the same molecules X 1 and X 2 (Fig. 10 (a)). Each molecule is not necessarily symmetric, but the dimer is assumed to have a symmetry such as C i , C 2 , or C s .
For simplicity, only the HOMOs and LUMOs of X 1 and X 2 are considered. The energy gap between the HOMO and LUMO is expressed as ǫ. The hopping integrals for the HOMO-HOMO, LUMO-LUMO, and HOMO-LUMO between X 1 and X 2 are denoted by t 1 , t 2 , and t 3 , respectively. The Coulomb interactions for the HOMO-HOMO and HOMO-LUMO within X 1 or X 2 are denoted by U 1 and U 2 , respectively. Considering only a single excitation as done by the TD-DFT calculations, there are 9 electronic configurations ( Fig. 10 (b)) where Figure 10: (a) Hubbard model of a system consisting of X 1 and X 2 . ǫ is the energy gap between the HOMO and LUMO. t 1 , t 2 , and t 3 are the hopping integrals for the HOMO-HOMO, LUMO-LUMO, and HOMO-LUMO, respectively, between X 1 and X 2 . U 1 and U 2 are the Coulomb interactions for the HOMO-HOMO and HOMO-LUMO, respectively, within X 1 or X 2 . (b) Electronic ground and excited configurations assuming a single excitation.

21
The model Hamiltonian for the basis defined in Fig. 10 (b) is given by Employing the Rayleigh-Schrödinger perturbation theory considering t 1 , t 2 , and t 3 as perturbations, the electronic states of the excited states are expressed as where |Ψ s 1 and |Ψ s 2 are symmetric and |Ψ a 1 and |Ψ a 2 are antisymmetric electronic states. |Ψ s 1 and |Ψ s 2 belong to different symmetries from |Ψ a 1 and |Ψ a 2 . The electronic states of the ground state is expressed as which depends on t 3 unlike the electronic states of the excited states.

22
Employing the Rayleigh-Schrödinger perturbation theory, the energies of the ground and excited states are given by The energy difference between E s 1 and E a 1 as well as E s 2 and E a 2 is |4t 1 t 2 /U 2 |. Since 4t 1 t 2 /U 2 is assumed to be small, |Ψ s 1 and |Ψ a 1 as well as |Ψ s 2 and |Ψ a 2 are pseudo-degenerate. The orders of the excited states are E s for t 1 t 2 < 0: i.e. the symmetry of excited states changes according to the sign of the product t 1 t 2 determined by the alignment of molecules. The excited state degenerates for t 1 t 2 = 0.
When t 1 , t 2 , and t 3 are no longer small compared to U 1 and U 2 , namely the perturbation theory is not valid, the degeneracy occurs in the case of t 1 = 0 or t 2 = 0 (see Fig. S10).
When the system has C i symmetry and t 1 t 2 > 0, S 1 is symmetry-forbidden A g and S 2 is symmetry-allowed A u . This is the case for Dimer Model 1 of the CNPPE solid. The electron density differences between S 1 (|Ψ s 1 ) and S 0 (|Ψ 0 ) as well as S 2 (|Ψ a 1 ) and S 0 are given by Thus, both ∆ρ 20 and ∆ρ 10 are expressed as the difference between the LUMOs and HOMOs 23 of X 1 and X 2 . In contrast, the electron density difference between S 2 and S 1 is expressed as Therefore, ∆ρ 21 is cancelled because ∆ρ 20 and ∆ρ 10 are the same due to the pseudodegeneracy of S 1 and S 2 . The overlap density between S 2 and S 0 as well as between S 2 and S 1 are given by ρ 20 is expressed as the non-vanishing product of HOMOs and LUMOs. In contrast, ρ 21 is cancelled because it is expressed as the difference between the square of the HOMOs and LUMOs. The other pairs of electron density differences and overlap densities in the pseudodegenerate electronic system are summarized in Table S9 and Table S10, respectively. HOMOs and LUMOs is negative, the first excited state is fluorescent (J-type aggregation).
In contrast, the product of the hopping integrals is positive, the first excited state is dark and the second one is fluorescent (H-type aggregation). Whether the first or second excited states are fluorescent, the AIEE could occur as long as the excimer forms in the aggregation phases because the excimer formation gives rise to the decrease of the diagonal VCCs.
In this study, we discussed a dimer with C i site symmetry. Other cyano-substituted compounds exhibiting AIEE, such as cyano-substituted bis(4-bromophenyl)-fumaronitrile, bis(3trifluoromethylphenyl)fumaronitrile, bis(4-methoxyphenyl)-fumaronitrile, 39 and cyano-substituted oligo(para-phenylene vinylene) (CN-DPDSB) 40    does not exist at the inversion center and where an atom exists at the inversion center. When an atom does not exist at the inversion center, the reducible representation of vibrational modes is decomposed as where N denotes the number of atoms. Half of the vibrational modes belongs to A g and the other half to A u .
On the other hand, when an atom exists at the inversion center, the number of vibrational modes is The numbers of vibrational modes belonging to A g and A u are different.
The numbers of vibrational modes belonging to A and B are not the same. When N a atoms exist on a rotational axis, the number of vibrational modes is The number of vibrational modes belonging to B increases with N a .
Half of the vibrational modes belongs to A ′ and the other half to A ′′ . When N p atoms exist on a mirror plane, the number of vibrational modes is The number of vibrational modes belonging to A ′ increases with N p .