Inclusion of Asymptotic Dependence of Reorganization Energy in the

Mar 8, 2016 - We recently developed an intuitive multistate parabolic model (MPM), based on the Marcus two-state model, to describe the redox and opti...
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Inclusion of Asymptotic Dependence of Reorganization Energy in the Modified Marcus-Based Multistate Model Accurately Predicts Hole Distribution in Poly‑p‑phenylene Wires Marat R. Talipov, Maxim V. Ivanov, and Rajendra Rathore* Department of Chemistry Marquette University P.O. Box 1881, Milwaukee, Wisconsin 53201-1881, United States S Supporting Information *

ABSTRACT: We recently developed an intuitive multistate parabolic model (MPM), based on the Marcus two-state model, to describe the redox and optical properties and spin/charge distribution in the cation radicals of different poly-p-phenylene wires and showed that MPM predictions closely matched, in most cases, with the experimental and computational findings. Application of MPM to different classes of poly-p-phenylene-based wires led us to recognize that its performance is not optimal in certain cases, especially in describing the hole distribution in second excited states of poly-p-phenylene wires due to the quadratic shape of the reorganization function. In this work we show that a revised multistate model (MSM), where parabolas were replaced with a composite quadratic/reciprocal function, successfully addresses these issues by taking into account the differing energetic requirement for near and distant units by quadratic and reciprocal components of the reorganization function, respectively. Moreover, the necessity of usage of the reciprocal component of the (composite) quadratic/reciprocal function was consistent with the Marcus equations for describing the reorganization energy and further supported by the constrained DFT calculations. The revised model (MSM) accurately describes the spin/charge distribution in the ground and excited states of various poly-p-phenylene wires and is expected to serve as a versatile and powerful predictive tool for the design and synthesis of next-generation charge-transfer materials for photovoltaic applications.

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INTRODUCTION Different classes of π-conjugated molecular wires are routinely employed as the medium for the charge and energy transfer in modern photovoltaic devices.1−4 The design and synthesis of next-generation long-range charge-transfer materials would require the development of a detailed understanding of charge-transfer mechanisms and structure−function relationship in various π-conjugated molecular wires of different topologies.5−13 In this context, we have recently carried out a combined experimental, DFT, and theoretical modeling study of the optoelectronic properties of a number of well-defined series of poly-p-phenylene (PPn) wires (Figure 1).14,15 For example, combined DFT/experimental study revealed that the spin/ charge distribution in uncapped HPPn+• gravitates toward the middle of the oligomer chain and is confined to seven pphenylene units in all HPPn+• with n ≥ 7, while in the endcapped RPPn+• (R = iA, RO) the hole lies toward the end of the chain and is relatively compact (Figure 1).14,15 Curiously, we found that the hole distribution in cyclic poly-p-phenylene cation radicals (CPPn+•) was similarly limited to seven pphenylene units (Figure 1). These experimental/DFT findings of hole distribution and unusual evolution of the redox and optoelectronic properties of linear/cyclic PPn were well reproduced using a multistate parabolic model (MPM) developed based on the Marcus’s two-state model.14,15 The versatility and predictive power of MPM and its subsequent application to different classes of molecular wires17 © XXXX American Chemical Society

Figure 1. Hole distribution in the linear uncapped (HPP10+•), alkyl(iAPP10+•), and alkoxy-capped (ROPP10+•) as well as cyclic (CPP10+•) poly-p-phenylenes [B1LYP-40/6-31G(d)+PCM(CH2Cl2)].14−16

led us to question whether a simple quadratic function18,19 can accurately describe the long-range behavior of the reorganizaReceived: January 16, 2016 Revised: March 8, 2016

