Article
Reference
Modeling Infrared Spectral Dynamics upon Symmetry Breaking of a Photo-Excited Quadrupolar Dye
NAZAROV, Alexey E., IVANOV, Anatoly I., VAUTHEY, Eric
Abstract
A significant number of quadrupolar dyes with a D-$pi$-A-$pi$-D or A-$pi$-D-$pi$-A structure, where D and A are electron donor and acceptor groups, were shown to undergo symmetry breaking (SB) upon optical excitation. During this process, the electronic excitation, originally distributed evenly over the molecule, concentrates on one D-$pi$-A branch, and the molecule becomes dipolar. This process can be monitored by time resolved infra-red (TRIR) spectroscopy and causes significant spectral dynamics. A theoretical model of excited-state SB developed earlier ( extit{J. Phys. Chem. C}, extbf{2018}, extit{132}, 29165) is extended to account for the temporal changes taking place in the IR spectrum upon SB. This model can reproduce the IR spectral dynamics observed in the $-mathrm{C}equiv mathrm{C}-$
stretching region with a D-$pi$-A-$pi$-D dye in two polar solvents using a single set of molecular parameters. This approach allows estimating the degree of asymmetry of the excited state in different solvents as well as its change during SB. Additionally, the relative contribution of the different mechanisms responsible for the [...]
NAZAROV, Alexey E., IVANOV, Anatoly I., VAUTHEY, Eric. Modeling Infrared Spectral
Dynamics upon Symmetry Breaking of a Photo-Excited Quadrupolar Dye. Journal of Physical Chemistry. C, 2020, vol. 124, no. 4, p. 2357-2369
DOI : 10.1021/acs.jpcc.9b10565
Available at:
http://archive-ouverte.unige.ch/unige:132980
Disclaimer: layout of this document may differ from the published version.
Modelling IR Spectral Dynamics upon Symmetry Breaking of a Photo-Excited Quadrupolar Dye
Alexey E. Nazarov,
†Anatoly I. Ivanov,
∗,†and Eric Vauthey
∗,‡†Volgograd State University, University Avenue 100, Volgograd 400062, Russia
‡Department of Physical Chemistry, University of Geneva, 30 Quai Ernest-Ansermet, CH-1211 Geneva 4, Switzerland
E-mail: Anatoly.Ivanov@volsu.ru; Eric.Vauthey@unige.ch
Abstract
A significant number of quadrupolar dyes with a D-π-A-π-D or A-π-D-π-A struc- ture, where D and A are electron donor and acceptor groups, were shown to undergo symmetry breaking (SB) upon optical excitation. During this process, the electronic excitation, originally distributed evenly over the molecule, concentrates on one D-π-A branch, and the molecule becomes dipolar. This process can be monitored by time resolved infra-red (TRIR) spectroscopy and causes significant spectral dynamics. A theoretical model of excited-state SB developed earlier ( Ivanov, A. I.J. Phys. Chem.
C, 2018, 122, 29165-29172) is extended to account for the temporal changes taking place in the IR spectrum upon SB. This model can reproduce the IR spectral dynamics observed in the−C≡C−stretching region with a D-π-A-π-D dye in two polar solvents using a single set of molecular parameters. This approach allows estimating the degree of asymmetry of the excited state in different solvents as well as its change during SB.
Additionally, the relative contribution of the different mechanisms responsible for the splitting of the symmetric and antisymmetric −C≡ C− stretching bands, which are both IR active upon SB, can be determined.
Introduction
1
Over the past few years, there has been an increasing interest for symmetric molecular
2
systems, for which photoexcitation triggers a charge transfer process along one of several
3
energetically-equivalent pathways, causing a breaking of the symmetry.1 These systems in-
4
clude molecules containing two or more identical chromophores where the excited subunit can
5
act either as electron donor (D) or acceptor (A).2–13 Such symmetry-breaking (SB) charge
6
separation process could be advantageously exploited for applications in photovoltaics and ar-
7
tificial photosynthesis.9,14,15Another class of compounds comprises multibranched molecules
8
with a D(-π-A)n or A(-π-D)n (n = 2,3) motif, that are attracting considerable interest for
9
their promising two-photon absorption properties.16–23Here SB manifests itself by a large flu-
10
orescence solvatochromism, indicative of a dipolar excited state, despite a symmetric ground
11
state, as evidenced by a negligible absorption solvatochromism and by the relative intensity
12
of the S1 ←S0 and S2 ←S0 bands in the one- and two-photon absorption spectra.19,24–38
13
Such excited-state symmetry breaking (ES-SB) involves a transition from a quadrupolar or
14
octupolar Franck-Condon excited state to a dipolar relaxed excited state, and, thus corre-
15
sponds to an increase of the excitation density on one branch of the molecule at the expense
16
of the other(s).1,39–42
17
This ES-SB process could be visualised in real time in quadrupolar molecules using
18
femtosecond time-resolved infrared (TRIR) spectroscopy. This was achieved by monitoring
19
either −C≡N vibrations located on the acceptor ends of A-π-D-π-A molecules consisting of
20
a pyrrolopyrrole D core and two cyanophenyl acceptors,43–45 or−C≡C−vibrations in the
21
π linkers of D-π-A-π-D molecules.46,47 Upon ES-SB, the TRIR spectrum transforms from a
22
spectrum with a single −C≡N or −C ≡ C− stretching band, as expected for a symmetric
23
excited state, to a spectrum with two bands, pointing to an asymmetric excited state. In the
24
case of the A-π-D-π-A dyes, the splitting of the−C≡N bands was shown to increase with the
25
extent of SB. This process was found to be very sensitive to the solute-solvent interactions.
