HÃ¼ckel Theory + Reorganization Energy = MarcusâHush Theory

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Hückel Theory + Reorganization Energy = MarcusHush Theory — Breakdown of the 1/n Trend in #Conjugated Poly-p-phenylene Cation Radicals is Explained Maxim V. Ivanov, Marat R. Talipov, Anitha Boddeda, Sameh H. Abdelwahed, and Rajendra Rathore J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b12111 • Publication Date (Web): 03 Jan 2017 Downloaded from http://pubs.acs.org on January 7, 2017

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The Journal of Physical Chemistry

Hückel Theory + Reorganization Energy = Marcus-Hush Theory — Breakdown of the 1/n Trend in πConjugated Poly-p-phenylene Cation Radicals is Explained Maxim V. Ivanov, Marat R. Talipov, Anitha Boddeda, Sameh H. Abdelwahed, and Rajendra Rathore* Department of Chemistry, Marquette University, P.O. Box 1881, Milwaukee, WI 53201-1881 ABSTRACT: Amongst the π-conjugated poly-p-phenylene wires, fluorene-based poly-p-phenylene (FPPn) wires have been extensively explored for their potential as charge transfer materials in functional photovoltaic devices. Herein, we undertake a systematic study of the redox and optical properties of a set of FPPn (n = 2-16) wires. We find that, while their absorption maxima (νabs) follow a linear trend against cos[π/(n+1)] up to the polymeric limit, redox potentials (Eox) show an abrupt breakdown from linearity beginning at n ~ 8. These observations prompted the development of a generalized model to describe the unusual evolution of redox and optical properties of poly-pphenylene wires. We show that the cos[π/(n+1)], commonly expressed as 1/n, dependence of the properties of various π-conjugated wires has its origin in Hückel Molecular Orbital (HMO) theory, which, however, fails to predict the evolution of the redox potentials of these wires, as the oxidation-induced structural/solvent reorganization is unaccounted for in the original formulation of HMO theory. Accordingly, aided by DFT calculations, we introduce here a modified HMO theory that incorporates the reorganization energy (Δα) and coupling (β), and show that the modified theory provides an accurate description of the oxidized FPPn wires, reproducing the breakdown in the linear cos[π/(n+1)] trend. A comparison with the Marcus-based multistate model (MSM), where reorganization (λ) and coupling (Hab) are introduced by design with the aid of empirically adjusted parameters, further confirms that the structural/solvent reorganization limits hole delocalization to ~8 p-phenylene units and leads to the breakdown in the linear evolution of the redox properties against cos[π/(n+1)]. The predictive power of the modified HMO theory and MSM offer new tools for rational design of the next-generation, long-range charge transfer materials for photovoltaics and molecular electronics applications.

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Introduction π-Conjugated poly-p-phenylene molecular wires have been extensively explored as materials for molecular electronics and photovoltaic applications, as they provide an effective medium for long-range charge and exciton transfer.1-3 Our recent studies of the poly-p-phenylene wires (RPPn: R = H, iA, iAO and iA2N, where n is the number of phenylene units, Figure 1) showed that the experimental redox potentials (Eox) of RPPn and the optical properties (νmax) of the resulting cation radicals (i.e. RPPn+•) saturate with n ≤ 8.4-8 The saturation point moves to smaller n as the electron-donor strength of end-capping group increases, i.e. saturation occurs at n = 5 for iAOPPn and n = 3 for iA2NPPn. Using DFT calculations and theoretical modeling, we showed that the saturation of the redox and optical properties occurs due to the hole migration towards one end of the molecule in order to continually engage the end-capped p-phenylene unit in hole stabilization (Figure 1).4 H

H

O

n-2

n

H

+• PPn

O

n-2

iA

N

N

n-2

+•

iAO

PPn

n-2

+•

iA2N

PPn

+• PPn

2

1.22

0.74

-0.08

3

1.09

0.80

0.15

4

1.04

0.83

0.27

5

1.00

0.85

0.27

6

0.97

0.86

0.27

7

0.96

0.86

Figure 1. Per-unit barplot representation of the spin/charge distributions in RPPn+•: R = H, iA, iAO and iA2N (n = 2-7), where n is the number of p-phenylene units [B1LYP-40/6-31G(d)+PCM(CH2Cl2)] and the corresponding experimental oxidation potentials (V vs Fc/Fc+). Note, that RPPn with larger n are unavailable due to the solubility issues.6

The observed breakdown from the expected linear trend of the RPPn/RPPn+• properties against cos[π/(n+1)], commonly expressed as 1/n,9 trend was experimentally confirmed for alkoxy- and dialkylamino-capped (i.e. R = iAO and iA2N) wires, however it could not be verified for uncapped or alkyl-capped (i.e. R = H and iA) wires due to solubility issues with higher homologues.4-8 The deleterious effect of the strong electron-donating end-capping groups and unavailability of the longer virgin HPPn and alkyl-capped iAPPn hampered our attempts to develop a full understanding of the evolution of the redox and optical properties of the π-conjugated poly-p-phenylene wires and true origin of the breakdown of the cos[π/(n+1)] or 1/n trends.9 Amongst the π-conjugated poly-p-phenylene wires, mixtures of oligomers of 9,9’-dialkyllfluorene, referred to as OFn, with n ≤ 54 (where n is the number of fluorenes) have been demonstrated to be highly soluble;10 and their potential as charge-transfer materials in actual photovoltaic devices has been extensively explored.11-16 Accordingly, in this study we use oligomers of 9,9’-dihexalfluorene as models for longer homologues of poly-pphenylene wires. These OFn oligomers will be referred to hereafter as fluorene-based poly-p-phenylenes, i.e. FPPn, where n is the number of phenylene units. For example, the structure of OF8 shown below contains 16 phenylene units and thus is named FPP16.

