In this sub-chapter, we will briefly present a simple theoretical model of symmetry breaking. We hosted a theorist, Prof. Anatoly Ivanov (Volgograd State University), for few months in our group and as a result we devised this model which we refer to as the Ivanov SB model188 (in contrast to the Painelli-Terenziani SB model presented in Chapter 1).
We consider a linear and symmetric AL–D–AR molecule with identical electron-accepting groups, AL and AR. Although we explicitly refer to an AL–D–AR structure, because it was parameterized for the current ADA system, the model is also valid for DL–A–DR systems. Excitation of the molecule leads to the population of a quadrupolar Franck-Condon excited state with equal amount of charge transfer on both acceptor arms, i.e.
dL = dR = 𝑒/2. The transferred charge 𝑒 can be less than a full electron charge. Afterward, symmetry breaking takes place, and dL ≠ dR, while conserving the magnitude of the displaced charge (dL + dR = 𝑒). The full two-step reaction can be described using the following scheme:
AL–D–AR → AL-e/2–D+e–AR-e/2 → AL-dL–D+e–AR-dR
For ADA in vacuum, the two lowest excited states are degenerate with the electron localized either on the left or the right acceptor, 𝜑¶ and 𝜑…, approach is in full agreement with the Painelli-Terenziani excitonic SB
model which is frequently used to discuss the absorption spectra of quadrupolar molecules with charge transfer excitons.30,36 Here V is equal to the dipole-dipole interaction energy and could be determined by comparing the one- and two-photon absorption spectra from the Davydov splitting between the bands.
where 𝑅¶… is the center-to-center distance between the two acceptors.
Since the model describes the variation of the energy of a many-body system in terms of the variation of the Coulomb interaction between two point charges, the ‘intramolecular’ dielectric constant, 𝜀"d, has to be considered as a model parameter. Since symmetry breaking conserves the total charge on the acceptors, dL + dR = 𝑒, the total interaction energy between the charges on the acceptors and that on the donor remains constant and is not considered. The model does not account for possible structural distortions of the molecule due to the asymmetric charge distribution that could result in different AL–D and D–AR distances.
The interaction of the molecule with the surrounding medium is considered within the framework of the linear response theory. The point charge polarizes the surrounding dielectric medium, which in turn generates a reaction field. The response of the medium close to the acceptors is obtained with the Kirkwood model that considers a point charge in the center of a cavity with an electric permittivity of 𝜀", embedded in a homogeneous, isotropic dielectric medium of electric permittivity 𝜀‚. The reaction field potential inside the cavity of radius 𝑟B is
Φ…• = − 𝑒 𝑟B𝜀"
𝜀‚− 𝜀"d
𝜀‚ (5.5)
leading to the expression for the solvation energy 𝜆 in the state Ψ given by
𝜆 = −𝑒Φ…• (5.6)
The response of the medium further from the acceptors is described by the Onsager’s reaction field. The dipolar parameter, D, characterizes the degree of asymmetry of the electronic distribution in ADA as
D = 𝑎¶/− 𝑎…/ (5.7)
The total average dipole moment of the molecule is expressed as
µ = Ψ µ Ψ = µD (5.8)
The solvation energy is
𝜆 = 2𝜇/∆𝑓
𝑟B< = 𝜆.∆𝑓 (5.9)
where the solvation energy of the quadrupolar moment is neglected and Δ𝑓 is the Onsager dipolar reorientation function that we used frequently throughout the previous parts of this thesis.
The Hamiltonian for the interaction with the solvent and the Coulomb repulsion between the charges is then written in the mean-field approximation with the help of projection operators for the states 𝜑¶ and 𝜑…. The stationary states of the zwitterionic forms of ADA are determined using the stationary Schrödinger equation. After solving nonlinear secular equations, three solutions for the energy are found for such a quadrupolar system: i) symmetric with 𝑎¶ = 𝑎… and the energy 𝐸v= 𝑉 − (𝜆 − 𝛾)/2;
The dependence of the energies 𝐸v, 𝐸6, and 𝐸A‚ on the electronic coupling for a given value of the solvation energy 𝜆 and the Coulomb repulsion energy 𝛾 is illustrated in Figure 5.17. The asymmetric solutions exist only as long as the inequality 𝑉 ≤ (𝜆 + 𝛾)/2 is fulfilled. One can see from this figure that, in this region, the asymmetric states are below the symmetric and antisymmetric ones. Consequently, they are the most stable states of the system. From (5.10) one obtains a simple expression for the dipolar parameter:
D = 2𝑎¶/− 1 = 1 − 4𝑉/
𝜆 + 𝛾 / (5.11)
As a consequence, the degree of asymmetry is determined by a single dimensionless parameter, 𝑟 = 𝑉/(𝜆 + 𝛾).