A

DOI: 10.1021/acs.jpcc.6b00514 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C tion term in π-conjugated molecular wires. This study will attempt to answer this question. Importantly, we show that the reorganization term plays an important role in governing the extent of hole distribution in PPn wires. Because the reorganization energies are expected to saturate in long-range regime rather than quadratically increase for the distant units, it necessitated the use of a different function having the correct asymptotic dependence. Accordingly, the modified multistate composite model (MSM), which makes use of a composite quadratic/reciprocal function, not only describes the spin/ charge distribution in ground (D0) and first excited (D1) states with high accuracy for different classes of poly-p-phenylenebased wires but also allows the description of higher excited states (e.g., D2), which was not feasible using the MPM. Moreover, we will show that the use of quadratic/reciprocal function is in complete accord with Marcus theory and it is further supported by the constrained DFT calculations.20−22

XΛ can vary from zero to one as exemplified for XΛ = 0.5, where the reorganization is distributed equally over both units (B′ in Figure 2B).25 Both structural (λi) and solvent (λo) reorganization terms can be described individually or together by a quadratic function,18,19 that is, eq 1 Gλ = (λi + λo)(XQ − XΛ)2

(1)

where |XQ − XΛ| represents the extent of separation between the charge and reorganization. Note that |XQ − XΛ| = 0 corresponds to a minimum on the free-energy surface (i.e., A or B), while |XQ − XΛ| = 1 corresponds to a vertically excited state (e.g., B*), Figure 2. The electronic coupling between the reactant (i.e., XQ = 0) and product (XQ = 1) diabatic states can be taken into account by expressing the model Hamiltonian in the form of a matrix, whose diagonal elements correspond to the free energies of the diabatic reactant/product states for a given XΛ, that is, parabolas, and the off-diagonal elements correspond to the electronic coupling (Hab) between these states (eq 2).14

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RESULTS AND DISCUSSION To provide a clear rationale for the inner workings of the MPM, let us briefly examine the Marcus two-state model,18,19,23,24 which involves two charge-delocalized (adiabatic) states produced by the interaction of two charge-localized (i.e., diabatic) states. The diabatic states can be uniquely defined by two independent parameters, one indicating the position of the charge XQ (i.e., XQ = 0/1 denotes charge localized on the left/ right chromophore) and another indicating the reorganization XΛ (i.e., XΛ = 0/1 denotes the structural and solvent reorganization localized on the left/right unit); see Figure 2.

⎡ λ(X − X )2 ⎤ Hab Λ Q0 ⎢ ⎥ H(XΛ) = ⎢ 2⎥ λ(XQ1 − XΛ) ⎦ Hab ⎣

(2)

where XQ0 = 0 and XQ1 = 1 are the values of XQ corresponding to the reactant/product diabatic states and λ is the reorganization energy (eq 1). Diagonalization of the Hamiltonian matrix in eq 2 produces the eigenvalues and the eigenvectors, which provide free energies and the charge distribution, respectively, for the ground and excited adiabatic states for a given XΛ. A systematic variation of XΛ in eq 2 at a given value of Hab easily generates the free-energy profiles of charge transfer in both ground and excited states. Extension of the two-state model (Figure 2) to the multistate parabolic model is depicted in Figure 3 on the example of a five-state system, where XΛ= 0, 1, 2, 3, 4 for states A−E, respectively. Figure 3 shows the free-energy profiles of the charge transfer together with the accompanying reorganization along the reaction coordinate in panel A, the transfer of reorganization without transferring the charge in panel B, and the vertical transfer of charge in panel C. As such, this depiction allows one to visualize the quadratic evolution of the reorganization energy term (Figure 3B) as well as provides the values of reorganization energies for various states (Figure 3C) to be used in model Hamiltonian, as follows. Similar to the two-state model (eq 2), the electronic coupling between various diabatic states can be represented by the Hamiltonian matrix, whose diagonal elements (Hi) are shown in Figure 3C,26 and the off-diagonal elements represent the electronic coupling (Hab) between the adjacent states (eqs 3a−3c)14

Figure 2. Schematic representation of the Marcus model (i.e., two uncoupled harmonic Born−Oppenheimer surfaces) showing groundstate free-energy profile of the charge transfer between two chromophores at Hab = 0 (A) and change in the reorganization that follows from a vertical charge-transfer state B* (represented by empty circle) to the ground-state B denoted on the red parabola (B).

