Mesyltoe 3P9

3.P9 Photochemistry

A summary of a lecture course given by Prof. James Durrant and Dr. Saif Haque.

Section 1

Photochemical Reactivity

Photochemistry is chemistry induced by light: that is reactions that are allowed to happen by absorption of light. Absorption of a photon of the correct energy promotes an electron from the ground state to produce an excited state. The excited state is one that is not usually thermodynamically accessible. The energy distribution in the excited state is not thermal, allowing more specific reaction paths than those generated by thermal excitation. Excited states are both stronger oxidants and stronger reductants than their ground states.

Uses of photochemistry

• Photosynthesis
• Photovoltaics
• Cis/trans isomerisation
• Probing reaction dynamics

Nomeclature

We have the highest occupied molecular orbital (HOMO), from which the electron is (usually) promoted. The lowest unoccupied molecular orbital (LUMO) is the orbital into which the electron is promoted. A singlet state is one in which the two highest energy electrons have opposing spins. A triplet state is one in which the two highest energy electrons have the same spin. A singlet state is denoted by the symbol S, the ground state by S₀ and the excited state by S₁. The triplet excited state is denoted by T₁. A molecule in the S₀ state is denoted by M, in the S₁ state by ¹M* and in the T₁ by ³M*.

Fundamental Processes

The excited state ¹M* can decay (release its energy to the environment) in loads of different ways.

• M + hv : light emission w_R = k_r[¹M*]
• M + heat : non-radiative decay w_IC = k_ic[¹M*]
• ³M* : intersystem crossing w_ISC = k_isc[¹M*]
• M' : isomer or fragment of M w_PD = k_OD[¹M*]
• M⁺ + Q^- : electron transfer w_Q = k_Q[¹M*][Q]
• M + ¹Q* : energy transfer

The first three are photophysics, and the rest photochemistry. Where there is no available photochemistry, the concentration of the excited state shows first order exponential decay caused by the first three, photophysical, decay paths.

Section 2

Electron Transfer

Imagine two potential wells, one at ¹A* (occupied) and one at B (unoccupied). They are of different depths, but the levels are constantly fluctuating. When the two wells are of the same depth, it is possible for an electron to jump from one to the other.

To understand this picture, we need to use some theory. We use the Born-Oppenheimer approximation, which allows the separation of the atomic wavefunction into its nuclear and electronic components, that is, to separate the vibrational wavefunction from the electronic wavefunction. The Franck Condon principle states that the probability of energy transfer is directly proportional to the overlap in vibrational wavefunctions. The principle of energy conservation states that transfer must occur between states of the same energy with fixed nuclear positions. We ignore the Franck-Condon principle here, and focus on 'weak coupling' cases, where there is no energy overlap.

Fermi's Golden Rule says: $k(s^{-1})=\frac{2\pi }{h}FC\cdot V^2$

Where k is the rate constant, FC is the Franck Condon factor, the measure of the probability that nuclei will be in a position such that isoenergetic electron transition is possible, and V is the electronic coupling, the measure of rate of the electronic transition when the nuclei are in the correct orientation. FC can be treated either classically (as a harmonic oscillator) or quantum mechanically (using vibrational wavefunctions). V must be treated quantum mechanically, as it involves tunnelling (see 3P3).

We can model these transfers using 1D potential energy surfaces. The two orbitals are modelled as anharmonic oscillators that cross. This model is discussed in more detail in 3P3 and 3O4.

With specific relevance to light absorption, the light excitation line between ground and excited PES is vertical. As there is often some offset between the surfaces (owing to the difference in structure), meaning that the energy required for electron promotion constantly fluctuates (due to thermal energy causing vibration in the system). This results in broadening of absorption lines. The Franck Condon factor determines how likely it is that the system will be promoted by a particular wavelength of light. In addition to electronic transitions, photons can also promote systems to higher vibrational levels. This is called a vibronic transition. For example, promotion from S₀⁰ to S₁³ means a transition from ground state vibration to v=3 vibration in addition to the optical excitation.

Section 3

Light absorption

We've met a description of light absorption before in the Beer Lambert law: A = εcl. The empirical strength of absorption bands is given by integration of the extinction coefficient over the entire band. We call this the oscillator strength, f.

$f\approx 4\cdot 10^{-9}\underset{v_1}{\overset{v_2}{\int }}ϵdv\approx 4\cdot 10^{-9}{ϵ}_{max}\mathrm{\Delta }{v}_{1/2}$

Where ε is the extinction coefficient, ν is the transition frequency in wavenumbers, ε_{max} is the peak extinction coefficient and Δν_1/2 is the transition full width at half the maximum (FWHM). The 4 x 10^-9 term is so that the strongest observed transition has an oscillator strength of 1.

