Emission Rate, Vibronic Entanglement, and Coherence in Aggregates

Article pubs.acs.org/JPCA

Emission Rate, Vibronic Entanglement, and Coherence in Aggregates of Conjugated Polymers Kinshuk Banerjee and Gautam Gangopadhyay* S. N. Bose National Centre for Basic Sciences, Block-JD, Sector-III, Salt Lake, Kolkata 700098, India ABSTRACT: Here we have studied a dimer model of conjugated polymer aggregates based on the traditional J and H structures, with the extension in treating the electronic and vibrational degrees of freedom at par. We have considered various exchange symmetries corresponding to the parameters of the excited state Hamiltonian in assigning the symmetry of the vibronic states of the aggregate, going beyond the homodimer case. The emission rates are determined as a function of system parameters at low temperature for both types of aggregates. We have also determined the vibronic entanglement as a measure of the coupled electronic and vibrational motion as well as the exciton coherence number in the emitting state. As a function of interchain interaction strength, emission rate and entanglement grossly follow similar trends for the J-aggregate and opposite trends for the H-aggregate in totally symmetric as well as asymmetric cases. Variation of other system parameters, like electronic excitation energy and electron−vibration coupling parameter are also thoroughly investigated in governing these quantities. The role of symmetry of the wave function in governing the spectra and the exciton coherence are also analyzed thoroughly, which offers a way to realize the connection between such macroscopic and microscopic quantum features.

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Recently, a nonadiabatic interchain exciton transfer model34 is introduced in a dimer of identical conjugated polymer chains that explains the dual emission in terms of a strong nonCondon sideband spectrum from the lower dipole forbidden interchain state.35 Along with the steady state spectra, experimental studies on ultrafast exciton dynamics in these systems also indicate significant nonadiabatic effect.36,37 For example, photoluminescence anisotropy decay on femtosecond time scales in a polydiphenylenevinylene derivative38 established the presence of interchain interaction as well as strong electronic coupling to nuclear vibrations in conjugated polymers. This feature is thought to be highly important in preserving the quantum phase coherence during intrachain energy transfer.39,40 Now such coupling between electronic and vibrational motion can be characterized in terms of entanglement between these degrees of freedom using theoretical measures of quantum information theory.41,42 Some recent theoretical studies report on such vibronic entanglement in molecular systems.43−45 The correlation between the luminescence and exciton coherence size is also established in a recent study on J-aggregates46 by Spano and Yamagata47 in terms of the generalized coherence function.48,49 With this background, here we have analyzed in detail the dimer model investigating the J- and H-aggregates, the basic morphologies in elucidating the structure−function relationships in molecular assemblies. The theory is constructed in

INTRODUCTION Organic π-conjugated polymers have emerged as revolutionary molecular materials generating immense interest from both experimental and theoretical viewpoints.1−5 They are organic semiconductors with high luminescence efficiency combined with processing flexibility of a polymer that ensures their special place in organic optoelectronics.6−12 The light emitting species in these polymer chains are generally believed to be the exciton species, which can be intrachain13,14 as well as interchain3 in nature. Formation of aggregates and luminescence from interchain exciton species are ubiquitous in conjugated polymer systems, as demonstrated by experimental as well as theoretical quantum-chemical studies.3,15−17 Interchain states having lower energy compared to the unaggregated case can be seen in the bathochromic shift in the spectra with reduced or quenched luminescence.18−20 Dual luminescence generated due to the presence of both intra- and interchain excitons are reported in poly(p-phenylenevinylene)−Si nanocomposites21 and for highly regioregular poly(3-hexylthiophene).22 The effect of structural disorder in the arrangement of monomer chain-units on the spectra are thoroughly analyzed with lamellar and herringbone morphologies in various oligophenylenevinylene and oligothiophene polymer aggregates.23−27 These studies ensure a special role of aggregate formation in governing the photophysics of conjugated polymers. The well-characterized field of molecular aggregates28−31 is revived in the context of these conjugated polymer aggregates, particularly focusing on the role of electron−vibration nonadiabatic coupling.32 The well-resolved vibronic progressions in the absorption and luminescence spectra of various conjugated polymers are also indicators of substantial electron−phonon coupling.33 © 2013 American Chemical Society

Special Issue: Structure and Dynamics: ESDMC, IACS-2013 Received: March 26, 2013 Revised: June 26, 2013 Published: June 28, 2013 8642

dx.doi.org/10.1021/jp4029535 | J. Phys. Chem. A 2013, 117, 8642−8650

The Journal of Physical Chemistry A

Article

the Hamiltonian,34 the off-diagonal electron−phonon coupling terms can be eliminated. The transformation is given as

terms of an aggregate excited state Hamiltonian taking into account both electronic and vibrational motion. The role of the various parameters of the dimer Hamiltonian in governing the nature of symmetry of the interchain dressed excitonic states and consequently the emission spectra are explored. We have also calculated the variations of vibronic entanglement and exciton coherence in the emitting state at low temperature as a function of these parameters. As these quantities are hard to realize experimentally, it is worthwhile to investigate any possible connection between these properties and the macroscopic spectral data.