OF8 = FPP16 We have successfully synthesized a series of OFn, n = 1-8 (or FPPn, n = 2-16) and recorded their absorption and emission spectra, determined their precise electrochemical redox potentials, and obtained the optical data of their cation radicals generated by careful redox titrations using robust aromatic oxidants.17,18 The availability of the experimental data on FPPn/FPPn+• with relatively large n now allows us to show that the evolution of the optical properties of neutral FPPn follows a linear trend, while the properties of their cation radicals (i.e. Eox and νmax) show a breakdown of the linear trend against cos[π/(n+1)]. The study herein will demonstrate, for the first time, with the aid of accurate experimental data, DFT calculations, and theoretical modeling, that the breakdown of the cos[π/(n+1)] trend reflects the hole localization19-23 in poly-p-phenylene cation radicals onto a limited number of p-phenylene units (i.e. n ≈ 8), and occurs due to the interplay between the energetic gain from the hole delocalization and the concomitant energetic


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penalty from structural/solvent reorganization. Importantly, we will show that the well known Hückel molecular orbital (HMO) theory, which is readily applicable for the description of neutral FPPn, can be applied to FPPn cation radicals if the structural reorganization is appropriately accounted for. Accordingly, we have developed a modified HMO theory that includes structural reorganization, and show that this theory can well reproduce the experimental properties of various poly-p-phenylene cation radicals. Indeed, the performance of the modified HMO model is remarkably similar to our recently developed Marcus-based multistate model (MSM), which includes the reorganization energy by design. The rationale development of new molecular wires for application in photovoltaic devices and organic electronics requires predictive chemical models, and we suggest that the modified HMO theory and MSM are valuable new tools for elucidation of these properties.

Results and Discussions Synthesis/spectroscopy/electrochemistry of FPP n and generation/spectroscopy of FPP n + • . The synthesis of FPP2-FPP16 was accomplished using readily available precursors by standard Suzuki coupling reactions,24,25 and the detailed synthetic schemes/procedures and characterization data (i.e. 1H/13C NMR, MALDI, and X-ray crystallography of representative oligomers) are compiled in Section 1 in the Supporting Information. The resulting FPPn oligomers were rigorously purified by column chromatography and were found to be freely soluble in dichloromethane, chloroform, toluene, THF, and similar solvents. The electronic absorption spectra of dichloromethane solutions of neutral FPPn are compiled in Figure 2A. The position of the characteristic absorption band of FPPn shows a clear red shift (i.e., shift to longer wavelength or lower energy) with increasing n (Figure 2A). The energies of the absorption maxima of FPPn tracked linearly with cos[π/(n+1)] (Figure 2B). Interestingly, the point for the absorption maximum of FPP108 at 388 nm (indicated by a pink symbol in Figure 2B)10 clearly fell onto the linear trendline, thus confirming that the cos[π/(n+1)] dependence holds up to the polymeric limit (vide infra).

B

A n

16 14 12 10 8 6 4 2

D

C n

2

16 14 12 10 8 6 4

Figure 2. A: Compilation of the normalized absorption spectra of neutral FPPn in CH2Cl2 at 22 ºC. B: Plot of energies of absorption maxima (νabs, Table 1) of neutral FPPn against cos[π/(n+1)]. C: Compilation of the emission spectra of FPPn in CH2Cl2 at 22 ºC excited at the absorption maxima. D: Plot of energies of emission maxima (νem, Table 1) against cos[π/(n+1)], also see Figure S42 in the Supporting Information. The position of emission maxima shifts red in emission spectra of FPPn only up to approximately 10 pphenylene units (Figure 2C, Table 1). Interestingly, a plot of the energies of emission maxima (νem) against cos[π/(n+1)] showed a linear dependence up to n = 10, followed by a breakdown, as the position of the emission band remained unchanged beyond n = 10 (Figure 2D). This contrasting evolution of the emission and absorption characteristics of FPPn with increasing n suggests that the exciton delocalization does not extend beyond 10 pphenylene units.



The redox properties of the FPPn were evaluated by electrochemical oxidation at a platinum electrode as a 1 mM solution in dichloromethane containing 0.1 M tetra-n-butylammonium hexafluorophosphate as the supporting electrolyte. Except FPP2, which underwent irreversible electrochemical oxidation,26 the higher homologues (i.e. FPP4–FPP16) showed reversible cyclic voltammograms at ambient temperatures and provided first oxidation potentials (Eox1), which were referenced to ferrocene as an internal standard (Figure 3A, Table 1). A plot of Eox1 of FPPn showed a linear dependence against cos[π/(n+1)] up to 8 p-phenylene units, followed by the breakdown or saturation (Figure 3B). It is noted that evolution of the second and higher oxidation potentials of FPPn with increasing n varied drastically when plotted against cos[π/(n+1)] as compared to Eox1 (Figure S5 in the Supporting Information). Unlike the exciton delocalization, which does not extend beyond 10 p-phenylene units (Figure 2D), the saturation of the Eox1 at 8 p-phenylene units (Figure 3B) suggests that hole delocalization in FPPn+• must not extend beyond 8 p-phenylene units.

