Theory of Vibrational Spectra of Excited Quadrupolar Molecules with

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Theory of Vibrational Spectra of Excited Quadrupolar Molecules with Broken Symmetry Anatoly I. Ivanov J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b10985 • Publication Date (Web): 10 Dec 2018 Downloaded from http://pubs.acs.org on December 14, 2018

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The Journal of Physical Chemistry

Theory of Vibrational Spectra of Excited Quadrupolar Molecules with Broken Symmetry Anatoly I. Ivanov∗ Volgograd State University, University Avenue 100, Volgograd 400062, Russia E-mail: [email protected]

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Abstract Quadrupolar molecules of the structure AL –D–AR , where D is an electron donor and AL and AR are identical acceptors are investigated. A general theory of vibrational spectra of such molecules undergoing symmetry breaking after photoexcitation is developed. The approach is based on the expansion of the electron-vibrational interaction in a series in powers of the displacements of the vibrational modes and the dissymmetry parameter. The theory provides a possibility to calculate the intensities of the IR bands and the splitting of the frequencies of local vibrations associated with vibrations of small groups of atoms symmetrically located in the left and right arms of the molecule. The theory explains the regularities of influence of symmetry-breaking on the IR band intensities and the splitting of the frequencies observed recently using time-resolved infrared spectroscopy.

Introduction Symmetry breaking (SB) photoinduced charge separation proved to be a widespread phenomenon observed in many multichromophoric systems containing two or three identical molecular units. 1–21 A peculiar feature of multibranched symmetric chromophores of DAn (ADn ) structure with n = 2, 3 (where A and D represent electron-acceptor and electrondonor groups, respectively) is a weak solvent dependence of the electronic absorption spectra and a strong solvatochromism of fluorescence spectra. 22–33 This clearly indicates the presence of a significant dipole moment in the excited state of such chromophores, in contrast to the ground state in which there is only a quadrupolar or octopolar moment. 13,25,34,35 This feature was explained by the assumption of SB in the excited state. 36–40 The idea of SB in excited quadrupolar molecules was firmly confirmed in the experiments with femtosecond time-resolved infrared spectroscopy which directly visualized the SB and provided unique information on its dynamic characteristics and magnitude. 17,18,41 In the experiments quadrupolar A-π-D-π-A molecules with −C≡N groups in both arms are explored. The 2


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stretching vibrations of a −C≡N group are appeared to be very sensitive to the SB and its extent. SB brightly manifests itself in the splitting of the −C≡N band and the magnitude of the splitting is a good measure of the SB extent. The initial Franck-Condon state created by photoexcitation of the chromophore is symmetric that is reflected in the absence of the splitting. SB was associated with solvent fluctuations since the time evolution of the splitting correlates well with the solvation dynamics. 18 Recently, extremely high sensitivity of SB to the interaction between the solute and solvent was reported. 42 Measurements of the SB extent in a large series of solvents allowed detecting the effects of quadrupolar, halogen-bond, dipole-dipole, and hydrogen-bond interactions. In particular, it was possible to quantify the interaction of polar solutes with non-dipolar and quadrupolar solvents which is strong evidence that the excited state symmetry breaking is a powerful tool for studying the solute-solvent interactions. 42 The model of quadrupolar molecules, developed in ref 36, explained the SB in terms of interaction of the molecule with the asymmetric intramolecular vibrations and solvent polarization. It includes three essential states: the neutral ground state and two excited degenerate zwitterionic states with localization of a charge on different arms of the molecules. The model successfully describes the one- and two-photon electronic spectra and the effect of SB on them. The model was successfully applied to describe a wide range of symmetry breaking effects on optical spectra in many charge transfer chromophores. 11,12,36–39 Later, the model was modified by reducing the number of electronic states to two (two excited degenerate zwitterionic states) and including the direct Coulomb interaction between the charges in asymmetrical state. 43 Such a simplified model has an analytical solution connecting the degree of SB with few parameters. The solution well describes an increase in the frequency splitting of −C≡N bands in an A − π − D − π − A molecule (ADA) consisting of a pyrrolopyrrole D core and two cyanophenyl acceptors with increasing the solvent polarity. 43 Although it was recognized that electron-vibrational interaction can promote SB and affect the vibrational motion of the molecule, 11,12,36 still there is no theoretical approach

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which could explain in simple terms the regularities derived from analysis of the experimental data provided by time-resolved infrared spectroscopy. The aims of this article are: (i) to develop a theory of influence of SB on IR spectra, (ii) to simulate the effect of SB on the IR absorption spectra of excited quadrupolar molecules, and (iii) to compare the simulated IR spectra with available experimental data.

