We are going to discuss the processes that an organic molecule undergoes when in an excited state. The Jablonski diagram presented in Figure 1.1 is a convenient way to proceed with introducing elementary radiative (photon absorption and emission) and nonradiative (internal conversion, intersystem crossing and vibrational relaxation) processes of a
S
photophysical cycle. No explicit photochemical processes are considered here.
Figure 1.1 | Jablonski diagram for a typical organic molecule. Adapted from Ref. 75.
Absorption. Electromagnetic waves can interact with a molecule promoting it to an excited state. Depending on the energy of the light, it might be, among others, an electronic or vibrational excited state.
Electronic excitation is normally achieved with ultraviolet or visible light, while vibrations are excited with less energetic infrared radiation.
Absorption occurs only if the electric field of the electromagnetic wave interacts with a transient molecular charge distribution characterized by the transition dipole moment 𝑀"→$.
𝑀"→$= Ψ" 𝜇 Ψ$ = 𝜒"𝜒$ 𝜓" 𝜇 𝜓$ (1.1) where 𝜇 is the dipole moment operator, Ψ is the total molecular wavefunction, while 𝜒 and 𝜓 are vibrational and electronic wavefunctions respectively. The square modulus of the integral 𝜒"𝜒$ is called the Franck-Condon (FC) factor and is a measure of the overlap between vibrational wavefunctions of initial and final state at the geometry of the initial state.75,76 The separation of the two integrals in (1.1) is valid within
the Born-Oppenheimer approximation. It follows from the Franck-Condon principle stating that electronic transitions occur on a much faster timescale than nuclear motions and thus should be considered vertical, i.e.
occurring without geometrical changes in the molecule.77
The probability of the transition 𝑖 → 𝑓 is determined by the Einstein coefficient for induced absorption 𝐵./ which is related to 𝑀.
𝐵./= 2𝜋
3ℏ/ Ψ" 𝑀"→$ Ψ$ / (1.2) The Einstein coefficient of absorption is also proportional to the absorption spectrum divided by the wavenumber:78
𝐵./∝ 𝜈6.𝜀(𝜈) (1.3) Eq. (1.2) and (1.3) lead to:
𝑀"→$ /∝ 𝜈6.𝜀(𝜈) (1.4)
Once the molecule appears on the excited-state potential energy surface various de-excitation processes come into play.
Emission. Emission of a photon is one of the main possible channels of electronic excited-state deactivation. It occurs due to the coupling of the excited state Ψ$ to the ground state Ψ" via the electronic transition dipole moment in analogy with absorption and is related to the Einstein coefficient for spontaneous emission, 𝐴/., which itself is proportional to the Einstein absorption coefficient:
𝐴/.=8𝜋𝜈<
𝑐< 𝐵./ (1.5)
Although Eq. (1.5) rigorously stands for atoms, the cubic frequency factor is also valid for polyatomic molecules.78 There are two different types of emission – fluorescence and phosphorescence – depending on whether the spin multiplicity of the excited state is preserved or altered upon the radiative transition. However, within the scope of this thesis we will encounter only singlet-singlet fluorescence and therefore we discuss only this type of emission hereafter.
The Einstein coefficient for emission is proportional to the
which is only strictly valid when the ground and excited states possess the same geometry.
In order to represent correctly absorption and emission spectra together, the absorption spectrum has to be divided by the first power of the wavenumber and the emission one by the third. This representation is called the transition dipole moment representation.78
Internal Conversion. Internal conversion (IC) is a nonradiative transition between two electronic states of the same multiplicity. It occurs as an isoenergetic horizontal transition in the Jablonski diagram (Figure 1.1) from a vibrational level of the higher electronic state to a higher vibrational level of the lower electronic state. Internal conversion from the electronic states higher than S1 generally occurs ultrafast leading to the population of the lowest singlet excited state from where the photochemistry and light emission occurs (Kasha’s rule80). However, with the advent of ultrafast spectroscopy, which is able to track the fate of higher excited states down to few fs time resolution, it became evident that this statement is valid statement only in the long-time limit.
From experimental results, it has been concluded that, for example, for aromatic hydrocarbons the radiationless S1→S0 transition is negligible if the energy difference ΔE between S1 and S0 states is larger than 60 kcal/mol but it becomes increasingly more important when the energy
difference decreases.81 These observations were summarized by the relationship
𝑘HI = 10.<𝑒6LMN (1.9) which shows the dependence of the rate constant on the energy gap between lowest singlet excited and ground states and which is referred to as the energy-gap law.82
This law has been successfully applied extensively to the internal conversion of various classes of compounds, e.g. carotenoids,83 xanthene dyes,84 azulene85 and linear polyenes,86 aromatic thiones87 and radicals88 as well as to charge transfer processes of numerous metal complexes89 and in many other systems.90
In addition to the energy gap between electronic states, large displacements between potential energy surfaces lead to large FC factors facilitating an efficient radiationless transition (Figure 1.2).81 In fact, the energy-gap law is not obeyed when the energy surfaces couple strongly or when large rearrangements take place, e.g. in vicinity of state crossings.