A

n=2

B

n=4

n=6

C

n 16

n=8

14

n = 10

5 x 104, M-1cm-1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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12 10 8 6 4

n = 12

D

n = 14

n = 16

Figure 3. A: Cyclic voltammograms of 1 mM FPPn in CH2Cl2 (0.1 M n-Bu4NPF6) at a scan rate of 200 mV s-1 and 22 ºC. B: Plot of first oxidation potentials (Eox1, Table 1) of FPPn against cos[π/(n+1)]. C: Compilation of the


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absorption spectra of FPPn+• in CH2Cl2 at 22 ºC. D: Plot of energies of the absorption maxima (red dots) of the FPPn+• (νabs, Table 1) against cos[π/(n+1)]. Absorption maximum of FPP2+• was obtained by laser flash photolysis.27,28 Reproducible spectra of cation radicals of FPP4-FPP16 in dichloromethane displayed in Figure 3C were obtained by quantitative redox titrations using three different aromatic oxidants, i.e. THEO+•SbCl6– (THEO = tetrasubstituted p-hydroquinone ether, Ered1 = 0.67 V vs Fc/Fc+, λmax = 518 nm, εmax = 7300 cm-1 M-1),29 NAP+•SbCl6– (NAP = cycloannulated naphthalene derivative, Ered1 = 0.94 V vs Fc/Fc+, λmax = 672 nm, εmax = 9300 cm-1 M-1),30,31 and TRUX+•SbCl6– (TRUX = cycloannulated truxene derivative, Ered1 = 0.78 V vs Fc/Fc+, λmax = 1400 nm, εmax = 9216 cm-1 M-1), see structures in the Section 2 in the Supporting Information. As an example, absorption spectra obtained by an incremental sub-stoichiometric addition of a concentrated solution of FPP6 to a solution of NAP+•SbCl6– in CH2Cl2 are displayed in Figure 4A. Careful quantification by a simple deconvolution procedure17,32 performed at each titration point confirmed a 1:1 stoichiometry of the redox reaction, i.e. FPP6 + NAP+• → FPP6+• + NAP (Figures 4B and 4C). The cation radical spectra of other FPPn in Figure 3C were similarly generated using NAP+• and THEO+• (Figures S7-S19 in the Supporting Information). Absorption spectra of FPPn+• in Figure 3C were identical across all oxidants used (i.e. NAP+•, THEO+• and/or TRUX+•), and did not change either upon a tenfold increase in their concentration or temperature lowering to 40 ºC.33 Due to the transient nature of FPP2+•, this cation radical was generated by laser flash photolysis using photoexcited chloranil as an oxidant; it absorbs at λmax = 700 nm (Table 1).27,28 Importantly, a plot of the energies of the absorption maxima of FPP2+•–FPP16+• (Figure 3C, Table 1) showed a linear cos[π/(n+1)] dependence up to ~10-12 p-phenylene units (Figure 3D), similarly to that observed for exciton delocalization in Figure 2D. A

B

8

C 1.00

+. FPP6

-1

M cm-1 4

ε (10

●

4 +.

0.00 500

1500

2500 500

Wavelength, nm

2500

1500

● +.

FPP6

● ●

●

0

● ●

●

●

NAP

2

● ●

Mole fraction

6

(

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


●

+.

● ●

●

0.0

NAP ● ●

1.0

0

●

2.0

Equiv. of FPP6

Wavelength, nm

Figure 4. A: Spectral changes observed upon the reduction of 0.038 mM NAP+• in CH2Cl2 (3 mL) by addition of 30-μL increments of 0.25 mM solution of FPP6 in CH2Cl2. B: Deconvolution of each UV-VIS absorption spectrum from Figure A into its component spectra, i.e. NAP+• and FPP6+• (as indicated). C: Plot of the mole fractions of NAP+• (blue) and FPP6+• (red) against the added equivalents of FPP6. Symbols represent experimental points, while the solid lines show best-fit to experimental points using ΔG1 = Eox(FPP6) Ered(NAP+•) = -122 mV.17,32 Table 1. Experimental Eox1 (V vs Fc/Fc+), λabs (nm)/εabs (M-1cm-1) and λem of FPPn, and λabs (nm)/εabs (M-1cm-1) of FPPn+•. FPPn+•

FPPn n Eox1

λabs

εabs

λem

λabs

2

1.27

265

16200

308

690

4

0.90

330

51900

365

1240

40700

6

0.79

352

89900

394

1680

76100

8

0.75

363

113600

406

2012

63800

10

0.74

370

156300

414

2230

54300

12

0.74

374

171500

416

2320

45300

14

0.73

377

194000

416

2410

44100

16

0.73

378

201000

414

2410

43400


εabs


DFT modeling of FPPn/FPPn+•. The experimental absorption (νabs) and emission (νem) energies, oxidation potentials (Eox1) of neutral FPPn and absorption energies of D0→D1 transitions of their cation radicals FPPn+• (Table 1) were reproduced by electronic structure calculations (DFT/TD-DFT) of FPPn/FPPn+• using B1LYP-40/6-31G(d)+PCM(CH2Cl2) level of theory.4 Computed absorption energies (i.e. HOMO-LUMO gap from TD-DFT) of FPPn followed a linear cos[π/(n+1)] trend, while the calculated emission energies,34 oxidation energies, and D0→D1 transitions in FPPn+• showed a breakdown from cos[π/(n+1)] trend (see Section 3 in the Supporting Information for additional details). Isovalue plots and per-unit barplots of HOMOs of FPPn and the spin/charge distributions in the corresponding cation radicals (FPPn+•) obtained by DFT calculations are displayed in Figure 5.

lot

FPPn HOMOs arp

2

ep lot

un it b

4 6 8

Per -

A

u val Iso

10 12 14 16 18

2 4

un

lot ep

it b

arp

lot

FPP Spin Densities

Per -

B

+. n

u val Iso

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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6 8 10 12 14 16 18

Figure 5. A: Isosurface (0.02 au) representations of HOMOs (𝝋𝐇𝐎𝐌𝐎 = 𝒌 𝒄𝒌 𝝌𝒌 where 𝒄𝒌 is a coefficient of basis function 𝝌𝒌 ) of neutral FPPn (n = 2-18) with corresponding barplot representations of the per-unit HOMO densities calculated as 𝒒𝒎 = 𝒏 𝒄𝟐𝒎𝒏 , where m is index of p-phenylene unit and n is an index of the basis function in unit m. B: Isosurface representations (0.001 au) of the spin-density distributions of FPPn+• with corresponding per-unit barplot representations of the Natural Population Analysis (NPA)35 spin density distributions.