Theory The two-level model of symmetry breaking in a linear and symmetric AL –D–AR molecule with identical electron-accepting groups, AL and AR was described in ref 43. Here we briefly outline its most important concepts. The model deals with a few lowest excited states, including a symmetric and antisymmetric stases, which can be populated by oneor two-photon absorption, respectively. Thus, the initial, Franck-Condon, excited state is quadrupolar with equal amount of charges on both acceptors, δL = δR = e/2. The symmetry breaking associates with the transition to a state with unequal charges δL 6= δR conserving the sum δL + δR = e. The model includes two basis states which are the lowest excited states corresponding to the electron localization either on the left or the right acceptor and are described by the wave functions ϕL and ϕR , respectively. The Hamiltonian of ADA in this basis is 



 EL

H0 =  

V

V  ER

. 

(1)

with EL = ER as follows from the symmetry of the molecule which are set EL = ER = 0. The single parameter of the model, V , determines the magnitude of the Davydov splitting between the symmetric and antisymmetric states. Since both states are seen in one- or twophoton absorption spectra the magnitude of V is a measurable quantity. Such an approach was earlier used to discuss the absorption spectra of quadrupole molecules. 25,44

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An arbitrary state of the system

Ψ = a L ϕL + a R ϕ R

(2)

is fully determined by two amplitudes aL and aR which are connected with the charges on the acceptors by equations: δL = e|aL |2 and δR = e|aR |2 . In what follows, the amplitudes aL and aR are assumed to be real and subjected to the normalization condition a2L + a2R = 1

(3)

If the wave functions ϕL and ϕR were one-electron, then the Coulomb interaction between the charges δL and δR would not exist, since this would be an electron self-action. In a multielectron molecule, the appearance of a charge on the right acceptor is not the transfer of an electron from the left acceptor, but rather the successive displacement of many electrons by a short distance. In this case, the charges δL and δR are associated with different electrons and a direct electrostatic interaction between charges of acceptors should be included in the model K = a2L a2R

e2 . εim RLR

(4)

The operator of this interaction can be written in a form h

i

HC = 2γC hΨ|PˆL |ΨiPˆR + hΨ|PˆR |ΨiPˆL ,

(5)

here PˆL and PˆR are the projection operators to the states ϕL and ϕR , correspondingly, RLR is the distance between the centers of charge localization on AL and AR , and

γC =

e2 . 4εim RLR

(6)

Here the "intramolecular" dielectric constant, εim , is introduced to somehow account for the

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molecular and solvent polarizabilities since the polarizations reduce the Coulomb interaction due to a partial screening the charges of the acceptors. Since the total charge on the acceptors, δL + δR = e, is assumed to be invariable in the course of SB, the total interaction energy between the charges on the acceptors and donor is also invariable and, thus, can be omitted. For SB the interaction of a molecule with a polar medium plays a crucial role. Supposing that the surrounding medium mainly interacts with the dipole moment of the molecule and the medium response is linear, the energy of the interaction, W , is proportional to the square of the dipole moment of the molecule, W ∼ µ2 . For estimations of the interaction energy the Onsager’s model can be used. Ignoring the interaction of the quadrupole moment of the molecule with polar medium and using the Onsager’s model, the energy of interaction of a dipolar molecule with polar solvent is

W0 = −

D2 µ2 ∆f , rd3

(7)

where the dipolar parameter (which is also dissymmetry parameter), D, is introduced

D = a2L − a2R ,

(8)

µ is the magnitude of the dipole moment of ADA in the state with D = 1, rd is the cavity radius, and ∆f = f (εs ) − f (n2 ) with f (x) = 2(x − 1)/(2x + 1). Here εs and n are the static dielectric permittivity and the refractive index of the solvent, respectively. The dipolar parameter, D, being the measure of the molecular asymmetry, determines the mean dipole moment by the equations hΨ|ˆ µ|Ψi = µD. The Hamiltonian for the interaction of the ADA molecule with the solvent can be expressed as h

i

HS = −λD PˆL − PˆR , 6


(9)