26
It does not take place in apolar solvents, but occurs not only in dipolar solvents, but also in
27
quadrupolar as well as in halogen- and hydrogen-bonding solvents.45 Moreover, its timescale
28
is similar to that of solvent relaxation.
29
Theoretically, ES-SB could be successfully described in terms of an essential state model.39–42,48,49
30
It was assumed to be induced by the interaction of the solute molecule with the solvent po-
31
larization and by electronic interactions with antisymmetric vibrational modes. This model
32
could reproduce the effects of ES-SB on one- and two-photon absorption spectra in quadrupo-
33
lar and octupolar donor-acceptor chromophores.39–42,48,49 More recently, a two-level model
34
describing the effect of SB on the vibrational spectrum of a quadrupolar dye with an exact
35
solution has been derived.50 This model was able to successfully reproduce the IR absorp-
36
tion spectrum in the −C≡N stretching region of a A-π-D-π-A dye after ES-SB in dipolar
37
solvents of varying dielectric constant. The two-level model was recently shown to predict
38
results very similar to those obtained within the essential state model when the S1-S0 gap
39
is significantly larger than the S2-S1 energy splitting.51 The −C≡N stretching vibrations
40
were described in terms of two normal modes, an IR active antisymmetric mode and an IR
41
inactive symmetric mode, with an intrinsic frequency splitting. Upon ES-SB, the dye is no
42
longer centro-symmetric and, thus the symmetric stretching mode has a non-zero IR cross
43
section, and the frequency splitting changes. The increase of this frequency splitting with
44
increasing polarity could be perfectly accounted for by the model.
45
Here we present an approach to describe the effect of solvent relaxation on the SB dy-
46
namics, as well as a method to model the accompanying IR spectral dynamics. It will be
47
applied to recent results obtained with a quadrupolar D-π-A-π-D dye (1, Figure 1) consisting
48
in a central benzothiadiazole electron acceptor flanked by two alkoxyphenyls donors, linked
49
through −C ≡ C− bridges.47 In non polar solvents, the TRIR spectrum exhibits a single
50
−C≡C−stretching band, in agreement with a purely quadrupolar symmetric excited state.
51
In polar solvents, the TRIR spectra show complex dynamics including the appearance of a
52
second −C ≡ C− stretching band and a frequency splitting that decreases with time. We
53
will show that the TRIR spectra recorded at different times and in two different solvents
54
can be perfectly reproduced with this extended model using the same set of parameters to
55
describe the dye. This approach allows determining the degree of asymmetry in different
56
solvents and its change during ES-SB. It also provides rich information on the different
57
mechanisms responsible for the band splitting. Finally, it confirms that the appearance of a
58
second vibrational band in the TRIR spectrum is an unambiguous evidence of ES-SB.
59
Theory
60
Overview of the ES-SB model
61
The theoretical model of ES-SB based on a two-essential-states approach was presented in
62
ref. 52 and its extension that includes the effect of electron-vibration coupling can be found
63
in ref. 50. Here, we present a brief overview of this model and the most relevant results.
64
The wavefunction of the excited states of a quadrupolar dye like 1 is expressed as:
65
Ψ =aLϕL+aRϕR (1)
whereϕL,R are the wavefunctions of the ’essential’ zwitterionic states with the positive charge
66
localized entirely on left or right branch, respectively, andaL,R are the associated coefficients
67
witha2L+a2R = 1. This quantities determine the positive charges of the donorsδL,R =−a2L,Re
68
with e being the electron charge. The total Hamiltonian accounting for symmetry breaking
69
and accompanying variation in the vibrational frequencies is39,50
70
H =HM+Hs+Hc+Hv0+Uint (2)
where
71
HM=
0 V V 0
(3)
includes the coupling between the two zwitterionic states, V, which determines the split-
72
ting of the one- and two-photon absorption bands, Hs describes the dipolar solute-solvent
73
interactions, Hc accounts for the Coulombic interactions, Hv0 and Uint are intramolecular
74
vibration and electron-vibration interactions, respectively. Within the self-consistent field
75
approximation, the Hamiltonian of the interaction of a polar molecule with a polar medium
76
is39,52
77
Hs =−λDDˆ (4)
where the dissymmetry parameter, D,
78
D=a2L−a2R = 2a2L−1 (5)
quantifies the extent of ES-SB. It varies from 0 in the absence of SB to 1 when the excitation
79
is entirely localised on one branch, i.e. when the excited state is purely dipolar. The
80
dissymmetry operator is defined as
81
Dˆ = ˆPL−PˆR (6)
with the projection operators ˆPL and ˆPR
82
PˆR(L)Ψ = ˆPR(L)(aLϕL+aRϕR) = aR(L)ϕR(L) (7)
The mean value of the dissymmetry operator is equal to the magnitude of the dissymmetry
83
parameter D = hΨ|D|Ψi. According to the Onsager model,ˆ 53 the interaction between a
84
molecule with a permanent dipole moment and a polar solvent is described as50
85
λ= µ20∆f
r3d =λ1∆f (8)
whereµ0 is the solute dipole moment,rd the cavity radius, and ∆f =f(εs)−f(n2) with the
86
solvent polarity function f(x) = 2(x−1)/(2x+ 1), the static dielectric constant εs and the
87
refractive index n. Eq 4 assumes that a molecule in a state with a dissymmetry parameter
88
D has an average dipole moment µ=Dµ0. The Hamiltonian of the Coulombic interactions
89
is defined as follows39,50
90
Hc=−γCDD, γˆ C = e2
4εimRLR (9)
where RLR is the distance between the charges located on the two donor sub-units, and εim
91
is the ”intramolecular” dielectric constant. The vibrational Hamiltonian of a quadrupolar
92
molecule with inversion symmetry has the form50
93
Hv0 = 1 2
X
i
p2si+ωsi2x2si
+X
j
p2aj+ωaj2 x2aj
(10)
where psi, paj, xsi, xaj, ωsi, ωsj are the momenta, the coordinates and the frequencies of the
94
symmetric and antisymmetric vibrational modes, respectively. The electron-vibration inter-
95
action is introduced through its formal expansion in powers of the dissymmetry parameter
96
and vibrational displacements up to the second order50
97
Uint =Uint(1)+Uint(2) (11)
Uint(1) = ˆDX
j
ζjxaj (12)
Uint(2) = ˆDX
ij
δijxajxsj +DDˆ 2
X
ik
[αikxaixak+βikxsixsk] (13)
where only the terms that are invariant under the inversion symmetry transformation are
98
kept; ζj is the linear electron-vibration interaction parameter, δij describes the coupling
99
between the antisymmetric and symmetric vibrational modes upon SB, whereas the param-
100
eters αik and βik quantify their frequency shifts due to SB. The last term, Uint(2), describes
101
the frequency variations of the normal molecular vibrations and the Duschinsky effect upon
102
SB. Since the frequency variations are small in real molecules, Uint(2) is small as well and its
103
effect on SB can be neglected. It should be noted that in the adiabatic approximation, the
104
linear electron-vibration interaction transforms into50
105
U¯int(1) =−ζDD, ζˆ =X
j
ζj2
ω2aj. (14)
Eq. 14 describes the effect of the electron-vibration interaction on the electronic subsystem
106
of the molecule.