⎡ H (X ) H 0 ab ⎢ 1 Λ ⎢ Hab H2(XΛ) Hab. ⎢ ⎢ Hab H3(XΛ) H(XΛ) = ⎢ 0 . . . ⎢ 0 0 ⎢ 0 ⎢ 0 0 ⎣ 0

Thus, Figure 2A shows the Marcus’s two-state plot depicting the initial (A, i.e., XΛ = 0, XΛ = 0) and final (B, i.e., XΛ = 1, XQ = 1) diabatic states arising from the charge transfer, while Figure 2B shows a state arising from the vertical charge transfer without accompanying structural/solvent reorganization (B*, i.e., XΛ = 0, XQ = 1; denoted by an empty circle on the red parabola). Relaxation of B* to B is achieved by the structural/ solvent reorganization; that is, XΛ changes from 0 to 1 at constant XQ = 1. It is noted that XQ is always an integer, while

. . . ... . .

⎤ ⎥ 0 0 ⎥ ⎥ 0 0 ⎥ . . ⎥ ⎥ Hi(XΛ) Hab ⎥ ⎥ Hab Hn(XΛ)⎦ 0

0

(3a)

Hi(XΛ) = λ(XQi − XΛ)2 B

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Figure 3. Schematic representation of MPM. (A) Free-energy profile of the charge transfer along the chain in a five-state system. (B) Free-energy profile of the reorganization transfer without transferring the charge. (C) Free-energy profile of vertical transfer of charge, which also provides diagonal elements Hi to be used in eq 3a.

XQi = i − 1

range component of the reorganization energy term in a multistate system. As it follows from the previous discussion, realistic reorganization energy function Hi(XΛ) should asymptotically approximate to a finite value that corresponds to the free energy of noncommunicating charge and reorganization, have one minimum, and be even, that is, Hi(−x) = Hi(x). Such a reorganization energy function would approximate to an inverted bell curve (well curve). The rationalization that reorganization term in multistate systems should be described by an inverted bell curve also emerges from the constrained DFT (CDFT) calculations,20−22 as follows. The CDFT calculations were performed on a representative poly-p-phenylene wire HPP10+•, as displayed in Figure 5. The gas-phase calculations of various diabatic states of HPP10+• at fixed XΛ = 2 (Figure 5A) showed that the structural reorganization energy does not change beyond the adjacent unit (Figure 5B, filled circles), while the solvent reorganization energies follow a reciprocal (1/|XQ − XΛ|) trend (Figure 5B, empty circles).28,29 The CDFT analysis clearly suggests that a reciprocal function will appropriately describe the reorganization term in a multistate system, which turns out to be in complete accord with the Marcus theory. The structural and solvent reorganization energies for the Marcus two-state model are expressed by eqs 4a and 4b, respectively

(3c)

Diagonalization of the Hamiltonian matrix in eqs 3a−3c produces the free energies and charge distribution for adiabatic states for a given XΛ, and systematic variation of XΛ in eqs 3a−3c at a given value of Hab produces the free-energy profiles of charge transfer in ground and excited states. Position of the global minimum on the ground state of the resulting profile could then be used for the prediction of the hole distribution as well as redox and optical properties at given values of Hab and λ. The multistate parabolic model previously discussed served well in reproducing the hole distribution in various poly-pphenylene wires (Figure 1) obtained by the experimental/DFT studies (Figures S1−S3 in the Supporting Information); however, the question remains as to how well the quadratic function takes into account the long-range component of reorganization term from distant units. For example, Figure 4, a

Figure 4. Showing the quadratic growth of the reorganization energy term upon separation of the charge and reorganization in a polychromophoric molecule. Thus, H3(x) = λ(XQ − XΛ)2 = λ(1 − 2)2 = λ, H4(x) = λ(1 − 3)2 = 4λ, H5(x) = λ(1 − 4)2 = 9λ, and H6(x) = λ(1 − 5)2 = 16λ, and so on.

