Modelling Kinetics of Ultrafast Photoinduced Intramolecular Proton

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C: Energy Conversion and Storage; Energy and Charge Transport

Modelling Kinetics of Ultrafast Photoinduced Intramolecular Proton-Coupled Electron Transfer Tatyana V. Mikhailova, Valentina A. Mikhailova, and Anatoly I. Ivanov J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b09097 • Publication Date (Web): 17 Oct 2018 Downloaded from http://pubs.acs.org on October 23, 2018

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The Journal of Physical Chemistry

Modelling Kinetics of Ultrafast Photoinduced Intramolecular Proton-Coupled Electron Transfer Tatyana V. Mikhailova, Valentina A. Mikhailova, and Anatoly I. Ivanov∗ Volgograd State University, University Avenue 100, Volgograd 400062, Russia E-mail: [email protected]

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The Journal of Physical Chemistry 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

Abstract A model of photoinduced intramolecular proton-coupled electron transfer is derived. The model includes three states: the ground, excited, and product states. The charge transfer is associated with both stages, photoexcitation and product formation. A larger part of the model parameters can be extracted from the stationary absorption and fluorescence spectra of a particular fluorophore. Two different reaction coordinates are associated with the two stages, which are not independent. The angle between the reaction coordinates strongly influences on the kinetics of ultrafast product formation. The stochastic multichannel approach is exploited for simulations of the kinetics. The simulations well reproduce the kinetics of ultrafast intramolecular proton-coupled electron transfer in 2-((2-(2-hydroxyphenyl)benzo[d]oxazol-6-yl)methylene)malononitrile in a few solvents. The transfer is shown to occur totally or partly, depending on the solvent, in nonequilibrium regime. Analysis of the kinetics of the excited state decay has uncovered a significant decrease in the magnitude of the reorganization energies of slow nuclear modes with increasing the solvent polarity. Such an unusual behavior of the total reorganization energy can be rationalized under the assumptions: (i) a slow intramolecular reorganization of a significant magnitude associates with the transition between excited and product states and (ii) intramolecular slow reorganization is accompanied by a change in the dipole moment of the fluorophore.

INTRODUCTION Proton and electron transfer are the simplest elementary and at the same time ubiquitous reactions in chemistry having diverse applications in emerging technologies. 1–9 Such kind of reactions also plays a key role in biochemical transformations, including photosynthesis, respiration, the energy transfer in cells, enzyme reaction. 1,8,9 Excited state charge and proton transfer reactions may find their applications in a variety of modern technologies, for example, for designing suitable chemical and/or biological sensors and fluorescence senscing and imaging. 7,9 2


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The number of compounds in which ultrafast photoinduced proton and electron transfers are observed is enormous. For the most part the electron and proton transfer reactions are interrelated. In the literature, sequential and concerted mechanisms of proton-coupled electron transfer (PCET) are singled out. 5,6,10–12 Sequential mechanism of PCET includes a stable intermediate product, when either the electron transfer (ET) proceeds before proton transfer (PT), or conversely the proton transfer takes place prior the electron transition. 13 The simultaneous transfer of the electron and proton in a single step without a stable intermediate product is interpreted as the concerted mechanism. 5,6,11,12 However, in ultrafast processes, the discrimination of these mechanisms is impossible, since the determination of a stable intermediate compound is difficult. Thus, a fast proton-coupled electron transfer is a more complex process which attracts much attention of the scientific community. 4,6,8,11,12,14–16 Typical and well-studied examples of molecular structures in which such processes can proceed are salicylates, benzoazoles, and hydroxyflavones. 4,14,17–19 Among the last group of compounds, one of the most studied molecules where PCET takes place is 3-hydroxyflavone (3HF). 14,18 In this molecular structure an exited-state intramolecular proton transfer (ESIPT) occurs between the oxygens of hydroxyl and carbonyl groups, which leads to the formation of a zwitter-ionic (bipolar) tautomer. Extensive studies of 3HF molecule and some of its derivatives provided the quantitative data on the dynamics of the populations of excited states. 4,14,18 The investigation of the detailed mechanism of the intramolecular PCET reactions proceeding after photoexcitation of molecule is still one of the fundamental issues. 4,14,18,20–23 P.-T. Chou and co-workers 14,18,20–23 have synthesized and presented several examples of PCET molecular compounds. 14,18 The first type of PCET system is 4’-N,N-diethylamino-3hydroxyflavone (further in text system I). The processes induced by optical excitation in I were interpreted within the three-level model assuming the excited state with considerable intramolecular electron transfer to be directly populated by the pump pulse and then proton transfer to occur. 14,18 The processes proceeding differently were observed in another com-

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pound, 2-((2-(2-hydroxyphenyl)benzo[d]oxazol-6-yl)methylene)malononitrile, or diCN-HBO (further in text system II). It is possible that here, the excited state proton transfer precedes the electron transfer. 14,18 Stationary optical spectra of I and II show dual emission associated with transitions from the initial excited state and PCET reaction product. 14,18 It should be noted that the fluorescence maximum position in the short-wavelength region (emission from the initial excited state) for system I strongly depends on the solvent polarity, and practically does not depend for the system II. A directly opposite dependence is demonstrated for the stationary fluorescence from the PCET product state. For the system II a fluorescence peak position strongly shifts with increasing the solvent polarity, while for the system I such a shift was not noticed. 14,18 A detailed study of PCET systems and their excited state kinetics is important for understanding the fundamental mechanism of such reactions, so that supposes a development of theoretical approaches providing quantitative description of the phenomenon. Success of Marcus’s theory of pure ET has strongly motivated the development of the theory of proton transfer and PCET. 24,25 For example, a detailed description of tunneling proton transfer reactions in polar environment was developed. 26–31 Since here only pure proton migration is reviewed, these models are not sufficient to describe the PCET kinetics. Hammes-Schiffer and co-workers 5,6,32 have extended theory and established a theoretical PCET model consisting of four diabatic states corresponding to different ET and PT configurations. They introduce two collective solvent coordinates corresponding to electron and proton transfer and describe reaction in terms of nonadiabatic transitions between these diabatic mixed electron-proton vibronic states. This model describes sequential mechanism of PCET, when the proton transfers prior the electron or the electron is transferred prior the proton, as well as the concerted mechanisms, where the electron and proton transfers occur simultaneously. The theories describing proton/electron transfer in terms of the thermal rate constant are applicable for description of the reactions which are significantly slower than the vibrational