107
To determine the resulting stationary excited states and energies of the molecule inter-
108
Figure 1: Energy-level scheme of a quadrupolar D-π-A-π-D molecule predicted by the model for the case where ES-SB is operative (D = 1), together with the structure of molecule 1. The distribution of the excitation in each state is represented by the red shading. The different shading for the state at E− highlights the antisymmetric nature of its associated wavefunction. OPA and TPA stand for one- and two-photon absorption, respectively.
acting with the solvent, the wavefunction eq 1 is inserted into the Schr¨odinger equation
109
HΨ = EΨ (15)
where H = HM+Hs +Hc+Hv0 + ¯Uint(1). It is important to note that eq 15 has two sets
110
of the solutions. The first set of solutions is associated with the nonpolarized medium and
111
describes the symmetric (aL =aR) and an antisymmetric (aL=−aR) ’excitonic’ states with
112
the energies, E+ and E−, respectively (Figure 1). The energy splitting of these states is
113
equal to 2V, and is equivalent to the Davydov splitting. The second set corresponds to the
114
polarized medium and includes two degenerate asymmetric states with the charge localized
115
predominantly on either of the two arms (|aL| 6= |aR|). Neglecting the quadratic electron-
116
vibrational interaction Uint(2), whose effect on the IR spectra is treated perturbatively, the
117
energy of these states is given by a simple expression
118
Eas =−(λ+γ), γ =γc+ζ. (16)
These states only exist when Eas < E−. This condition is fulfilled when V < λ+γ. In this
119
case, optical population of the antisymmetric electronic state upon one photon absorption
120
is followed by SB. The dipolar character of the asymmetric states is given by
121
D= s
1− V2
(λ+γ)2 (17)
One can see that the D is determined by a single dimensionless parameter V /(λ+γ).
122
Symmetry Breaking and IR Absorption Spectrum
123
To see how the vibrational spectrum of the excited molecule is affected by ES-SB, we consider
124
two equivalent vibrational modes, here the −C ≡ C− stretching vibrations, localised on
125
each of the two arms of the molecule with the coordinates xL and xR. The interaction
126
between these two local modes results into antisymmetric and symmetric normal modes, with
127
coordinatesxaandxsand frequenciesωaandωs. If the two−C≡C−groups are separated by
128
several atoms, as it is the case here, direct coupling between the two local modes is negligible,
129
and ωa and ωs can be expected to be very close to each other.54 However, anharmonic
130
interactions between the vibrational modes as well as electron-vibration interactions can
131
lead to a splitting of these frequencies.50Both mechanisms are fairly universal, so a splitting
132
is expected in most molecules, although its magnitude can vary widely. In the model, this
133
splitting, in the absence of any ES-SB (D= 0), is given by ∆ω0 =ωs0−ωa0orω2s0−ωa02 =κ2,
134
whereωs0 =ωs(D= 0),ωa0=ωa(D= 0), andκis connected to the linear electron-vibration
135
coupling parameter, κ2 =ζ2/V50 (Figure 2, top).
136
The experimental TRIR spectra reveal that the changes in the vibrational frequencies of
137
the −C ≡ C− groups upon SB is about 1%. This weak effect indicates that the electron-
138
vibration interaction of these modes is weak as well. Therefore, the effect of SB on these
139
vibrational modes can be described in terms of perturbation theory, and the effect of these
140
vibrational modes on the electronic state of the solute can be ignored. On the other hand, the
141
Figure 2: Schematic illustration of the frequency splitting of the symmetric and antisym- metric−C≡C−stretching vibrations of 1in an IR absorption spectrum. In the absence of SB (D= 0), the splitting, is predominantly due to electron-vibration coupling, but only the antisymmetric vibration is IR active (top). Upon SB (D >0), other mechanisms associated with the parameters α, β and δ come into play. Both vibrations are IR active (bottom).
Here, α and δ are taken as positive, whereas β < 0. The functions f(x) should not be confused with the polarity functions.
linear electron-vibration coupling with the many other vibrational modes of the solute that
142
are not experimentally monitored can be significant, and was thus included in the electronic
143
Hamiltonian.