H̃ = eSH e−S where 2

S=

H̃ =

∑ i,j=1 i≠j

ℏVij(|i⟩⟨j|)

H vib

(3)

Hel − vib

⎜

k=1

2

2

∑

ℏVij|i⟩⟨j| ∏ Dk (αijk)

i,j=1 i≠j

k=1

2

|ψ a⟩ =

⎛ ℏgik 2 ⎞ † ⎜ ⎟ = ∑ |i⟩⟨i| ∑ ⎜ℏgik (ak + ak ) + ωk ⎟⎠ i=1 k=1 ⎝

2

∑ ℏωk⎛⎝ak†ak +

1 ⎞⎟⎤⎥ 2 ⎠⎥⎦

(6)

with i , j , k = 1, 2

(7)

The argument αijk is defined as αijk = (gik − gjk)/ωk. The ath eigenstate of the composite dimer Hamiltonian is given as

and M

(5)

Dk (αijk) = exp(αijk(ak† − ak ))

(2)

M

⎛ 1⎞ = ∑ ℏωk ⎜ak†ak + ⎟ ⎝ 2⎠ k=1

(ak† − ak )

Here σz = |2⟩⟨2| − |1⟩⟨1|. The detuning between the electronic transition energies is given by Δ = ε2 − ε1, which gives the free energy change due to the exciton transfer from one monomer unit to the other. Dk(αijk) is the Glauber displacement operator50 defined as

M

i=1

ωk

⎡Δ ε1 + ε2 + ⎢ σz + 2 ⎣⎢ 2 +

Here Hel defines the electronic part of the Hamiltonian, Hvib denotes the vibrational degrees of freedom, and Hel−vib defines the coupling between electronic and vibrational motion. They are given as

∑ |i⟩⟨i|εi +

gik

and the transformed Hamiltonian, H̃ becomes

AGGREGATES OF POLYMER CHAINS: VIBRONIC DIMER The aggregate excited state Hamiltonian is constructed as34 H = Hel + H vib + Hel − vib (1)

M

|i⟩⟨i|

i,k=1

■

Hel =

∑

∞

∑ ∑ i = 1 n1, n2 = 0

M

Cia, n1, n2|i , n1 , n2⟩ (8)

The basis |i, n1, n2⟩ is taken as the one excitonic state |i⟩, dressed with phonons with “n1” quanta in the vibrational mode of chain-1 and “n2” quanta in the vibrational mode of chain-2. We again point out that these are the excitonic states of the dimer. The corresponding ground state is taken as |G⟩ ≡ |g, vac⟩ where |g⟩ ≡ Π2j=1|gj⟩ and |vac⟩ ≡ Π2j=1|nj = 0⟩, i.e., all the monomer chain units in the ground vibrational state of their ground electronic state. The curvatures of the ground and excited state nuclear potentials of the chains are taken to be the same. All the terms having the dimension of energy are scaled with respect to the vibrational frequency, ℏω.

(4)

Here M is the number of monomer units, i.e., the number of polymer chains forming the aggregate. The state |i⟩ in eq 2 denotes that the ith monomeric unit, i.e., the ith chain, is electronically excited, which involves vibrational excitation as well. The remaining units are considered to be in their respective ground electronic states but can be vibrationally excited.23 Hence the Hamiltonian is written in the site representation with |i⟩ ≡ |1g, 2g, ..., ie, ..., Mg⟩. In eq 2, εi is the vertical electronic excitation energy of the ith unit relative to its ground state in the unaggregated system. For conjugated polymers, this is generally taken to be the S0 → S1 transition energy. Vij is the interchain interaction that transfers the excitation from one chain to the other and it is generally taken as the nearest-neighbor coupling with j = i ± 1. In eq 3, a†k and ak are the creation and annihilation operators, respectively, for the kth vibrational mode with frequency ωk. Here we have considered a single vibrational mode on each chain which is coupled to the corresponding electronic transition. For conjugated polymers, this mode is generally taken as the C C vinylic stretching. Equation 4 represents the electron− vibration coupling with gik being the electron−vibration coupling parameter taken linear in the vibrational coordinate with the excited state nuclear potential being modeled as a displaced harmonic oscillator.50 The total Hamiltonian in eq 1 is thus of the Holstein Hamiltonian form.51 The term ℏgik2/ωk in eq 4 represents the reorganization energy which is also a measure of the Stokes shift. Here we consider the case of the simplest aggregate, i.e., the dimer (M = 2). Applying a standard polaron transformation on

