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The Functional Mode Electron Transfer Theory Hanning Chen J. Phys. Chem. B, Just Accepted Manuscript • Publication Date (Web): 12 Jun 2014 Downloaded from http://pubs.acs.org on June 17, 2014

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The Journal of Physical Chemistry

The Functional Mode Electron Transfer Theory Hanning Chen* Department of Chemistry the George Washington University 725 21st Street, NW, Washington, DC 20052 Fax: 202-994-5873 Phone: 202-992-4492 Email: [email protected]

ACS Paragon Plus Environment


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Abstract A solid approach has been developed to ascertain the correlation of electron transfer with molecular vibration in a quantitative manner. Specifically, the reaction coordinate is identified by maximizing the linear Pearson’s correlation coefficient between atomic displacement and diabatic energy gap. In the limit of fast molecular vibration, the rates of electron transfer driven by multiple vibrational modes have been derived respectively under the strong and weak vibronic coupling conditions. Our functional mode electron transfer theory is then justified by investigating the electron transfer of a betaine-30 molecule from its first excited state to its ground state when being solvated in glycerol triacetate. Among the 210 available vibrational modes of betaine-30, only 7 are essential to the electron transfer by cumulatively accounting for more than 60% of the total reorganization energy. Since all essential vibrational modes are significantly faster than thermal fluctuation, the electron transfer is primarily driven by intramolecular quantum tunneling. Interestingly, the calculated reaction driving force of 1.95 eV is substantially greater than the reorganization energy of 0.58 eV, placing the reaction in the inverted Marcus region. Nevertheless, a sizable Franck-Condon factor of 1.58×10-3 eV-1 is still achieved due to the large vibronically weighted zero-point energy of the essential vibrational modes. After determining the electronic coupling strength as 0.14 eV by the constrained density functional theory, the overall electron transfer rate at 300 K is found to be 0.30 ps-1, which agrees nearly perfectly with experimental values.

Keywords:

electron transfer, vibronic coupling, functional mode analysis, QM/MM

simulations, constrained density functional theory, tight-binding density functional theory

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1. Introduction: Electron transfer is an essential indicator of a chemical reaction, which usually leads to oxidation state change in all reactants. An interesting example that even conserves the reaction stoichiometry is the self-exchange electron transfer between two hexaaquaruthenium complexes:

[Ru(H 2O)6 ]2+ + [Ru(H 2O)6 ]3+ → [Ru(H 2O)6 ]3+ + [Ru(H 2O)6 ]2+

(1)

At 298.15K, the reaction occurs rapidly1 at a rate of 20 M −1s −1 without the need of any external stimulus such as light irradiation. Since the water exchange on both hexaaquaruthenium complexes is significantly outpaced by the electron transfer,2 the process can be regarded as an unambiguous outer-sphere redox reaction that involves charge transfer between two non-bonded chemical species. Therefore, it is not surprising that its rate, ket , nearly perfectly conforms with the Marcus theory:3 − 2π 2 1 ket = J e ! 4πλ kBT

( λ +ΔG0 )2 4 λ kBT

(2)

where kB is the Boltzmann constant, T is the temperature, J is the electronic coupling strength, λ is the reorganization energy, and ΔG 0 is the driving force, which is zero for self-exchange

reactions.

Apparently,

the

facile

charge

self-exchange

in

hexaaquaruthenium aqueous solution can be ascribed to a small λ , which in turn stems from the structural similarity between [Ru(H 2O)6 ]2+ and [Ru(H 2O)6 ]3+ . In the latter, the

Ru-O bond length is 2.03Å and is slightly elongated to 2.12Å upon one-electron addition,4 thus substantially diminishing the energy penalty due to the mismatch between solute charge distribution and solvent dielectric response. In contrast to the abovementioned solvent-aided electron transfer mechanism, some redox centers can