144
To calculate the IR absorption spectrum in a state with broken symmetry, we use the clas-
145
sical theory of the electromagnetic wave absorption. The interaction energy of the vibrations
146
with the electric field of an electromagnetic wave is written as
147
HintEM =−µviba E(t)−µvibs E(t) (18)
where E(t) is the projection of the electric component of the electromagnetic wave on the
148
direction of the oscillator dipole moment, µviba and µvibs are the dipole moments associated
149
with the antisymmetric and symmetric vibrations, respectively, which are assumed to be
150
parallel. In the electronic antisymmetric state, µvibs = 0 because the symmetric vibration
151
is not IR active. However, the dipole moments of the −C ≡ C− groups that link the
152
donor and acceptor sub-units of the molecule depends on D. This results in a difference
153
in the absorption coefficients of these two groups in the asymmetric electronic state. This
154
can be formally described by expanding the interaction energy eq 18 in a power series of
155
the asymmetry parameter D up to the second order. Considering the invariance of the
156
interaction energy eq 18 with respect to the inversion transformation and the fact that the
157
electric field E(t) is an odd function of this transformation, we obtain
158
µviba E(t)'F0cos(ωt)xa(1−ξaD2), µvibs E(t)'F0cos(ωt)xsξsD (19)
Here we have supposed that E(t) = E0cos(ωt) and included the amplitude of the electrical
159
field E0 in the quantity F0. It is shown below that the quantities ξa and ξs are measures of
160
the difference in dipole moment of the−C≡C−groups in the two branches of the molecule.
161
162
Furthermore, we assume that the electronic,τe, vibrational,τv, and medium,τm, timescales
163
satisfy the conditionτeτvτm. If the medium controls SB, then the dissymmetry param-
164
eterDis a much slower function of time than the high frequency vibrations, and consequently
165
the intramolecular high-frequency vibration modes follow adiabatically the variation of D.
166
Under these assumptions, the equations of motion of−C≡C−stretching vibrations in the
167
presence of an IR optical field at frequencyω have the following form
168
¨
xa =−∂Hv
∂xa −ηax˙a +Facos(ωt) (20)
¨
xs =−∂Hv
∂xs −ηsx˙s+Fscos(ωt) (21)
where Hv =Hv0+Uint, the dots above the vibrational coordinates denote time derivatives,
169
ηa and ηs are the damping frequencies of the antisymmetric and symmetric normal modes
170
and determine the homogeneous width of the IR absorption bands. The amplitudes of the
171
interaction between the vibrations and the IR field are
172
Fa =F0(1−ξaD2), Fs =F0ξsD (22)
Equation 22 is a simplest approximation of the dependence of the dipole moment of the
173
−C ≡ C− groups on the dissymmetry parameter. The comparison of the simulated and
174
experimental spectra described below evidences that the dipole moment of the −C ≡ C−
175
groups varies considerably upon changing the dissymmetry parameter.
176
A particular solution to the system of linear differential eqs 20–21 can be found in the
177
form of a sum complex conjugate solutions xa(s)(t) +x∗a(s)(t) where xa(s)(t) = ¯xa(s)eiωt. The
178
complex amplitudes ¯xa and ¯xs are given by a system of linear algebraic equations
179
(ωa2−ω2+iηa)¯xa+Dδ¯xs+D2α¯xa =Fa (23) (ωs2−ω2+iηs)¯xs+Dδx¯a+D2βx¯s =Fs (24)
Here, the equations are written without the linear electron-vibrational interaction, since it
180
does not affect the form of the bands. The solutions of eqs 23 and 24 are
181
¯
xa = Fa∆s−FsDδ
∆a∆s−D2δ2, x¯s = Fs∆a−FaDδ
∆a∆s−D2δ2 (25)
with
182
∆a = ¯ωa2−ω2+iηaω, ∆s = ¯ωs2−ω2 +iηsω (26)
183
¯
ωa2 =ω20−κ2+D2α, ω¯2s =ω02+D2β (27) where ω0=ωs0.
184
According to the classical theory of absorption, the amount of energy absorbed per unit
185
time is equal to the work of the external force during the same time interval. It can be
186
calculated as
187
I(ω) =A(ω) = 1 T
Z T 0
Fadxa
dt +Fsdxs dt
cos(ωt)dt (28)
where
188
xa(s)(t) = 1
2 x¯a(s)eiωt+ ¯x∗a(s)e−iωt
(29) and T = 2π/ω is the period of the electromagnetic wave. This results in the IR spectral
189
intensity50
190
I(ω) =−1
2ω[FaIm¯xa+FsIm¯xs] (30) In the symmetric state, D = 0, eq 30 predicts a spectrum with a single band, which
191
corresponds to the absorption of the antisymmetric vibrational mode, the other one being
192
Raman active only (Figure 2, top). However, in a state with broken symmetry, D 6= 0,
193
the calculated spectrum exhibits two bands at the frequencies of the antisymmetric and
194
symmetric vibrations (Figure 2, bottom). In the case of narrow bands, ηa, ηs ω0, the
195
magnitude of the band splitting can be well estimated using eq 31:
196
∆ω= q
[(α−β)D2−κ2]2+ 4D2δ2
2ω0 (31)
The model distinguishes four frequency splitting mechanisms illustrated in Figure 2, three
197
of which being due to SB. The first splitting mechanism is also operative in the symmetric
198
state of the quadrupolar molecule and its magnitude is related to the parameter κ (blue).