Figure 5.17 | Energy level scheme of ADA as a function of coupling strength 𝑉.
The dependence of the dipolar parameter D on the solvent dielectric constant for different values of 𝑟 shows that there is a threshold value of 𝜀‚ below which symmetry breaking is not possible (Figure 5.18A). This is one of the most important predictions of the model. The Onsager and Kirkwood models give qualitatively similar results but have quantitative differences. The threshold vanishes when 𝑟 < 0.5 is fulfilled in the least polar solvent. In this case, symmetry is broken in the whole range of dielectric constants and the dipolar parameter is close to unity. As the ratio 𝑟 increases, the threshold shifts toward a higher dielectric constant and, when 𝑟 > 0.5, symmetry breaking is no longer possible at any dielectric constant (Figure 5.18A).
The physical meaning of these trends is fairly transparent. The energy splitting of the symmetric and antisymmetric states, which have no dipolar character, increases with the electronic coupling and the stabilization energy of the antisymmetric state is equal to 𝑉 (Figure 5.17). On the other hand, the interaction between the solute and the surrounding polar medium stabilizes the state with the maximum dipole moment, i.e., the asymmetric state (we neglect for the moment the Coulomb interaction).
Which of these two effects dominate depends on the relative magnitudes of the electronic coupling and solvation energy. For small 𝑉, solvation energy does not need to be large to favour symmetry breaking. On the other hand, for large 𝑉, solvation does not suffice to break symmetry.
Figure 5.18 | A. Dependence of the degree of asymmetry on the solvent dielectric constant for various values of the V/(λ0+γ) ratio, using the Onsager (solid lines) or the Kirkwood (dashed lines) reaction field, and assuming n2 = 2 and γ = 2λ0. B. Best fits of the square of the dipolar parameter vs. the Onsager polarity function to the experimental frequency splitting of the C≡N stretching bands of ADA (black triangles) using different γ/λ1 ratios (the solvents are explicitly shown). The figures are adapted from Ref. 188.
These conclusions can be discussed more quantitatively. The energy of the isolated molecule without the Coulomb interaction E0 = Ψ Hg Ψ =
= 2𝑉𝑎¶𝑎… amounts to −𝑉 for the antisymmetric state, 𝑎¶ = −𝑎…, whereas E0 = 0 for the totally asymmetric state (considered as a limiting case) with 𝑎¶ = 1, 𝑎… = 0. Consequently, the energy of the asymmetric state is larger than that of the antisymmetric state by 𝑉. On the other hand, the interaction energy with a solvent, 𝐸‚= Ψ H‚ Ψ , for the symmetrical states is smaller than that for the asymmetrical state by 𝜆 + 𝛾 /2
(Figure 5.17). Therefore, the asymmetric state is below the antisymmetric state as long as 𝑉 < 𝜆 + 𝛾 /2, and thus symmetry breaking is possible. At first sight, this result might appear surprising because, as solvation and Coulomb repulsion have an opposite effect on the total energy, one could expect a difference 𝜆 − 𝛾 in the denominator instead of a sum. This apparent contradiction disappears if one considers that the solvation and the Coulomb repulsion energies have an opposite dependence on the dipolar parameter, D. Indeed, at D = 0, solvation energy is the smallest, whereas the Coulomb interaction is the largest.