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where λ=

µ2 ∆f = λ1 ∆f rd3

(10)

The effects of Coulomb interaction and interaction of the solute dipole moment with the polar solvent on SB are very similar. To stress the deep analogy, we use the identity 4a2L a2R = 1−D2 which allows eq 5 to be recast in the form h

i

HC = −γC D PˆL − PˆR + γC ,

(11)

where the inessential constant γC , the last summand, can be omitted. Here the normalization condition, eq 3, is used. Symmetry breaking leads to a redistribution of electron density inside the molecule. The redistribution affects the vibrational subsystem of the molecule due to the electronvibrational interaction. Within the harmonic approximation we may expect the equilibrium positions and the frequencies of the intramolecular vibrations to change as it is upon optical excitation of a molecule. Of course, the symmetry of the molecule imposes some restrictions on these changes. Hamiltonian of a quadrupolar molecule must possess the inversion symmetry. The symmetry leads to the division of intramolecular vibrations into symmetrical and antisymmetrical modes. So, within the harmonic approximation, vibrational Hamiltonian of a symmetrical molecule has to have the form

Hv0 =

i i 1 Xh 2 1 Xh 2 2 2 xaj , psi + ωsi2 x2si + paj + ωaj 2 i 2 j

(12)

where psi , paj , xsi , xaj , ωsi , ωaj are the momenta, coordinates, and the frequencies of the symmetrical and antisymmetrical vibrations, respectively. Then electron-vibrational interaction is introduced through its formal expansion in powers of the dissymmetry parameter and vibrational displacements. Keeping terms, which are invariant under the inversion symmetry transformation, up to the second order, we get for the energy of the electron-vibrational

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interaction

Uint = D

X j

ζj xaj + D

X

δij xaj xsi +

ij

D2 X [αik xai xak + βik xsi xsk ] 2 ik

(13)

where the quantities ζj , δij , αik , and βik are considered to be phenomenological constants, but they can be calculated by quantum chemical methods. Their physical meaning is fairly transparent. The parameters αik and βik are responsible for changes in the vibrational frequencies and the Duschinsky effect, while the parameters δij describe the interaction of symmetrical and antisymmetrical vibrational modes. This interaction results in a possibility of the molecule to absorb IR radiation not only on the frequencies of the antisymmetric vibrations but also on the frequencies of the symmetrical modes. Equation 13 assumes that the dissymmetry parameter D is small. For non-small values of D, one can assume that the quantities D2 αik and D2 βik should be replaced by some even functions αik (D) and βik (D). Analogically, Dζj and Dδij should be replaced by odd functions of the parameter D. The total Hamiltonian of intramolecular vibrations including electron-vibrational interaction is Hv = Hv0 + Uint

(14)

The first summand, linear in D and xaj , was introduced earlier in ref 36 as an interaction that together with solute-solvent interaction causes symmetry breaking. Time resolved monitoring of symmetry breaking has shown that it occurs on a timescale that greatly exceeds the period of intramolecular vibrations. 42 In this extreme, the average value of the vibrational coordinate adiabatically follows the variations in the dissymmetry parameter D. For a given value of D, the average values of xaj can be found from the condition of the potential energy minimum. It is x¯aj = −

8

Dζj 2 ωaj


(15)

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The mean energy of this interaction is (1)

Uint = D

X

¯ 2 ζj x¯aj = −ζD

(16)

where ζ¯ =

X ζj2 j

2 ωaj

(17)

Comparing eq 16 with eq 9, one can see that the interactions of the molecule with solvent polarization and antisymmetric intramolecular vibrations play very similar roles in symmetry breaking. 36 The quadratic in vibration displacements terms in eq 13 describe variations of the frequencies of vibrational modes and their interactions. They are expected to be much smaller than the linear term and their effect on symmetry breaking can be neglected in the first approximation. The stationary states of the molecule are determined by the stationary Schrödinger equation HΨ = EΨ,

(18)

(1)

with H = H0 + HC + HS + Uint being the total Hamiltonian of a quadrupolar molecule interacting with the surrounding medium. Its explicit form is:

−(λ + γ)(a2L − a2R )aL + V aR = EaL ,

(19)

(λ + γ)(a2L − a2R )aR + V aL = EaR .