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undergo notable structural adaptations to accompany charge redistribution within themselves. One famous example of the so-called inner-sphere electron transfer is the Creutz-Taube ion:5 ⎡⎣ Ru ( NH 3 )5 − N 2C 4 H 4 − Ru ( NH 3 )5 ⎤⎦

5+

(3)

wherein two ruthenium complexes are bridged by a pyrazine ligand that serves as an electron shuttle to provoke an ultrafast interchange between ⎡⎣ Ru ( NH 3 )5 ⎤⎦ ⎡⎣ Ru ( NH 3 )5 ⎤⎦

2+

3+

and

, eventually resulting in a completely delocalized mixed-valence

compound.6 As shown by a reflectance infrared spectroelectrochemical study on linked ruthenium clusters,7 the electron transfer in a molecular system can be primarily driven by its intramolecular vibrations, which are typically much faster than the solvent reorganization. For instance, the ket in a pyrazine-bridged triruthenium complex with 4dimethylaminopyridine ligands nearly reaches its adiabatic limit,7 which is also the nuclear frequency factor. If the driving vibrational modes (the ones that couple most strongly with a given electron transfer event) are even faster than the random thermal motion, i.e., !ω vib ≫ k BT , the semi-classical Marcus theory3 is no longer applicable and the Jortner formula8 must be adopted by accounting for the full quantization of nuclear wavefunction. Under the approximation of single driving vibrational mode, the Jortner formula for strong vibronic coupling, i.e., λ ≫ "ω vib , is given by:8 ( ΔG

+λ )

2π J 2 − 2 λ!0ω vib ket = e ! λ !ω vib

2

(4)

whereas in the limit of weak vibronic coupling, i.e., λ ≪ "ω vib , it is analogous to the radiationless energy-gap law:8

3


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ket =

2π J 2

! − ( ΔG0 + λ ) !ω vib

e

−

λ !ω vib

( ΔG0 + λ ) ⎛ ln⎛ −( ΔG0 + λ ) ⎞ −1⎞

e

!ω vib ⎜⎝ ⎜⎝

λ

⎠⎟

⎟ ⎠

(5)

In general, the electron transfer in condensed phases is driven by both solvent reorganization and intramolecular vibration. For some molecules, the relative importance of these two mechanisms can be readily modulated by varying their solvents. For example, the charge separation rate, ket , in betaine-30 (B30) after photoexcitation (Fig. 1) was found to be 2.0 ps −1 when being solvated by the small and thus swift acetonitrile molecules, whose average relaxation rate, τ s , is also as fast as 2.0 ps −1 .9 The excellent agreement between ket and τ s reveals that the electron transfer in the B30 dyad is mainly controlled by the solvation dynamics of acetonitrile. By contrast, when the bulky and thus sluggish glycerol triacetate (GTA) is selected as solvent, ket is at least ten times faster than τ s across a wide rage of temperatures,9 suggesting the shift of electron transfer mechanism to the intramolecular vibrations, whose fitted characteristic wavenumbers are nearly temperature invariant. More interestingly, a later picosecond time-resolved antiStokes Raman study10 on GTA solvated B30 unambiguously identified a vibrational mode at 1600 cm-1 whose rise time of 3.8 ps is in line with the electron transfer characteristic time of 3.5 ps at room temperature, further justifying the predominance of intramolecular vibrations of a dyad molecule in electron transfer mechanism when slowly relaxing solvents are used. In fact, even when the GTA solvent molecules are nearly entirely frozen by lowering the temperature to 228 K, which is approximately 50 K below the melting point of GTA, the electron transfer in B30 can still proceed at an appreciable rate of 0.18 ps −1 ,9 apparently through vibrational quantum tunneling.

