Article pubs.acs.org/JPCA
Polarons in π‑Conjugated Polymers: Anderson or Landau? William Barford,*,† Max Marcus,†,‡ and Oliver Robert Tozer†,§ †
Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford, Oxford, OX1 3QZ, United Kingdom ‡ Magdalen College, University of Oxford, Oxford, OX1 4AU, U.K. § University College, University of Oxford, Oxford, OX1 4BH, U.K. S Supporting Information *
ABSTRACT: Using both analytical expressions and the density matrix renormalization group method, we study the fully quantized disordered Holstein model to investigate the localization of charges and excitons by vibrational or torsional modesi.e., the formation of polaronsin conformationally disordered π-conjugated polymers. We identify two distinct mechanisms for polaron formation, namely Anderson localization via disorder (causing the formation of Anderson polarons) and self-localization by selftrapping via normal modes (causing the formation of Landau polarons). We identify the regimes where either description is more valid. The key distinction between Anderson and Landau polarons is that for the latter the particle wave function is a strong function of the normal coordinates, and hence the “vertical” and “relaxed” wave functions are different. This has theoretical and experimental consequences for Landau polarons. Theoretically, it means that the Condon approximation is not valid, and so care needs to be taken when evaluating transition rates. Experimentally, it means that the self-localization of the particle as a consequence of its coupling to the normal coordinates may lead to experimental observables, e.g., ultrafast fluorescence depolarization. We apply these ideas to poly(p-phenylenevinylene). We show that the high frequency C−C bond oscillation only causes Landau polarons for a very narrow parameter regime; generally we expect disorder to dominate and Anderson polarons to be a more applicable description. Similarly, for the low frequency torsional fluctuations we show that Anderson polarons are expected for realistic parameters. localizes the particle and “spontaneously” breaks the translational symmetry. This is a self-localized (or autolocalized) “Landau polaron”.7,8 Notice that self-trapping is a necessary but not sufficient condition for self-localization. As explained below, self-localization occurs in the limit of vanishing oscillator frequency (i.e., the adiabatic or classical limit) and vanishing disorder.9 A second mechanism for particle localization in disordered systems was proposed by Anderson. In this mechanism the particle, acting as a wave in a random potential, undergoes multiple, random scattering that exponentially localizes the particle wave function.10,11 We denote a particle localized by disorder as an “Anderson polaron”. States in the Lifshitz tail of the density-of-states of disordered one-dimensional systems form superlocalized “local ground states”,12 which for excitons define chromophores.13,14 Although not exact, the distinction between Anderson and Landau polarons is not merely a semantic one, as they lead to different experimental observables, as discussed in section IV.
I. INTRODUCTION The one-dimensional nature of π-conjugated polymers means that charges and excitons are highly susceptible to localization. An understanding of the mechanisms that cause this localizationand in particular, the consequences of themare necessary prerequisites to the development of models of charge and energy dynamics in conjugated polymer systems. There are two broad mechanisms of wave function localization. First, a particle (i.e., an electron or hole, or an exciton center-of-mass particle) that couples to a set of harmonic oscillators (e.g., normal modes of vibration, and torsional and polarization modes) becomes “self-trapped”. Selftrapping means that the coupling between the particle and oscillators causes a local displacement of the oscillator that is proportional to the local particle density.1−5 Alternatively, it is said that the particle is dressed by a cloud of oscillators. As there is no barrier to self-trapping in one-dimensional systems,6 there is always an associated relaxation energy (defined below). If the particle and oscillators are treated quantum mechanically, then in a translationally invariant system the self-trapped particle forms a Bloch state and is not localized. However, if the oscillators are treated classically, the nonlinear feedback induced by the particle-oscillator coupling self© XXXX American Chemical Society
Received: September 8, 2015 Revised: January 8, 2016
A
DOI: 10.1021/acs.jpca.5b08764 J. Phys. Chem. A XXXX, XXX, XXX−XXX
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We solve eq 1 using the Density Matrix Renormalization Group (DMRG) method,19−22 treating the particle and harmonic oscillators quantum mechanically. DMRG is an extremely accurate variational, truncation method. For a recent general review, see ref 23. A discussion of the DMRG parameters used for solving the Holstein model is given in refs 9 and 18. In the adiabatic (i.e., Born−Oppenheimer) limit, where the harmonic oscillators are treated classically, the Holstein model is
They also require different theoretical approaches when, for example, calculating transition rates. This is because the Landau polaron wave function is a function of the normal coordinates and thus the “vertical” wave function is different from the “relaxed” wave function, rendering the Condon approximation invalid. This is generally not the case, however, for Anderson polarons, except close to the crossover to the Landau polaron regime. Since π-conjugated polymers typically exhibit conformational and site disorder, and the coupling to vibrational and torsional modes is strong, charges and excitons in conjugated polymers generically form polarons. It is the purpose of this paper to explore the regimes of validity of both descriptions of polarons and to make predictions for the most appropriate descriptions in π-conjugated polymers. To quantify the discussion, in the next section we employ the fully quantized disordered Holstein model and discuss the properties of Landau and Anderson polarons. Our calculations apply to isolated polymer chains; however, we also expect them to be applicable to polymers in the condensed phase. We give analytical expressions for the crossover between the Landau and Anderson polaron regimes. In section III we apply these rules to describe the most appropriate description of polarons in poly(p-phenylenevinylene) (PPV) and confirm them by numerical (Density Matrix Renormalization Group) calculations. We conclude in section IV.
