Chapter 2 Fundamentals of Photophysics and
2.2 Photophysical Processes
𝜇!" = 𝜓!𝜇𝜓!𝑑𝜏 (2.6)
where 𝜓! and 𝜓! are the wavefunctions of the initial and final states, respectively, and 𝜇 the molecular dipole operator. A transition between two states occurs if it is associated with a non-zero transition dipole moment 𝜇!".
The square of the transition moment is proportional to the oscillator strength (𝑓):1
𝑓=8𝜋!𝑚!𝜈!"
3ℎ𝑒! 𝜇!"! ≈4.7×10!!𝜈!"𝜇!" ! (2.7)
where 𝑚! and 𝑒 are the mass and the charge of the electron, respectively, 𝜈!"
is the frequency of the transition. The oscillator strength is an important quantity in spectroscopy that allows to measures the intensity or probability of an optical transition. Also, it is related to the integrated absorption band of a transition, and thus can be determined experimentally through the molar extinction coefficient (𝜀):
𝑓=4.3×10!! 𝜀(𝜈)𝑑𝜈 (2.8)
2.2 Photophysical Processes
Thus, according to equation 2.2 the energy acquired by a molecule due to absorption of a photon causes a molecule to be promoted to a higher electronic energy level called excited state. The general scheme that illustrates processes, which are taking place after molecule has been excited, is called Jablonski diagram. This diagram represents the electronic states of a system and the transitions between them, like one depicted in Figure 2.1.
Here, the electronic states are represented by thick horizontal lines, while
vibrational states by thin lines.* Singlet (S) and triplet (T) states refer to the spin multiplicity of the electronic state and are collected in separate columns.** All states are arranged in vertical order and indicate the relative energies each, and are labeled consecutively by increasing energy, where S0
is the singlet ground state, and v = 0 is the lowest vibrational state.
Figure 2.1 Jablonski energy diagram describing molecular energy levels and possible transitions between different singlet and triplet states: absorption, internal conversion, vibrational relaxation (VR), intersystem crossing (ISC), fluorescence and phosphorescence processes.
The first process is the absorption of a photon of a particular energy by the molecule. The following photophysical processes can be radiative or radiationless transitions between different states, and commonly indicated as straight and wavy arrows on Jablonski diagram, respectively.
* Usually only a fraction of these vibrational states are presented due to a vast number of possible vibrations levels in a molecule. Additionally, in the gas phase each of these states can be subdivided even further into rotational energy levels, however, usually such levels are omitted in Jablonski diagrams.
** Note that horizontal displacement does not indicate a change in structure, and is only used to group states by their spin multiplicity.
2.2 Photophysical Processes
2.2.1 Non-radiative Processes
Let us follow now the processes that are taking place after a system has been excited to a higher electronic and vibrational state. Such state usually called “hot”. Once the system is there, it will slowly relax to the lowest vibrational level by giving the excess energy away to other vibrational modes within the same molecule, or transferred to the surrounded molecules. In a condensed medium, there are many collisions between the molecule and its environment, so the excess vibrational energy is converted into heat, which can be easily absorbed by neighboring “cold” solvent molecules. This process is very fast and typically occurs on the order of 10-12 s, thus it has a high probability to occur first after a photon has been absorbed.3
Further relaxation requires the intramolecular transformation of electronic energy into vibrational energy, and takes place when the lowest vibrational level of a higher state overlaps with the highest vibrational levels of the lower state. This process called internal conversion (IC) when the molecular spin remains the same, and intersystem crossing (ISC) when it changes. In most cases the separation between energy states, which are greater than the lowest excited sate, is small, and gives a high degree of coupling between vibrational and electronic levels. That is why a spin-allowed IC occurs with a high probability and in the same time frame as vibrational relaxation (VR). The rate constants of such transitions have been established both theoretically and experimentally and known as the energy gap law, which states that non-radiative decay rate increases exponentially as the energy gap decreases.4
At the same time, ISC is usually forbidden by the rule of conservation of angular momentum, as it involves a change of spin multiplicity. As a result, this process generally occurs on very long time scales. However, El-Sayed’s rule states that the rate of ISC is relatively large if the radiationless transition involves a change of molecular orbital type, e.g.
the nπ* → ππ* transition (Figure 2.2).5 In such case the change of spin is compensated by change of the orbital angular momentum keeping the total angular momentum constant. The required spin flip is achieved by the action of a magnetic field invoked by spin-orbit coupling.6 Spin-orbit coupling is enhanced in the presence of heavy atoms that is known as the heavy atom
effect.
Figure 2.2 Schematic representation of the El-Sayed’s rules for ISC.
2.2.2 Radiative Processes
As discussed above, the separation between the lowest electronic state and the ground state is usually greater than between higher states, additionally there is a poor overlap between vibrational and electronic states, thus IC and ISC to the ground state becomes relatively slow. Such slow down recovery permits other processes compete with IC (or ISC) from the lowest excited state, one of which is the return to the ground state by emission of light. This process is called fluorescence when it occurs between two states of same multiplicity, and phosphorescence when the multiplicity is different. The ultrashort lifetime of upper electronic excited states and the long lived lowest excited state underlie the Kasha-Vavilov’s rule that states that emission takes place from the lowest electronic excited state, namely the S1 and T1 states, and that the emission quantum yields do not depend on the excitation wavelength.7 There are many exceptions, however, like azulene that emits mostly from the S2 state.
The efficiency of emission is characterized by the quantum yield (QY) of luminescence.8 It is defined as the ratio of photons absorbed to photons emitted. The QY of fluorescence can come close to 1, while the phosphorescence is usually weak because it is spin-forbidden process thus its rate is much smaller than the rate of radiationless deactivation.
Also, luminescence QY can be defined by the rate of excited state decay:
Ф= 𝑘! 𝑘!
! (2.9)