The concept of symmetry breaking in multipolar compounds was treated theoretically by the group of Prof. Anna Painelli and Prof.
Francesca Terenziani within the essential-state model formalism.36–39 This model is briefly presented below and is referred to as the Painelli-Terenziani SB model hereafter.
Essential-state models offer an efficient theoretical tool to investigate optical properties of conjugated molecules composed of several electron donating (D) and accepting (A) groups linked by p-bridges. These compounds undergo charge-transfer transition(s) upon photoexcitation in the absorption band(s) at low energy. Charge resonance between D and A moieties governs their low-energy physics. In chemical language, this phenomenon is described using several resonating forms as basis states.
Mulliken was the first to apply this formalism116 to describe the optical spectra of dipolar D-p-A chromophores, for which two states are necessary: neutral DA and zwitterionic D+A-.
For quadrupolar chromophores, three resonating states are necessary:
D+A-D⟷DAD⟷DA-D+ (Figure 1.6A). Although we are explicitly considering the DAD structure here and hereafter, the same discussion applies to ADA chromophores as well, provided that the role of D and A is interchanged. The DAD structure is designated as |𝑁 (neutral state), whereas |𝑍. and |𝑍/ are the two degenerate zwitterionic states. The energy difference between the zwitterionic states and the neutral state is defined as 2𝜂, so that for positive 𝜂, the neutral form is lower in energy than zwitterionic ones. The mixing between the states is described by an off-diagonal matrix element of the Hamiltonian 𝑁 𝐻 𝑍. = 𝑁 𝐻 𝑍/ =
= − 2𝑡, that measures the probability of electron transfer from D to A and backward. The direct mixing between 𝑍. and 𝑍/ is set to zero, because represents the hopping between non-nearest neighbor sites. The dipole moments of the two zwitterionic states have the same magnitude 𝜇g, but point in the opposite directions. This is the only matrix element of the dipole moment operator that is considered in the chosen basis.
The zwitterionic states can be combined in the following symmetric and antisymmetric wavefunctions: |𝑍v = 1/ 2 |𝑍. + |𝑍/ and |𝑍6 =
Here, 𝜎 is the mixing operating for the two gerade states, 𝜌 is the iconicity measuring the average charge on the central A site, while 𝛿 measures the unbalance of the charge on the two external D units. The electronic Hamiltonian and dipole operator can be written as:
𝐻Tn= 2𝜂𝜌 − 2𝑡𝜎 (1.20)
𝜇 = 𝜇g𝛿 (1.21)
The eigenstates can be obtained:
|𝑔 = 1 − 𝜌|𝑁 + 𝜌|𝑍v (1.22)
|𝑐 = |𝑍6 (1.23)
|𝑒 = 𝜌|𝑁 − 1 − 𝜌|𝑍v (1.24)
where 𝜌 measures the weight of |𝑍v in the ground state, or equivalently it defines the amount of charge separation in the ground state, and therefore the quadrupolar moment of the molecule. It is determined by the model parameters:
𝜌 = 0.5(1 − 𝜂/ 𝜂/+ 4𝑡/) (1.25) The transition dipole moments and transition frequencies, i.e. the relevant quantities for spectroscopy, can be expressed in terms of 𝜌:
ℏ𝜔S} = 𝐸}− 𝐸S = 2𝑡 1 − 𝜌
𝜌 , 𝜇S} = 𝑔 𝜇 𝑐 = 𝜇g 𝜌 (1.26) ℏ𝜔ST = 𝐸T− 𝐸S = 2𝑡 1
𝜌(1 − 𝜌), 𝜇ST = 𝑔 𝜇 𝑒 = 0 (1.27) ℏ𝜔}T = 𝐸T− 𝐸} = 2𝑡 𝜌
1 − 𝜌, 𝜇}T = 𝑐 𝜇 𝑒 = −𝜇g 1 − 𝜌 (1.28) Figure 1.6 summarizes the electronic three-state model for quadrupolar chromophores. From Eq. (1.26)-(1.28), it follows that odd |𝑐 state is one-photon accessible from the ground state, whereas the transition to the even
|𝑒 state is one-photon forbidden (but two-photon allowed). On the other hand, |𝑒 is one-photon accessible from |𝑐 state.
Figure 1.6 | A. Basis states for quadrupolar DAD molecules. B. Energy level sketch of the three-state model. Adapted from Ref. 117.
In quadrupolar chromophores, two equivalent molecular vibrations must be introduced, one for each branch of the molecule. These two vibrational coordinates have the same vibrational frequency and the same relaxation energy 𝜖•, i.e. the energy gain when the molecule is allowed to relax upon excitation due to different geometries of neutral and zwitterionic states. The symmetric and antisymmetric combination of vibrational coordinates are defined respectively as 𝑞v= 1/ 2 𝑞.+ 𝑞/ and 𝑞6= 1/ 2 𝑞.− 𝑞/ with their corresponding conjugate momenta, 𝑝v and 𝑝6.