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Historically, the significant coupling of electron transfer to intramolecular vibrational modes was called “phase phonons”,11 whose spectroscopic and kinetic effects were investigated by Raman scattering,12 radiative charge recombination13 and holetransfer absorption.14 With the advance of ultrafast laser-based spectroscopic techniques,15 it is now feasible to monitor molecular vibrational dynamics of excited-state species on a real time basis.16 In a recent femtosecond stimulated Raman study on a donor-bridge-acceptor dyad molecule,17 several new ring-stretching modes between 1000 and 1700 cm-1 were discovered on the acceptor moiety upon photoexcitation. Since all those new vibrational modes are parallel to the long molecular axis, they are believed to be the primary driving force for the charge recombination processes. In another surfaceenhanced Raman scattering experiment on interfacial charge transfer between a dye molecule and a TiO2 semiconductor surface, a large number of bending and torsional vibrational modes mostly involving TiO2 atoms were found to exhibit substantial Raman activity enhancement after photocurrent is detected.18 Therefore, a perspicuous correlation between molecular vibration and electron transfer is of particular interest to the solar energy research community who are striving to minimize the wasteful thermal dissipation of harvested photonic energy in order to achieve the thermodynamic limit of solar energy conversion efficiency as predicted by the Shockley-Queisser detailed balance analysis.19 In addition, a better understanding of the vibronic coupling may also greatly benefit many fields in biomedical research, such as ultraviolet-induced DNA repair,20 bimodal diagnostic imaging,21 and in vivo monitoring of heme perturbation.22 Most recently, the quantized phonons of trapped ions have even been exploited to construct high-fidelity quantum bits whose vibronic states can be readily manipulated by

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a laser beam,23 laying the foundations for the next-generation quantum information processing. In this novel approach,23 the number of possible values that can be stored in a single quantum bit is only limited by its available vibrational states up to the anharmonicity limit, thus substantially reducing the transistor size and power consumption compared to silicon-based semiconductors. In light of the importance of the interplay between nuclear motions and electronic transitions, we will present a statistically reliable and computationally efficient method, namely the functional mode electron transfer theory (FMET), to ascertain their correlations in condensed phases. The remainder of this paper is organized as follows. In section 2, the protocol to construct diabatic states by the constrained density functional theory24 will be introduced. Then, the diabatic energy gaps of a sufficient number of snapshots extracted from molecular dynamics trajectory will be projected onto a selected set of vibrational modes to identify their relative contributions to the electron transfer reorganization energy by means of functional mode analysis (FMA).25 Subsequently, both the reorganization energy and driving force will be calculated by the thermodynamic integration26 before the relative strength of vibronic coupling is determined for the evaluation of overall electron transfer rate. In section 3, our proposed FMET will be justified by numerical simulations on the GTA-solvated B30 dyad model system (Fig. 2) through the comparison with experimental observations and other theoretical studies. Finally, we will summarize our results and briefly discuss possible applications of our theory in section 4.

2. Theoretical Derivations of the Functional Mode Electron Transfer Theory:

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According to the Holstein polaron theory,27 the strong electron-phonon interaction results in the formation of self-trapped polarons whose electronic wavefunctions are well localized on a lattice site. If the hopping between two neighboring polaron sites is much slower than the vibration of the driving phonon mode, i.e., γ ≪ ω vib , the charge transfer must proceed in a diabatic manner that accounts for the recrossing of dividing surface.28 A computationally convenient way to construct diabatic states with localized charges is the constrained density functional theory (CDFT),24 which adds a position-dependent Hartree potential to the original Kohn-Sham Hamiltonian to achieve a desired charge distribution. For example, the charge distribution in the ground diabatic state of B30, S0 , can be expressed as C ( S0 ) = C pyr − Cola = Cg

(6)

where C pry and Cola are the charges of the 2,4,6-triphenylpyridinio moiety and the 2,6dipheyl-phenolate moiety (Fig. 1), respectively. Please note that the value of Cg can be easily determined from adiabatic calculation, and it may vary with different levels of quantum theory and charge population analysis methods. Similarly, the charge distribution in the excited diabatic state, S1 , is given by C ( S1 ) = C pyr − Cola = Cg − 2