Molecular basis of light absorption

An absorption transition is possible if the photon energy = the difference in energy of electronic states. A typical HOMO-LUMO gap is around λ = 333nm, which is in the UV range. The greater the conjugation in a molecule (as a rule), the smaller the HOMO-LUMO gap, the longer the wavelength of light, and the more likely it is that the molecule is coloured. The presence of d-orbitals (such as in transition metal complexes) can also result in lower energy, visible transitions. We also mustn't forget M→L charge transfer or L→M charge transfer, both of which are light activated.

Because these wavelengths are much greater than the dimensions of molecules, the molecules 'feel' the oscillating electric field: E = E₀cos(2πνt). The electric field that the light induces causes an oscillating molecular dipole, meaning that the orbitals are distorted and so are no longer eigenstates of the Hamiltonian operator, Ĥ₀ + Ĥ’ . There is an additional contribution to electron energy in the form of Ĥ’ = -e.E.r, where r is the distance from the electron to the centre of charge, and E is the electric field vector. This can drive a change in electronic state according to Fermi's Golden Rule.

The Transition Dipole

$k(s^{-1})=\frac{2\pi }{h}FC\cdot V^2=\frac{2\pi }{h}FC\cdot <{\psi }_f|eEr|{\psi }_i{>}^2$

The last part represents the extent to which the light field is trying to push the electron from i to f. If we focus purely on the ability of the molecule to respond to the light field, we define the transition dipole to be:

${\mu }_{fi}=-<{\psi }_f|er|{\psi }_i>$

Where r is still the distance between the electron and the centre of charge. This is the rate of transition between i and f in the presence of a light field, which is the strength of the light absorption, meaning that:

$|{\mu }_{fi}{|}^2\propto f$

Where f is the empirical oscillator strength.

Factors controlling the transition dipole include electron spin. For a transition to be allowed, there must be no change in spin. This means that photo-excited states are normally singlets, as it is impossible for a singlet state to be excited to a triplet state.

Another factor is the electronic orbital symmetry. All orbitals have an angular quantum number l. l=0 for an s orbital, 1 for a p orbital, 2 for a d orbital, and so on. Δl = ±1 for a transition to be allowed. μ is large if i and f overlap spatially and exhibit a large change in orbital symmetry.

Absorption Lineshapes

Vibronic transitions result in side bands on the higher energy side of the electronic transition. In the gas phase it can result in complex absorption line shapes. In solution a multitude of low energy vibrations result only in band broadening: only high frequency vibrations resolve as discrete bands.

The strength of the vibronic transitions is dependent on the spatial overlap of the nuclear wavefunctions. This means that $FC=<{\psi }_i|{\psi }_f{>}^2$. The larger this number, the stronger the bands. The magnitude of this integral depends on the wavefunction symmetry and the difference between the S₀ and S₁ equilibrium nuclear positions. The Franck Condon factor doesn't change the overall oscillator strength, but shifts the strength between vibronic transitions. We can see what the strongest transition will be by drawing a line vertically from the maximum equilibrium ground state position and seeing where it meets another maximum. This obviously depends on how different the two structures are from one another. The greater the difference in structure, the higher the vibrational state of the strongest transition. That means that excitation will result in a 'vibrationally hot' S₁, meaning, in extreme cases, that photodissociation occurs, so great is the vibration.

Generating excited states

To initiate photochemistry, light must be absorbed and electrons promoted. High oscillator strength transitions increase photochemical activity, as do low energy transitions (in the visible spectrum) as this overlaps with solar energy.

Light emission

All the same rules apply for light emission as apply for light absorption with regard to allowed transitions. Emission between singlet states is fluorescence. Some relaxations occur through non-radiative decay, that is, energy relaxing through vibrations in the bonds. The emission spectrum is normally the mirror image of the absorption spectrum. Where S₀ absorbs to S₁^v, S₁⁰ goes to S₀^v. There is splitting between S₀⁰→S₁⁰ absorption and S₁⁰→S₀⁰ emission due to solvent relaxation.

Solvent relaxation is cause of the red-shift effect on the S₁⁰→S₀⁰ transition in an emission spectrum. When the charge distribution in a molecule changes, the solvent molecules rearrange to stabilise this new state. This lowers the S₁⁰ state in energy, and raises the ground state's energy. This makes the energy gap smaller, which means that the light emitted is lower in energy, further 'red'. This is called the Stokes shirt. The magnitude of the effect depends on the polarity of the solvent (greater for more polar solvents).