199
It originates from weak linear electron-vibration interaction50 and vibrational anharmonic
200
coupling. The second and third mechanisms are the frequency shifts of the antisymmetric and
201
symmetric vibrations due to SB (red). Their magnitudes are determined by the parameters
202
α and β. The fourth mechanism is associated with the interaction between the symmetric
203
and antisymmetric vibrations in an asymmetric electronic state and is described by the
204
parameter δ (green). Although, the molecule is no longer centro-symmetric upon SB, we
205
still refer to the vibrations as symmetric and antisymmetric to emphasize their origin.
206
The IR spectrum calculated with eq 30 only depends on the dissymmetry parameter
207
through the αD2, βD2, δD, ξaD2, and ξsD products. Therefore, it is an invariant of the
208
transformation:
209
α0 =αg2, β0 =βg2, δ0 =δg, ξa0 =ξag2, ξs0 =ξsg, D0 =D/g (32)
with an arbitrary factor g. As a result, only relative D values can be determined from the
210
fit of eq 30 to the experimental spectra. It should be emphasized thatg is not an adjustable
211
parameter of the theory, since the simulated spectra does absolutely not depend on it.
212
The theory leading to eq 30 operates with normal vibrational coordinates of the quadrupo-
213
lar molecule in the symmetric electronic state. These coordinates have to be symmetric or
214
antisymmetric with respect to inversion. SB changes the electronic state of the molecule and
215
should result in a variation of the vibrational frequencies and to the Duschinsky effect. As
216
a result, normal vibrations in a state with broken symmetry are neither symmetric nor anti-
217
symmetric. To describe these variations, the vibrational Hamiltonian is formally expanded
218
in a power series of D up to the second order.50 This implies that the variations in the
219
natural frequencies should be small. However, this does not require the dissymmetry param-
220
eter, D, to be small also, the theory is applicable when the conditions including additional
221
multipliers:
222
αD2/ω02, βD2/ω20, δD/ω02, ξaD2, ξsD1 (33) are fulfilled. This is because, in fact, the power series contains the terms listed in eq 33.
223
Symmetry Breaking Dynamics
224
The TRIR spectra recorded with the D-π-A-π-D molecule 1 in polar solvents show pro-
225
nounced dynamics upon ES-SB that can be assigned to the increase of D with time.47 The
226
theoretical model can be generalized to describe the time dependence ofDreflected by these
227
spectral dynamics.50,52 A comparison of the theoretical model with the experimental data
228
enables the estimation of important parameters related to the solute and to its interaction
229
with polar solvents.
230
The magnitude of the dissymmetry parameter at equilibrium, Deq is given by eq 17.50,52
231
To include the time evolution of D in the theory, we start by noting that the operator of
232
the interaction energy of the solute with the solvent polarization described by eq 4 includes
233
the productDD. In this expression, the multiplier ˆˆ Daccounts for the dipole moment of the
234
solute, whileDis proportional to the magnitude of the solvent polarization. WhereasDand
235
the mean value of ˆD are equal at equilibrium, they can differ in a nonequilibrium state. For
236
an arbitrary nonequilibrium state, this interaction energy can be written as
237
Wint =hΨ|Hs|Ψi=−λDDm (34)
where Dm is a new quantity characterizing both the equilibrium and nonequilibrium polar-
238
ization of the solvent.
239
To justify eq 34, we note that, in the Onsager model, the interaction energy between a
240
solute dipole moment, ~µ, and a polar solvent is Wint =−~µ ~E, where E~ =~µ∆f /rd3 is the so-
241
called equilibrium solvent reaction field.53 By principle, the nonequilibrium solvent reaction
242
fieldE~m < ~E and corresponds to the field generated by an electric dipoleµ~m< ~µ. The energy
243
of the solvent reaction field Wm can be calculated as follows. In the linear approximation
244
of the solvent response, Wm has to be a quadratic function of Dm. The exact expression of
245
Wm is determined by the condition that the energy Wint+Wm has a minimum at Dm=D.
246
This gives Wm=λDm2/2.
247
The quantity Dm is thus directly associated with the solvent reaction field. For example,
248
Dm = 0 in an unpolarized solvent. As long as the solvent polarization is not equilibrated
249
with the field of the solute dipole moment, Dm is determined by the equality µm = µ0Dm.
250
Dm plays the same role in the theory of SB electron transfer as the reaction coordinate in
251
the electron transfer theory.
252
Solvent relaxation is usually expressed in terms of a relaxation function X(t).55 By defi-
253
nition, this function decreases from unity to zero. It reflects the decay of the solvent polari-
254
sation from its initial value in the presence an external field to zero after the external field
255
has been switched off. Consequently, the temporal evolution of Dm can be approximated as
256
Dm(t) = Deq(1−X(t)). This equation assumes a linear solvent response. The solvent po-
257
larisation, henceDm, is initially zero and then increases to its equilibrium value. As solvent
258
relaxation is assumed to be slow relative to electronic and intramolecular vibrational motion,
259
the electrons and nuclei of the solute follow adiabatically the changes in solvent polarization.