The frequency splitting of the C≡N stretching bands of ADA depends strongly on the Onsager dipolar reorientation function of the solvent, Δf (Figure 5.8). As this dependence is apparently similar to that presented in Figure 5.18A, the fit of the model was first performed assuming that the band splitting, ∆𝜈ESA, is proportional to the dipolar parameter, namely, that
∆𝜈ESA = 𝐴𝐷, where 𝐴 is an arbitrary constant. A fit of the model based on the later assumption proved not to be possible, because the predicted increase of 𝐷 with Δf in the region of moderate to highly polar solvents is much weaker than observed. However, a good fit can be obtained assuming that ∆𝜈ESA is proportional to the square of the dipolar parameter, ∆𝜈ESA =
= 𝐴𝐷/ (Figure 5.18B), and that the Coulomb repulsion energy, 𝛾, is independent of Δf. Such a quadratic dependence of the band splitting on 𝐷 is in good agreement with the correlation between the [C≡N]d- stretching frequency and the fractional charge, d-, recently calculated for 1,2-dicyanobenzene.164 Equally satisfactory fits can be obtained with different 𝛾/𝜆g ratios. A variation of 𝛾/𝜆gfrom zero to eight has only a weak impact on the quality of the fit. This is simply due to the fact that all the experimental points are located in regions where a variation of 𝛾/𝜆g affects only weakly the Δf dependence of the dipolar parameter, 𝐷. However, an increase of 𝛾/𝜆g leads to a larger value of 𝐴 and, hence, to a smaller 𝐷. For 𝛾 < 𝜆g, 𝐷 can be as large as 0.8, whereas for 𝛾 ≫ 𝜆g, 𝐷 does not exceed 0.3.
The dependence of the band splitting on the Onsager function in Figure 5.18B emphasizes its threshold character. The band splitting is zero for cyclohexane, as symmetry-breaking is not operative in nonpolar
solvents, whereas it takes place in di-n-butyl ether. Therefore, the threshold is located at 0 < Δf < 0.2. According to the present model, the absence of symmetry breaking in ADA in nonpolar solvents indicates that 𝑉 > 𝛾/2. For a molecule with 𝑉 < 𝛾/2, symmetry breaking should take place even in nonpolar solvents. Whereas both 𝑉 and 𝛾 decrease with increasing distance between the donor and acceptor units, both quantities vary oppositely with increasing conjugation of the p-bridges between the D and A units. Indeed, better conjugation favours 𝑉 as well as the screening of the charges on the acceptors via the ’intramolecular’ dielectric constant, 𝜀"d, and, thus, 𝛾 should decrease.197–199 As a consequence, symmetry breaking in nonpolar solvents might be difficult to achieve with p-bridges, but could be realized in AD2 or DA2 molecules with saturated bridging units. In such systems, however, the electronic coupling between the D and A units might be too small for the electronic transition from the ground to the quadrupolar excited state to have a significant oscillator strength.
The experimental data allow a relatively precise determination of the threshold polarity above which symmetry breaking takes place. According to the model, the polar solvation energy at the threshold is such that 𝑉 = 𝜆 + 𝛾 /2. In the case of ADA, the experimental results point to a threshold between CHX and DBE. The magnitude of the electronic coupling is estimated from the splitting of the one-photon and two-photon absorption band maxima that is equal to 2𝑉. Estimation of this quantity from experimental spectra results in 𝑉 = 1800 cm-1. Additional information on the magnitudes of other model parameters can be derived from the ∆𝐸 = 1600 cm-1 shift of the fluorescence spectrum of ADA observed by going from the apolar cyclohexane to acetonitrile (Δf = 0.62).
The average energy of the molecule expressed in terms of the dipolar parameter 𝐷 is
E = Ψ H Ψ = −V 1 − 𝐷/− 𝜆 + 𝛾
2 𝐷/+ 𝑐𝑜𝑛𝑠𝑡 (5.12) where the constant is independent of 𝐷. The shift of the fluorescence band reflects the change of the total average energy of the molecule upon variation of the solvent polarity and can be estimated as
∆E = E D = 0 − E D = V 1 − 𝐷/− 1 + 𝜆 + 𝛾
2 𝐷/ (5.13) By using Eq. (5.11), this equation can be rewritten as
𝜆 + 𝛾 = 2(∆E + 𝑉) (5.14) according to Eq. (5.4), this Coulomb repulsion energy is equivalent to that between two electrons at 10 Å in a medium with a dielectric constant of 𝜀"d = 2. This distance is equal to that between the centers of the two benzonitrile rings of ADA. Furthermore, according to the Onsager model, the value of 𝜆 corresponds to the solvation energy of a dipole of about 10 D.
Finally, the polar solvation energy required for symmetry breaking to take place amounts to 490 cm-1, corresponding to Δf = 0.08, i.e., somewhere between CHX and DBE, as observed experimentally. No quadrupolar solvation is taken into account in this model, which as we demonstrated above is an important contributor to SB. Only qualitative agreement with the experiment can be expected from the model, whose main aim is to determine solute and solvent parameters playing the most vital role in symmetry breaking.