G λi = λi(XQ − XΛ)2

(4a)

⎛ 1 1 1 ⎞⎛ 1 1⎞ G λo = (Δe)2 ⎜ + − ⎟⎜⎜ − ⎟⎟ 2a 2 R ⎠⎝ Dop Ds ⎠ ⎝ 2a1

(4b)

where Δe corresponds to the extent of solvent polarization, a1 and a2 are the effective radii of the chromophores (where one of them carries reorganization and another carries positive charge), Dop and Ds are the optical and static solvent dielectric constants, and R is the distance between the two chromophores. In Marcus’s two-state model, the term 1/R in eq 4b is constant, while the term Δe changes in proportion to |XQ − XΛ|; however, the extension of eq 4b to the case of a multichromophoric system where charge and reorganization are separated (i.e., |XQ − XΛ| ≥ 1, see, e.g., Figure 5A) turns the term 1/R into a variable (R ∝ |XQ − XΛ|) while the term Δe becomes a constant. In a similar manner, the Gλi term in eq 4a, which accounts for the structural reorganization and depends quadratically on the |XQ − XΛ| in the Marcus’s two-state model,

hybrid of Figure 3B,C, shows the profile of the energetic penalty of separation of charge (XQ) and reorganization (XΛ)27 from the second unit at which XΛ = XQ = 1. Such an evaluation of the reorganization term in context of a polychromophoric system by placing the charge at a distant monomer unit (i.e., |XQ − XΛ| > 1) clearly shows that the reorganization term beyond the adjacent (λ) and next-to-adjacent (4λ) units rapidly becomes too energetically demanding, that is, 9λ for fourth unit, 16λ for fifth unit, and so on. This analysis suggests that complete separation of the hole- and reorganization-bearing units would require infinite energy (Figure 4). In other words, parabolic function does not appropriately account for the longC

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nent shown as inset in Figure 6A suffices for the description of the two-state model.30 Thus, a composite function to be used in the model Hamiltonian (i.e., eq 3a) is defined in eq 5 ⎧ λ(X − X )2 , if |X − X | ≤ t Qi Λ Qi Λ ⎪ Hi(XΛ) = ⎨ ∞ a , if |XQi − XΛ| ≥ t ⎪λ − |XQi − XΛ| ⎩

(5)

where λ is the structural reorganization, λ∞ is the free energy of completely separated charge and reorganization (|XQ − XΛ| → ∞), and the requirement of continuity of Hi(XΛ) and its first derivative at |XQ − XΛ| = t is used to define the parameters a = 2λt3 and t = λ∞/3λ . Note that the parameter t corresponds to a distance of switching from the short-range (i.e., quadratic) to the long-range (asymptotic reciprocal) regimes of the interaction between the charge- and reorganization-bearing units. Application of the composite function (eq 5) for constructing the effective Hamiltonian (eq 3a) in MSM (Figure 6B) showed very similar description of the hole distribution and, in turn, optoelectronic properties of the ground and first excited states of various PPn wires, as compared with the original MPM (Figure 7A and Figure S2 in the Supporting Information). Interestingly, MSM predicts a more delocalized hole distribution in the excited (D1) state of ROPPn+• as compared with MPM, which resembles more closely to the perunit hole distributions obtained by DFT calculations (Figure 7, column 3). Such an improved description of the hole by MSM originates from the fact that inclusion of the appropriate

Figure 5. (A) Schematic representation of various charge- and reorganization-localized states of HPP10+• used in the CDFT calculations. (B) Reorganization energies from the CDFT calculations [B1LYP-40/6-31G(d)+PCM(CH2Cl2)]14−16 versus |XQ − XΛ| (left) and 1/|XQ − XΛ| (right). Filled blue circles show relative electronic energies (ΔE) in the gas phase, while the empty red circles show relative Gibbs free energies (ΔG) in solution. Filled red circle in the right panel indicates the reorganization term extrapolated to an infinitely long poly-p-phenylene wire. Also see the Supporting Information for the computational details and usage of the ωB97XD functional in Figure S7.