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SYMMETRY OF THE WAVE FUNCTION Based on the nature of the monomer chain units and their orientation, the vibronic states of the aggregate can have different kinds of symmetry.35,52 In this respect, two basic dimer aggregate structures are well-known in literature: H- and J-aggregates.53,54 The interaction between the identical monomer units is modeled as the interaction between their transition dipoles in the point dipole approximation.55 In Haggregates, the two monomer units are placed cofacially with their transition dipoles parallel, which results in the positive sign of the interaction energy. This makes the lowest excitonic state antisymmetric, and transition between this state and the totally symmetric ground state is forbidden. In J-aggregates, the two monomer units are placed in-line with their transition dipoles in the same direction, resulting in a negative sign of the interaction energy. Then the lowest excitonic state becomes symmetric with allowed transition to or from the ground state. These two basic geometries of the dimer have been successfully applied to interpret the spectra of various molecular aggregates 8643



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in terms of simple transition dipole vector addition diagrams.53 Now the conjugated polymers have vibronic transitions with closely spaced energy levels that require theories beyond the simple picture of electronic degrees of freedom or vibrations taken at most at the Born−Oppenheimer level. Still in many cases, the conjugated polymer aggregate spectra can be broadly characterized in terms of J- or H-like behavior. For example, the dimer of polydiacetylene chains exhibit J-aggregate-like spectra whereas π-stacks of regioregular poly(3-hexylthiophene) act mostly as H-aggregates. It is worth mentioning that subtle variations and preparation dependence of the luminescence in these systems are recently explained by a complex model combining J- and H-aggregate properties based on the relative strengths of intrachain and interchain interactions.56 However, here we do not consider such detailed description of polymer modeling and instead focus on the simple dimer problem. In this section, we will investigate the symmetry of the vibronic states given in eq 8 in J- and H-aggregate geometries for various combinations of system parameters. We point out that as J- and H-aggregates are generally considered to be composed of identical monomer units, the situations studied here are more general in treating asymmetric cases. The characterization of the aggregate as J or H is thus based on the sign of the interchain interaction, V12 = V. We have taken the vibrational frequencies of the two modes to be equal, i.e., ω1 = ω2 = ω; all other parameters are scaled with respect to ω. The symmetry will be explored with respect to variations in the purely electronic parameter, Δ, and the electron−vibration coupling parameter, αijk. Here we further define α121 = (g11 − g21)/ω ≡ α1 and α122 = (g12 − g22)/ω ≡ α2 with α211 = −α1 and α212 = −α2. We set g12 = g21 = 0 for calculating the emission spectra and other properties of the emitting state. The role of symmetry of the vibronic states in governing the spectra, vibronic entanglement and exciton coherence in the emitting state are studied as a function of these parameters as well as the interchain interaction, V in the next section. Now we study different cases of the vibronic state symmetry. Identical Electronic Excitation Energy Parameter. We take the electronic excitation energies of the two chains to be identical, i.e., Δ = 0 but consider g11 ≠ g22. An exchange operator, pe12, in the electronic subspace of the system can be utilized that exchanges the electronic indices.35 Although the effective electron−vibration coupling parameters are different, because the modes are coupled to each excitonic state (see eq 4), we can also apply the parity operator, Π, acting as a permutation operator in the vibrational subspace.55 It is defined as Π = exp[∑2k=1iπa†k ak]. Now applying these two operators successively on the dimer excitonic state(eq 8), we get e e p12 Π|ψ a⟩ = p12

+ =

C1,a n1, n2 e iπ(n1+ n2)|1, n1 , n1, n2 C2,a n1, n2 e iπ(n1+ n2)|2, n1 , n2⟩

∑

P12|ψ a⟩ = ± ∑ C1,a n1, n2|1, n1 , n2⟩ + C2,a n1, n2|2, n1 , n2⟩ n1, n2

(10)