An important revelation obtained from this model is that the Coulomb interaction of the charges located on the acceptors can be as important for the symmetry breaking as the interaction of the charges with the polar solvent. In some cases, this contribution can be large enough to enable symmetry breaking even in nonpolar solvents. It should be noted that this model could in principle be applied to any spectroscopic quantity that reflects the extent of asymmetry of the electronic distribution of a molecule
and is not limited to a vibrational frequency splitting. However, unambiguous spectroscopic signatures of symmetry breaking are still scarce.
The main limitation of the model is the assumption that the charge on the central D unit, which is equal to the sum of the absolute values of the charges on the two A moieties, is a phenomenological parameter. This quantity is considered to be a constant when the solvent polarity is varied.
Partial charge transfer with 𝑒 substantially smaller than the charge of an electron can be expected in most quadrupolar molecules. This problem could be alleviated with a more elaborate description of the AL–D–AR
system including at least three states: a locally excited AL–D*–AR state and two degenerate states with charge transfer to the left and right branches of the molecule, AL-–D–AR and AL–D–AR-. The two-level model is only applicable if 𝑉ÇF ≪ ∆𝐺IÊ , where 𝑉ÇF is the electronic coupling between the donor and acceptor and −∆𝐺IÊ is the free energy of charge separation. If this condition is not fulfilled, a variation of the parameters 𝑉, 𝜆 and 𝛾 results in a change of the charge on the donor and, hence, of the total charge on the acceptors. This situation is encountered and discussed in the following chapter.
The effect of the intramolecular vibrational degrees of freedom on the symmetry breaking is also not included in this model. The interaction of electronic degrees of freedom with antisymmetric intramolecular vibrations is predicted to result in symmetry breaking.30,36 However, the experimental data suggest that in the systems for which symmetry-breaking dynamics were directly monitored, such as the ADA molecule discussed here, the process is mainly driven by solute-solvent interactions, as its timescale coincides with that of solvent relaxation and it is not operative in apolar solvents.
Chapter 6
Symmetry Breaking in Time and Space
Let us step into the night and pursue that flighty temptress, adventure.
J. K. Rowling hile in Chapter 3 we concentrated on the ethynyl p-spacers connecting the donor and acceptor moieties in the quadrupolar DAD and, in Chapters 4 and 5, we focused on the nitrile acceptor fragments in the quadrupolar ADA, we have always had either one or the other in a single molecule. It is time to combine them together in the same quadrupolar rod in this chapter.
We introduce the ethynyl p-bridges in the molecular structure of the ADA rod that we discussed before and the obtained molecule will be referred to as A-p-D-p-A (Figure 6.1). This molecule, like ADA, was also synthesized by the group of Prof. Daniel Gryko (Institute of Organic Chemistry, Polish Academy of Sciences).26,200 We will discuss throughout this piece how incorporation of the p-bridge affects the symmetry breaking phenomena compared to ADA, which are now well understood.
The geometry of the molecule is not altered upon incorporation of the p-spacer apart from a lengthening of the charge-transfer branches (Figure 6.2), which allows the separation between the donor and acceptor units to be increased. Table 6.1 lists the key geometrical parameters of both quadrupolar rods. The D-A distance was estimated both as a center-to-center (from the center-to-center of the pyrrolopyrrole bicycle to the center-to-center of the
W
phenyl ring of the cyanophenyl fragment) and as a center-to-terminus distance (from the center of the pyrrolopyrrole to the terminal nitrogen atom of the cyanophenyl fragment). Introduction of the phenylethynyl spacers increases the separation between donor and each of the acceptors by 6.9 Å. It makes this molecule about 80 % longer than ADA. The dihedral angles between the plane of the heterocycle and both phenyls (i.e.
those in the p-spacer and the lateral tolyl groups) stay intact upon incorporation of the spacer.
Figure 6.1 | Molecular structures and absorption spectra of ADA and of A-p-D-p‑A in THF (black) as well as fluorescence spectra in cyclohexane (blue), tetrahydrofuran (green) and acetonitrile (red). Pump wavelength for all time-resolved experiments is indicated.
Such lengthening influences electronic absorption and fluorescence spectra dramatically. Figure 6.1 demonstrates the absorption spectrum in THF and a few emission spectra in nonpolar cyclohexane, medium polar THF and highly polar acetonitrile for both compounds. The redshift of A-p-D-p-A compared to ADA absorption by 1550 cm-1 is explained by the
extension of the conjugated p-system. However, the solvent dependence of its fluorescence is strikingly much more enhanced. Indeed, upon going from cyclohexane to acetonitrile, the emission band of A-p-D-p-A and ADA downshifts by 5150 and 1630 cm-1 respectively. Such a strong solvent dependence of the fluorescence is indicative of significantly enhanced symmetry breaking in the longer rod.