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and solvent relaxations. Ultrafast reactions proceed on timescale of nuclear relaxation and nonequilibrium of the solvent and intramolecular vibrational modes can be of paramount importance. 33–42 Thus, theoretical treatment of ultrafast reactions has to include nonequilibrium of the nuclear subsystem formed by an excitation pulse and specific stages of a complex reaction. Ultrafast stages of the reactions proceed in parallel with the relaxation of the nonequilibrium state created at the previous stage, and, hence, the memory about it is preserved. If the reaction coordinates corresponding to the two sequential stages are dependent on each other, the memory effect can be important. 43–45 The measure of the correlation between the reaction coordinates is the deviation from their orthogonality. 46 The angle between the reaction coordinates can be calculated if the reorganization energies of all transitions are known. 45 The mentioned features of the PCET reactions can be incorporated within the well-tested stochastic multichannel model developed for ET reaction including multistage processes. 36 Since for I and II molecular systems, both fluorescence spectra from electronically excited and from product states are observed, the PCET model under consideration must involve at least three diabatic states corresponding to the ground, electronically excited states and proton-coupled electron transfer state (product state) of proton/electron transfer (PE) systems. 14,18 This establishes a very deep analogy between the PCET reaction and the multistage intramolecular photoinduced ET in donor-acceptor electron molecular systems. A correct description of PCET as a multistage process (photoexcitation and PCET stage) requires that each stage its own reaction coordinate should be assigned. 44,45 The main aims of this article are: (i) to develop a model of photoinduced intramolecular proton-coupled electron transfer in which the nonequilibrium of the nuclear subsystem formed by the excitation pulse is taken into account, (ii) to simulate proton-coupled electron transfer kinetics and to estimate an effective rate constant of PCET, (iii) to investigate the PCET kinetics dependence on the angle between the directions of photoexcitation and PCET stages, and (iv) to compare the simulated PCET kinetics with the available experimental

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data. 14

Theory and Computational Details In a series of works it was shown that PCET kinetics can be successfully described in terms of nonadiabatic transitions between several diabatic states. 5,6,32 The approach 5,6,32 is exploited here to simulate kinetics of the coupled proton/electron transfer in I and II systems (which are denoted as PE). The multichannel stochastic approach which had previously been successfully applied for multistage electron transfer kinetics simulation in donor-acceptor complexes and dyads is used in this work. 35,41,42,47–49 It takes into account the solvent relaxation dynamics and reorganization of intramolecular high-frequency oscillations of PE, and is well suited for ultrafast nonequilibrium kinetics modelling. For PCET kinetics description, three electronic states of the PE system are considered: a ground state, PE, an electronically excited state, PE*, and a final product state, PCET*. Here, the symbol * denotes an excited state. For brevity, the states listed above are numbered as 1, 2, 3, respectively. Photoexcitation of a molecule leads to a 1 → 2 transition, which is associated with charge transfer/redistribution, leading to an alteration in the dipole moment. The next step is the photoinduced process, the transition from 2 → 3, which includes both the transfer of a proton and an electron. It should be emphasised that the analysis implemented in ref 18 showed that in the system I the excited state intramolecular ET (ESIET) takes place prior to ESIPT while in the system II ESIPT proceeds concomitantly with the charge transfer. The description of all possible transitions between these three states requires involving of two coordinates, Q1 and Q2 . 45 A new distinctive aspect of the model is an angle between the directions of reaction coordinates corresponding to the stages of PE system photoexcitation and PCET from PE* state (see Figure 1). The quantitative description of the PCET kinetics in the molecular structures I and II within the multichannel stochastic model requires knowledge of the basic energy parame-

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Excited state of PE system (PE*)

|

G

12

Product state

|

|

G

23

|

(PCET*)

fluorescence

absorbtion fluorescence

P C ET

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


|

G

13

|

Ground state of PE system

Figure 1: Electronic states and photochemical processes in the PE compounds. Thick horizontal black, blue, and red lines denote a ground, excited and product states, respectively. Absorbtion and fluorescence processes are indicated from each state by identical-colored arrows. The free energy changes between the ground and excited states, −∆G12 , the excited and product states, −∆G23 , and the ground and product states, −∆G13 , are indicated by black dashed arrows, respectively. ters, such as the free energy change and the medium reorganization energy for each stage, the reorganization energy of high-frequency intramolecular vibrational modes and also their frequencies. A larger part of parameters can be returned from the fitting of stationary ab(12) sorption and fluorescence spectra. The medium reorganization energy, Erm , and the free

energy gap, −∆G12 (see Figure 1), at the stage of photoexcitation can be found out from the stationary absorption and the fluorescence spectra associated with the transition between PE and PE*. 14,18 The fluorescence spectra corresponding to the transition from the product (13) to the ground state determines the values of the energy parameters Erm and −∆G13 .

For the fitting of the absorption and fluorescence spectra, 14,18 the equations are used 50

A1 (ω) = Cω

∑ S n e−S n

Fj (ω) = Cω 3

n!

∑ S n e−S n

n!

{

exp −

(12) (|∆G12 | + Erm + n¯ hΩ − h ¯ ω)2 (12)

}

4Erm kB T { } (1j) + n¯hΩ + h ¯ ω)2 (−∆G1j + Erm exp − (1j) 4Erm kB T

(1) (2)

Here j = 2, 3, h ¯ is the Planck constant, kB is the Boltzmann constant, T is the temperature, S = Erv /¯hΩ, Ω, Erv , and n = 0, 1, 2, ... are the Huang-Rhys factor, the frequency of intramolecular quantum vibrational mode, its reorganization energy, and the vibrational quantum number, respectively. 7


2.0

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absorption

fluorescence 1

fluorescence 2

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

Intensity (arb. units)


2.5

h

eV

3.0

3.5

Figure 2: Fitting of the static absorption and dual fluorescence spectra of I in ACN solution. 14 The experimental data and the result of the fitting are depicted by the dots and (12) solid lines, respectively. The parameters at the stage of photoexcitation, Erm and −∆G12 , are defined by the stationary absorption and the fluorescence spectra associated with the (13) PE → PE* transition. The Erm and −∆G13 values are determined by fluorescence spectra corresponding to the PCET* → PE transition. The results of a fit of eqs 1 and 2 to the experimental stationary absorption and fluorescence spectra of I in ACN are shown in Figure 2. The free energy gaps and the total reorganization energies returned from such a fitting are reliably determined, because their values are determined by the positions of the maximum of the absorption and fluorescence bands. The basic energy parameters of PCET reactions for molecular systems I and II in a few solvents (acetonitrile (ACN), dichloromethane (DCM), and benzene) obtained from the fitting are listed in Tables 1 and 2. Here also the values of the longitudinal dielectric relaxation time τL , and solvent polarity function ∆f = (ε0 −1)/(2ε0 +1)−(ε∞ −1)/(2ε∞ +1), where ε∞ and ε0 are the optical and static medium permittivities, are written down. Table 1: The energy parameters of the PCET reaction for I (4’-N,Ndiethylamino-3-hydroxyflavone) complex in a few solvents.