(7)

to reflect the photo-induced charge recombination. ! For a given nuclear configuration, r , its diabatic energy gap, ΔED , is defined as:

! ! ! ΔED ( r ) = ES1 ( r ) − ES0 ( r )

(8)

! ! where ES1 ( r ) and ES0 ( r ) are the energies of the S1 and S0 states, respectively. It turns

out that the variance of ΔED along a vibrational mode can be used to probe the mode’s

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correlation with electron transfer. For instance, the variance is zero for an entirely uncorrelated mode, whereas it is maximized in the case of single driving mode. In this sense, the reorganization energy can be projected onto all vibrational modes under a linear approximation: N vib

N vib

i=1

i=1

λ = ∑ λi = ∑ ci2 λ

(9)

where the coefficient, ci2 , indicates the relative contribution of the i th vibrational mode,

!" V i , to the degeneracy of vibronic energy as demanded by a radiationless transition . Inspired by the FMA method,25 ci2 , can be systematically determined by identifying the maximally correlated motion (MCM) vector: N vib !" " V MCM = ∑ ciVi

(10)

i=1

through the maximization of the Pearson’s correlation coefficient, R , which is defined as: R=

(

cov pV!" MCM (t), ΔED (t)

σ V!"

MCM

σ ΔED

)

(11)

! ! where pV!" MCM (t) is the projection of instantaneous nuclear configuration r ( t ) on VMCM ,

" " !" i.e., pV!" MCM (t) = r (t) − r i V MCM , ΔED ( t ) is the instantaneous diabatic energy gap,

(

(

)

)

cov pV!" MCM (t), ΔED (t) is the covariance function between pV!" MCM (t) and ΔED ( t ) , and their standard deviations are denoted as σ V!"

MCM

and σ ΔED , respectively. Mathematically, the

search for the maximum value of R , is simplified to solving the following coupled linear equations:

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∑ c cov( p N vib

i

!" V i (t )

)

(

)

, pV!" j (t ) = cov ΔED (t), pV!" j (t ) , j = 1,..., N vib

i=1

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(12)

With ci2 in our hands, we are in a good position to derive ket driven by multiple vibrational modes. According to the Fermi’s golden rule,29 the radiationless transition rate is given by: ket =

2π 2 J F ( 0 ) !

(13)

where F ( E ) is the energy-dependent line shape function that can be expressed as the Fourier transform of the time-dependent generating function, f (t) : +∞

F(E) =

iEt − 1 ! f (t)e dt ∫ 2π ! −∞

(14)

Under the approximation of harmonic oscillations,8

log f (t) =

N vib N vib iΔG0t Nvib − ∑ Dk (2 n" k + 1) + ∑ Dk (n" k + 1)eiω kt + ∑ Dk n" k e−iω kt ! k=1 k=1 k=1

(15)

where ω k , Dk and n! k are the angular frequency, Huang-Rhys parameter and thermally averaged vibrational quanta of the i th vibrational mode, respectively. If a vibrational mode is much faster than thermal fluctuation, i.e, !ω vib ≫ k BT , n! k =

1 e

"ω k /kBT

−1

≈0

(16)

Then, within a short time limit, Eq. 15 can be simplified as: log f (t) =

N iΔG0t Nvib iΔG0t Nvib ⎛ 1 ⎞ i ( ΔG0 + λ ) t 1 2 vib + ∑ Dk ( eiω kt − 1) ≈ + ∑ Dk ⎜ iω k t − ω k2t 2 ⎟ = − t ∑ Dkω k2 ⎝ ⎠ ! ! 2 ! 2 k=1 k=1 k=1

and its Fourier transform yields

9


(17)