A close look at the HOMOs of neutral FPPn showed that they are delocalized over the entire poly-p-phenylene chain (Figure 5A). Notably, the spin/charge distribution in smaller homologues (n ≤ 8) of FPPn is similar to the HOMO distribution; however in higher homologues (n > 8) the spin/charge distribution is limited to only ~8 pphenylene units (Figure 5B). In this context, we note that the X-ray crystal structures of a large number of neutral aromatic hydrocarbons and their cation radicals have established that the spin/charge distribution and accompanying structural reorganization track in accordance with the nodal structure of HOMO, i.e. bonds with bonding HOMO lobes undergo elongations whereas the bonds with antibonding lobes undergo contractions.36 Indeed, an examination of the structural reorganization in FPPn+•, i.e. elongation and contraction of C-C bonds and decrease in dihedral angles between the adjacent fluorenes, collectively referred to as a quinoidal distortion (Figure 6A), clearly tracks the spin/charge distributions (Figures 6B and 6C), but not the expected HOMO distribution in neutral FPPn (Figure 5A).


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A

Structural reorganization (quinoidal distortion)

H

H

H

B

H

Structural Reorganization

C

Charge

2

2

4

4

6

6

8

8

10

10

12

12

14

14

16

16

18

18

Figure 6. A: Schematic representation of the quinoidal distortion induced by 1-e- oxidation. B: Per-unit barplot representation of the charge-density distribution, which is identical to the spin-density distribution in Figure 5B. C: Per-unit barplot representation of the distribution of the quinoidal distortion, see Section 3 in the Supporting Information for details. The lack of close correspondence between the HOMO and spin/charge density distributions arises due to the interplay between the energetic gain from the hole delocalization and the countervailing energetic penalty from the structural/solvent reorganization in longer FPPn+• wires. As such, this suggests that the precursor HOMOs for the FPPn+• must be different from the original HOMOs of neutral FPPn. Indeed, analysis of the electronic structures of neutral FPPn at the cation-radical (structurally reorganized) geometries shows that their HOMO densities closely follow the spin/charge and structural reorganization distributions in FPPn+• (compare Figures 5, 6 and 7).

HOMOs

2 4 6 8

Per -

lot

ep

un it

ba

rpl

ot

neutral FPPn @ cation radical geometry u val Iso

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10 12 14 16 18

Figure 7. C: Isosurface (0.02 au) representations of HOMOs of neutral FPPn (n = 2-18) calculated at the equilibrium geometries of FPPn+• and corresponding barplot representations of the per-unit HOMO densities.

As the HOMO distribution in π-conjugated poly-p-phenylene wires and the cos[π/(n+1)] dependence of their properties can be accounted for using simple tight-binding approaches such as Hückel molecular orbital (HMO) theory, we examined the HOMOs of original and reorganized FPPn wires in the context of HMO theory, in order to uncover the underlying cause of the breakdown of the cos[π/(n+1)] dependence of the properties of FPPn+• cation radicals, as follows.

Hückel Molecular Orbital (HMO) Theory Originally, HMO theory was developed to provide energies and a composition of the molecular orbitals (MOs) in π-conjugated hydrocarbons, where each MO was represented as a linear combination of the atomic orbitals (AOs).37-40 The energies of the MOs are determined from the energies of the AOs (α) and resonance integral (or electronic coupling) β between each pair of the adjacent AOs, while the interactions between non-adjacent AOs are neglected. For example, Figure 8A depicts the prediction of HMO theory (eq. 1), where the evolutions of HOMO and LUMO energies follow cos[π/(n+1)] trend with n being the number of atomic orbitals. 𝜀!"#" = 𝛼 − 2𝛽 cos

! !!!

eq. 1



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

A

B

Figure 8. A: Schematic representation (𝛼 = 0 and 𝛽 = 1) of the evolution of HOMO and LUMO energies with n. B: Schematic representation (𝛼 = 0 and 𝛽 = 1) of the HOMO-LUMO gap, which provides the energy of absorption maximum (νabs), against cos[π/(n+1)]. Furthermore, the HOMO-LUMO energy gap that provides the energies of the electronic absorption maxima (νabs) also follows the linear cos[π/(n+1)] trend (eq. 2 and Figure 8B), which is often approximated as the 1/n trend (see Section 4 in Supporting Information for additional details).9,41-46 𝜈!"# ≈ 𝜀!"#$ − 𝜀!!"! ∝ cos

! !!!