The Hamiltonian accounting for the electron-vibration coupling is expressed as
𝐻 = 𝐻Tn− 𝜖•𝜔𝑞v𝜌 − 𝜖•𝜔𝑞6𝛿 +1
2 𝜔/𝑞v/+ 𝑝v/ +1
2 𝜔/𝑞6/+ 𝑝6/
(1.29)
The symmetric coordinate 𝑞v is coupled to 𝜌 that mixes the two symmetric states |𝑁 and |𝑍v . This means that the 𝑞v mode drives a symmetric displacement of the charge. In contrast, 𝑞6 is coupled to the antisymmetric vibration mixing |𝑍v and |𝑍6 thus driving unbalance of the charge distribution in the molecule and is responsible for symmetry breaking.
Figure 1.7 shows the phase diagram where quadrupolar systems are classified according to their values of the ground-state quadrupolar moment (𝜌) and the strength of electron-vibration coupling (𝜖•). Three different regions exist in this diagram corresponding to three classes of centrosymmetric molecules. Class I chromophores are characterized by low quadrupolar moment (ground state is almost pure |𝑁 state) and essentially almost any value of 𝜖• could induce symmetry breaking in their first excited state since |𝑍v and |𝑍6 are almost degenerate. In this case, system localizes in one of the two equivalent minima corresponding to states with equal and opposite dipole moments. In the opposite limit (class III chromophores), the charge instability already takes place in the ground state which almost coincides with the |𝑍v state. This is the case because the optically allowed |𝑍6 state is again almost degenerate with the ground state and the |𝑁 state is higher in energy. In the central region (class II chromophores) featuring compounds with intermediate quadrupolar moments, all three states have a single minimum that characterizes nondipolar ground and both excited states. Importantly, Figure 1.7 and the Hamiltonian in Eq. (1.29) describe the systems in nonpolar environment.
Therefore, according to this model, SB is a purely intramolecular process determined by the structural characteristics of the centrosymmetric molecule.
Additionally, the effects of dipolar solvation are considered in the Painelli-Terenziani SB model118–120 as the ones that are able to facilitate and enhance symmetry breaking. The coupling between electronic degrees of freedom of the solute with the slow orientational motion of the solvent molecules is considered within the framework of the reaction-field approach. The Hamiltonian for this type of interaction is120
𝐻‚ƒn„ = −𝜇g𝐹…𝛿 + 𝜇g/
4𝜖ƒ@𝐹…/ (1.30)
Where 𝐹… is the reaction field, whose equilibrium value is proportional to the dipole moment of the solute. The parameter 𝜖ƒ@ measures the solvent relaxation energy. It is clear, that 𝐹… couples to the same dipolar operator, 𝛿, as the 𝑞6 coordinate.
Figure 1.7 | Phase diagram for quadrupolar chromophores, describing the stability of the different states as a function of the quadrupolar character (𝜌) and the electron-vibration coupling (𝜖„) in the mixing units between neutral and zwitterionic states. The potential energies are shown (in order of increasing energy) for the ground, one- and two-photon allowed states (according to Figure 1.6). Adapted from Ref. 36.
In this treatment, dipolar solvation becomes relevant only upon initial symmetry breaking due to structural changes, which are at the origin of the process and increases the depth of one of the minima where system localizes for class I chromophores. Instead, for class II chromophores, the systems with a small 𝜖•, which should not break the symmetry of their lowest excited state, could undergo SB in highly polar solvents with a high value of 𝜖ƒ@. It would translate into an effective downward shift of the boundaries in the phase diagram in Figure 1.7. Therefore, as far as symmetry breaking is concerned, it is the total relaxation energy (vibrational and solvation contributions together) that determines whether the process takes place. It should be underlined, that only dipolar solvation is considered within the Painelli-Terenziani model and the higher-order terms are neglected (e.g. solvation due to solvent quadrupole moment).
The authors of the model specify that, although vibrations and solvation cooperate, they have different timescales. The vibrational coordinate is purely quantum mechanical, whereas the solvation can be
described in terms of a slow classical coordinate.121 The authors conclude that if the height of the barrier between the two minima is low, then false symmetry breaking may occur,122 which means that tunneling between the two symmetry-broken states would result in their constant interconversion.
Instead, in polar solvents the SB is true and interconversion between the two states is too slow to take place appreciably.
In a similar fashion, the authors extend their model to octupolar chromophores comprising three charge-transfer branches linked in a C3
symmetry (D-(p-A)3 or A-(p-D)3). In this case, one neutral and three zwitterionic states are necessary for the essential-state model (Figure 1.8A). The development of the model follows the same lines as for quadrupolar molecules. The four basis wavefunctions are determined to be of A- and E-symmetry: |𝐴. = |𝑁 , |𝐴/ = 1/ 3 |𝑍. + |𝑍/ + |𝑍< ,
|𝐸. = 1/ 6 2|𝑍. − |𝑍/ − |𝑍< , |𝐸/ = 1/ 2 |𝑍/ − |𝑍< . The two E-states are degenerate and stay unmixed, while the A-states intermix:
|𝑔 = 1 − 𝜌|𝐴. + 𝜌|𝐴/ (1.31)
|𝑐. = |𝐸. (1.32)
|𝑐/ = |𝐸/ (1.33)
|𝑒 = 𝜌|𝐴. − 1 − 𝜌|𝐴/ (1.34)
Naturally, for this type of molecules, 𝜌 quantifies their octupolar moment.