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1 ⎛ ΔG0 + λ −E ⎞ !ω F ⎟⎠

2π − 2 ⎜⎝ F (E) = e 2π !ω F

2

(18)

where ω F is the functional-weighted vibrational frequency defined as

λ k 2 λ Nvib 2 ω k = ∑ ck ω k ! k=1 k=1 !ω k

N vib

N vib

ω F2 = ∑ Dkω k2 = ∑ k=1

Since

(19)

1 1 !ω k is the zero-point energy of a given vibrational mode, !ω F can be also 2 2

considered as the vibronically weighted zero-point energy that reflects the quantum tunneling strength. Now, it is easy to recognize that for strong vibronic coupling, 1 ⎛ ΔG0 + λ ⎞ !ω F ⎟⎠

J 2 2π − 2 ⎜⎝ ket = 2 e ! ωF

2

(20)

By contrast, when the weak vibronic coupling is encountered, i.e., λ ≪ "ω vib , the short time limit applied in Eq. 17 is no longer valid. Thus, the integral in Eq. 14 must be evaluated explicitly in the complex plane. Since a highly oscillatory phase shift is expected on f ( t ) due to the relatively fast molecular vibrations, F ( E ) can be calculated by the steepest decent method that expands the integrand near its saddle point: N vib

iΔG0t ∑ Dk ( e 1 ! F(E) = e e k=1 ∫ 2π ! −∞ +∞

iω k t

)

−1

e

−

iEt !

−

dt =

e

N vib

∑ Dk k=1

+∞

2π !

−∞

∫e

i( ΔG0 −E )t vib + Dk eiω k t ! k=1

−

N

∑

dt =

e

N vib

∑ Dk k=1

+∞

2π !

−∞

∫e

z(t )

dt

(21)

Apparently, at the saddle point, t 0 , z ' ( t ) satisfies the following condition: N vib

z ' ( t ) t=t0 = 0 → ( ΔG0 − E ) + ∑ Dk !ω k eiω kt0 = 0

(22)

k=1

and it is also easy to find that N vib

z '' ( t ) t=t0 = − ∑ Dkω k2 eiω kt0 k=1

10


(23)


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Then, the second-order Taylor’s expansion of e z(t ) yields −

F(E) =

e

N vib

∑ Dk

e 2π !

k=1

z(t 0 ) +∞

∫e

1 2 z ''(t 0 )( t−t 0 ) 2

−

dt =

−∞

N vib

∑ Dk

−

z(t 0 )

N vib

∑ Dk (1+iω kt0eiω kt0 −eiω kt0 )

e e e k=1 = N vib ! −2π z ''(t 0 ) ! 2π ∑ Dkω k2 eiω kt0 k=1

(24)

k=1

Since the line shape function is physically observable, F ( E ) must be a real number that

(

requires t 0 being present on the imaginary axis of the complex plane, i.e., t 0 = it 0' t 0' ∈R

)

. Consequently, the physically meaningful root of Eq. 23 can be readily determined through nonlinear fitting: N vib

ΔG0 + λ ∑ ck2 e−ω kt0 = 0 '

(25)

k=1

Finally, the radiationless transition rate in the weak coupling limit is given by:

ket =

J

2

2π e !

−λ

N vib

c2

∑ !ωk k ⎛⎝ 1−ω kt0'e− ω kt0 −e− ω kt0 ⎞⎠ '

'

k=1

N vib

(26)

λ ∑ c !ω k e 2 k

− ω k t 0'

k=1

Up to now, there are only three undetermined parameters, namely, ΔG0 , λ and J in Eq. 20 and Eq. 26. In the present study, all those quantities will be calculated by the thermodynamic integration method,26 which defines a mixed Hamiltonian, H mix (η ) , as a linear combination of the ground and excited diabatic states: H mix (η ) = (1− η ) H S0 + η H S1

(27)

After sufficiently sampling the gradient of the mixed Hamiltonian along the reaction coordinate, η , ΔG0 can be obtained by the discrete summation: η ⎛ ∂H mix (η ) ⎞ ΔG0 = ∑ ⎜ Δηi ∂η ⎟⎠ ηi i=1 ⎝