(20)

where γ = γC + ζ¯ is a parameter characterizing the intensity of intramolecular nonlinear interactions. The set of nonlinear eqs 19 and 20 has (i) symmetric (aL = aR ), (ii) antisymmetric (aL = −aR ), and (iii) asymmetric solutions with D 6= 0 and the energy 43

Eas = −(λ + γ). 9


(21)


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The asymmetric solution is stable if V < λ + γ. 43 In this state the dipolar parameter is v u u D = t1 −

V2 . (λ + γ)2

(22)

It should be noted that λ and γ used here differ by a factor of 2 from that used in ref 43.

Discussion To calculate the IR absorption spectrum in a state with broken symmetry, we use the classical theory of the electromagnetic wave absorption. From eq 14, we obtain the equations of motion for molecular vibrations interacting with an electromagnetic wave with a frequency of ω

2 x¨aj + ηaj x˙ aj + ωaj xaj + D

X

δij xsi + D2

i

x¨si + ηsi x˙ si + ωsi2 xsi + D

X

αij xai = Fj eiωt

(23)

βij xsj = 0

(24)

i

X

δij xaj + D2

j

X i

where the dots above the vibrational coordinates denote time derivatives, ηaj and ηsi are the friction coefficients, which are the reciprocal of the relaxation time of the corresponding vibration, Fj is the interaction strength of the j-th vibration with the electromagnetic wave, ω is the electromagnetic wave frequency. According to the selection rule, only antisymmetric vibrations interact with IR radiation, since it can be absorbed only if the vibrations lead to a change in the dipole moment. The amplitudes of driven oscillations with the frequency ω are determined by the set of linear algebraic equations

2 ωaj − ω 2 + iηaj ω x¯aj + D

X

δij x¯si + D2

X

i

ωsi2 − ω 2 + iηsi ω x¯si + D

X

δij x¯aj + D2

j

10

αij x¯ai = Fj

(25)

βij x¯si = 0

(26)

i


X i

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For real force Fj (t) = Fj (eiωt + e−iωt ), the mean energy absorbed by j-th vibration is D

E

Ij (ω) = Fj (t)(x˙ aj + x˙ ∗aj ) = −2ω

X

Fj =¯ xaj

(27)

j

where the brackets stand for time average. The full IR absorption spectrum is determined by the sum of the spectra of independent absorbers I(ω) =

P

Ij (ω).

From eqs 23 and 24, two general conclusions follow. Firstly, symmetry breaking leads to a change in the frequencies of antisymmetric and symmetric vibrations. Secondly, a molecule in a state with broken symmetry can absorb at the frequencies of antisymmetric as well as symmetric vibrations due to the interaction between them (the terms with couplings δij ). The general theoretical description is illustrated by the simplest example, but the most important for applications. First of all, we note that the relatively weak interaction of antisymmetric and symmetric vibrational modes with different frequencies leads to insignificant shifts of their frequencies and very weak absorption at the frequency of the symmetric mode. A much more pronounced effect is expected when there are two modes with the same frequency in the symmetric state. In this case, splitting of one absorption band into two bands should be observed. This phenomenon was experimentally investigated in details and reported in a series of articles. 17,18,41,42 In the molecules studied, there are two −C≡N or −C≡C− groups located close to the edges of the arms. Two well localized stretching vibrations are associated with these groups. Because of the large distance between the groups, their vibrations interact weakly, so that there are two mutually decoupled effective coordinates, xL and xR , describing the vibrational motion in each branch. The two coordinates are equivalent by symmetry and have the same natural harmonic frequency ω0 . Limiting the consideration by these two vibrational modes, we can write Hamiltonian of the vibrations in terms of the local and symmetrized coordinates as

Hv0 =

i i i i 1 Xh 2 1 Xh 2 1h 2 1h 2 pL + ω02 x2L + pR + ω02 x2R = pa + ω02 x2a + ps + ω02 x2s 2 2 j 2 2 j

11


(28)


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where 1 xa = √ (xL − xR ) 2 1 xs = √ (xL + xR ) 2

(29) (30)

are the coordinates of the antisymmetric and symmetric vibrational modes. The energy of electron-vibrational interaction is

Uint = Dζ xa + Dδ xa xs +

i D2 h 2 α xa + β x2s 2

(31)

and eqs 25 and 26 are reduced to a couple of equations

∆a x¯a + Dδ x¯s = F

(32)