eq. 2

Indeed, the HOMO-LUMO energy gap obtained from TD DFT calculations (Figure S21 in the Supporting Information) and the experimental absorption maxima of neutral FPPn follow a linear cos[π/(n+1)] trend (Figure 2B, Table 1), suggesting that HOMO/LUMO are delocalized up to the polymeric limit.47-53 However, a linear evolution of the experimental oxidation potentials against cos[π/(n+1)] trend was not observed (Figure 3B, Table 1). Moreover, as shown above, oxidation-induced structural/solvent reorganization leads to a mismatch between HOMO of neutral FPPn and spin/charge density distributions in longer (n > 8) FPPn+• wires (compare Figures 5, 6 and 7). Thus, the energies of the precursor HOMOs for neutral FPPn at cation radical geometry are different than at the neutral geometry. In the context of the Koopmans’ theorem36,54-56 one should expect a linear evolution of HOMO energies (eq. 1 and Figure 8A) with vertical ionization energies – not with adiabatic oxidation energies. Thus, a description of both FPPn and FPPn+• within the HMO theory/Koopmans’ paradigm would require appropriate modifications to account for the structural/solvent reorganization accompanying 1-eoxidation. Application of the HMO theory to FPPn. In order to describe properties of the FPPn wires within the framework of HMO theory, MOs of FPPn are represented as a linear combination of the HOMOs of each monomeric p-phenylene unit, an approach referred to hereafter as a coarse-grained HMO theory. Unlike the simple poly-p-phenylene wires (RPPn, Figure 1), in FPPn wires the dihedral angle between p-phenylenes within one fluorene (~0.08 ± 0.05°) and adjacent fluorenes (~37.15 ± 0.11°) are different and, therefore, the values of the coupling (𝛽) will alternate for each pair of the adjacent p-phenylenes as depicted in Figure 9. A H H

B

Figure 9. A: The structure of FPP4 with the highlighted alternation in the dihedral angles. B: Isosurface (0.02 au) of ‘bisallylic’ HOMO of benzene and its HOMO energies (α) used to model the monomeric units in FPPn. The alternating couplings between the monomeric units are shown as βin and βout. The interactions between p-phenylenes, following the Hückel’s approximations, can be represented using the tight-binding 𝑛×𝑛 Hamiltonian matrix:


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𝛼 𝛽!" 0 ⋯ 0 0 0 𝛽!" 𝛼 𝛽!"# ⋯ 0 0 0 0 𝛽!"# 𝛼 ⋯ 0 0 0 ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ , eq. 3 𝐇= ⋮ 0 0 0 ⋯ 𝛼 𝛽!"# 0 0 0 0 ⋯ 𝛽!"# 𝛼 𝛽!" 0 0 0 ⋯ 0 𝛽!" 𝛼 where 𝛼 is the HOMO energy of the p-phenylene unit, βin is the coupling between two p-phenylenes within a fluorene and βout is the coupling between p-phenylenes in two adjacent fluorenes (Figure 9A). Numerical diagonalization of the Hamiltonian 𝐇 will produce the MO energies 𝜀! (i.e. eigenvalues) and the corresponding MO density distributions (i.e. eigenvectors), where the eigenvalue with the smallest magnitude corresponds to the HOMO energy 𝜀!"#" , while the coefficients 𝑐!! in the eigenvector represent HOMO densities at the corresponding units (Figure 10A).

A Hückel Hamiltonian matrix

A linear combination of monomeric HOMOs

HOMO energy HOMO density

B

Hückel Molecular Orbital Theory of FPPn

Figure 10. A: Numerical diagonalization of the Hückel Hamiltonian matrix provides HOMO energy and HOMO density distribution as a linear combination of the monomeric HOMOs on the example of FPP4. B, left: Couplings βin and βout are obtained as a splitting between ‘bisallylic’ MOs from the angular scan of biphenyl. B, right: Linear dependence of HOMO energies of FPPn against cos[π/(n+1)] and their HOMO density distributions shown as per-unit barplot representations. The HOMO energy for each p-phenylene unit in FPPn for the construction of Hückel Hamiltonian matrix was used as the HOMO energy of benzene,57 i.e. α = -7.4 eV. Couplings βin = -1.0 eV and βout = -0.8 eV were estimated as a splitting between ‘bisallylic’ MOs from the angular scan of biphenyl (Figure 10B, left and see Section 5 in the Supporting Information for additional details). Numerical diagonalization of the resulting Hückel Hamiltonian matrices provided the HOMO energies (eigenvalues) and HOMO densities (eigenvectors) for various FPPn, which were in excellent agreement with those obtained from the DFT calculations (compare Figures 10B, right and 5A). The observation of the linear cos[π/(n+1)] dependence of the HOMO energies (Figure 10B right) and energies of absorption maxima (νabs in Table 1, Figure 2B) are consistent with the prediction of the HMO theory. In sharp contrast, however, oxidation potentials of FPPn show a breakdown when plotted against cos[π/(n+1)] (Figure 3B). The cause of such breakdown lies in the fact that the HMO theory does not account for the structural/solvent reorganization accompanying 1-e- oxidation. We hypothesized that an accounting of the reorganizational energy in the Hückel Hamiltonian matrix should allow an accurate description of the redox properties of FPPn. It is expected that the energy of HOMO of a neutral aromatic donor at the (cation-radical) reorganized geometry will be elevated by a certain value Δα as shown on the example of benzene and its cation radical (Figure 11A). Unlike the benzene cation radical, which bears one full charge, in FPPn+• the amount of charge is different



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for each p-phenylene unit (Figure 6B). Thus, the HOMO energies (αi) for p-phenylene units in neutral FPPn at the reorganized geometries were elevated by qiΔα, where qi is the value of the charge at the ith unit (Figure 11B). [Note that the DFT calculations have demonstrated that the charge, spin and normalized structural reorganization in each p-phenylene unit in FPPn+• show identical distributions (Figures 5 and 6).]