Light absorption to higher excited states (such as S₂ or S ₃) typically results in immediate relaxation to S₁. The strength of the S₀⁰→S₁⁰ absorption means that there is a large rate constant for the corresponding emission, but the emission is in kinetic competition with other decay pathways, such as internal conversion, intersystem crossing and photochemistry. The absorption and emission processes share the same transition dipole, so radiative decay should be related to absorption strength, and indeed it is:

$k_r\propto |\mu {|}^2\propto f\propto \int ϵ(v)dv$

This is the Strickler-Berg relationship. For a typical optical transition in the visible region, $k_r\approx 3{ϵ}_{max}\mathrm{\Delta }{v}_{1/2}\approx 10^{9}f$

Fluorescence Lifetime and Yields

The rate constant for the decay of ¹M* is the sum of all decay pathways. If we ignore photochemistry, this means $\frac{d{[}^{1}M*]}{dt}=-(k_R+k_{ic}+k_{isc}){[}^{1}M*]$.

This is turn means ${[}^{1}M*]={[}^{1}M*{]}_0{e}^{k_{0}t}={[}^{1}M*]e^{\frac{-t}{{\tau }_f}}$

The quantum yield is the number of photons emitted over the number of photons absorbed, that is the fraction of excited stated that undergo fluorescent decay.

${\mathrm{\Phi }}_F=\frac{k_R}{k_R+k_{IC}+k_{ISC}}=\frac{k_R}{k_0}=\frac{{\tau }_f}{{\tau }_R}$

Where τ_R is the natural radiative lifetime, and τ_f is the experimental decay time. If radiative decay were the only pathway, τ_R = τ_f. On a plot of emission decay, we take the τ_f to be the time taken for approximately half the excited state to decay.

Singlets vs. Triplets

According to Hund's rule, for two unpaired electrons, the lowest energy state is when the two spins are parallel. This means that the S₁ state is higher in energy than the T₁ state, and both are higher in energy than S₀.

Rates for Non-Radiative Transitions

Fermi's golden rule is $k=\frac{2\pi }{h}FC\cdot V^2$.

Internal conversion and intersystem crossing derive from molecular interactions that the Born-Oppenheimer approximation assumes to be zero. IC derives from the vibronic coupling of the electronic wavefunction to nuclear vibrations, V_vib. ISC derives from spin-orbit coupling V_SO. To refresh, IC is the conversion of electronic energy to vibrational excitation, with no change in electron spin. ISC is conversion of electronic energy into hot vibrational excitation accompanied by change of electron spin. Vibrational relaxation is dissipation of 'hot' vibrational energy into surroundings, which is fast relative to IC and ISC.

Phosphorescence is emission during relaxation from a triplet to a singlet state. The lifetime of phosphorescence is much greater than that of fluorescence, owing to the forbidden nature of the relaxation. As T₁ is lower in energy than S₁, this emission is red-shifted from fluorescence.

ISC is enhanced by a large FC factor (large, flexible molecules, lots of polarity, small energy gaps) and a large spin/orbit coupling (heavy atoms, high charge). Large spin-orbit coupling also results in more phosphorescence (makes sense, as phosphorescence is dependent on ISC).

IC is also known as non-radiative decay. It results in electronic energy being converted into vibrational energy. Like in ISC, it is fastest in flexible molecules with lots of polarity and small energy gaps. It's typically fast from higher excited states to the first excited state, and it prevents significant emissions from higher electronic excited states.

Because k_ISC is spin forbidden, it slower than k_IC for comparable energy gaps. It's much faster for S₁→T₁ than for T_{1</sub→S0, owing to the smaller energy gaps involved.}

Section 4

Emission Quenching

This is photochemistry involving a single molecule or complex, so it isn't reliant on diffusional collision to occur. Photodissociation, photoisomerisation, intramolecular energy and electron transfer are all examples of this. It's a first order reaction pathway. The photochemistry can be observed as either a loss of fluorescence intensity or as an acceleration of decay (lower φ_f' or τ_expt). We can obtain the Φ_PC (quantum yield of photochemistry) and the k_PC (rate constant of photochemistry).