Dδ x¯a + ∆a x¯s = 0

(33)

where ∆a = ω ¯ a2 − ω 2 + iηa ω, ∆s = ω ¯ s2 − ω 2 + iηs ω, ω ¯ a2 = ω02 + D2 α, ω ¯ s2 = ω02 + D2 β. The solution of eqs 32 and 33 is x¯a =

∆s ∆a ∆s − D2 δ 2

(34)

The IR absorption spectrum is defined by the relation

I(ω) = −2ω=¯ xa

(35)

The spectrum has two maxima with a split between them q

∆ω =

(α − β)2 D4 + 4D2 δ 2 2ω0

(36)

Here the change in the frequencies ω ¯ a and ω ¯ s , accompanying symmetry breaking, is assumed to be much less than the initial frequency, ∆ω ω0 . For parameters that satisfy the 12


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inequality (α − β)2 δ 2 , eq 36 predicts a nearly quadratic dependence of the frequency splitting, ∆ω on D, that agrees well with experimental data. 43 Typical IR absorption spectra are pictured in Figure 1. It shows that an increase in the magnitude of the dissymmetry parameter leads to the expected increase in the band splitting. In the calculations, it is assumed that the values of a and b have opposite signs, since the charge of one of the acceptors increases and the other decreases with increasing D, and quantum-chemical calculations, reported in ref 45, have showed that the stretching frequency of −C≡N group decreases monotonically with increasing the magnitude of fractional negative charge in this group. These calculations also showed that with a change in the partial charge in a −C≡N group, the frequency shift of the symmetric vibrational mode is several times smaller than that for the antisymmetric mode. This is the reason why the magnitude of the parameter a is significantly less than b. The coupling of the antisymmetric and symmetric modes, δ, also affects the magnitude of the band splitting (compare the solid and dashed lines of the same color in Figure 1). The ratio of the peak values of the smaller and larger absorption bands, r, is govern mostly by this coupling. The ratio decreases with decreasing Dδ and the lesser band disappears in the limit Dδ → 0. There is one important difference between the calculated and experimental spectra. Namely, the experimental ratio, r, increases with increasing the splitting being lesser 0.1 for the investigated molecule and solvents, and the calculated ratio decreases. There are at least two reasons of the discrepancy. First, some effects might be missed in the model, for example, the frequencies of symmetric and antisymmetric vibrations may slightly differ in symmetric molecule due to the indirect interaction of local oscillations of the C≡N groups. The second reason can be connected with the fact that the theory takes into account only the homogeneous broadening of the spectral bands caused by the finite lifetime of the excited vibrational states. In real systems, there is an inhomogeneous broadening, which is larger than the homogeneous broadening and the magnitude of the inhomogeneous broadening increases with D. 42 Since the broadening of the band does not change its area, an increase in the broadening leads to a decrease in

13



4

3

A

1 2

Absorption, arbitrary 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

4

B

3

1 2

0.96

0.98

1.00

1.02

/

Figure 1: IR absorption spectra for four values of the dissymmetry parameter D = 0.1 (1, black lines), 0.3 (2, red lines), 0.6 (3, green lines), and 0.8 (4, blue lines). The parameters are: a/ω02 = 0.02, b/ω02 = −0.08, ηa /ω0 = ηs /ω0 = 0.006, δ/ω02 = 0.03 (frame A, solid lines) and δ/ω02 = 0.05 (frame B, dashed lines)

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the band maximum. If the width of the larger band increases with D faster than that of the weaker, then the inhomogeneous broadening should result in stronger suppression of the larger maximum. This can also change the dependence of the ratio r on the dissymmetry parameter. To get a more visual idea of the mechanism of splitting an IR spectral band, let us consider another simple physical model. In the symmetric state, the natural frequencies of oscillations of the −C≡N groups coincide due to the symmetry of the molecule. In a state with broken symmetry, these frequencies have to be different. Assuming that symmetry breaking leads only to a difference in the frequencies of the local vibrations, we can write the potential energy of the vibrations of the −C≡N groups in terms of the local and symmetric coordinates as follows

Uv1 =

i i 1h 2 2 1h 2 ωL xL + ωR2 x2R = (ωL + ωR2 )(x2a + x2s ) + 2(ωL2 − ωR2 )xa xs 2 4

(37)