(12) solvent Erm , eV ACN 0.25 DCM 0.19 Benzene 0.14

−∆G12 , eV 2.70 2.73 2.82

(13) Erm , eV −∆G13 , eV ∆f τL , ps 51 0.14 2.34 0.6108 0.26 0.14 2.33 0.4342 0.56 0.16 2.33 0.1375 2.10

For the system II, both reorganization energies, as expected, decrease with decreasing (13) (12) , while Erm polarity of the solvent. For the system I, this trend is performed only for Erm

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Table 2: The energy parameters of the PCET reaction for II (diCN-HBO) complex in a few solvents.

(12) , eV solvent Erm ACN 0.27 DCM 0.21 Benzene 0.18

−∆G12 , eV 3.03 3.00 3.00

(13) , eV −∆G13 , eV Erm ∆f τL , ps 51 0.40 2.35 0.6108 0.26 0.39 2.36 0.4342 0.56 0.31 2.40 0.1375 2.10

is practically independent of the solvent polarity. This indicates that a slight change in the dipole moment of the fluorophore accompanies transitions between the 1 and 3 states. The conclusion is also confirmed by the independence of the change in the free energy ∆G13 of the solvent polarity. Apparently, the reorganization energy for the 3 → 1 transition is associated with the reorganization of intramolecular low-frequency modes. Such a reorganization can be related to the motion of a large amplitude, so that the reorganization time can correlate with the solvent relaxation time, since the reorganization requires a rearrangement of the solvent molecules surrounding the fluorophore. In all the cases, the alteration of the reorganization energy is much weaker than the variation of the solvent polarity ∆f . This may also indicate that a considerable part of the total reorganization energy is associated with intramolecular reorganization that is expected to be weakly dependent on ∆f . It is appeared that the fitting to the stationary absorbtion and fluorescence spectra is weakly sensitive to division of the total reorganization energy on the reorganization energies of intramolecular high-frequency vibrational and low-frequency relaxation modes. The fairly strong influence of the solvent polarity on the stationary spectra suggests that the reorganization of the solvent inputs considerably in the total reorganization energy. This is the reason why the reorganization energies of intramolecular high-frequency vibrational modes, Erv , and its frequency are taken fixed and are equal to Erv = 0.1 eV, and h ¯ Ω = 0.1 eV. The free-energy surfaces for the diabatic states involved in the PCET reaction are (see

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U 2

3

Coordinate Q

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

Page 10 of 41

U

U

1

2

Coordinate Q

1

Figure 3: Isolines of the free energy surfaces, Uj (j = 1, 2, 3), on the coordinate plane, Q1 , Q2 . Black, blue, and red circles correspond to the ground, excited and product states of PE molecular systems in solvents, respectively. The red dashed lines show the intersections of the terms (including their vibrational repetitions) of the excited and product states. The blue arrows depict the movement of the wave packet along the U2 surface to its equilibrium position. Here φ is the angle between the reaction coordinate directions of photoexcitation (1 → 2) and the subsequent PCET stage (2 → 3). Figure 3): 45 (

U1 =

Q21 Q22 + , 2 2 (

(m) ⃗

U3

Q1 −

=

(⃗ n)

U2

√

(13) Erm

Q1 −

= )2

(

cos θ

√

(12) Erm

2 Q2 −

+

2

)2 N ∑ Q22 + ∆G12 + nα h ¯ Ωα 2 α=1

+ √

(13) Erm

(3)

)2

sin θ + ∆G13 +

2

M ∑

mα h ¯ Ωα

(4)

α=1

The vector indexes ⃗n={n1 , n2 , ..., nα , ..., nM } and m={m ⃗ 1 , m2 , ..., mα , ..., mM } for freeenergy surfaces U2 and U3 stand for the set of the vibrational sublevels of the excited and product states (blue and red circles in Figure 3). The quantum numbers nα and mα for an αth intramolecular vibrational mode with a frequency Ωα are equal to 0, 1, 2, ... An angle between the reaction coordinates, corresponding to the transitions 1 → 2 and 1 → 3, is denoted as θ (see Figure 3). (23) can not be determined Unfortunately, the magnitude of the reorganization energy Erm

from the stationary spectra. The angle θ is also unknown being connected by eq 5 46 √ (23) Erm

=

(12) Erm

+

(13) Erm

(12)

(13)

− 2 Erm Erm cos θ

10


(5)

U

U

3

2

U

1

|

G

23

|

excitation

free of Cut

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


energy

Page 11 of 41

Q

(3)

1

Q

(2)

1

Coordinate Q

1

Figure 4: Cuts of the free energy surfaces Uj (j = 1, 2, 3). Black, blue, and red curves correspond to the ground, excited and product states of PCET compounds. The vibrationally excited sublevels of product state are shown by red dashed lines. The red vertical arrows stand for intramolecular vibrational relaxation. (12) (13) (23) with the medium reorganization energies, Erm , Erm , and Erm . In what follows, the angle

θ is considered to be a variable parameter. (23) Dependences of the medium reorganization energy, Erm , in the systems I and II and

angle between the reaction coordinate directions of the photoexcitation and the subsequent PCET stage, φ, on angle θ are presented in Figure 5. It is easy to see that in both systems (23) as the angle θ grows, the energy Erm increases and the angle φ decreases.

Nevertheless, there is fundamental difference of parameters determining PCET kinetics in systems I and II. As can be seen, the medium reorganization energy for the PCET stage, (23) Erm , in system II is larger and is subjected to stronger changes in different solvents than

that in the system I, if θ > 90◦ . The second difference is related to the magnitude of the angle between the reaction coordinate directions of the photoexcitation and the subsequent PCET stage, φ. For the system I the value of this angle can not exceed 90◦ in strong polar solvents. The PCET kinetics will be described in the framework of the stochastic approach 52 generalized to account the angle between the solvent coordinates in the multidimensional configuration space of nuclear degrees of freedom corresponding to two successive transitions (photoexcitation and proton coupled electron transfer). A set of equations for the probability

11



E

Page 12 of 41

1,6 (23)

ACN (blue)

rm

DCM (red)

1,2

a)

Benzene (green)

II

0,8 I 0,4

0,0

0

120

40

80

120

160

200

b)

II

100 80

I

ACN (blue)

60

DCM (red) Benzene (green)

40 20 0

0

40

80

120

160

200

(23) , (frame a) and angle between Figure 5: Dependencies of medium reorganization energy, Erm the reaction coordinate directions of the photoexcitation and the subsequent PCET stage, φ, (frame b) on angle θ for a molecular systems I (solid lines) and II (dashed lines) for a few mediums: ACN (blue lines), DCM (red), and benzene (green).