also becomes a constant in a multichromophoric system upon a separation of charge and reorganization. This analysis of Marcus equations, applied to a multistate system, suggests that the solvent reorganization term should follow a reciprocal function, while the structural reorganization term should saturate at |XQ − XΛ| = 1 (Figure 5B). As it follows from the previous discussion, a composite quadratic/reciprocal function would be a more realistic choice for the description of the reorganization energy term in the multistate model than a simple quadratic function. Figure 6A depicts the composite function, where the quadratic compo-

Figure 7. (A) Comparison of the per-unit hole distribution in the ground (D0) and excited (D1) states in (representative) RPP10+• obtained by the DFT calculations [B1LYP-40/6-31G(d)+PCM(CH2Cl2)],14−16 by MPM (Hab = 9λ, and ε = 0λ, −3.7λ, and −8.5λ for RPPn+• with R = H, iA, and RO, respectively), and by MSM (λ∞ = 13.8λ, Hab = 7.6λ, and ε = 0λ, −3.7λ, and −8.9λ for R = H, iA, and RO, respectively). (B) Free-energy profiles of RPP10+• with respect to the reorganization coordinate, obtained by MPM and MSM. Also see Figures S4−S6 in the Supporting Information.

Figure 6. Schematic representation of the composite quadratic/ reciprocal function (A) and the MSM (B). D

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The Journal of Physical Chemistry C reorganization term allows for the more delocalized charge distribution and leads to the modified free-energy surfaces, especially in the excited state, where the barriers of charge transfer between end-capped units are somewhat reduced as compared with MPM (Figure 7B). A logical modification of MPM to MSM and its application to various RPPn+• in their ground (D0) and first excited (D1) states (Figure 7) demonstrated that the MSM provides the same or better level of accuracy as MPM in the description of hole and optoelectronic properties. More importantly, the MSM affords an accurate description of higher excited states in cyclic poly-p-phenylenes, which was not possible with the MPM. For example, DFT calculations predicted that energy of the D0 → D2 transition in CPPn+• is essentially identical to νD0→D1 in CPP5+• and CPP6+•, which have uniform hole distribution in the ground state, and is larger than νD0→D1 in the case of CPPn+• with n > 6 (Figure 8A, see also Figure S3 in the Supporting Information). The second excited state (D2) for larger CPPn+• (n > 6) has a charge-transfer character because the hole transfers to the opposite side of the ring during the D0 → D2 transformation (Figure 8C). As shown in Figure 8C on the example of CPP11+•, the hole distribution in D0, D1, and D2 states is accurately reproduced by MSM but not by MPM. A close examination of the hole distribution by MPM and MSM in Figure 8C shows that the distribution of hole in D0, D1, and D2 states, predicted by MPM, closely resembles density distribution of the corresponding states of the quantum harmonic oscillator (QHO).15 In contrast, MSM predicts that only D0 and D1 states are similar to the QHO states, while in D2 state it is akin to that expected based on the charge-transfer excitation.15 Because the hole distribution determines optoelectronic properties of molecular wires, the excited-state energies obtained by MSM and MPM (Figure 8B) again showed that only MSM produced excitation energies that closely corresponded to those obtained by the DFT calculations or by experiment.15 Failure of the MPM to accurately describe higher excited states (e.g., D2) of CPPn+• is understandable in hindsight because the quadratic shape of the reorganization function is expected to produce a QHO-like distribution, as it completely neglects involvement of the distant p-phenylene units due to the prohibitively large reorganization term. On the contrary, usage of the composite function in MSM allows a QHO-like distribution of D0 and D1 states, similar to MPM, due to the quadratic component of reorganization energy term, while its reciprocal component allows for the hole transfer to the opposite side of the CPPn cycle because of the reciprocality of the reorganization term. The observation that hole distribution in the D2 state of medium-sized cyclic poly-p-phenylenes is delocalized on the opposite side of the ring from the hole distribution in the ground (D0) state is in stark contrast with linear HPPn, where the second excited state (D2) resembles that of the QHO.32 At the same time, it is clear that properties of all electronic states of cyclic and linear PPn should converge to the same polymeric limit because the effect of cyclic topology and associated structural deformation of CPPn, which differentiate cyclic versus linear poly-p-phenylenes, vanishes as n → ∞.15 Thus, the hole distribution in the D2 state of large (n → ∞) CPPn should become identical to the D2 state in the linear HPPn, see Figure 8.