This potential energy is a special case of that considered above. It leads to the splitting of the IR absorption band into two bands of equal intensity with a split of ωL − ωR between them. Thus, this intuitively clear model explains the nature of the interaction of symmetric and antisymmetric vibrational modes arising from symmetry breaking, but it fails to predict the correct ratio of intensities of IR absorption bands observed in the experiment. Let us discuss the effect of the linear electron-vibrational interaction D

P

j

ζj xaj on the

IR spectrum. To do this, we neglect the electron-vibrational interaction, which is quadratic in the vibrational displacements. Supposing the dissymmetry parameter D to be fixed or to vary much slower than the vibrations, one can expect that the interaction has little effect on the vibrational frequencies, since the linear interaction changes only the position of the potential minimum and does not affect the potential energy curvature, which determines the natural frequencies of the vibrations. Apparently, this can be the case when SB is governed by solvent polarisation which usually quite slow. In such conditions, changes in

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the vibrational frequencies are created by the terms in eq 13, quadratic in the vibrational displacements. In the opposite limit of fast electronic subsystem, we can use adiabatic separation of variables. Within the adiabatic approximation, the adiabatic potential for the vibrations can be calculate as follows. The mean energy in an arbitrary electronic state given by eq 2 is determined by eq 38 h

ET = hΨ|H0 + PˆL − PˆR

iX

ζj xaj +

j

= −V

√

1 − D2 − D

X

ζj xaj +

j

1X 2 2 ω x |Ψi 2 j aj aj

1X 2 2 ω x 2 j aj aj

(38)

where D is arbitrary. According to the variational principle, in the stationary state the function ET has a minimum. For given values of xaj , the minimum is reached at P

j ζj xaj D=q P V 2 + ( j ζj xaj )2

(39)

Inserting eq 39 into eq 38, we obtain the potential energy of the vibrational subsystem for arbitrary values of the coordinates r

UAd = − V 2 +

X

ζj x2aj +

1X 2 2 ωaj xaj 2

(40)

One can see that the linear interaction greatly changes the potential energy of the vibrations if the parameters ζj are not small. Since the antisymmetric vibrations interact with IR radiation, a cardinal reconstruction of the IR spectrum of a molecule is expected in the course of SB. In the case of small ζj , the linear interaction leads to a renormalization of the natural frequencies of antisymmetric intramolecular vibrations. Further, we show that the linear interaction, even being weak, can lead to observable effects in the IR spectra. We again consider only the simplest model with a couple of equivalent vibrations in the two arms of the molecule. So, we have the only antisymmetric vibrational mode and for the weak 16


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electron-vibrational coupling V 2 ζ 2 x2a , we obtain from eq 40 q 1 2 2 1 UAd = − V 2 + ζ 2 x2a + ω02 x2a ' −V + ω ˜ x 2 2 a a

(41)

where ω ˜ a2 = ω02 −

ζ2 V

(42)

Since only antisymmetric vibration is involved in the linear electron-vibrational interaction, a difference arises between the frequencies of symmetric and antisymmetric vibrations, ωs2 − ω ˜ a2 = ζ 2 /V , even if the dissymmetry parameter is zero. When the linear electron-vibrational interaction is weak, its effects should be combined with the effects created by the terms quadratic in the displacements of the oscillations. This combination changes the dependence of the frequency splitting on the dissymmetry parameter. Now we obtain ∆ω = ω0

q

[(α − β)D2 − ζ 2 /V ]2 + 4D2 δ 2 2ω02

(43)

instead of eq 36. For the parameters adopted in Figure 3, the dependence of ∆ω on the parameter D2 is also close to linear except for the region of small values of D (see the red and black lines in Figure 2). This agrees with the experimental data that show linear dependence of the splitting on D2 . 43 At the same time the dependence of ∆ω on the parameter D2 can strongly deviate from linear. For small values of the parameter δ, the dependence becomes nonmonotonic. In ADA, the splitting is quite large to be distinctly detected. 43 In this molecule, the quantities β − α and ζ 2 /V have opposite signs, that decreases the splitting. It is expected that a significantly larger splitting can be detected in molecules with β − α and ζ 2 /V of the same sign. The manifestation of this splitting in the IR spectra is shown in Figure 3. In the calculations of I(ω), in the equation for ∆a the shifted frequency ω ˜ a2 instead of ω02 is used, that is ω ¯ a2 = ω ˜ a2 + D2 α. The appearance of an additional splitting of the frequencies of symmetric