Comparing eqs 9 and 10, we get the relation C1,a n1, n2 = ±e iπ(n1+ n2)C2,a n1, n2

(11)

The “+” sign corresponds to symmetric state and “−” sign to antisymmetric state. Identical Electron−Vibration Coupling Parameter. Here we take g11 = g22 but Δ ≠ 0. Then exchange in the electronic indices is not permitted, but exchange is valid in vibrational indices. Thus Π|ψ a⟩ =

∑ C1,a n ,n |1, n2 , n1⟩ + C2,a n ,n |2, n2 , n1⟩ 1

2

1

2

n1, n2

=

∑ C1,a n ,n eiπ(n + n )|1, n1, n2⟩ 1

1

2

2

n1, n2

+ C2,a n1, n2 e iπ(n1+ n2)|2, n1 , n2⟩

(12)

Comparing the two lines of eq 12, we obtain the relation Cia, n1, n2 = e−iπ(n1+ n2)Cia, n2 , n1i = 1, 2

(13)

Totally Symmetric Case. In the case of no asymmetry in either the purely electronic or the vibronic parameters, one has Δ = 0 and g11 = g22. Then both the above symmetry relations, eqs 11 and 13, are simultaneously valid. This leads to the symmetry relation C1,a n1, n2 = ±C2,a n2 , n1

(14)

This relation can also be obtained directly by applying the overall two-particle exchange operator, P12, on the excitonic states with the permutations in the electronic and vibrational indices being permitted on an equivalent basis. The above three relations thus represent the interplay of the parameters of the dimer Hamiltonian in governing the symmetry of the excitonic states at different levels of electronic and vibrational degrees of freedom. We have schematically shown the connections between the wave function coefficients based on the exchange symmetry in Figure 1 for different cases considered here.

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AGGREGATE EMISSION SPECTRA: ENTANGLEMENT AND COHERENCE LENGTH IN THE EMITTING STATE Emission spectra are calculated using the lowest energy eigenstate of the Hamiltonian (see eq 8) denoted by |ψ(em)⟩. This physically corresponds to a low temperature case where

n 2⟩

∑ C1,a n ,n eiπ(n + n )|2, n1, n2⟩ 1

1

2

2

n1, n2

+ C2,a n1, n2 e iπ(n1+ n2)|1, n1 , n2⟩

(9)

As the exchange symmetry is present in both the electronic and the vibrational subspaces, the excitonic states are also eigenstates of a two-particle permutation operator, P12, of the whole system with eigenvalues ±1. It is easy to follow that P12 = pe12Π and then

Figure 1. Schematic showing the relation between the wave function coefficients based on the exchange symmetry for different cases of electronic and electronic−vibrational parameters. 8644



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any kind of thermalized emission is avoided.23 The expression of the emission rate per frequency interval can be simply written as23,50 S(ω) =

1 μ2

eigenvalues will be identical in both cases as is well-known for a two-level system. Therefore, the opposite symmetry of the emitting state for J- and H-aggregates will have no effect on the vibronic entanglement and EVN will be the same in both systems. Another useful concept in determining the extent of exciton coherence and delocalization in the vibronic states of the aggregate is the coherence function.47,48 The coherence function for the emitting state of a cyclic N-mer is defined as47

∑ |⟨ψ (em)|μ(|g ⟩⟨i| + hc) n1, n2 2

×

∏ |gi , ni⟩|2 Γ(ω − ε(em) + nTω0)

(15)

i=1

∏2i=1|gi⟩,

Here |g⟩ ≡ ε is the energy of the state |ψ ⟩, Γ is the Lorentzian line shape function, μ is the magnitude of the transition dipole moment, and nT is the total number of vibrational quanta in the dimer ground electronic state. The sum in eq 15 is performed with the constraint n1 + n2 = nT. The vibrational levels in the ground state are thus of a twodimensional isotropic harmonic oscillator with degeneracy of the nth level being (n + 1). The damping factor, γ, of the line shape function is taken as γ = 0.35ω0. Inserting the explicit form of |ψ(em)⟩ in eq 15, one can easily show that the nature of symmetry of the emitting state results in complete cancellation of the 0−0 emission in the case of the H-aggregate in the totally symmetric case as defined earlier. Here we have focused on the emission rate over the whole frequency range given as (em)

(em)

A(em)(s) =

∑ ⟨ψ (em)|Bi†Bi+ s |ψ (em)⟩, i

⎛N ⎞ s = 0, ±1, ..., ±⎜ − 1⎟ , even N ⎝2 ⎠

(19)