Figure 1.2 | Effect of the energy gap and relative displacement of the potential energy surfaces on the Franck-Condon factors for a radiationless transition. Adapted from Ref. 81.
Intersystem Crossing. Intersystem crossing (ISC) is a radiationless process between two isoenergetic vibrational levels of electronic states of different multiplicities. The most encountered scenario for organic molecules is a transition between a molecule in the zero vibrational level of the S1 state to the isoenergetic vibrational level of the Tn triplet state from where vibrational relaxation and internal conversion in the triplet manifold bring it to the zero vibrational level of the T1 state. In principle, such a singlet-triplet transition is forbidden, but if mixing of spin states due to magnetic interactions is available, it might become allowed. An electronic transition that involves a change of spin angular momentum requires some coupling with another source of angular momentum that can both trigger the transition and allow conservation of the total angular momentum and energy of the two interacting systems. For organic molecules, the most important interaction that couples two spin states and that provides a means of conserving the total angular momentum of the system is the coupling of the electron spin with the orbital angular momentum (spin-orbit coupling).77
The rates of the ISC process can be fast enough to compete with other de-excitation pathways, such as fluorescence and internal conversion. The El-Sayed’s rule91 states that the probability and the rate of a spin-forbidden radiationless transition is much larger when it involves two orbital configurations of different type (such as 1p-p* and 3n-p*, for example) when compared to the orbital configurations of the same type (such as
1p-p* and 3p-p*). The spin-orbit coupling increases in the presence of atoms with high nuclear charge (heavy atoms) capable of causing electrons to accelerate strongly and thereby create a large magnetic moment as the result of their orbital motion.77 It causes so called internal and external heavy-atom effects which promote ISC rates.75 Finally, the rate of this process is inversely proportional to the energy gap between the S1 and the triplet state to which intersystem crossing actually occurs (i.e. T1 or some upper triplet Tn).81
Vibrational Relaxation (VR). The Franck-Condon nature of the electronic excitation of a polyatomic molecule dictates that some vibrational modes are excited during the electronic transition thus
imparting it a vibronic character. Since the FC factors differ largely for various normal modes in the molecule, only some – Franck-Condon active – modes end up being populated. These high-frequency modes transfer the energy rather quickly to the low-frequency part of vibrational spectrum through anharmonic coupling. This process is usually very fast taking place on a subpicosecond timescale and is called intramolecular vibrational redistribution (IVR). The population of the intramolecular low-frequency modes (LFM) influences (through anharmonic coupling) the frequencies and intensities of the high-frequency vibrations often leading to their redshift. This gives a characteristic shape (redshift and asymmetrical broadening on the low-frequency side) to a vibrationally hot band (Figure 1.3).92,93
Figure 1.3 | Simulated high-frequency infrared absorption bands of different vibrational temperatures. A. Absorption spectrum A(T). B. Difference spectrum ΔA(T) = A(T) – – A(300 K). The plots are taken from Ref. 92.
This thermalization within the molecule itself leads to the establishment of a Boltzmann distribution of the LFM population and is followed by the vibrational energy transfer to the solvent called vibrational cooling (VC). It is often stated that IVR occurs much faster than VC and
the two are well separated in time.94 In the Landau-Teller description of vibrational energy relaxation in liquids, the rate of relaxation is indeed proportional to the square of the coupling multiplied by the power spectral density of the time correlation function describing the fluctuating forces exerted by solvent molecules on the solute vibration of interest.95,96 The latter term corresponds to friction exerted by the solvent on solute vibrations. Since the spectral density for molecular liquids decreases exponentially with increasing frequency, the friction is the greatest in the low-frequency part of vibrational manifold. This fact underlies the conventional wisdom that VC occurs primarily from the low-frequency modes of solute to the low-frequency modes of solvent. This indeed necessitates IVR to be faster than VC. However, since IVR occurs on a multitude of timescales from tens of femtoseconds97 to 1-2 ps,98 its slower components might overlap with the fastest part of intermolecular vibrational cooling and the two processes might be entangled in time.98 Additionally, in a few studies, solvent was evidenced to facilitate99,100 or to hinder IVR.101 Therefore, the IVR in condensed phase is not necessarily a purely intramolecular process.