Figure 1.8 | A. Basis states for octupolar D-(p-A)3 molecule. B. Energy level sketch of the four-state model. Adapted from Ref. 37.
Figure 1.8B illustrates the arrangement of energy levels and one- and two-photon transitions. States |𝑔 and |𝑒 are nondipolar and the |𝑔 →|𝑒 transition is one-photon forbidden, but two-photon allowed. On the other hand, states |𝑐. and |𝑐/ have non-vanishing dipole moment along the x-direction (Figure 1.8A) and are thus one-photon allowed from the ground state. They are also two-photon allowed but with weaker intensity.
Additionally, the |𝑐. →|𝑒 and |𝑐/ →|𝑒 transitions are also one-photon allowed.
To account for vibrational motion, three vibrational coordinates (𝑞., 𝑞/, 𝑞<) with identical frequencies (𝜔) and relaxation energies (𝜖•) have to be introduced. The symmetry adapted vibrational coordinates are defined as 𝑞F= 1/ 3 𝑞.+ 𝑞/+ 𝑞< , 𝑞N‡ = 1/ 6 2𝑞.− 𝑞/− 𝑞< and 𝑞Nˆ =
= 1/ 2 𝑞/− 𝑞< . The first one (𝑞F) is totally symmetric and the other two have E-symmetry. If the Hamiltonian similar to Eq. (1.29) is written explicitly, it can be seen that a totally symmetric vibration couples to 𝜌 that modulates symmetric charge transfer from the donor to the acceptors (for D-(p-A)3 systems), while the E-type vibrations couple to 𝛿 operators leading to unbalanced charge distribution on the three peripheral groups.
This result is in line with what was shown for quadrupolar chromophores.
Figure 1.9 displays potential energy surfaces as functions of the 𝑞N‡ and 𝑞Nˆ coordinates while 𝑞F is kept at its equilibrium value. The most striking feature of this phase diagram is the absence of region II where the symmetry is preserved. States |𝑐. and |𝑐/ are degenerate only at the origin (𝑞N‡ = 𝑞Nˆ = 0) and any perturbation removes the degeneracy and leads to symmetry breaking. In sharp contrast to quadrupoles, where SB is conditional, i.e. determined by the values of 𝜌 and 𝜖•, for octupoles the symmetry preservation simply does not exist. Class III octupolar chromophores in this model have both the ground state |𝑔 and the first excited state |𝑐. states multistable with triple minima.
Addition of polar solvation to this treatment leads to the same conclusions as for quadrupolar molecules. Solvent reorientation is substantially slower compared to vibrational motion and should be treated
as a classical coordinate trapping the symmetry-broken octupolar system in one of the minima irreversibly.
Figure 1.9 | Phase diagram for octupolar chromophores as a function of octupolar character (𝜌) and electron-vibration coupling strength (𝜖„) (in units of 2𝑡). Right panel shows a magnified region of the conical intersection near the degeneracy point. Adapted from Ref. 37.
This model was tested and validated with steady-state absorption and fluorescence spectra as well as with TPA data for a number of quadrupolar and several octupolar organic chromophores showing an overall excellent agreement with these experiments.
Chapter 2
Ultrafast Spectroscopy in Practice
Perilous to us all are the devices of an art deeper than we possess ourselves.
J.R.R. Tolkien n this chapter we describe experimental spectroscopic techniques and practices used for the work presented in this thesis. The main emphasis of this dissertation is placed on ultrafast vibrational spectroscopy where the author believes he gained much of his expertise.
Modification of the TRIR setup to accommodate multiple different pump pathways was also one of the goals of this PhD program. Nevertheless, we are fortunate to host many other time-resolved techniques in our laboratories. We believe that the use of not a single method but a combination thereof is critical to gain sufficient physical insight into the excited-state dynamics of organic molecules. This proves to be particularly true for the systems investigated in the last two chapters of this thesis.
Therefore, we will first briefly describe other complementary broadband time-resolved ultrafast techniques namely electronic UV-vis transient absorption and fluorescence upconversion spectroscopy (FLUPS). A brief introduction is also given to a few single-wavelength techniques such as time-correlated single photon counting (TCSPC) and two-photon-excited fluorescence spectroscopy (TPEF), which played an important supportive role in this work.
The importance of steady-state spectroscopy is frequently underemphasized, especially in the ultrafast community, and it is often
I
done as a matter of habit. However, when applied systematically and with great care it is able to provide significant pieces of information. It was chosen not to discuss it here as this chapter is dedicated to time-resolved methods only, but the information regarding these methods and procedures is given in Appendix 1.