12


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(⃗ n)

(m) ⃗

distribution functions ρ2 (Q1 , Q2 ) and ρ3 (Q1 , Q2 ) is 33,52–62 (⃗ n)

∂ρ2 ∂t

(m) ⃗ ∂ρ3

∂t

=

n) ˆ 2 ρ(⃗ L 2

−

∑

(

k⃗nm ⃗

(⃗ n) ρ2

−

(m) ⃗ ρ3

)

+

⃗ ˆ 3 ρ(3m) = L +

k⃗nm ⃗

(

(⃗ n) ρ2

−

(m) ⃗ ρ3

)

(nα +1)

+

∑ α

⃗ n

(⃗ n′ )

ρ2 τv

α

m ⃗

∑

∑

−

∑ ρ2(⃗n) α

(m ⃗ ′)

ρ3

(mα +1)

τv

−

(6)

(nα )

τv

⃗ ∑ ρ(3m) α

(7)

(mα )

τv

ˆ j (j = 2, 3), depict the diffusion on the free energy Here the Smoluchowski operators, L surfaces Uj :

(

2 ∑ ∂ ∂2 (j) ˆj = 1 1 + (Qk − Qk ) L + kB T τL k=1 ∂Qk ∂Q2k (1)

(1)

(2)

√

where Q1 = Q2 = 0, Q1 =

(12)

(2)

(3)

Erm , Q2 = 0, Q1 =

√

)

(8) (3)

(13)

Erm cos θ, and Q2 =

√

(13)

Erm sin θ.

The medium relaxation is characterized by a single effective relaxation time, τL . 63 (m) ⃗

The transitions between vibrational sublevels of PCET state, U3 , and the excited state, (⃗ n)

36,55,58 Due to the fast vibrational relaxU2 , are described by the Zusman parameters k⃗nm ⃗.

ation/redistribution in the PE* state to the ground vibrational state (solid blue line in Figure 4), it is assumed that in the excited state only the ground vibrational sublevel is populated, (⃗ n̸=⃗0)

that is, the probability distribution function is ρ2

= 0. Moreover, pump pulse carrying

frequency is usually close to the red edge of the excitation band that leads only to population of the ground states of all the high-frequency vibrational modes. For the product state, the vibrational sublevels are populated during the solvent relaxation. The points of the transitions between vibrational sublevels of the product state and the excited state are denoted by the blue empty circles in Figure 4. Therefore, the Zusman parameters are denoted as 36,55,58 (

k⃗0m ⃗ F⃗0m ⃗

2πV232 F⃗0m (⃗0) (m) ⃗ ⃗ δ U2 − U3 = h ¯ ∏ e−Sα Sαmα = mα ! α

)

(9) (10)

Here δ(Q) is the Dirac delta function, V23 is the matrix element of transitions on the stage of product formation. The Huang-Rhys factor Sα = Erv /¯hΩα corresponds to αth high

13



Page 14 of 41

frequency vibrational mode with the frequency Ωα . The excited vibrational sublevels of the PCET state rapidly decay with the rate constant 1/τv(mα ) , τv(mα ) = τv /mα due to a single quantum irreversible vibrational transition mα → mα − 1. 64 In the simulations the rather typical value of τv = 100 fs is used since its real value is not known for the molecules considered here. A variation of the vibrational relaxation time constant, τv can noticeably change the rate constant of the nonadiabatic transition, but its effect on the dependences of the rate constant on the model parameters is not strong. 40,56 It was shown in earlier investigations 59 that charge transfer rate constant weakly depends on the spectral density (set of frequencies and weights of high-frequency vibrational modes) if the total reorganization energy defined as Erv =

∑ α

Ervα is kept constant and the number of

the high-frequency vibrational modes, M, exceeds or is equal to 5. As the universal spectral density, we accept the high-frequency vibrational spectrum of the complex which consists phenylcyclopropane (PhCP) as the electron donor and tetracyanoethylene (TCNE) as the electron acceptor (see Table 3). 65 This data was previously successfully used for electron transfer reactions dynamics description. Table 3: Frequencies and weights of high-frequency vibrational modes for charge transfer in PhCP/TCNE complexes. 65

PhCP/TCNE No h ¯ Ωα , eV Ervα /Erv 1 0.1272 0.07923 2 0.1469 0.08898 3 0.1823 0.10397 4 0.1935 0.49838 5 0.1993 0.22944 The initial conditions for the set of eqs 6 and 7 are defined as

(⃗0)

[

ρ2 (Q1 , Q2 , t = 0) = A exp −

    

(

(⃗0) U2

] h ¯ ωe − + U1 U1 exp −  kB T 2¯ h2   

14


)2



  τe2      

(11)

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(⃗ n̸=⃗0)

ρ2

(Q1 , Q2 , t = 0) = 0,

(m) ⃗

ρ3 (Q1 , Q2 , t = 0) = 0

(12)

Here A is the normalization factor, the parameters ωe and τe are the carrier frequency and duration of the Gaussian form pump pulse. The pump pulse duration is assumed to be short, so a solvent is considered to be frozen during the excitation. The equations 6 and 7 with the initial conditions 11 and 12 are solved numerically by the Brownian simulation method. 47,62,66 The PCET rate constant, kPCET , and population of the exited state of PE system, PPE∗ , are determined by equations: −1 kPCET

∫

∞

=

PPE∗ (t)dt

(13)

ρPE∗ (Q1 , Q2 , t)dQ1 dQ2

(14)

0

∫

P

PE∗

(t) =

In eq 13, the value of kPCET is defined as the average rate constant of the excited state decay, combining non-equilibrium and thermal regimes. The decay of P E ∗ , in principle, along with the PCET may include other channels that are not accompanied by the transfer of the population from the PE* state to the PCET* state. Since in the literature there is no strict evidence of the important role of these channels in the excited state decay, in the following, we identify kPCET with the PCET rate constant.