For n + s > N, n + s is replaced by n + s − N, whereas for n + s < 1, n + s is replaced by n + s + N. B†i is the local exciton creation operator defined as47 B†i = |i,vac⟩⟨G| = |i,vac⟩⟨g,vac|. Thus the coherence function takes into account the vibrational degrees of freedom explicitly. Now for N = 2, s = 0, 1. Then we get 2

A(em)(0) =

∑(∑ i=1

i1, l1, l 2

∑

×

Ci(em) ⟨i , l1 , l 2|Bi†Bi | 1 , l1 , l 2 1

Ci(em) |i , n1 , n2⟩) 2 , n1 , n 2 2

i 2 , n1, n2

I=

∫ dω S(ω)

2

(16)

=

In what follows, we call this integrated emission rate simply emission rate. We have numerically calculated the contributions of 0−0 emission and sideband emissions to the total emission rate. They are written as I ≡ Itot = I0 − 0 + Isb

i = 1 l1, l 2 , n1, n2

(20)

and (em) C1,(em) l1, l 2C 2, n1, n2⟨l1 , l 2|0, 0⟩⟨0, 0|n1 , n 2⟩

l1, l 2 , n1, n2

In the numerical calculation, the sideband emission rates are determined by taking up to 0−5 emission peaks in the spectra. Now complete description of the quantum state of a molecular system containing both electronic and vibrational degrees of freedom is given in terms of a density operator ρ̂ in the product space of the vibrational and electronic Hilbert spaces, Hvib ⊗ Hel. The reduced density operators for the vibrational and electronic subsystems are obtained by taking the trace over the Hilbert spaces of the electronic system and the vibrational motion, respectively, as ρ̂vib = Trel[ρ̂] and ρ̂el = Trvib[ρ̂]. To characterize the coupled electronic and vibrational motions in the dimer, the entanglement between the electronic and vibrational degrees of freedom of the composite state of the system can be expressed using the von Neumann entropy of entanglement41,42,57 as k

∑

A(em)(1) =

(17)

E VN = −Trel(ρel̂ log 2 ρel̂ ) = −∑ γk log 2 γk

(em) Ci(em) , l1, l 2Ci , n1, n2⟨l1 , l 2|0, 0⟩⟨0, 0|n1 , n 2⟩

∑ ∑

+

∑

(em) C2,(em) l1, l 2C1, n1, n2⟨l1 , l 2|0, 0⟩⟨0, 0|n1 , n 2⟩

l1, l 2 , n1, n2

(21)

The coherence number, Nc, for the emitting state is defined as47 1

Nc =

∑s = 0 |A(em)(s)| A(em)(0)

(22)

It can be easily understood from the above expressions of coherence function that the coherence number, Nc, for the emitting states will be same for both J- and H-aggregates. Role of Interchain Interaction, V. At first we study the effect of the interchain interaction parameter, V on the emission rate and entanglement in the emitting state for various cases as detailed below. We have taken the magnitude of V to be the same for comparison between J- and H-aggregates with the sign being opposite: V is positive for H and negative for J. We have thus shown the variations of state properties as a function of |V|. In all the numerical studies, the parameters, V, g11, g22, and Δ are scaled with respect to the vibrational energy, ℏω. Totally Symmetric Case: g11 = g22, Δ = 0. We have shown the emission rates for J- and H-aggregates in Figure 2 as a function of interchain interaction, V taking the absolute value. The electron−phonon coupling parameters are g11 = g22 = 1.0. According to the symmetry relation in eq 14, the oscillator strength of the 0−0 emission will be zero for the H-aggregate with positive sign of interaction, V. Thus only the sideband emission rate,58 Isb is shown in Figure 2b, which decreases with

(18)

Here γk are the eigenvalues of ρ̂el. Now the von Neumann entropy42 of the system is defined as −Trel(ρ̂el log ρ̂el). For unentangled states the entropy is zero and so is EVN. On the other hand, the maximally entangled state with state space of dimensionality D has the maximum entropy log D. So for the dimer with two electronic states, the maximum entropy will be log 2 and the maximum entanglement will be EVN = 1. For the dimer aggregate, the matrix elements of the reduced electronic density matrix, ρ̂el for the emitting state are given as (em) (em) (ρ̂el)ij = ∑n1,n2Ci,n C . Now although the off-diagonal 1,n2 j,n1,n2 elements will have opposite signs for J- and H-aggregates, the 8645



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Figure 2. (a) Emission rates and the ratio, I0−0/Isb for the J-aggregate and (b) the sideband emission rate, Isb, for the H-aggregate, as a function of |V| for the totally symmetric case.