A

B

neutral FPP4 at cation radical geometry

0.0 H H

neutral geometry

cation radical geometry

Figure 11. A: Comparison of HOMO energies of neutral benzene at neutral geometry with neutral benzene at cation radical geometry. B: Schematic representation of the structural reorganization of FPP4 upon 1-e- oxidation with the highlighted alternation in the dihedral angles and corresponding elevations in the HOMO energies of the monomeric units. Thus, the Hückel Hamiltonian matrix was modified (eq. 4) to take into account structural reorganization by elevated HOMO energies (αi) of various p-phenylene units and different couplings (βout) owing to the reduced dihedral angle in oxidized FPPn (Figure 11B). 𝛼! 𝛽!" 0 ⋯ 0 0 0 𝛽!" 𝛼! 𝛽!"# 𝜃 ⋯ 0 0 0 0 𝛽!"# 𝜃 𝛼! ⋯ 0 0 0 ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ , eq. 4 𝐇= ⋮ 0 0 0 ⋯ 𝛼!!! 𝛽!"# 𝜃 0 0 0 0 ⋯ 𝛽!"# 𝜃 𝛼!!! 𝛽!" 0 0 0 ⋯ 0 𝛽!" 𝛼! A numerical diagonalization of the resulting Hamiltonian matrix in eq. 4 provided the HOMO energies and HOMO densities for FPPn at reorganized geometries. Figure 12A shows that HOMO energies (eq. 4, eigenvalues) at reorganized geometries show a deviation from the linearity, whereas the HOMO energies (eq. 3, eigenvalues) at non-reorganized (neutral) geometries follow a linear dependence against cos[π/(n+1)]. Indeed, the HOMO energies obtained from the modified Hamiltonian matrix (eq. 4, eigenvalues) follow a linear trend with the experimental oxidation potentials of FPPn (Figure 12B, red line), attesting that the Koopmans’ paradigm does hold true if the structural reorganization accompanying 1-e- oxidation is appropriately accounted for in HMO theory.

A

neutral FPPn @ neutral geometry

neutral FPPn @ CR geometry

B

neutral FPPn @ CR geometry

neutral FPPn @ neutral geometry

Figure 12. A: Evolution of HOMO energies of neutral FPPn using Hückel Hamiltonian without reorganization (i.e. @ neutral geometry) in eq. 3 (blue) and neutral FPPn using modified Hückel Hamiltonian that includes reorganization (i.e. @ cation-radical geometry) in eq. 4 (red). B: Plot of the HOMO energies from Figure 12A against the experimental oxidation potentials (Eox1) of FPPn. Thus, we demonstrated that incorporation of the structural/solvent reorganization, available from the DFT calculations, into the HMO theory allows an accurate prediction of the evolution of the oxidation potentials of πconjugated poly-p-phenylene wires (e.g. FPPn) as well as the HOMO density distributions, which parallel the spin/charge density distributions obtained from DFT calculations. In this context, it is noteworthy that the


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Marcus theory, by its construction, takes into account the structural/solvent reorganization and electronic coupling in a parameterized form without the necessity of DFT calculations. Accordingly, next, we examine the applicability of the Marcus-based58-61 multistate model (MSM)4 for the description of FPPn+• and compare with the modified HMO approach.

Multistate Model The multistate model (MSM) utilizes a bell-shaped composite quadratic/reciprocal function (rather than a simple parabolic function as in the Marcus two-state model) for the realistic description of the charge distribution along the reaction coordinate 𝑥 (Figure 13A).62 The application of MSM to FPPn+• required a simple !" !" modification where alternating couplings 𝐻!" and 𝐻!"# were used to account for the different dihedral angles between p-phenylenes within a fluorene and between adjacent fluorenes (Figure 13B). l ve n A s t H + structural/solvent reorganization

H

hole

B H

n

H

Figure 13. A: Schematic representation of the bell-shaped diabatic state, where structural/solvent reorganization is fixed at the single p-phenylene and the position of the hole is varied along the reaction coordinate 𝑥. In the vicinity of the reorganized unit H(x) is quadratic and at larger separation distances it follows Coulomb law of !" electrostatic interaction. B: Representation of several diabatic states in FPPn+• with alternating couplings 𝐻!" and !" 𝐻!"# . Thus, the MSM Hamiltonian matrix applicable to FPPn+• can be expressed as a tight-binding 𝑛×𝑛 matrix that depends on the reaction coordinate 𝑥 (eq. 5). Numerical diagonalization of the Hamiltonian matrix 𝐇 𝑥 for each 𝑥 results in the adiabatic potential energy surface with the lowest-energy surface 𝐸! 𝑥 corresponding to the ground state of FPPn+•. The minimum on the ground state surface 𝐸! 𝑥 defines the position 𝑥!"# of the center of the hole distribution and the energy at this point 𝐸! 𝑥!"# directly corresponds to the oxidation energy of FPPn. 𝐻! 𝑥 !" 𝐻!" 0 𝐇 𝑥 = ⋮ 0 0 0

!" 𝐻!" 𝐻! 𝑥 !" 𝐻!"# ⋮ 0 0 0

0 !" 𝐻!"# 𝐻! 𝑥 ⋮ 0 0 0

⋯ ⋯ ⋯ ⋱ ⋯ ⋯ ⋯

0 0 0 ⋮ 𝐻!!! 𝑥 !" 𝐻!"# 0

0 0 0 ⋮ !" 𝐻!"# 𝐻!!! 𝑥 !" 𝐻!"


0 0 0 , eq 5 ⋮ 0 !" 𝐻!" 𝐻! 𝑥


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!" Numerical solution of the MSM Hamiltonian matrix (eq. 5) for FPPn+• with parameters 𝜆 = 2, 𝜆! = 12, 𝐻!" = !" 19, 𝐻!"# = 15 provided the oxidation energies (eigenvalues), vertical D0→D1 excitation energies and hole distributions (eigenvectors) in the ground and excited state (see Section 6 in the Supporting Information for additional details). The plots of the oxidation energies obtained by MSM against the cos[π/(n+1)] trend reproduce the breakdown from the linearity for n > 8 as observed with experimental oxidation potentials of FPPn (compare Figures 14A and 3B). Moreover, the hole distributions in FPPn+• obtained from MSM closely resemble those obtained from DFT calculations (compare Figure 14B and 6).