260
Within this approximation, the interaction energyWint =−λ(Deq)2(1−X(t)) increases with
261
time from zero to its equilibrium value. Formally, this equation can be obtained from eq 34
262
with the substitution
263
λ→λ(1−X(t)) (35)
Applying this substitution to eq 17 results in eq 36, which accounts for time dependence of
264
the dissymmetry parameter due to the varying solvent response:
265
D(t) = s
1− V2
[γ+λ(1−X(t))]2 (36) Eq. 36, designated from now on approach 1, is a relatively crude estimation of the time
266
dependence ofD. A better description (approach 2) can be obtained by solving the stationary
267
Schr¨odinger equation with a fixed value ofDm (adiabatic approximation). Eqs 18 – 20 from
268
ref 50 can be recast in the form
269
(λDm+γxy)x−(V −E)y = 0
(V +E)x+ (λDm+γxy)y = 0 (37)
x2+y2 = 2
wherex=aL−aR, andy=aL+aR. Forγ = 0, a simple equation relating the dissymmetry
270
parameter and solvent polarization is obtained:
271
D= λDm
pV2+λ2Dm2 (38) Here D = xy is the equilibrium value of the dissymmetry parameter of the molecule in
272
a frozen solvent with a given polarization Dm. When γ 6= 0, the functional dependence
273
D = D(Dm) can be calculated by solving eqs 37 numerically. The time dependence of the
274
dissymmetry parameter can then be calculated assumingDm(t) = Deq(1−X(t)). To describe
275
the temporal evolution of SB completely self-consistently, it is necessary to solve a dynamic
276
problem. We plan to do this in the future.
277
The physics underlying the above description is as follows. Immediately after photoex-
278
citation into the antisymmetric electronic excited state (Figure 1), the solute is purely
279
quadrupolar, D = 0, and is surrounded by non-polarized solvent. Therefore, the dipolar
280
solute-solvent interaction considered here is zero, corresponding formally to Dm= 0. Ther-
281
mal fluctuations of the solvent polarization rapidly induces a dipole moment on the solute,
282
which in turn polarises the solvent. Such a concurrent increase in both solvent polarization
283
and dipolar character of the solute proceeds until equilibrium is reached. This type of re-
284
laxation could be expected to be slower then the solvent response to a quasi instantaneous
285
change of permanent dipole moment of the solute, as determined for example from time-
286
resolved fluorescence measurements with push-pull molecules.56,57 The relaxation function
287
X(t) deduced from such measurements is usually approximated by eq 39:55
288
X(t) = X
i
aiexp(−t/τi), X
ai = 1 (39)
Figure 3 illustrates the influence of the parameters γ and λ on the time dependence of
289
D, assuming a single exponential solvent relaxation function X(t) = exp(−t/τ) with τ = 1
290
ps. A few important conclusions can be drawn from the analysis of eq 36 (approach 1) and
291
0 2 4 6 0 . 0
0 . 2 0 . 4 0 . 6 0 . 8
Dissymmetry parameter
t i m e / p s
γ/ V = 0 . 7 , λ/ V = 0 . 5 γ/ V = 0 . 9 , λ/ V = 0 . 5 γ/ V = 0 . 9 8 , λ/ V = 0 . 5 γ/ V = 1 . 2 , λ/ V = 0 . 5
Figure 3: Time dependence of the dissymmetry parameter D for different values of γ and λ, calculated with eq 36 (solid lines) and by solving eqs 37 (dashed lines), and assuming monoexponential solvent relaxation with a 1 ps time constant.
the solution of eqs 37 (approach 2). As mentioned above, SB is not possible at any time
292
when (γ +λ)/V < 1, and D(t) ≡ 0 (not shown in Figure 3). When γ/V is significantly
293
smaller than one and (γ +λ)/V slightly larger than unity, approaches 1 and 2 predict
294
strongly different D(t) (compare the red solid and dashed lines). Since approach 2 is more
295
consistent, the first one can be considered as inapplicable for such parameters. At the same
296
time, when γ/V is close to but larger than unity, both approaches lead to similar results
297
(compare the green solid and dashed lines). In this case, the much simpler approach 1 can be
298
applied. It should be noted that whenγ/V >1, the dissymmetry parameter differs from zero
299
already at time zero. This is the case when the intramolecular interactions, assumed here
300
to have an instantaneous response, suffice to break symmetry without solvent polarization.
301
Both approaches show that, immediately after the onset of SB, D rises very rapidly before
302
reaching a plateau. This fast increase ofDoccurs on a shorter timescale than that of solvent
303
relaxation due to the presence of a threshold in the dependence ofD on solvent polarization.
304
Results and Discussion
305
Analysis of TRIR Spectra
306
The above model is now put to the test with TRIR spectra recorded in the−C≡C−stretch-
307
ing region upon S1 ←S0excitation of the D-π-A-π-D molecule1in the non polar cyclohexane
308
(CHX), in the medium polar tetrahydrofuran (THF) and in the highly polar dimethylfor-
309
mamide (DMF).47 In CHX, the spectra show a single excited-state absorption band, whose
310
shape and position remain essentially independent of time. Two excited-state absorption
311
bands are observed in the polar solvents, with positions and relative intensities changing
312
significantly with time (Figure 4).47 These spectra have been recorded upon red-edge exci-
313
tation of the S1 ←S0 transition. Therefore, any spectral dynamics due to the relaxation of
314
vibrational excess energy can be safely excluded.
315
The experimental TRIR spectra of the D-π-A-π-D molecule 1 show that the bandwidth
316
of the −C≡C− stretching vibrations increases noticeably with the solvent polarity. In gen-
317
eral, such trend originates from the direct interaction of the polar bond vibrations of the
318
dye with solvent polarization fluctuations. However, for molecules undergoing symmetry
319
breaking, at least two specific band-broadening mechanisms should additionally be consid-
320
ered. Both are due to the interaction of the solvent polarization fluctuations with the overall
321
dipole moment of the solute that results in fluctuations of the dissymmetry parameter D.
322
These fluctuations of D lead to (i) a modulation of the intramolecular vibration frequencies
323
and (ii) a modulation of the dipole moment of the intramolecular polar bonds (see eq 19).