N

11


(28)

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Without any additional simulation, the reorganization energy can be also deduced from the following equation:

λ = H mix (η = 1) − H mix (η = 0 )

η =0

+ ΔG0

(29)

In addition, the value of J will be determined by following the CDFT protocol.24

3. Thermally-activated Charge Separation in GTA-solvated B30: B30 has been widely investigated by both experiments30 and simulations31 as a model dyad system whose intramolecular electron transfer rate after photoexcitation is prone to its solvent. In fact, even its S0 → S1 optical gap varies significantly from 1.42 eV in diphenyl ether to 2.74 eV in water.32 The sensitivity of the optical gap is primarily ascribed to the solvent polarity and rigidity,32 which certainly have an impact on the charge transfer rate too. Since the present study is focused on the effect of intramolecular vibrations, the bulky and slowly relaxing GTA was chosen as our solvent in the hope of suppressing molecular orientational fluctuations. Unless otherwise specified, all simulations were performed using the CP2K package.33 Our simulation system (Fig. 2) was constructed as follows. At first, a B30 molecule was solvated by 558 GTA molecules in a cubic box. Then, the generalized Amber force field (GAFF)34 was adopted to ascertain the atom types for both B30 and GTA molecules before the determination of their bonding and Van der Waals parameters. In addition, their atomic partial charges were derived according to the restricted electrostatic potential (RESP) fitting scheme.35 Subsequently, the solvated system was equilibrated by molecular dynamics simulation for 2.0 ns under the NPT condition at 1.0 atm and 300K to eventually achieve a 55.81Å × 55.81Å × 55.81Å simulation box with a

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mass density of 1.17 g/cm 3 , which is less than 1% off the experimental value of

1.16 g/cm 3 .36 As shown in the derivation of FMET, its success heavily relies on the quality of the vibrational normal modes and their correlation with diabatic energy gap. Since the vibrational modes are identified by diagonalizing the covariance matrix of mass-weighted atomic displacements in FMA,25 a large number of nuclear configurations must be sampled by molecular dynamics simulations to ensure the convergence of the eigenvectors and eigenvalues. On the other hand, the accurate construction of the diabatic states is also desired for a precise picture of charge transfer that definitely demands quantum mechanical treatment. In order to achieve a compromised balance between efficiency and accuracy, the hybrid quantum mechanics/molecular mechanics (QM/MM) method37 was applied by modeling the B30 solute using the self-consistent-charge tightbinding density functional theory (SCC-DFTB)38 and by describing the GTA solvent using GAFF.34 Moreover, an empirical dispersion correction39 was added to the SCCDFTB parameter set,38 while the Coulomb scheme was selected to account for the electrostatic coupling between the QM and MM subsystems. Under the NVT condition at 300 K, a 1.0 ns QM/MM trajectory was obtained using a time step of 1.0 fs. Then, a total of 10,000 snapshots were extracted from the trajectory to form the training set of FMA. After examining the Mulliken charge distribution40 of all snapshots in the training set, the extent of charge separation in the ground state B30, which is quantified as C ( S0 ) in Eq. 6, is found to be +1.16. Accordingly, the value of C ( S1 ) in Eq. 7 is set to -0.84 to complete our definition of diabatic states, which can be justified by their corresponding dipole moments, µ . As shown in Fig. 3, the calculated µ for the S0 and S1 states are 16.1