N−1 n=1 N
−
†
|Ψ⟩ =
∑ ℏωn⎛⎝bn̂ bn̂ + ⎜
n=1
†
∑ Ψn|n⟩
(4)
where Ψn is the particle wave function. The equilibrium oscillator displacements satisfy
Q̃ n = A n|Ψn|2
(5)
which implies that self-trapping vanishes for a delocalized particle on an infinite chain. Equation 5 enables an iterative solution of eq 2 for disordered chains. A. Landau Polaron. We first consider a uniform chain (with the disorder σ = 0) when the harmonic oscillators are treated classically. Thus, assuming that {Q̃ n} are classical variables, substituting eq 5 into eq 2 yields a nonlinear Schrödinger equation for Ψn. In the continuum limit eq 2 has the exact solution1,2,4
+ bn̂ )|n⟩⟨n|
1 ⎞⎟ 2⎠
(2)
(3)
n
n=1
N
+
n=1
where Kn is the force constant. (Q̃ n is the classical expectation value of (b̂†n+ bn̂ )/√2.) The general one-particle eigenstate of eq 2 is
n=1
∑ A nℏωn(bn̂
N
∑ ℏωnQ̃ n2
⎛ K ⎞1/2 Q̃ n = ⎜ n ⎟ Q n ⎝ ℏωn ⎠
∑ tn(|n⟩⟨n + 1| + |n + 1⟩⟨n|) N
1 2
where Q̃ n is the dimensionless harmonic oscillator displacement, defined in terms of the dimensionful displacement, Qn, via
N−1
1 − 2
∑ A nℏωnQ̃ n|n⟩⟨n| + n=1
II. POLARONS IN THE DISORDERED HOLSTEIN MODEL The Holstein model is widely used to describe charges or excitons in molecular J-aggregates15 and π-conjugated polymers16−18 that couple to local harmonic oscillators,4,5 e.g., vibrational or torsional modes. A derivation of the Holstein model to describe charge or exciton coupling to vibrational modes in polymers is given in ref 18, while a derivation of it to describe charge or exciton coupling to torsional modes is given in Appendix B of the Supporting Information. On a linear chain of N moieties the fully quantized disordered Holstein model is defined as Ĥ =
∑ tn(|n⟩⟨n + 1| + |n + 1⟩⟨n|)
̂ = HBO
Ψn =
⎛ λ ⎞1/2 ⎜ ⎟ sech(λ(n − n0)) ⎝2⎠
(6)
Sω̃ 2
(7)
where
(1)
where |n⟩ is the ket representing a charge or Frenkel exciton on moiety n, e.g., a phenylene ring or vinylene unit. b†n̂ (b̂n) creates (destroys) an Einstein harmonic oscillator (e.g., a local vibrational or torsional mode) of energy ℏωn on moiety n. tn is the particle (e.g., a charge or exciton) transfer integral between moieties n and (n + 1), which is taken to be a Gaussian random variable with a mean t0 and standard deviation σ. Thus, the first term on the right-hand-side of eq 1 describes the particle transfer between neighboring moieties with off-diagonal disorder. An is the particle-oscillator coupling parameter, and thus the second term describes a coupling between the particle and the linear displacement of the local harmonic oscillator. The final term on the right-hand-side of eq 1 is the oscillator kinetic and elastic energy.