Figure 6.2 | Optimized ground-state geometry of A-p-D-p‑A and of ADA as seen from the top (top) and side (bottom) of the pyrrolo[3,2-b]pyrrole bicycle. Carbons are grey, nitrogens are blue and hydrogens are white.
Table 6.1 | Key geometrical parameters of ADA and of A-p-D-p‑A obtained from the optimized geometries calculated by DFT (CAM-B3LYP201 level theory with 6‑31G(d,p) basis set as implemented in Gaussian 09154).
Geometrical parameter ADA A-p-D-p-A D-A distance (center-to-center) / Å 5.0 11.9 D-A distance (center-to-terminus) / Å 9.0 15.9 Dihedral angle (D/p-spacer) / º 38.4 39.2 Dihedral angle (heterocycle/lateral tolyl group) / º 56.6 55.1
Figure 6.3 displays stationary electronic spectra of this longer quadrupolar rod in the majority of solvents used. Figure 6.4A demonstrates solvent polarity dependencies of A-p-D-p-A and ADA absorption and
emission maxima. Absorption of neither of them demonstrates any dependence on the dipolar reorientation function (Figures 6.3A, 6.4A (squares); Figure 4.5), but its position is linearly dependent on the solvent dispersion (Figure 6.4B). Moreover, the slope j of this dispersion dependence is the same for both compounds (Table 6.2). It confirms the nondipolar electronic ground state whose energy is determined by the dispersive interaction with the environment. Additionally, the one-photon forbidden S2←S0 transition pops up in the two-photon absorption data available from the literature200 similarly to ADA, thus indicating the quadrupolar nature of the ground and Franck-Condon excited states.
Therefore, introduction of the p-spacer does not alter the nature of the ground state of the molecule.
Figure 6.3 | Steady-state electronic absorption (A) and fluorescence (B) spectra of A-p-D-p‑A.
Figure 6.4 | A. Solvent polarity dependence of the absorption (squares) and emission (circles) maxima of A-p-D-p‑A (orange) and ADA (blue). B. Dependence of the absorption maximum of A-p-D-p‑A (orange) and ADA (blue) on the solvent polarizability function f(n2) = 2(n2-1)/(2n2+1) and best linear fits.
The fluorescence behavior is completely different from the absorption.
Figure 6.4A shows a substantial dependence of the fluorescence on the dipolar function already in weakly polar solvents, and it becomes even steeper from moderately polar to highly polar ones (Δf ≥0.34). Such dependences demonstrate a good linearity with the exception of two outliers highlighted with a black contour (vinyl acetate and chloroform).
They appear off-trend because of the substantial contribution of quadrupolar (vinyl acetate) (Figure 5.5) and weak H-bonding (chloroform) interactions which are not accounted for by Onsager dipolar function, but are important for symmetry breaking as we have seen in the previous chapter. The non-zero fluorescence dependence on solvent polarity function indicates a dipolar nature of the excited state, and the bilinear appearance indicates an increase of the dipolar character in medium and highly polar solvents. The polarity dependences for both regimes are substantially stronger for A-p-D-p-A compared to ADA as indicated by their slopes (j1/j2) (Figure 6.4A and Table 6.3). It evidences that introduction of the p-spacer in the charge-transfer branches not only enhances the dipolar character of the symmetry-broken excited-state, but leads to a more pronounced increase of the dipolar character upon going
They appear off-trend because of the substantial contribution of quadrupolar (vinyl acetate) (Figure 5.5) and weak H-bonding (chloroform) interactions which are not accounted for by Onsager dipolar function, but are important for symmetry breaking as we have seen in the previous chapter. The non-zero fluorescence dependence on solvent polarity function indicates a dipolar nature of the excited state, and the bilinear appearance indicates an increase of the dipolar character in medium and highly polar solvents. The polarity dependences for both regimes are substantially stronger for A-p-D-p-A compared to ADA as indicated by their slopes (j1/j2) (Figure 6.4A and Table 6.3). It evidences that introduction of the p-spacer in the charge-transfer branches not only enhances the dipolar character of the symmetry-broken excited-state, but leads to a more pronounced increase of the dipolar character upon going