13


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D and 5.3 D, respectively, both of which are in good agreement with the experimental values of 15 D and 6 D.41 Evidently, the substantial reduction of µ by 11.8 D (Fig. 3) upon photoexcitation is due to the intramolecular electron transfer within B30 from its 2,6-dipheyl-phenolate moiety to its 2,4,6-triphenylpyridinio moiety (Fig. 1). Since a B30 molecule has 72 atoms, there are 210 vibrational modes available in total. Among them, only a few are essential to the S1 → S0 charge separation as suggested by some earlier experimental studies.9-10, 42 For example, an intramolecular vibrational mode of 1800 cm-1 was proposed after fitting the static absorption spectra of B30 in GTA to a line shape mode that includes one additional low-frequency solvent relaxation mode.9 In general, all vibrational modes can result in quantum dephasing for charge transfer to some extent, and their relative contributions are reflected in the variation of reorganization energy, a sensitive indicator of vibronic coupling strength. If the vibrational coherence is neglected due to the rapid thermal phase randomization, the electron transfer reaction coordinate can be ascertained by maximizing the linear correlation between diabatic energy gap and atomic displacement. After following the

! FMA protocol25 to project ΔED onto Vvib , the profile of the FMA coefficients, ci2 , was determined and is shown in Fig. 4. It is interesting to note that there are only 7 vibrational modes whose ci2 are greater than 0.05, and they cumulatively account for 60.5% of the reorganization energy. By contrast, the 194 vibrational modes, whose ci2 are lesser than 0.01, only contribute 25.3%. Moreover, the 7 essential modes are substantially faster than the majority of their non-essential counterparts. For instance, the nitrogen-carbon bonding stretching mode with the largest ci2 of 0.203 vibrates at 2634 cm-1, outpacing 202 others. Again, the rather few essential modes of B30 with their distinctively faster

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Page 16 of 30

vibrations than the solvent relaxation confirms the long-standing speculation that in a sluggish solvent such as GTA, the electron transfer in a dyad molecule can be mainly driven by intramolecular vibrations through quantum tunneling when !ω vib ≫ k BT .

! Specifically, the most probable tunneling channel is along VMCM , which is shown in the inset of Fig. 4 as a moiety-stretching mode to distend the electron donor and acceptor. N vib "! !" !" ! Since VMCM = ∑ V i and V i i V i = #ω i−1 for a quantum harmonic oscillator at its ground i=1

state, an effective angular frequency, ω MCM , can be conveniently defined as

ω

−1 MCM

N vib

= ∑ ci2ω i−1

(30)

i=1

It turns out that our calculated ω MCM is 1660 cm-1, which is close to the experimental charge transfer timescale.10 Prudently, the quality of ci2 was also examined by projecting them onto 5,000 snapshots that were extracted from a separate 0.5 ns QM/MM trajectory. As shown in Fig. 5, the cross-validated Pearson’s coefficient, Rc , is 0.73, which is only slightly smaller than the training coefficient R of 0.78, suggesting that our sampling is sufficient. According to the Fermi’s golden rule,29 the Franck-Condon (FC) factor also depends on ΔG0 and λ , both of which were determined by the thermodynamic integration method26 using a linearly interpolated Hamiltonian (Eq. 27) that spans over 11 100-ps sampling windows. After a discrete summation of the thermally averaged Hamiltonian gradient, ΔG0 was found to be 1.95 eV, which is significantly greater than

λ by 1.37 eV (Fig. 6). Consequently, we will limit our discussions from now on to the strong coupling limit since our calculated λ of 0.58 eV is still notably greater than the

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!ω vib of any essential vibrational mode. Apparently, the electron transfer process is in the Marcus inverted region wherein the increasing exergonicity imposes additional activation barrier, which thankfully can be effectively flattened by a large ω F as shown in Eq. 20. For example, in our B30 model system, ω F was determined by Eq. 19 as 3101 cm-1, which is of the same order of magnitude as ΔG0 and λ , leading to a rather large FC factor of 1.58 × 10 −3 eV-1 . Moreover, in order to validate our assumption of diabatic electron transfer mechanism, the electronic coupling strength, J , was also calculated by CDFT on 200 snapshots. It turns out that its standard deviation of 0.02 eV is much smaller than its mean value of 0.14 eV, also justifying the Condon approximation that assumes a constant J along the reaction coordinate. After all, J is still smaller than