λ=
S = A2 /2
(8)
is the Huang−Rhys parameter for a single moiety, and
ω̃ = ℏω/t0
(9)
is the ratio of the harmonic oscillator energy to the particle transfer integral. Equation 6 describes the well-known Landau polaron self-localized at n0, whose spatial extent is (in units of the repeat distance) Sc = λ−1 = B
2 Sω̃
(10) DOI: 10.1021/acs.jpca.5b08764 J. Phys. Chem. A XXXX, XXX, XXX−XXX
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trapping, it is more appropriate to consider the polaron as Anderson localized (i.e., predominately localized by disorder). In regime II the polaron effective mass is so large that the self-trapped polaron Anderson localization length is smaller than the classical Landau polaron size, i.e., S s < S c. In this regime the polaron is self-localized by self-trapping, and a Landau polaron description is more appropriate. The crossover between regimes I and II is defined by S s = S c, i.e., when
In practice, the position of the spontaneous symmetry breaking, n0, is determined by disorder or defects on the polymer chain (and we remind the reader that eq 6 is only valid in the adiabatic, classical limit, defined by eq 2). The Landau polaron relaxation energy, Er, defined as the difference in energy between a particle with and without coupling to the normal modes, is1,2,4
Er =
ℏωS 6Sc
⎛ σ ⎞(1) (2Sω̃ )3/2 ⎜ ⎟ = ⎝ t0 ⎠ 8 + Sω̃1/2
(11)
B. Anderson Polaron. Setting A = 0 in eq 1 defines the disordered Anderson model for a “free” or “untrapped” particle (i.e., a particle not coupled to a set of harmonic oscillators). Disorder in one-dimensional systems always localizes a free particle. According to single-parameter scaling theory,24 the localization length of a free particle at the band edge subject to Gaussian random disorder is (in units of the repeat distance) ⎛ t0 ⎞2/3 ⎜ ⎟ ⎝σ ⎠
S0 ≃
(18)
Regime I is valid for (σ/t0) ≤ (σ/t0)(1). Thus, for fixed disorder, as ω̃ is decreased (i.e., as the oscillators behave more classically) there is a crossover from Anderson polarons to self-localized Landau polarons, as indicated in Figure 1.
(12)
Now let us consider the properties of the quantized disordered Holstein model (eq 1) with A ≠ 0. Since the particle bandwidth W = 4t0 and the effective mass, m*, of a particle at the bottom of a parabolic band satisfies m* ∼ W−1, we can use the effective mass of the polaron as a proxy for its bandwidth and assume that in general the Anderson localization length of a self-trapped polaron satisfies9
Ss =
⎛ m0 ⎞2/3 ⎜ ⎟ S ⎝ m* ⎠ 0
(13)
where m0 is the free particle mass, i.e., S s(m* = m0) = S 0. Evidently, S s vanishes as m* diverges. According to weak-coupling perturbation theory, the effective mass of a self-trapped particle described by eq 1 satisfies25 m* Sω̃1/2 =1+ m0 8
Figure 1. Phase diagram of the fully quantized disordered Holstein model (eq 1) for C−C bond oscillations, using eq 18 and eq 19 with PPV parameters for excitons, i.e., Sexciton C−C = 2.53. The vertical-axis is the dimensionless strength of the disorder (i.e., σ/t0) and the horizontalaxis is the ratio of the oscillator energy to the particle transfer integral (i.e., ω̃ = ℏω/t0). Regime I is where the Anderson polaron is a more applicable description, regime II is where the Landau polaron is a more applicable description, and in regime III, disorder dominates. Note that for PPV, ω̃ exciton(=ℏω/t0) = 0.088.