17



0.04

0.03

/

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2

0.02 3

1

0.01

0.00 0.0

0.2

0.4

0.6

0.8

1.0

2

D

Figure 2: Frequency splitting versus squared asymmetry parameter D2 for a/ω02 = 0.0, b/ω02 = −0.07. The rest of the parameters are: δ/ω02 = 0.035, ζ 2 /V ω02 = 0.025 (1, red line), δ/ω02 = 0.035, ζ 2 /V ω02 = 0 (2, blue line), δ/ω02 = 0.005, ζ 2 /V ω02 = 0.025 (3, green line), dashed line is the linear approximation to line 1.

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Figure 3: Normalized IR absorption spectra for four values of the dissymmetry parameter D = 0.0 (1, black lines), 0.2 (2, green lines), 0.4 (3, blue lines), and 0.6 (4, red lines). The parameters are: a/ω02 = 0.0, b/ω02 = −0.07, ηa /ω0 = ηs /ω0 = 0.005, δ/ω02 = 0.035, ζ 2 /V ω02 = 0.025

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and antisymmetric vibrations, which is independent of the dissymmetry parameter, produces qualitative changes in the spectra. Now the intensity of the weak maximum increases with increasing the splitting between the maxima as it is observed in the experiments. The simulated spectra pictured in Figure 3 are very similar to those obtained by experimental measurements of IR absorption spectra in photoexcited ADA (see Figure S7 in Supporting Information to ref 42). In the calculations the value of ζ 2 /V ω02 = 0.025, corresponding to the frequency splitting of about 1.2% is used. For smaller values of this parameter, the intensity of the weak maximum decreases with increasing the splitting between the maxima, whereas for its larger values, the splitting becomes larger than that observed in experiments. 42 It should be noted that local vibrations associated with the left and right arms of the molecule in a symmetric state can interact with each other, at least indirectly, through the bonds with the atoms between the groups. Such an interaction also leads to the splitting of the frequencies of symmetric and antisymmetric vibrations in the symmetric state of the molecule. The effect of such splitting on IR spectra is expected to be the same as that studied in this paragraph (produced by weak linear electron-vibrational interaction).

Conclusion We have studied the effect of the symmetry-breaking charge transfer in the excited state of quadrupolar molecules on intramolecular vibrational spectrum. Although we explicitly refer to the structure A − π − D − π − A, but the model is also applicable to the triads of the form D−π−A−π−D. A theory of the effect of symmetry breaking on the vibrational spectrum has been developed. The expansion of the electron-vibrational interaction in a series in powers of the displacements of the vibrational modes and the dissymmetry parameter, which takes into account the inversion symmetry of the molecule, underlies the proposed approach. The first term of the expansion, which is the product of the displacements of the antisymmetric intramolecular vibrational modes and the dissymmetry parameter, was previously introduced

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into the theory as one of the interaction responsible for the symmetry breaking. 36 Here we show that the next terms of the series, quadratic in the displacements of the intramolecular vibrational modes and the dissymmetry parameter, should be taken into account to describe the effect of symmetry breaking on the vibrational spectrum observed in experiments using time-resolved infrared spectroscopy. 17,18,41,42 The theory reveals the mechanism of splitting the frequencies of local vibrations in small groups of atoms symmetrically located in the left and right arms of the molecule. The normal coordinates of such vibrations are symmetric and antisymmetric combinations of local coordinates in quadrupolar molecules. In a symmetric molecule, only the antisymmetric vibrational mode can absorb IR radiation, and in the experiment one IR absorption band is observed. In a state with broken symmetry, the frequencies of symmetric and antisymmetric vibrational modes become different and interaction of symmetric and antisymmetric vibrational modes occurs, that leads to the appearance of a new absorption band at a frequency of symmetric oscillation. The intensity of this band increases with increasing the strength of the interaction of symmetric and antisymmetric vibrational modes. The theory predicts the strength to be proportional to the dissymmetry parameter. A qualitative comparison of the calculated IR spectra dependence on the dissymmetry parameter has shown that the theory explains the regularities of the symmetry-breaking effect on the IR band intensities and its splitting observed recently in the experiments. 17,18,41,42

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

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