!ω vib for all essential modes, suggesting a non-negligible probability of dividing surface recrossing. Finally, ket was evaluated by Eq. 13 at 300 K that yields a value of 0.30 ps-1, which is in excellent agreement with the experimental rates of 0.29 ps-1 at 295 K and 0.35 ps-1 at 303 K.9

4. Discussions: In recent years, the optical manipulation43 of vibrational modes has received increasing attentions as an emerging technique to modulate electron transfer rate,44 thanks to the rapid development of quantum control spectroscopy.45 In a coherent control study using shaped laser pulses, the visualization and controlling of vibrational wave packets can be even refined to a single molecule at a time by adapting the temporal and spatial distribution of excited electronic wavefunctions to the dynamics of individual molecules.46 In spite of the encouraging achievements on the experimental side, the

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theoretical ascertainment of the critical vibronic coupling has remained a challenging task in condensed phases primarily due to the great amount of participating vibrational modes, most of which actually have negligible contributions to a given electron transfer process. To tackle the challenge on the theory side, we developed a statistically reliable approach in the present study to unambiguously identify the vibrational modes that are essential to the transition between two diabatic states, paving the way for the systematic design of novel molecular devices whose electrical or optical properties can be easily controlled by thermal resonance. For example, in a recent femtosecond optical-pulse shaping experiment47 that selectively excited an electronic transition and three phonon modes simultaneously in a polycrystalline tetracene film, a 20% quantum yield enhancement has been achieved on the triplet formation. The substantial performance improvement stems from singlet fission,48 an interesting photochemical process that can turn two singlet molecules into two triplet molecules upon the absorption of a single incident photon, suggesting the feasibility of circumventing the long-standing Shockley-Queisser limit19 for solar energy conversion. Thus, the extension of our FMET theory to investigate the rapid spin exchange in singlet fission is highly desired for ultra-efficient photovoltaic applications in future.

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Acknowledgements The research was supported by the start-up grant and the University Facilitating Fund of the George Washington University. Computational resources utilized in this research were provided by the Argonne Leadership Computing Facility (ALCF) at Argonne National Laboratory under Department of Energy contract DE-AC0206CH11357 and by the Extreme Science and Engineering Discovery Environment (XSEDE) at Texas Advanced Computing Center under National Science Foundation contract TG-CHE130008.

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Figures:

Figure 1. The photo-induced charge recombination and thermally activated charge separation in a betaine-30 molecule. The carbon, nitrogen, oxygen and hydrogen atoms are colored cyan, blue, red and white, respectively.

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Figure 2. A betaine-30 molecule (highlighted) solvated by 558 glycerol triacetate molecules in a 55.81Å × 55.81Å × 55.81Å simulation box. All solvent molecules are intentionally blurred for visual clearance.

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Figure 3. The histogram of betaine-30 dipole moments for the ground (S0) and first excited (S1) diabatic states.

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Figure 4. The profile of FMA coefficients, ci2 , for the vibrational modes of betaine-30 molecule. The 7 essential driving modes with ci2 > 0.05 are colored red, while the others are colored black. The maximally correlated motion (MCM) ascertained by FMA is also shown in the inset.

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Figure 5. The scatter plot for the cross validation of FMA coefficients, ci2 , on the diabatic energy gap, ΔED .

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Figure 6. The free energy diagram of the S0 and S1 diabatic states of a betaine-30 molecule.

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47. Grumstrup, E. M.; Johnson, J. C.; Damrauer, N. H., Enhanced Triplet Formation in Polycrystalline Tetracene Films by Femtosecond Optical-Pulse Shaping. Phys. Rev. Lett. 2010, 105 (25), 257403. 48. Smith, M. B.; Michl, J., Singlet Fission. Chem. Rev. 2010, 110 (11), 6891-6936.

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