Results and Discussion For the numerical modelling of PCET kinetics, the parameters of molecular compounds I (13) (12) , , Erm and II (see Tables 1 and 2) are chosen as the basic model energy parameters (Erm

−∆G12 and Erv ), since they are typical values for systems with proton-coupled electron transfer. The free energy gap for the stage 2 → 3, −∆G23 , and the angle between the reaction coordinate directions of the stages 1 → 2 and 1 → 3, θ, are varied. Since the (23) , can not be determined magnitude of the reorganization energy for the PCET stage, Erm

15



from the stationary spectra, it was defined from eq 5 for each preset value of the angle, θ, (see Figure 5a). The results of numerical simulations of the ultrafast PCET kinetics in a few solvents (ACN and DCM, see Table 1) are pictured in Figures 6 and 7. They show how strongly the kinetics can alter with variation of the free energy gap −∆G23 , the angle θ, and the solvent polarity. The dependences of the PCET rate constant, kPCET , on the free energy gap, −∆G23 , and the angle, θ, presented in Figures 8, 9, 10, demonstrate how the solvent dynamic properties (the solvent relaxation time τL ) and the angle between the reaction coordinates θ influence on the free energy gap law. It turns out that for the ultrafast PCET process occurring under nonequilibrium conditions, the influence of these parameters on the free energy gap law, in contrast to the thermal reactions, can be important. Analysis of the results allows us to formulate a few trends inherent in the PCET. Firstly, the excited state population kinetics, PPE∗ , in both compounds presented in Figures 6 and 7 is strongly nonexponential. For weakly exergonic PCET, −∆G23 ≤ 0.2 eV, (black and green lines) one can see: (i) a relatively fast decay at timescale of a few hundred fs (nonequlibrium regime) and (ii) a slower decay at larger timescale (thermal regime). This is detected for any angles θ in both systems (see all frames in Figures 6 and 7). One can see that the system II reaches the equilibrium stage much more quickly than the system I. With rising −∆G23 , the role of nonequilibrium mode increases and in strong exergonic region, −∆G23 > 0.4 eV, PCET proceeds primarily in nonequilibrium regime except for the case θ = 120◦ in the system II (frame c, Figure 7), where a noticeable part of the reaction still proceeds in the equilibrium regime. There is a quasi-plateau at early times (red dashed and solid lines in frames a, b in Figures 6 and 7). The quasi-plateau at early times reflects a reaction delay associated with delivering the particles from the initial position to the reaction zone (the region of localization of transitions is represented by the blue circles in Figure 4). With a growth of the angle θ the equilibrium stage is achieved faster (compare the lines in frames a and c in Figures 6 and 7). This reflects approaching the most effective sinks (the points of the crossings of the U2 and U3 terms) to the initial position of the wave packet on the

16


Page 16 of 41

Page 17 of 41 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


U2 term. The data for different solvents are indicated by solid (ACN) and dashed (DCM) lines in Figures 6 and 7. The Figures show that the decay of the photoexcited PE* state occurs faster (compare solid and dashed lines in Figures 6 and 7) in solvent with a greater polarity (ACN). The two-stage kinetics was observed in experiments 18 that supports the inferences obtained here. Thus, the angle between the reaction coordinates, corresponding to the transitions 1 → 2 and 1 → 3, θ, is an important parameter that strongly affects the PCET kinetics. Its increase can transform the reaction regime from nonequilibrium to equilibrium one. Since the seminal work of Marcus, 24 the dependence of the rate constant on the reaction free energy gap (known as free energy gap law) is explored. Here, we demonstrate the dependence of the free energy gap law on the angle θ. In Figure 8 the dependence of PCET rate constant, kPCET , on the free energy gap, −∆G23 , is shown. It is bell-shaped with two distinct regions. In the Marcus normal region the PCET rate constant grows with increasing the exergonicity parameter, while in Marcus inverted region the rate constant decreases monotonically so that there is a pronounced maximum in between. As the angle θ increases, the maximum of kPCET (−∆G23 ) shifts toward larger values of exergonicity. This is due to the fact that the position of the maximum depends on the reorganization energy (23) (23) Erm . As we noted earlier, Erm increases with the θ growth (see Figure 5), so the values

of the free energy gap, −∆G23 , at which kPCET is maximum, are also shifted towards larger values. Besides, on the ascending branch of the free energy law the magnitude of the PCET rate constant is higher in DCM than in ACN solution for both compounds while on the descending branch the opposite trend is predicted (see frames a and b in Figure 8). The most important characteristics of the solvent which influence on the charge transfer rate are the solvent polarity and relaxation timescale. 36 The effect of each of these parameters on the PCET rate constant, kPCET , is separately shown in Figure 9. The numerical simulations of the kinetics represented by same-colored lines are carried out at fixed energy parameters

17



Page 18 of 41

1 (a)

=60

P

PE*

0.6

0,1

0.1

0.4

0.2

0.3 0,01 0,0

0,5

1,0

1,5

2,0

t, ps 1

(b)

=100

P

PE*

0.6

0,1 0.2

0.3

0.4

0,01 0,0

0,5

1,0

1,5

2,0

t, ps

1 (c)

P

=120

PE*

0.2

0,1

0.3

0.6

0.4

0,01 0,0

0,5

1,0

1,5

2,0

t, ps

Figure 6: Excited state population kinetics, PPE∗ , in compound I in ACN (solid lines) and DCM (dashed lines) solvents. Time dependence of the PE* population decay is presented for a few values of the free energy change between the photoexcited (PE*) and product (PCET) states, −∆G23 = 0.1 eV (green lines), 0.2 eV (black), 0.3 eV (magenta), 0.4 eV (blue) and 0.6 eV (red). Other parameters used: V23 = 0.03 eV, θ = 60◦ (frame a), 100◦ (frame b), and 120◦ (frame c). 18


Page 19 of 41 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


1 =60

(a)

P

PE*

0.6

0,1

0.2 0.3 0.4 0,01 0,0

0,5

1,0

1,5

2,0

t, ps 1

(b)

= 90

P

PE*

0.2 0,1 0.3

0.4 0.6 0,01 0,0

0,5

1,0

1,5

2,0

t, ps 1

(c)

= 120 0.2

P

PE*

0.3 0.4

0,1

0.6

0,01 0,0

0,5

1,0

t, ps

1,5

2,0

Figure 7: Kinetics of excited state population, PPE∗ , in the compound II in ACN (solid lines) and DCM (dashed lines) solvents. Time dependence of the PE* population is presented for a few values of the free energy change between the photoexcited (PE*) and product (PCET) states: −∆G23 = 0.2 eV (black), 0.3 eV (magenta), 0.4 eV (blue) and 0.6 eV (red). Other parameters used: V23 = 0.03 eV, θ = 60◦ (frame a), 90◦ (frame b), and 120◦ (frame c). 19 ACS Paragon Plus Environment


a) = 120

5 -1

= 100

kPCET, ps

= 60

O

O

O

- circles - triangles

- stars

4

3

ACN

2

DCM

1 0,0

0,2

0,4

-

G

23

0,6

0,8

, eV

b)

5 = 120 = 90

4

O O

- circles

- triangles - stars

-1

= 60

O

kPCET, ps

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

Page 20 of 41

3

2

ACN DCM

1

0,2

0,4

0,6

-

G

23

0,8

, eV

Figure 8: Dependence of the PCET rate constant, kPCET , on the free-energy gap, −∆G23 , for a few values of θ angle in ACN (blue) and DCM (red) solutions. Data for different values of the angle is indicated by symbols: stars – 60◦ , triangles – 100◦ (90◦ for system II), and circles 120◦ . The value of the matrix element of transition, V23 , is 0.03 eV. The frames a and b show the data obtained with system I and system II, respectively.