Figure 8. Evolution of the first two vertical excitation energies of C PPn+• with respect to n as obtained by B1LYP-40/6-31G(d)+PCM(CH2Cl2)14−16 (A) and theoretical modeling using MPM and MSM (B). Per-unit hole distribution in the ground (D0) and excited (D1, D2) states of PPn+• on the examples of CPP11+• (C), HPP15+• (D), and C PP20+• (E), as denoted.31

Interestingly, MSM not only reproduces charge-transfer character of the D2 state in medium-sized CPPn but also predicts the identical hole distribution in the D2 state of linear (Figure 8D) and cyclic (Figure 8E) poly-p-phenylenes to the extent of the polymeric limit (Figure 8E).33 This prediction from MSM was further verified by the DFT calculations, which indeed demonstrated that in large CPPn+• (n > 15) the hole distribution in the D2 state becomes similar to that in linear H PPn+• (Figure 8D; see also Figure S13 in the Supporting Information). Thus successful performance of MSM in describing the hole distribution in D0, D1, and D2 states and, in turn, the optoelectronic properties of various poly-p-phenylenes demonstrates the importance of gradual saturation of the reorganization term, which is better described by a reciprocal rather than a E

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quadratic function, which neglects long-range reorganization term.33

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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.jpcc.6b00514. Computational details (Section S1), full results obtained by the MPM and MSM models (Section S2), details of the CDFT calculations (Section S3), comparison of the multistate models based on the parabolic and inverse Gaussian functions (Section S4), hole distribution in the cyclic poly-p-phenylenes in their ground and first and second excited cation radical states (Section S5), additional references (Section S6), coordinates and energies of the calculated equilibrium structures of H PP15+• and CPPn+• (n = 14−20) (Section S7), and Figures S1−S13. (PDF)

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REFERENCES

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SUMMARY AND CONCLUSIONS In conclusion, it has been shown using a combined experimental/DFT approach that confinement of a hole in different series of poly-p-phenylene cation radicals (Figure 1) to seven p-phenylene units originates from the interplay between the energetic gain from charge delocalization and concomitant energetic penalty from the structural/solvent reorganization. Because long-range solvent reorganization is expected to saturate due to the complete separation of charge and associated structural/solvent reorganization at distant units, it is apparent that the reorganization energy term should be quadratic only for short-range interactions, for example, as in the case of Marcus two-state model. Reinterpretation of the Marcus equations for the reorganization energy term (i.e., eqs 4a and 4b), together with the constrained DFT calculations, led us to the modified multistate model, MSM, which makes use of a composite quadratic/reciprocal function (eq 5). It is demonstrated that MSM properly describes the hole distribution in the ground (D0) and first excited (D1) states as well as allows accurate description of higher excited states (e.g., D2), which was not feasible using MPM. The studies are underway to demonstrate that the role of long-range reorganization term is expected to be essential in different classes of molecular wires including Geländer-like poly-p-phenylenes,34 where the hole distribution is relatively compact as compared with the PPn+•, and these experimental/ DFT results will be reported in due course.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the NSF (CHE-1508677) and NIH (R01HL112639-04) for financial support and Professors S. A. Reid and Q. K. Timerghazin for helpful discussions. The calculations were performed on the high-performance computing cluster Père at Marquette University funded by NSF awards OCI0923037 and CBET-0521602 and the Extreme Science and Engineering Discovery Environment (XSEDE) funded by NSF (TG-CHE130101). F