(14)
which is valid for ω̃ ≪ 1 (eq 9) and λ ≪ 1 (eq 7), as validated by the DMRG calculations of ref 9. Thus ⎛ ⎞2/3 8 ⎟ S Ss = ⎜ ⎝ 8 + Sω̃1/2 ⎠ 0
In regime III, the disorder is so strong that S 0 < S c, and strong Anderson localization is the appropriate description. In this regime the predictions of the fully quantized (eq 1) and the Born−Oppenheimer (eq 2) Holstein models are indistinguishable, because disorder determines the polaron size in both cases. The crossover between regimes II and III is defined by S 0 = S c, i.e., when
(15)
The Anderson polaron relaxation energy is14,18,27
Er =
ℏωS IPR
(16)
⎛ σ ⎞(2) ⎛ Sω̃ ⎞3/2 ⎜ ⎟ =⎜ ⎟ ⎝ 2 ⎠ ⎝ t0 ⎠
where IPR is the inverse participation ratio, defined as IPR = (∑ |Ψn|4 )−1 n
(19)
The phase diagram of the fully quantized disordered Holstein model is shown in Figure 1 for a particular choice of S, as discussed in the next section.26
(17)
Since IPR is proportional to the Anderson localization length, S s, eq 16 implies that the relaxation energy of an Anderson polaron vanishes as its size diverges. C. Crossover Conditions. In ref 9, three regimes of the fully quantized disordered Holstein model (eq 1) were identified. In regime I the self-trapped polaron Anderson localization length is larger than the classical Landau polaron size, i.e., S s > S c. In this regime, although S s < S 0 because of the enhanced effective mass of the polaron arising from self-
III. APPLICATION TO POLY(P-PHENYLENE VINYLENE) A. Benzenoid−Quinoid and Vinylene Vibrations. As shown in Appendix A of the Supporting Information, normal modes in which neighboring carbon atoms oscillate in antiphase couple most strongly to charges and excitons. These are predominately the collective phenylene C−C bond oscillations, that lead to the benzenoid-quinoid distortion, and C
DOI: 10.1021/acs.jpca.5b08764 J. Phys. Chem. A XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry A the vinylene C−C bond oscillation (both at ∼1800 cm−1). Using the expressions in Appendix A of the Supporting Information, we find values for the Huang−Rhys parameters that are very close to those found by experimental fitting, as described in ref.14 Namely, for excitons Sphenylene = 3.7 and Svinylene = 8.0. This further implies that the density-weighted mean Huang−Rhys parameter for a PPV monomer is Sexciton C−C = 2.53.14 Notice that the values of S for charges are four-times smaller, because the electron−phonon coupling constant A is two-times smaller.18 Figure 1 shows the phase diagram of the fully quantized disordered Holstein model (eq 1) as a function of the dimensionless strength of the disorder (i.e., σ/t0) and the ratio of the oscillator energy to the transfer integral (i.e., ω̃ = ℏω/t0). These results were obtained using eq 18 and eq 19 with Sexciton C−C = 2.53. Regime I is valid for light polarons (i.e., large ω̃ ) and weak disorder (i.e., small σ/t0). As the disorder increases or ω̃ decreases (so the oscillators are more classical and the polarons become heavier) there is a crossover to the Landau polaron regime. Finally, as the disorder increases and dominates, there is a re-entry to the Anderson polaron regime. Evidently, for these parameters relevant to PPV the regime where Landau polaron formation is expected is very narrow. In Figure 1, σ is the standard deviation of the transfer integral t0 and is thus an energetic measure of the disorder. We now convert σ to the standard deviation of the torsional angle between neighboring moieties (ϕ), namely to σϕ, via
σϕ = σ(dϕ/dt 0)
These theoretical arguments are confirmed by (essentially exact) DMRG calculations of the fully quantized disordered Holstein model, eq 1. Figures 2−4 show the exciton center-of-
Figure 2. Exciton center-of-mass density in disordered PPV, with torsional disorder σϕ = 1°. The blue dash-dotted curve is an untrapped Anderson localized polaron. The red dashed curve is the self-trapped Anderson localized polaron (obtained from the DMRG solution of the fully quantized Holstein model, eq 1). The black solid curve is a selflocalized Landau polaron (obtained from the Born−Oppenheimer Holstein model eq 2). This is regime I. Note that for excitons the untrapped and self-trapped Anderson-localized polarons correspond to the “vertical” and “relaxed” exciton densities, respectively. Thus, the Condon approximation is valid in this regime.