20


Page 21 of 41

-1

a ACN

4

DCM

kPCET, ps

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


3 b

2

1

0,2

0,4

0,6

-

G

23

0,8

, eV

Figure 9: Dependencies of the PCET rate constant, kPCET , on free-energy gap, −∆G23 in diCN-HBO compound 14,18 for θ = 90◦ . Data obtained with the same energy parameters are shown by same-colored a and b series of lines. Dashed lines reflect the impact of dynamic solvent effect without any energy parameter change: τL =0.26 (red dashed line) and τL =0.56 (blue dashed line). (see Table 2): red lines for DCM parameters and blue one for ACN. Comparison of samecolored dashed and solid lines shows the dynamic solvent effect (influence of τL on the rate constant). It is especially large in the strong exergonicity region. The strongest influence of (23) solvent polarity (reorganization energy Erm is varied) is detected in the weak exergonicity

region (compare the solid line with the dashed line of another color in Figure 9). Since there is a correlation between the values of −∆G23 and θ, the dependence of effective rate constant, kPCET , on θ is similar to that on −∆G23 (see Figure 10). The numerical simulations have shown that the value of the angle θ∗ at which the PCET rate constant reaches its maximum depends on both the solvent polarity and the PCET free energy gap, −∆G23 . The simulations show that the replacement of ACN by DCM solvent can lead to both an increase and decrease of kPCET in the system II. The first trend is predicted for small angles θ, while the second is expected for large values of θ. In the system I with −∆G23 = 0.6 eV

21



only the first trend is operative. To confirm the applicability of the model elaborated for quantitative description of the PCET kinetics, we examine a possibility of its fitting to experimental data obtained by P.-T. Chou and co-workers. 14 The time-resolved fluorescence decay of the excited state in diCN-HBO compound in ACN and DCM solvents is shown in Figure 11. The best fit of PCET kinetics in ACN solution is obtained for the value of the angle θ = 85◦ . The experimental kinetics in DCM can be reproduced only with a significantly larger value of the angle θ = 123◦ . It is easy to note a difference in decay kinetics for diCN-HBO in ACN and DCM solvents. Decay of PE* state in DCM solvent unlike ACN clearly demonstrates two stages with considerably different rate constants. The second stage with slower rate is the equilibrium stage which looks like a quasi-plateau at large times (see the red and blue lines in insert in Figure 11). At the same time, in ACN, the reaction includes a single stage, ie, completely occurs in a nonequilibrium mode. The underlying physics of these results is fairly transparent. The pump pulse vertically transfers a part of the ground state population to the excited state, which is visualized as the appearance of a wave packet (a dark bell on term U2 in Figure 4). Next, the wave packet begins to move downward along the term U2 and, when it reaches the intersections of the term U2 with high vibrational sublevels of the term U3 (red dashed parabolas) the transitions to the product state (U3 ) proceed. Such transitions occur in parallel to the motion of the wave packet to the bottom of the term U2 (the motion visualizes the relaxation of the solvent to its equilibrium with the new charge distribution in the fluorophore in the excited state) and is called nonequilibrium transitions. Nonequilibrium transitions can be very effective because of the large number of excited vibrational states in polyatomic molecules, so that only a small fraction of the particles on the U2 term can avoid nonequilibrium transitions. 49,62 In this case, the product formation is expected to have a single stage and the decay of the excited state is nearly exponential (the rate constant weakly dependents on time). The effectiveness of nonequilibrium transitions strongly depends on the characteristics of the

22


Page 22 of 41

Page 23 of 41

a)

4

kPCET, ps

-1

5

3 2 ACN

1 0

DCM Benzene

40

80

120

160

-1

200

b)

4

kPCET, ps

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


3 2 ACN

1

DCM Benzene

0

40

80

120

160

200

Figure 10: Dependence of the PCET rate constant, kPCET , on angle, θ, in ACN (blue), DCM (red), and benzene (green) solutions. The values of free energy gap, −∆G23 , are 0.4 eV (shown by triangles) and 0.6 eV (shown by circles). The value of the matrix element of transition, V23 is 0.03 eV. The frames a and b show the data obtained with the systems I and II, respectively.

23



P

ET*

600

Intensity, arb. units

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

Page 24 of 41

1

0,1

300 0,01 0

1

2

t, ps

3

ACN DCM 0 -2

0

2

t, ps

4

6

Figure 11: Time-resolved fluorescence decay of the excited state (PE*) in diCN-HBO molecular system (the system II) for ACN (blue lines) and DCM (red lines) solvents. The experimental data 14 are indicated by identically colored symbols. The numerical parameters are: V23 = 0.02 eV; −∆G23 = 0.68 eV (ACN) and 0.64 eV (DCM); θ = 85◦ (ACN) and 123◦ (DCM); τL = 0.19 ps (ACN) and 0.41 ps (DCM). The insert in the upper right corner demonstrates the time dependence of the excited state population kinetics, PPE∗ . All energy parameters are taken the same as in frame a. solvent. 67 In particular, a change in the solvent viscosity can lead to a suppression of the efficiency of nonequilibrium transitions, which leads to an increase or decrease in the number of particles in the U2 state that survive at the nonequilibrium stage. Such particles can also transit into the product state, but they have to overcome the free energy barrier between the minima of the terms U2 and U3 . According to Marcus theory, the activation energy is

Ea =

(23) 2 (∆G23 + Erm ) (23)

4Erm

(15)

and the rate constant at the equilibrium stage can be much smaller than that of the nonequi(23) holds. librium one. In the above speculations, we assume that the inequality −∆G23 < Erm

The kinetics with two stages, whose rate constants differ very strongly, was previously observed in charge recombination in excited donor-acceptor complexes consisting of 1,2,4trimethoxybenzene and tetracyanoethylene. 68 The second, much slower, stage was detected

24


Page 25 of 41

0.1

Free energy

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


U2 Ea

0.0

(ACN)

(DCM)

G23

U3

-0.6

(DCM)

G23

(ACN)