A

B

+.

hole distributions in FPPn 2 4 6 8

10 12 14 16 18

Figure 14. A: Plot of oxidation energies of various FPPn+• obtained by MSM against cos[π/(n+1)] trend. Oxidation energies were scaled to reproduce experimental oxidation potentials Eox1. B: Per-unit barplot representation of the hole distributions in various FPPn+• obtained by MSM. A close examination of the potential energy surfaces of various FPPn+• obtained from the MSM (Figure 15A) reveals the presence of several minima within ~10 meV of each other, with different hole distributions extending to ~8 p-phenylene units (see insets in Figure 15B). As such, this analysis suggests that in higher homologues of FPPn+• (n > 8) multiple isomeric electronic structures are feasible. Indeed, the DFT calculations identified two almost isoenergetic electronic isomers of FPP18+• and FPP16+• with different positions of the spin/charge distribution center in the poly-p-phenylene chain in excellent agreement with MSM predictions (Figure 15C).63

A

B

+.

FPP18

C

FPP + 16

FPP + 18

Figure 15. A: Ground state surfaces of FPPn+• (n = 2-18) with respect to the reaction coordinate x from MSM. B: Zoomed-in ground state surface of FPP18+• and hole distributions shown for three distinct minima (as indicated) obtained from MSM. C: Electronic isomers of FPP18+• and FPP16+• showing their spin-density distributions as isosurface plots and per-unit barplot representations. Energies of these electronic isomers for a given FPPn+• differ only by ~0.01 kcal/mol.

Modified Hückel Molecular Orbital Theory vs Marcus- based Multistate Model Hückel molecular orbital (HMO) theory has been extensively used as an effective model to predict energies and a composition of the molecular orbitals of π-conjugated hydrocarbons.37-40 The HOMO energy obtained by HMO theory can be used to approximate vertical ionization energy based on the Koopmans’ paradigm.54,55 However, HMO theory as originally formulated fails to predict adiabatic oxidation potentials, because the oxidation-induced structural/solvent reorganization is not accounted for. Here, on the example of the fluorene-based poly-p-phenylene cation radicals, we have introduced a modified HMO theory, where structural/solvent reorganization was incorporated by adjusting the HOMO energies (α) of interacting (via electronic coupling 𝛽) monomeric p-phenylenes with the aid of DFT calculations (Figure 16A). The degree of the HOMO energy elevation (Δα) of each unit was set in proportion to the amount of charge at the corresponding p-phenylene unit using benzene as a model monomeric unit. This modified HMO model


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provided the HOMO energies and density distributions and, in turn, predicts the evolution of the oxidation potentials of FPPn. Modified Hückel Molecular Orbital Theory A

H H

electronic coupling

H H

neutral FPPn at cation radical geometry

B

Multi-State Model

H H

electronic coupling

HOMO density

H H

FPP + n

charge density

Figure 16. Schematic representations of the modified Hückel Molecular Orbital theory (A) and Marcus-based multistate model (B). In this context, it is significant to note that the modified HMO model provides similar evolution of the oxidation potentials of FPPn, as the Marcus-based multistate model (MSM), which by its design, takes into account structural/solvent reorganization as a function of the charge transfer coordinate x with empirically adjusted parameters, i.e. reorganization energy (λ) and electronic coupling (Hab). An extension of the two-state model58-60 into a multistate model using the interaction amongst several diabatic states provides position and distribution of the hole and stabilization energy in FPPn+• (Figure 16B). We have, thus, demonstrated that the modified HMO model (where reorganization energy is introduced with the aid of DFT calculations) and MSM (where reorganization energies and electronic couplings parameters are adjusted until the experimental properties and/or spin/charge distributions in FPPn+• are reproduced) both provide an accurate description of the evolution of the redox properties and the charge distribution of the poly-p-phenylene cation radicals. This comparative analysis presented above highlights the importance of the structural/solvent reorganization in limiting the hole distribution to ~8 p-phenylene units in long poly-p-phenylene wires and, thereby, accounts for the breakdown of their redox and optical properties against cos[π/(n+1)] trend.

Conclusions In this contribution, we have examined the evolution of the redox and optical properties of the π-conjugated poly-p-phenylene wires, using fluorene-based poly-p-phenylene (FPPn) wires, where n = 2-16. Both experimental data and DFT clearly demonstrate that the absorption energies νabs of neutral FPPn follow a linear trend against cos[π/(n+1)] up to the polymeric limit, while oxidation potentials Eox, demonstrate a breakdown from the linearity for large n. In parallel, DFT calculations demonstrate that the HOMO densities of neutral FPPn are delocalized over the entire chain, while the spin/charge and structural reorganization in FPPn+• extend only to ~8 p-phenylene units. Application of the coarse-grained Hückel Molecular Orbital (HMO) theory on the example of neutral FPPn provides HOMO-LUMO energy gaps (or the energies of absorption maximum), which follow a linear cos[π/(n+1)] trend up to the polymeric limit. However, HOMO energies (i.e. vertical ionization energies) from HMO theory fail to reproduce evolution of the adiabatic oxidation potentials, because HMO theory, in its original formulation, does not account for the structural/solvent reorganization that accompanies 1-e- oxidation. To account for the structural/solvent reorganization in FPPn, we introduce reorganization energy into a HMO theory by elevating HOMO energies of each p-phenylene unit proportionally to the amount of charge at the unit in the corresponding cation radical (FPPn+•) with the aid of DFT calculations. Once the structural/solvent reorganization is included, the HOMO energies and HOMO density obtained from the modified HMO theory become consistent with the evolution of the experimental adiabatic oxidation potentials of FPPn. Thus, by contrasting the results from original and modified HMO theories against experimentally observed oxidation potentials, we have shown that the origin of the cos[π/(n+1)] trend breakdown, which limits the hole distribution to ~8 p-phenylene units, is due to the structural/solvent reorganization that accompanies 1-eoxidation. Furthermore, the modified HMO model predictions are fully consistent with our recently developed Marcus-based multistate (MSM) to FPPn+•, which by its construction, accounts for the structural/solvent reorganization. Thus, Huckel theory which incorporates reorganization energy equals Marcus-Hush Theory. We



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believe that the modified HMO model can serve as an effective tool for understanding the fundamental properties of poly-p-phenylene wires and will aid toward the rational design of the next-generation long-range charge and energy transfer materials for photovoltaics applications.