324
Both types of modulations affect the bandwidth and their contribution is expected to be
325
significant. Theoretical analysis of this phenomenon is very complex due to the nonlinear
326
coupling of the dissymmetry parameter fluctuations with the solvent fluctuations. Retar-
327
dation effects, i.e. the non-instantaneous response of intramolecular vibrations to changes
328
in solvent polarization, can also be important. This complex problem is not analysed here.
329
Instead, we refer to previous transient 2D-IR spectroscopy experiments, which showed that
330
the IR bands of a similar dye measured in a series of polar solvents can well reproduced by
331
a Voigt function with a homogeneous width of about 10 cm−1, similar to the value assumed
332
here (see Supporting Information in ref 58). Theory also shows that the IR band of solutes
333
in solvents with two different relaxation timescales can be well approximated with a Voigt
334
function.59
335
Equation 30 predicts the IR absorption bands to be close to Lorentzian since only ho-
336
mogeneous line broadening is considered. This is true as long as the widths ηa and ηs are
337
similar, as it is the case here. In the limit where ηa ηs, the weaker band has a strongly
338
asymmetric shape typical of a Fano resonance.60 However, comparison with experimental
339
spectra should take inhomogeneous broadening into account. For this, the TRIR spectra
340
with two clearly distinguishable bands were fitted with the sum of two Voigt functionsf(ω):
341
f(ω) =Z−1ω[V(ω−ω1, γ1, σ1) +V(ω−ω2, γ2, σ2)] (40) V(ω, σ, γ) =
Z ∞
−∞
G(ω0, σ)L(ω−ω0, γ)dω0 (41) G(ω, σ) = 1
σ√
2πe−ω2/(2σ2) (42)
L(ω, γ) = γ
π(ω2+γ2) (43)
whereZ−1is a normalization factor. The functionV(ω−ωi, γi, σi) in eq 40 is the convolution
342
of a Gaussian inhomogenous distribution, G(ω), of width σi, with a Lorentzian function,
343
L(ω), of width γi and centred atωi. A fitting off(ω) to experimental TRIR spectra returns
344
the deconvoluted spectrum fD(ω)
345
fD(ω) =Ze−1ω[L(ω−ω1, γ1) +L(ω−ω2, γ2)] (44)
which does not include the inhomogeneous broadening. Here Ze−1 is another normalization
346
factor and determines the ratio of the heights of the two Lorentz bands. The simulated
347
spectrum eq 30 is then fitted to the function fD(ω).
348
Fitting the sum of two Voigt functions with arbitrary inhomogeneous (σi) and homo-
349
geneous widths (γi) to the TRIR spectra measured at different time delays in polar sol-
350
vents leads to a rather large scattering in the magnitude of the homogeneous width. Since
351
such result is not meaningful, the analysis was repeated with a fixed homogeneous width,
352
γ1 = γ2 = 10 cm−1. The quality of all fits is excellent in the area of both band maxima
353
as illustrated in Figure 4. The best-fit parameters are listed in Table 1. However, the
354
low-frequency band is strongly asymmetric and its low-frequency shoulder is not so well re-
355
produced. This discrepancy is minor in THF but increases in the more polar DMF. This
356
reflects the evolution of the distribution of frequencies in a highly asymmetric potential
357
E +Wm (see eq 37). As expected, this deviation decreases with time. This band-shape
358
asymmetry demonstrates the important role of the specific band-broadening mechanisms.
359
However, this discrepancy has a minor effect on the positions of the band maxima and can
360
lead to an overestimation of the relative amplitude of the high-frequency band. The error
361
on the parameter specified in Table 1 is about 15% at early time and decreases at longer
362
time.
363
This fit of eq 40 can be considered as a procedure to remove the inhomogeneous broad-
364
ening from the experimental TRIR data. The resulting deconvolved spectra, depicted with
365
solid coloured lines in Figure 4, can now be fitted with eq 30 to determine the parameters
366
associated with both the solute (ω0,κ,α,β,ξa,ξs) and the solvent response, which are indi-
367
rectly reflected in the magnitude of the dissymmetry parameter {Di} at a given time delay
368
{ti}. This fit involves a large number of adjustable parameters. To improve the convergence,
369
the optimization procedure was split into three stages. (i) In the first stage, the {Di} values
370
at different time delays were considered as independent adjustable parameters, whereas the
371
solute parameters were fixed. The {Di} values were then found using the golden section
372
search method in the 0 to 1 range. (ii) In the second stage, the solute parameters α, β, κ,
373
ξa, andξs were optimized using the particle swarm optimization method. Here, ω0 was kept
374
constant, since it strongly influences the spectral behavior. Note that to reduce computation
375
2020 2080 2140
ω / cm−1
f ( ω ) /ω , f
D( ω ) /ω , a .u .
t=0.2 t=0.4 t=1.0 t=2.0 t=4.0 t=8.0 t=20
THF
2000 2060 2120 2180
ω / cm−1
DMF
Figure 4: TRIR spectra in the −C≡C− stretching region recorded at various times after excitation of 1in THF and DMF (dots) and best fits of the sum of two Voigt functions (eq 40, black dashed lines). The deconvolved spectra (eq 44) including only the homogeneous broadening are represented with solid coloured lines.
time, the particle swarm optimization method can be used as an initial approximation for
376
the Nelder-Mead optimization method. (iii) In the third and final stage, the value of ω0 was
377
determined using again the golden section search method, with the interval set within the
378
limits of the width of the weaker band. Thus, we formulated a triple embedded optimiza-
379
tion problem. To improve the accuracy in the determination of the solute parameters, it is
380
necessary to use sets of experimental data in at least two different solvents.