Figure 4. (a) Emission rates and (b) the ratio, I0−0/Isb as a function of |V| in the case of the J-aggregate for Δ = 0.1, g11 = g22 = 1.0. Same quantities are plotted for the H-aggregate in (c) and (d), respectively.

|V|. From Figure 2a one can see that for the J-aggregate (with negative sign of V), however, Isb as well as the 0−0 emission rate, I0−0, increase with |V|. This induces the rise of the total emission rate, Itot, as well as that of the ratio I0−0/Isb with |V|. In Figure 3a we have plotted the vibronic entanglement, EVN, in the emitting vibronic state as a function of |V|. As already

Figure 5. (a) Vibronic entanglement, EVN, and (b) exciton coherence number, Nc, in the emitting state as a function of |V| for Δ = 0.1, g11 = g22 = 1.0.

the H-aggregate behavior. The coherence number, Nc, starts from a value less than 2.0, as shown in Figure 5b, indicating the asymmetry, Δ = 0.1. With a rise in the interchain interaction, the effect of this asymmetry gets reduced and Nc rises toward its maximum value, 2.0. The growing symmetry is also reflected in the fall of 0−0 emission rate toward zero in the case of the H-aggregate, as already observed above. Asymmetry in Electronic−Vibrational Coupling Parameter: g11 ≠ g22, Δ = 0. Here we take the electronic−vibrational coupling parameters to be different: g11 = 1.0, g22 = 0.5 with Δ = 0.0. For the J-aggregate, Itot, I0−0, and Isb increase with |V| (Figure 6a); the ratio I0−0/Isb also rises steadily (Figure 6b) as

Figure 3. (a) Vibronic entanglement, EVN, and (b) exciton coherence number, Nc, in the emitting state as a function of |V| for the totally symmetric case, g11 = g22, Δ = 0.

mentioned, it does not depend on the symmetry of the wave function and hence is same for both J- and H-aggregates. The vibronic entanglement, EVN, increases steadily with |V|. This feature tallies with the variation of the various emission rates for the J-aggregate shown in Figure 2a but is opposite to that of the H-aggregate. In Figure 3b the exciton coherence number, Nc, in the emitting vibronic state is plotted as a function of |V|. The totally symmetric wave function results in a fully delocalized excitation along the length of the dimer with Nc = 2.0. As this symmetry does not depend on the magnitude of interchain coupling, Nc is independent of |V|. Asymmetry in Electronic Parameter: Δ ≠ 0, g11 = g22. In this case, the electronic excitation energies of the chains are taken different as Δ = 0.1 but g11 = g22 = 1.0. The emission rates for J- and H-aggregates are plotted in Figure 4 as a function of interchain interaction, |V|. For the J-aggregate, Itot, I0−0, and Isb as well as the ratio I0−0/Isb rise monotonically with |V| as shown in Figure 4a,b, respectively. For the H-aggregate, the case is again the opposite with all the quantities decreasing steadily with |V|; I0−0 actually becomes vanishingly small as shown in Figure 4c. This is due to the fact that rise in |V| reduces the asymmetry created by the nonzero Δ. In Figure 5a vibronic entanglement, EVN, in the emitting state is plotted with |V|. EVN increases with an increase in |V|; this feature again tallies with the variation of emission features of the J-aggregate in this case whereas it is opposite compared to

Figure 6. (a) Emission rates and (b) the ratio, I0−0/Isb as a function of |V| in the case of the J-aggregate for g11 = 1.0, g22 = 0.5, Δ = 0.0. The same quantities are plotted for the H-aggregate in (c) and (d), respectively. 8646



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the rate of increase of I0−0 is greater than that of Isb. For the Haggregate, again one finds all the above-mentioned emission rates to fall with |V| (Figure 6c). However, here the rate of fall of Isb becomes higher compared to that of I0−0 and this results in the rise of the ratio, I0−0/Isb, for the H-aggregate also, shown in Figure 6d. Now comparing the variation of EVN with |V| shown in Figure 7a with the emission features in Figure 6, one can say that in

Figure 8. (a) Emission rates and (b) the ratio, I0−0/Isb, as a function of Δ in the case of the J-aggregate for g11 = g22 = 1.0 and |V| = 0.3. The same quantities are plotted for the H-aggregate in (c) and (d), respectively.