ASSOCIATED CONTENT Supporting Information. Experimental and computational details; coordinates and energies of the calculated structures. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Author * E-mail: [email protected]

Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

ACKNOWLEDGMENT We thank the NSF (CHE-1508677) and NIH (R01-HL112639-04) for financial support and Professors Scott A. Reid and Qadir K. Timerghazin for helpful discussions. The calculations were performed on the high-performance computing cluster Père at Marquette University funded by NSF awards OCI-0923037 and CBET-0521602, and the Extreme Science and Engineering Discovery Environment (XSEDE) funded by NSF (TG-CHE130101).

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(47) The observation that (experimental) absorption νmax of FPP108 falls onto the cos[π/(n+1)] trend suggests that HOMO/LUMO in large FPPn are delocalized up to a polymeric limit possibly due to a narrow distribution of the dihedral angles 0-37o which allow sufficient electronic coupling and should be contrasted with poly-phenylenevinylenes where multiple dihedral angles between vinylenic and aromatic chromophores are responsible for localization of the HOMO/LUMO, see Refs. 47-52.



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(48) Grimm, S.; Tabatabai, D.; Scherer, A.; Michaelis, J.; Frank, I. Chromophore Localization in Conjugated Polymers: Molecular Dynamics Simulation. J. Phys. Chem. B. 2007, 111, 12053-12058. (49) Qin, T.; Troisi, A. Relation Between Structure and Electronic Properties of Amorphous MEH-PPV Polymers. J. Am. Chem. Soc. 2013, 135, 11247-11256. (50) Jackson, N. E.; Savoie, B. M.; Kohlstedt, K. L.; Marks, T. J.; Chen, L. X.; Ratner, M. A. Structural and Conformational Dispersion in the Rational Design of Conjugated Polymers. Macromolecules. 2014, 47, 987-992. (51) Woo, H. S.; Lhost, O.; Graham, S. C.; Bradley, D. D. C.; Friend, R. H.; Quattrocchi, C.; Brédas, J. L.; Schenk, R.; Müllen, K. Optical Spectra and Excitations in Phenylene Vinylene Oligomers. Synth. Met. 1993, 59, 13-28. (52) Camposeo, A.; Pensack, R. D.; Moffa, M.; Fasano, V.; Altamura, D.; Giannini, C.; Pisignano, D.; Scholes, G. D. Anisotropic Conjugated Polymer Chain Conformation Tailors the Energy Migration in Nanofibers. J. Am. Chem. Soc. 2016, 138, 15497-15505. (53) Padmanaban, G.; Ramakrishnan, S. Fluorescence Spectroscopic Studies of Solvent-and Temperature-induced Conformational Transition in Segmented Poly [2-methoxy-5-(2'-ethylhexyl) Oxy-1, 4-phenylenevinylene](MEHPPV). J. Phys. Chem. B. 2004, 108, 14933-14941. (54) Koopmans, T. Über Die Zuordnung Von Wellenfunktionen Und Eigenwerten Zu Den Einzelnen Elektronen Eines Atoms. Physica. 1934, 1, 104-113. (55) Richards, W. G. The Use of Koopmans' Theorem in the Interpretation of Photoelectron Spectra. Int. J. Mass. Spectrom. Ion. Phys. 1969, 2, 419-424. (56) Savoie, B. M.; Jackson, N. E.; Marks, T. J.; Ratner, M. A. Reassessing the Use of One-electron Energetics in the Design and Characterization of Organic Photovoltaics. Phys. Chem. Chem. Phys. 2013, 15, 4538-4547. (57) Alternatively, HOMO energy (α) of a monomeric unit could be approximated as the average between HOMO and HOMO-3 of FPP2. (58) Marcus, R. A. Theory of Oxidation-reduction Reactions Involving Electron Transfer. 4. A Statistical-mechanical Basis for Treating Contributions From Solvent, Ligands, and Inert Salt. Discuss. Faraday. Soc. 1960, 29, 21-31. (59) Hush, N. S. Intervalence-transfer Absorption. Part 2. Theoretical Considerations and Spectroscopic Data. Prog. Inorg. Chem. 1967, 8, 12. (60) Robin, M. B.; Day, P. Mixed Valence Chemistry-a Survey and Classification. Adv. Inorg. Chem. Radiochem. 1968, 10, 247-422. (61) 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. (62) Talipov, M. R.; Ivanov, M. V.; Rathore, R. Inclusion of Asymptotic Dependence of Reorganization Energy in the Modified Marcus-Based Multistate Model Accurately Predicts Hole Distribution in Poly-p-phenylene Wires. J. Phys. Chem. C. 2016, 120, 6402-6408. (63) The dynamic interconversion between FPPn+• isomers could contribute to the additional stabilization as indicated by the non-zero slope in the plot of almost invariant oxidation potentials of FPPn (n > 8) (Figure 3B) and may play an important role in the incoherent charge-transfer mechanism along the long poly-p-phenylene wires.


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HÃ¼ckel Theory + Reorganization Energy = MarcusâHush Theory

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HÃ¼ckel Theory + Reorganization Energy = MarcusâHush Theory