mass density for parameters relevant to PPV for various values of σϕ. Figure 2, for σϕ = 1°, is clearly in regime I, where the DMRG-calculated self-trapped Anderson-localized polaron density is very similar to the untrapped Anderson-localized particle density. The self-trapped polaron is more delocalized than the Born−Oppenheimer (classical) prediction. Figure 3, for σϕ = 5°, is now in regime II, where the DMRGcalculated Anderson-localized polaron density is very similar to
(20)
as σϕ has more physical significance and enables us to evaluate the crossover values in the torsional disorder. Table 1 shows the values of the torsional disorder for the regime I−II crossover, σ(1) ϕ , and the regime II−III crossover, Table 1. Values of the Torsional Disorder (deg) for the Regime I−II Crossover, σ(1) ϕ , and the Regime II−III a Crossover, σ(2) ϕ charges oscillator type A. C−C vibration B. torsionc
b
excitons
σ(1) ϕ
σ(2) ϕ
σ(1) ϕ
σ(2) ϕ
comments
3.3
3.4
5.0
5.5
regime II is very narrow predominately regime I for realistic disorder
9.0
12
24
88
a The transfer integral tn = t(ϕn). For charges, t0(ϕ0) = t0(ϕ0 = 0) cos ϕ0, where t0(ϕ0 = 0) = −2.4/√6 eV. For Frenkel excitons, t0 = JDD + JSE cos2 ϕ0, where JDD = −0.60 eV and JSE = −1.68 eV. ϕ0 = 15° and Ktor = 0.46 eV rad−2. The parameters are chosen for PPV.14 Regime II, (2) defined by σ(1) ϕ ≤ σϕ ≤ σϕ , is where a Landau polaron is the more appropriate description. σ ϕ = σ (dϕ/dt0), where the crossover σ is determined from eqs 18 and 19. bℏωC−C = 0.19 eV, Scharge C−C = 0.633, and c Sexciton C−C = 2.53. Ator is determined from eq 7 in Appendix B of the Supporting Information and ℏωtor = 0.01 eV.
Figure 3. As for Figure 2, with torsional disorder σϕ = 5°. This is regime II. Also shown by the green dotted curve is the adiabatic limit of the DMRG calculation, defined by taking ω̃ → 0 for fixed classical polaron size (i.e., fixed Sω̃ )9 in eq 1.
σ(2) ϕ . For charges t0(ϕ0) = t0(ϕ0 = 0) cos ϕ0, where t0(ϕ0 = 0) is the charge transfer integral and ϕ0 is the mean torsional angle. For Frenkel excitons t0 = JDD + JSE cos2 ϕ0, where JDD and JSE are the through-space and through-bond exciton transfer integrals, respectively.17 Table 1 confirms the prediction of Figure 1 that the Landau polaron regime is very narrow. For excitons it is defined by 5.0° ≤ σϕ ≤ 5.5°, whereas for charges it is essentially nonexistent.
the Born−Oppenheimer polaron density. Finally, Figure 4, for σϕ = 15°, is now in regime III, where the polaron localization length is determined by disorder and is smaller than the classical Landau polaron size given by eq 10. Since the validity of regime II is very narrow for these PPV parameters, regimes II D
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polaron’s Anderson localization length becomes smaller than the classical Landau polaron size. In this regime a Landau polaron is a more appropriate description. Physically, as the oscillator frequency is decreased the normal modes behave more classically and the polaron becomes heavier. A heavier polaron has a smaller Anderson localization length, and when this localization length is smaller than the classical Landau polaron size, a Landau polaron is the more appropriate description.9 Finally, as the disorder is increased still further, the Anderson localization length of the untrapped particle becomes smaller than the classical Landau polaron size, and again an Anderson polaron is a more appropriate description. It should be noted that the distinction between Anderson and Landau polarons is not exact: self-trapped Anderson polarons are more localized than “free” particles. Nevertheless, the distinction is a useful starting point when building theories of electronic and optical processes in conjugated polymers. The key distinction between Anderson and Landau polarons is that for the latter the particle wave function is a strong function of the normal coordinates, and hence the “vertical” and “relaxed” wave functions are different. This has theoretical and experimental consequences. Theoretically, it means that the Condon approximation is not valid, and so care needs to be taken when evaluating transition rates. Experimentally, it means that the self-localization of the particle as a consequence of its coupling to the normal coordinates may lead to experimental observables, e.g., ultrafast fluorescence depolarization.28 We applied these ideas to poly(p-phenylenevinylene). We showed that the high frequency C−C bond oscillation only causes Landau polarons for a very narrow parameter regime; generally we expect disorder to dominate and Anderson polarons to be a more applicable description. Similarly, for the low frequency torsional fluctuations we showed that Anderson polarons are expected for realistic parameters. In conclusion, the results presented in this paper indicate that charge and exciton polarons in π-conjugated polymers should generally be described as Anderson polarons, not self-localized Landau polarons. This has implications in the simplification of Fermi Golden rule rates.14,18,29 It also has implications for the interpretation of ultrafast fluorescence depolarization, where rather than being a consequence of self-localization by selftrapping, it is more likely caused by localization from higherenergy quasi-extended exciton states to low-energy local exciton ground states.16,30
Figure 4. As for Figure 2, with torsional disorder σϕ = 15°. This is regime III. Notice that the disorder-localized Born−Oppenheimer polaron is smaller than the classical Landau polaron size, given by eq 10.