all other emission rates and the ratio, I0−0/Isb decrease with rise in Δ. The increase of Isb with Δ is small. However, for the Haggregate, I0−0, Itot, Isb, and the ratio I0−0/Isb all increase with Δ; Isb again shows a small rise compared to the other quantities (Figure 8c,d). We have plotted EVN and Nc in the emitting state as a function of Δ in Figure 9. Both quantities show a decline with

Figure 7. (a) Vibronic entanglement, EVN, and (b) exciton coherence number, Nc, in the emitting state as a function of |V| for g11 = 1.0, g22 = 0.5, Δ = 0.0.

this situation, the variation of the ratio I0−0/Isb with |V| is similar to the variation of vibronic entanglement for both aggregates. The variation of Nc plotted in Figure 7b is interesting as it decreases slightly with rising |V|. This indicates that in the case of asymmetry in the electron−vibration coupling parameter (g11 ≠ g22), increase in interchain interaction actually enhances the asymmetry in the range studied here. This behavior is in contrast with the previous case of asymmetry in electronic transition energies, Δ ≠ 0 where the interchain interaction acts to restore the symmetry. This enhancement in asymmetry, characterized by a slight increase in exciton localization, can be the cause that slows down the rate of fall of the 0−0 emission in the H-aggregate with |V| and in turn, gives the enhancement, although small, in I0−0/Isb. To summarize the results so far, we have found that the effect of increasing interchain coupling, |V|, on the emission spectra is grossly the same for both the totally symmetric case and the various asymmetric cases. The emission rates for the J-aggregate increase with |V| whereas the emission rates for the H-aggregate fall with |V|. This is obviously due to the opposite symmetry of the emitting state in the two cases. Now the vibronic entanglement, EVN, also increases with |V| for both systems, as expected. The variation of the coherence number, Nc, with | V| is a more subtle issue. For asymmetry in the electronic parameter, Δ ≠ 0, Nc follows EVN as with increasing |V|, the system becomes more symmetric and exciton delocalization becomes easier. However, for asymmetry in the vibronic coupling parameters, g11 ≠ g22, Nc falls with |V|, showing the opposite trend of EVN. This small enhancement in asymmetry and hence exciton localization makes the ratio I0−0/Isb rise a little with increasing |V| for the H-aggregate, although all the emission rates individually decline. Role of Electronic Excitation Energy Parameter. We now investigate the role of the difference between the electronic excitation energies, Δ, in governing the state properties for fixed value of electron−vibration coupling strength and interchain interaction. These parameters are set at g11 = g22 = 1.0 and |V| = 0.3. From Figure 8a,b, it is evident that for the J-aggregate, except the sideband emission rate, Isb,

Figure 9. (a) Vibronic entanglement, EVN, and (b) exciton coherence number, Nc, in the emitting state as a function of Δ for g11 = g22 = 1.0 and |V| = 0.3.

rising Δ, characterizing the growing asymmetry and the consequent localization of excitation. This growing asymmetry affects the H- and the J-aggregate in opposite manner, as expected. However, the variation of the sideband emission rate, Isb, as a function of Δ shows similar trends for both systems. Role of Electron−Vibration Coupling Parameter. Finally, we study the effect of electronic−vibrational coupling parameters on the state properties. We set the parameters at g11 = 1.0, Δ = 0.0, and |V| = 0.3 and vary g22. In the case of the Jaggregate, I0−0, Itot, and I0−0/Isb decrease with increasing g22 but Isb rises (Figure 10a,b). Also I0−0 and Isb cross each other at g22 ∼ 1.1 and, at higher values of g22, Isb actually reaches a maximum. Now for the H-aggregate, I0−0 decreases with g22, becomes zero at g11 = g22 = 1.0 for symmetry reasons, and then increases again due to the asymmetry (Figure 10c). This feature is obviously reflected in the variation of the ratio I0−0/Isb with g22 shown in Figure 10d. Isb and Itot rise slowly at first and then steadily rise at a faster rate with g22, plotted in Figure 10c. However, as g22 goes on increasing (>1.3), Isb attains a maximum. The variations of EVN and Nc as a function of g22 are shown in Figure 11 for the emitting state. Both quantities pass through a 8647



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0, the exciton gets fully delocalized as there is no extra energy cost. For nonzero Δ, the exciton resides in the lower energy unit of the dimer with higher probability, indicating localization that can be partial or total, depending on the magnitude of Δ. However, in this case the strength of interchain coupling, |V|, also governs the extent of exciton delocalization. Increasing |V| opposes the effect of nonzero Δ and increases exciton delocalization. The dependence of exciton localization on vibronic coupling is weaker and |V| plays a minor role in this case.