U3

-0.7

-1.5

-1.0

Q23

0.0

0.5

Figure 12: Cuts of the free energy surfaces along the reaction coordinate of the 2 → 3 transition. Q23 is the coordinate along the line connecting the points of the U2 and U3 terms minimums (see 3). only in a fast solvent (acetonitrile), and it disappeared in slower solvents, valeronitrile and octanonitrile. Modelling the kinetics of charge recombination confirmed that its efficiency at the nonequilibrium stage increases with increasing the solvent relaxation time, that greatly reduces the yield of the thermalized excited state, and its decay becomes invisible in the experimental transition spectra in slow solvents. 69 Here we are faced with the opposite situation, the second slower stage, which, apparently, should be interpreted as a thermal stage, is observed only in DCM, which is slower than ACN. Obviously, the mechanism should be different. The values of the angle θ = 85◦ in ACN and 123◦ in DCM are returned from fitting the excited state kinetics in diCN-HBO to experimental data. Using the values of the angles, eq 5, and the data from Table 2, we get (23) = 0.61 eV and 0.91 eV in ACN and DCM, correspondingly. Taking into account the Erm

value of the free energy gap for the transition 2 → 3, −∆G23 = 0.68 eV, we conclude that it is slightly larger than the value of the reorganization energy of the classical nuclear modes in ACN that means that the thermal transitions occur in the Marcus inverted region (see the red curve in Figure 12). In this case, the wave packet motion to the bottom of the U2 curve, due to the solvent relaxation, has to result in increase of the rate constant of 2 → 3 25



Page 26 of 41

transition because the most effective term intersections are placed at the curve bottom (see Figure 12). This is the reason why the kinetics in ACN has the only stage. For 2 → 3 (23) transitions in DCM, the reorganization energy Erm noticeably exceeds the free energy gap

−∆G23 and particles that avoided the nonequilibrium 2 → 3 transitions, accumulate in the vicinity of the U2 curve minimum. After fast thermalization, the reaction proceeds in the thermal regime, which requires the passage of particles through the free energy barrier (see Figure 12), which significantly reduces the rate constant and, as a result, two-stage kinetics is visible. Although the activation barrier with these parameters is very small, Ea ≈ kB T , it is sufficient to reduce the rate constant approximately three-fold at the thermal stage in accord with the experiment. Further insights can be gained from the analysis of the reorganization energy dependence on the solvent polarity. First of all we notice that the alteration of the reorganization energy is much weaker than the variation of the solvent polarity ∆f . According to the Onsager reaction field model, the magnitude of solvent reorganization energy is Since Erm is proportional to ∆f , we may suppose that there is a component of Erm which weakly depends on ∆f . Apparently, it can be associated with intramolecular reorganization. We denote it (in) as Erm . Supposing independence of this component of the solvent polarity, we can write the

dependence of the total reorganization energy on the solvent polarity as follows:

(solv) (in) Erm = Erm ∆f + Erm

(16)

(solv) (in) The quantities Erm and Erm for each transition i → j can be determined using the total

reorganization energy values in two solvents ACN and DCM. They are (in eVs):

(12) Erm = 0.14∆f + 0.19 (13) = 0.02∆f + 0.39 Erm

26


Page 27 of 41 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


(23) Erm = −0.69∆f + 1.03

(17)

For the transitions 1 → 2 and 1 → 3, the values are physically reasonable, while for (solv) the transitions 2 → 3 a negative value of Erm is obtained. This unphysical result directly

follows from another surprising result returned from fitting: a decrease in the total reor(23) ganization energy, Erm , with increasing polarity of the solvent. Taken into account the

very small variation of the free energy gap, −∆G23 , (it is 0.36 and 0.40 eV in ACN and DCM, correspondingly) with the solvent polarity, the last trend is a direct consequence of the appearance of the second thermal stage in the fluorescence decay of the excited state. To rationalize the apparent contradiction of these results with an expected increase in the reorganization energy with increasing solvent polarity, we note that although the law of the reorganization energy addition, eq 16, is generally accepted, it is not universal. The law is applicable only if the degrees of freedom which are associated with the reorganization (solv) (in) energies Erm and Erm are independent. Only in this case the coordinates associated with (solv) (in) Erm and Erm are orthogonal, and eq 16 represents the Pythagorean theorem (the reorga-

nization energy is proportional to the square of the coordinate shift). In the general case, the motions along the coordinates are mutually dependent, for example, if the motion along the intramolecular coordinate is accompanied by a change in the dipole moment of the molecule, then the equilibrium polarization of the solvent also changes. For dependent coordinates the law of the reorganization energy addition has the form √

Erm =

(solv) Erm ∆f

+

(in) Erm

(solv)

(in)

− 2 Erm ∆f Erm cos ϑ

(18)

Equations 5 and 18 are very similar, but the angles ϑ and θ have completely different meanings. The angle ϑ is a measure of the correlation between motions along the solvent and intramolecular coordinates. Having two values of the total reorganization energy, we can (in) (solv) , and ϑ. It follows from eq 18 that the , Erm not determine three unknown quantities Erm (solv) are compatible with the obtained values of the total reorganization positive values of Erm

27



Page 28 of 41

energy only if the angle ϑ does not exceed 90◦ . For example, with ϑ = 70◦ , ϑ = 60◦ , ϑ = 45◦ (solv) and ϑ = 0◦ instead of eq 16 we obtain fairly reasonable parameters Erm = 0.18 eV, 0.06 eV, (in) and 0.03 eV, and 0.0144 eV, Erm = 0.69 eV, 0.74 eV, 0.75 eV, and 0.754 eV correspondingly. (solv) An increase in the angle in the range ϑ > 70◦ leads to a rapid increase in Erm , which ends

with the disappearance of the positive solution. The physical meaning of the angle ϑ can be demonstrated with a simple model. We introduce an intramolecular coordinate X and a solvent coordinate Q which are associated with different degrees of freedom and, hence, are independent by definition. In terms of these coordinates the free energies surfaces are written in the form X 2 Q2 + 2 2 (X − X0 )2 (Q − Q0 )2 + + ∆G23 = 2 2

U2 =

(19)

U3

(20)

where X0 and Q0 describe a change of equilibrium positions along the coordinates X and Q. By definition, the reorganization energy for the transition 2 → 3 is

Erm =

X02 Q20 + 2 2

(21)

where the first and second terms represent the reorganization energies of the intramolecular and solvent degrees of freedom, respectively. This result completely agrees with eq 16. Next, we assume that a variation of the intramolecular coordinate X leads to an alteration of the equilibrium state of the solvent due to, for example, a redistribution of charges in the molecule. That is, we assume that the dipole moments of a molecule in a non-equilibrium (arising from a pumping pulse or a non-adiabatic transition) and the equilibrium configurations are noticeably different. The intramolecular relaxation is assumed to be associated with low-frequency, large-amplitude vibrational motions (probably overdamped), which, in its turn, are accompanied by a change in the dipole moment of the solute. In other words, the solvent polarization is directly coupled to the dipole moment of the molecule and indirectly 28