381
The best-fit of eq 30 to the deconvolved TRIR spectra in THF and DMF are presented
382
Table 1: Parameters obtained from the fit of eq 40, assuming fixed homogeneous widths γ1 = γ2 = 10 cm−1, to the TRIR spectra of 1 recorded at various time delays in THF and DMF. The band widths, σi, and positions, ωi, are in cm−1.
solvent time / ps ω1 σ1 ω2 σ2
CHX 20 0.000 2072.3 3.33 – –
THF 0.2 0.038 2083.5 6.79 2145.5 0.00 0.4 0.054 2083.5 7.32 2144.4 0.00 1 0.081 2086.0 8.22 2143.1 0.00 2 0.106 2088.0 8.68 2142.2 1.14 4 0.129 2089.4 9.26 2141.8 0.81 8 0.143 2090.7 9.19 2141.5 0.01 20 0.143 2091.4 9.34 2141.5 0.00 DMF 0.2 0.150 2090.1 14.85 2142.1 4.23 0.4 0.203 2092.6 14.70 2141.4 2.41 1 0.287 2095.5 15.16 2139.8 0.00 2 0.419 2098.4 15.57 2139.2 0.00 4 0.603 2101.3 15.41 2138.8 0.00 8 0.829 2103.2 14.84 2138.6 0.00 20 1.000 2103.8 15.08 2138.4 0.00
in Figure 5 with the parameters listed in Tables 2. Due to the invariance of the spectrum
383
calculated with eq 30 relative to the transformation eq 32, the value of one of the parameters,
384
δ/ω02 = 0.01, was fixed. Such choice only determines the relative values of D, and does not
385
affect the quality of the fit. According to eq 32, the D values obtained from the fit are not
386
calibrated and can, thus, exceed 1. Figure 5 reveals that the fit of eq 30 to the experimental
387
TRIR spectra at different time delays in THF and DMF using a single set of solute parameters
388
is excellent. The TRIR spectra in CHX are practically independent of time and comprise
389
a single band that points to an absence of SB in this solvent. The spectrum calculated for
390
D = 0 with the solute parameters obtained from the fit in polar solvents coincides very
391
well with the experimental spectra of 1 in CHX, as shown in Figure S1. Moreover, the
392
value of ω0 obtained from the fit leads to a frequency of the antisymmetric vibrations of
393
ωa0 =ω0 −∆ω0 = 2073.6 cm−1, in perfect agreement with the band measured in CHX. It
394
is well known that the fit of a Voigt function to IR spectra does not allow determining the
395
Lorentz width with high precision. However, uncertainties in the homogeneous widths of the
396
2060 2100 2140
ω / cm−1
I ( ω ) /ω , a .u .
t=0.2 t=0.4 t=1.0 t=2.0 t=4.0 t=8.0 t=20
THF
2060 2100 2140
ω / cm−1
DMF
Figure 5: Best fits of eq 30 (black dashed lines) to the experimental TRIR data in different solvents after removal of the inhomogeneous broadening using eq 40 (coloured lines).
deconvoluted TRIR spectra do not have a significant effect on the solute parameters obtained
397
from the fit of eq 30. If the homogeneous bandwidth,γi, obtained from the fit overestimates
398
the true value by a factorh, the band-height is underestimated by the same factor. Therefore,
399
this imprecision affects neither the band positions nor the solute parameters.
400
The low-frequency band in the TRIR spectra of1in polar solvents is considerably broader
401
than the high-frequency band (Table 1). One of the reasons for this difference is that |α|
402
is significantly larger than |β|. Therefore, the position of the low-frequency band (anti-
403
symmetric mode) is much more sensitive to the degree of asymmetry than the position of
404
the high-frequency band (symmetric mode). In this case, it is natural to expect that fluc-
405
tuations in the dissymmetry parameter, themselves due to solvent fluctuations, lead to a
406
stronger inhomogeneous broadening of the low-frequency band.
407
Table 2 indicates that all parameters are within the limits of applicability of the theory
408
given by eq 33, except forξaD. The latter implies that the dipole moment of the −C≡C−
409
group varies strongly up SB. In reality, this dependence can significantly deviate from the
410
linear approximation assumed in eq 22.
411
Table 2: Solute parameters obtained from the fit of eq 30 to the deconvolved TRIR spectra measured with 1 in THF and DMF. The values of , 2γ1 =ηa, and 2γ2 =ηs are taken from Table 1.
parameter time, ps D(CHX) D(THF) D(DMF)
ω0, cm−1 2147.2 0.2 0.58 0.73
κ, cm−1 562.0 0.4 0.58 0.79
∆ω0, cm−1 73.5 1 0.64 0.84
δ/ω20 0.010 2 0.69 0.90
α/ω20 0.032 4 0.72 0.95
β/ω02 -0.012 8 0.74 0.98
ξa 0.604 20 0 0.76 1.0
ξs 0.190
Frequency Splitting and Dissymmetry Parameter
412
The best-fit parameters listed in Table 2 indicate that, in the absence of SB (D = 0), the
413
symmetric and antisymmetric −C≡C− stretching vibrations of 1 are split by ∆ω0 = 73.5
414
cm−1. In this case, only the antisymmetric vibration contributes to the IR spectrum (Figure
415
2). As SB takes place, i.e. as D increases, the ’symmetric’ vibration becomes also visible
416
in the IR spectrum. Figure 5 reveals that the splitting of the two bands decreases with
417
increasing D. This behaviour contrasts with that observed previously with an A-π-D-π-
418
A molecule consisting of a pyrrolopyrrole D core and two cyanophenyl acceptors where
419
the splitting was increasing with D. This opposite behaviour is due a different sign of the
420
parameterβ, which itself depends on the nature of the vibrational mode considered,−C≡C−
421