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SUMMARY AND CONCLUSION In this paper, we have characterized the symmetry of the vibronic states of a dimer of conjugated polymer chains treating the electronic and vibrational motions on equal quantum mechanical footing. The symmetry of the wave function governs the emission properties of the system as exemplified in terms of the two basic morphologies: J- and H-aggregates. However, to investigate the dependence of luminescence on various parameters, we have generally considered different asymmetric cases going beyond the homodimer. Analyses of the total emission rate and its 0−0 and sideband constituents as a function of interchain interaction, electronic excitation energy difference, and electron−vibration coupling parameter establish features of various complexity. The symmetry of the eigenstates dictates the vibronic entanglement and the coherence number to be the same in both J- and H-aggregate emitting states. For the J-aggregate, the emission rates follow a trend similar to that of the vibronic entanglement as a function of interchain coupling. In the case of the H-aggregate, the corresponding trends are opposite to each other. However, variation of the ratio of 0−0 and sideband emission rates shows more intricate behavior. Whether it will follow the same trend as that of entanglement depends on the source of asymmetry and can have qualitatively similar behavior for both types of aggregate. This is also linked to the nature of variation of the exciton coherence in various asymmetric scenarios and whether it follows the same trend as of the vibronic entanglement as a function of |V|. We have shown that both kinds of behavior are possible depending on whether the asymmetry is generated from a purely electronic parameter or from vibronic parameters. We have also analyzed the role of these electronic and vibronic coupling parameters for a fixed value of |V|, mainly focusing on the role of symmetry in governing the state properties. We have found that the sideband emission rates for both types of aggregates follow similar kinds of behavior as a function of these parameters although other properties are quite distinct, even opposite. Particularly, for different vibronic coupling, this results in a qualitative resemblance between the variation of sideband emission rate and the vibronic entanglement. Thus along with the thorough characterization of macroscopic emission properties in these systems in terms of wave function symmetry, understanding the quantum features of the emitting state gives new insight into the structure−function relationship in these systems. Connection of effects like entanglement and coherence with experimentally measurable properties like spectra is of huge importance in realizing their possible functional roles. This study can thus be extended in a straightforward manner considering larger systems with aggregate morphologies more complex than the J- and Hgeometries.

Figure 10. (a) Emission rates and (b) the ratio, I0−0/Isb, as a function of g22 in the case of the J-aggregate for g11 = 1.0, Δ = 0.0, |V| = 0.3. The same quantities are plotted for the H-aggregate in (c) and (d), respectively.

Figure 11. (a) Vibronic entanglement, EVN, and (b) exciton coherence number, Nc, in the emitting state as a function of g22 for g11 = 1.0, Δ = 0.0, |V| = 0.3.

maximum. For Nc the maximum occurs at g22 = 1.0 where full symmetry occurs and Nc = 2.0. EVN becomes maximum at g22 ∼ 0.8, showing the complex interplay between electronic and vibrational parameters besides the role of symmetry. So we see that only the variation of Isb with g22 shows some qualitative resemblance with the above-mentioned feature in having a maximum present for both J- and H-aggregates that occurs in the range of g22 = 1.7 − 1.8. The detailed comparison of various scenarios thus establishes the role of various system parameters in governing the spectral feature as well as the properties of the emitting state like vibronic entanglement and exciton coherence. The variations of emission rates with interchain interaction, |V|, follow opposite trends for J- and H-aggregates based on the nature of symmetry of the emitting state. This is true for the totally symmetric scenario as well as in the presence of asymmetry. The ratio of 0−0 and sideband emission rates, however, shows some nontrivial feature in the case of different electron−vibration coupling parameters, which is connected to the nature of variation of the coherence number. For a particular |V|, with different electronic excitation energies and different vibronic couplings, the asymmetry produced can explain the spectral and other properties. In these cases, we find that the sideband emission rates, Isb, show similar features for both J- and Haggregates. The exciton coherence number, Nc, gives a measure of the extent of exciton delocalization over the aggregate. Studying its variation in various circumstances considered here, we see that the delocalization of the exciton depends strongly on the detuning between the electronic transition energies, Δ. For Δ = 8648



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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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Emission Rate, Vibronic Entanglement, and Coherence in Aggregates

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