Page 29 of 41 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


to the intramolecular coordinate X. A similar supposition was made in ref 70 which states: ”ESIPT may be governed by low-frequency, large-amplitude vibrational motions associated with the hydrogen bond” (0)

In linear approximation, we get Q0 = Q0 + αX. Here the dimensionless parameter α describes the intensity of interaction of the solvent and intramolecular degrees of freedom. Appearance of the interaction results in a change in the reorganization energy and instead of eq 21 we obtain (0)

Erm =

X02 (Q0 + αX0 )2 + 2 2

(22)

Introducing the reorganization energies

(in) Erm =

X02 Q2 (solv) (1 + α2 ), Erm ∆f = 0 2 2

(23)

one obtains the same law of addition of the reorganization energies as in eq 18 with α cos ϑ = ± √ 1 + α2

(24)

where the ”−” sign is selected if the shifts of the term minima, X0 and Q0 , have the same sign and ”+” in the opposite case. This example shows that for X0 and Q0 of the same sign, the angle ϑ lesser than 90◦ corresponds to the negative values of the parameter α. A very similar result is obtained if one assumes that the relaxation of the solvent leads to a change (0)

in the equilibrium geometry of the solute, that is, X0 = X0 + βQ. The interaction of the solvent and slow (hence, classical) intramolecular degrees of freedom, derived from the fit of the stationary spectra and the excited state decay kinetics, allows us to draw some conclusions about the character of the processes in the fluorophore and solvent in the transition 2 → 3. The interaction is predicted to be fairly large, for example, the parameter α = 0.36 for ϑ = 70◦ and quickly increases with increasing the angle ϑ. Strong interaction means that after the transition between diabatic states 2 and 3, the relaxations of the solvent and intramolecular degrees of freedom to a new equilibrium state 29



Page 30 of 41

proceed in parallel. If the motion along the solvent coordinate is slower than the motion along the intramolecular coordinate, then the combined relaxation should be determined by the solvent relaxation timescale. Since the interaction considered assumes that the intramolecular relaxation is accompanied by intramolecular charge redistribution or charge transfer, we can conclude that the transition 2 → 3 in diCN-HBO involves to some extent a solvent driven (hence, slow) intramolecular charge transfer. Now we outline the mechanistic picture of the processes in the excited diCN-HBO. Let the dipole moments in the equilibrated states 2 and 3 be equal to ⃗µ2 and ⃗µ3 , respectively. Before the transition 2 → 3 the geometry of the solute and the state of the solvent are close to equilibrium (see Figure 12). Taking into account that the state of the nuclear subsystem does not change during the nonadiabatic transition, immediately after the transition the nuclear subsystem in the state 3 is far from the new equilibrium. Relaxation to the equilibrium includes both intramolecular and solvent reorganization, moreover its larger part associates with intramolecular reorganization. Apparently, such a slow intramolecular reorganization involves the movement of a large amplitude, such as conformational changes. Since separate parts of the molecule carry electric charges, intramolecular reorganization should be accompanied by a change in the dipole moment of the solute. Therefore one can expect that immediately after transition 2 → 3 the dipole moment of diCN-HBO differs from its equilibrium value ⃗µ3 . Let it be equal to ⃗µ∗3 . It is evident from above analysis, that a decrease of the total reorganization energy with increasing solvent polarity takes the inequality to be met (Q0 )2 |⃗µ2 − ⃗µ3 |2 (Q0 + αX0 )2 |⃗µ2 − ⃗µ∗3 |2 ∆f = > ∆f = a3 2 a3 2 (0)

(0)

(25)

where the well known equation

Erm

(⃗µi − ⃗µj )2 = ∆f a3

(26)

for the solvent reorganization energy for a nonadiabatic transitions between states of a solute

30


Page 31 of 41 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


with dipole moments ⃗µi and ⃗µj is used. Here a is the radius of the solute. Unfortunately, except for the special case of ⃗µ2 = 0, we can not say anything about the difference between the lengths of the vectors ⃗µ3 and ⃗µ∗3 , due to the lack of any information about the angles (23) between them and vector ⃗µ2 . Although the total reorganization energy Erm in diCN-HBO

varies from 0.6 to 0.9 eV, the solvent reorganization energy is estimated to be fairly small, it is of the order of 0.1 eV. This agrees with the estimates of the changes in the dipole moment in these systems of the order of 1 D. 21

Conclusions A theoretical approach for PCET kinetics description is proposed. It takes into account both the dynamics of solvent relaxation and reorganization of intramolecular low- and highfrequency vibrational modes. The model assumes the presence of three diabatic states of the PE system and requires involving two reaction coordinates associated with photoexcitation and subsequent PCET stages. The reaction coordinates are not independent and the magnitude of this dependence is quantified by a single parameter: the angle between them in the space of nuclear degrees of freedom. 46 This parameter is very important characteristic of ultrafast PCET, since the simulations within multichannel stochastic approach show strong dependence of PCET kinetics on the angle θ. Many parameters of the model can be determined from the stationary absorption and fluorescence spectra of a particular fluorophore. Unfortunately, the information containing in the stationary spectra of the systems I and II is not sufficient to calculate the angle θ. Its value can be obtained from fitting of the simulated time-resolved fluorescence decay to experimental data. The PCET is shown to occur totally or partly, depending on the solvent, in nonequilibrium regime. Fit of the diCN-HBO excited state decay kinetics to experimental data has uncovered, that there is a significant decrease in the magnitude of the reorganization energy of slow

31



nuclear modes with increasing the solvent polarity. This unusual behavior of the total reorganization energy can be compatible with an expected increase in solvent reorganization energy with increasing solvent polarity if we assume that (i) a slow intramolecular reorganization is associated with the 2 → 3 transition and (ii) intramolecular slow reorganization is accompanied by a change in the dipole moment of diCN-HBO. These findings well agree with the conclusion reported in Ref 18: ”diCN-HBO undergo excited state proton transfer, concomitantly accompanied with the charge transfer process, such that the excited state proton transfer reaction dynamics are directly coupled with solvent polarization effects”. Although it was theoretically shown two decades ago that two successive or parallel reactions, in general, are described in terms of dependent reaction coordinates and the extent of the dependence is quantified by the angle between them but according to our best knowledge, a manifestation of the effect in real systems was not demonstrated so far. The simulations of kinetics of ultrafast photoinduced intramolecular proton-coupled charge transfer in diCNHBO performed in this article clearly evidence that the two reaction coordinates are strongly dependent and moreover, the intramolecular and solvent reorganizations are also strongly coupled.

Acknowledgement The study was performed by a grant from the Russian Science Foundation (Grant No. 1613-10122).

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Modelling Kinetics of Ultrafast Photoinduced Intramolecular Proton

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