and III are essentially indistinguishable. Moreover, as already noted, in regime III the predictions of the fully quantized (eq 1) and the Born−Oppenheimer (eq 2) Holstein models are indistinguishable, because disorder determines the polaron size in both cases. Figure 3 also shows the adiabatic, classical limit of the DMRG calculation, defined by taking ω̃ → 0 for fixed classical polaron size (i.e., fixed Sω̃ )9 in eq 1. As expected, in this limit the polaron becomes a self-localized Landau polaron, and thus it agrees with the solution of eq 2. B. Torsional Modes. Torsional modes couple strongly to charges and excitons via the transfer integral, as described in Appendix B of the Supporting Information. They are also low frequency (i.e., deeper in the adiabatic, classical regime) and are therefore potential candidates to cause Landau polarons. However, their coupling is so strong that the Huang−Rhys parameter is large and hence the classical Landau polaron size is small. This implies that self-trapped Anderson polarons are the more appropriate description for both charges and excitons for realistic values of the disorder (as indicated in Table 1).
IV. CONCLUSIONS The Holstein model is widely used to describe charges or excitons in molecular J-aggregates15 and polymers16−18 that couple to local harmonic oscillators,4,5 e.g., vibrational or torsional modes. Using both analytical expressions and the density matrix renormalization group method, we have studied the fully quantized disordered Holstein model to investigate the localization of charges and excitons (i.e., the formation of polarons) in conformationally disordered π-conjugated polymers. We identified two distinct mechanisms for polaron formation, namely Anderson localization via disorder (causing the formation of Anderson polarons) and self-localization by self-trapping via normal modes (causing the formation of Landau polarons). Treating the harmonic oscillators quantum mechanically, we showed that there are crossovers between the regimes where either description is more valid. Qualitatively, for weak disorder and large oscillator frequencies an Anderson polaron is a more appropriate description. In this regime, the Anderson localization length, although reduced because of the polaron’s enhanced effective mass, is larger than the classical Landau polaron size. As the disorder is increased or the oscillator frequency decreased, however, the self-trapped
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.5b08764.
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Huang−Rhys parameters for vibrational modes and the “Holstein” model for torsional modes (PDF)
AUTHOR INFORMATION
Corresponding Author
*(W.B.) Telephone: +44 (0)1865 275162. E-mail: william.
[email protected]. Notes
The authors declare no competing financial interest. E
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occur for 0 ≤ σ/t0 ≤ (σ/t0)(2), whereas σ/t0 > (σ/t0)(2) is the strong disorder regime. (27) Fornari, R. P.; Troisi, A. Theory of Charge Hopping Along a Disordered Polymer Chain. Phys. Chem. Chem. Phys. 2014, 16, 9997− 10007. (28) Ruseckas, A.; Wood, P.; Samuel, I. D. W.; Webster, G. R.; Mitchell, W. J.; Burn, P. L.; Sundström, V. Ultrafast Depolarization of the Fluorescence in a Conjugated Polymer. Phys. Rev. B: Condens. Matter Mater. Phys. 2005, 72, 115214. (29) Barford, W.; Tozer, O. R. Theory of Exciton Transfer and Diffusion in Conjugated Polymers. J. Chem. Phys. 2014, 141, 164103. (30) Barford, W.; Boczarow, I.; Wharram, T. J. Ultrafast Dynamical Localization of Photoexcited State in Conformationally Disordered Poly(p-phenylenevinylene). J. Phys. Chem. A 2011, 115, 9111−9119.
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DOI: 10.1021/acs.jpca.5b08764 J. Phys. Chem. A XXXX, XXX, XXX−XXX