DOI: 10.1021/acs.jpcc.6b00514 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C (22) Van Voorhis, T.; Kowalczyk, T.; Kaduk, B.; Wang, L.-P.; Cheng, C.-L.; Wu, Q. The Diabatic Picture of Electron Transfer, Reaction Barriers, and Molecular Dynamics. Annu. Rev. Phys. Chem. 2010, 61, 149−170. (23) Robin, M. B.; Day, P. Mixed Valence Chemistry. A Survey and Classification. Adv. Inorg. Chem. Radiochem. 1968, 10, 247−422. (24) Brunschwig, B. S.; Creutz, C.; Sutin, N. Optical Transitions of Symmetrical Mixed-valence Systems in the Class II–III Transition Regime. Chem. Soc. Rev. 2002, 31, 168−184. (25) For example, values of XΛ outside of the range 0 to 1 are relevant to the Marcus’s inverted region. (26) Note that the corner elements H1 and Hn can be modulated by a constant ε to take into account the effect of electron-donating or electron-withdrawing end-capping groups; see ref 14. (27) The concept of reorganization utilized in this study closely corresponds to “phonon” in solid-state theory, while the concept of a positive charge considered together with the accompanying reorganization corresponds to “polaron”. See also: Reimers, J. R.; Hush, N. S. Hole, Electron, and Energy Transfer Through Bridged Systems. VIII. Soliton Molecular Switching in Symmetry-Broken Brooker (Polymethinecyanine) Cations. Chem. Phys. 1993, 176, 407−420. (28) In fact, studies of self-trapped charge in condensed matter, which are closely related to the problem of charge distribution in πconjugated molecular wires, also led to the recognition of the 1/R dependence of the potential well that arises due to the long-range Coulombic interactions; see: Emin, D. Polarons; Cambridge University Press, 2013. (29) It is also noted that Nelsen and coworkers augmented the reorganization energy function by quartic term to obtain a better fit of the experimental data; see: Nelsen, S. F.; Ismagilov, R. F.; Trieber, D. A. Adiabatic Electron Transfer: Comparison of Modified Theory with Experiment. Science 1997, 278, 846−849. (30) In Marcus model, reorganization and charge separation is limited to only two (adjacent) units, which can be described accurately by parabolas; however, the long-range separations of charge and reorganization in multistate system necessitate the modification of the function describing the reorganization term. (31) It is noted that the cos[π/(n + 1)] coordinate derived from the Hückel-like (i.e. tight-binding) model using the monomer-based frontier orbitals provides much better agreement with the experimental observables (e.g. redox potentials and excitation energies of neutral and cation radicals) than the often-used linear dependence of 1/n coordinate, derived from a particle-in-a-box model, which works reasonably well only for smaller n; see: Torras, J.; Casanovas, J.; Alemán, C. Reviewing Extrapolation Procedures of the Electronic Properties on the π-Conjugated Polymer Limit. J. Phys. Chem. A 2012, 116, 7571−7583. (32) Note that the D2 state of HPPn+• (n → ∞) is relatively more delocalized than predicted by QHO model due to the anharmonicity of the reorganization function. (33) It is noted that the choice of the function plays an important role in reproducing hole distribution in higher excited states (e.g., D2) of CPPn+•. For example, multistate model based on the inverted Gaussian function (which approaches to the asymptotic limit much faster than a reciprocal function) properly predicts charge-transfer hole distribution in the D2 state of medium-sized PPn, but it performs poorly in predicting the nature of the D2 state of large CPPn+• (n → ∞); see Figures S8-S12 in the Supporting Information. (34) Modjewski, M.; Lindeman, S. V.; Rathore, R. A versatile Preparation of Geländer-type p-Terphenyls from a Readily Available Diacetylenic Precursor. Org. Lett. 2009, 11, 4656−4659.

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DOI: 10.1021/acs.jpcc.6b00514 J. Phys. Chem. C XXXX, XXX, XXX−XXX