${{\mathrm{\Psi }}_{P^{\prime }C}}^{\prime }=\frac{k_{PC}}{k_0+k_{PC}}$

$k_{PC}=\frac{1}{{\tau }_{expt}}-\frac{1}{{\tau }_f}$

Bimolecular Kinetics

Bimolecular reactions require the collision of molecules, so they have a second order rate law, $Rate=k_Q[M*][Q]$, but since [Q]>>[M*] (typically), the reaction is pseudo first order, and $k_{e}ff=k_Q[Q{]}_0$. In the gas phase, we used collisional theory:

$k_Q\le k_{collision}\approx {\sigma }_c^{2}oll(\frac{8k_{B}T}{\pi \mu }{)}^{\frac{1}{2}}{e}^{\frac{-E_a}{k_{B}T}}$

And in the solution phase we use this:

$k_Q\le k_{diff}\approx \frac{8RT}{3\eta }$

Where η is solvent viscosity. The reason why k_Q is less than k_diff is that not every collision results in a reaction (due to energy barriers). Bimolecular reactions often originate from triplet states. The long relaxation time of triplet states means that they exist on a longer timescale than it takes for collisions to occur, meaning that collisions are the rate determining step.

In a bimolecular reaction, by plotting [Q] against Φ_F/Φ_F' (that is, quantum yield over quantum yield of photochemistry, we can obtain the lifetime and the rate constant.

${\mathrm{\Phi }}_F=\frac{k_r}{k_0}$ ${{\mathrm{\Phi }}_F}^{\prime }=\frac{k_r}{k_0+k_Q[Q]}$ $\frac{{\mathrm{\Phi }}_F}{{{\mathrm{\Phi }}_f}^{\prime }}=\frac{k_0+k_Q[Q]}{k_0}=1+\frac{k_Q}{k_0}[Q]=1+\frac{1}{k_0}{k}_Q[Q]=1+{\tau }_0{k}_Q[Q]$

Singlet vs. Triplet Photochemistry

Singlet state photochemistry requires chemistry on a 1-10ns timescale. This means that an efficient reaction requires high concentrations, because otherwise diffusion isn't fast enough for a reaction to occur. This can also work when the quencher is held next to the excited molecule, such as in supramolecular structures, or in a pigment or protein complex.

Triplet state photochemistry can happen on a much longer timescale: the lifetime of a triplet it typically >1ms. The downsides are that it requires ISC, so a lower quantum yield, that the triplet energy state is lower, and that reaction mechanisms may be spin sensitive, complicating matters.

Energy and Electron Transfer

Energy transfer has a trivial mechanism: an excited state emits light, which is then absorbed by another molecule to produce its own excited state. There are three distinct steps, and there is no encounter or really interaction between the excited state and the receiver molecule. It requires the spectral overlap of donor emission and acceptor absorption. Also required are strong absorption and emission dipoles, meaning this is limited to singlet/singlet transfer. The probability of this occuring depends on the quantum yield of emission by D*, the number of A molecules in the path of emitted light, the light absorbing ability of A, and the overlap in the spectrum of A and D*. Also important is the extinction coefficient of A at the overlapping wavelength.

Energy transfer by non-radiative mechanisms is also possible. This can be a Coulombic interaction (Forster), with the dipole in one molecule inducing a dipole in the other, or an electron exchange mechanism (Dexter), which is pretty self-explanatory. A Forster transfer is based on the Fermi Golden Rule, and can be considered analogous to VdW interactions.

$V_{DD}\propto \frac{{\mu }_s{\mu }_Q}{R^3}$

Where V is the electrostatic interaction energy, R is the distance between the two, and μs are the dipoles of the molecules. The Franck Condon factor governing this kind of transfer is related to the spectral overlap integral: that is the integral of the overlap in the spectra of the two molecules. $FC\propto J=\int I_S{ϵ}_{Q}dv$ where I is the intensity of the emitted light, and ε is the extinction coefficient of the acceptor. In total, we have:

$k_{entr}\propto V_{DD}^2\cdot FC\propto \frac{1}{R_{DA}^6}\int I_S{ϵ}_{Q}dv$

This can be simplified to:

$k_{entr}=\frac{1}{\tau }({\frac{R_0}{R_{DA}}}^6$

R₀ includes J, proportionality factor, etc. The energy transfer quantum efficiency is given by:

${\varphi }_{entr}=\frac{R_0^6}{R_0^6+R_{DA}^6}$

R₀ is defined such that when k_entr = k₀, R₀ = R_DA. Good (high) values for R₀ are 1-10nm.

Dexter energy transfer is based of coupled HOMO and LUMO electron exchange. The electron clouds of the donor and acceptor must overlap in space, and so the exchange occurs in the region of the overlap. As a result of this, it relies much more strongly on distance than the Forster transfer. The distance must be within 0.5nm, but there are no spin requirements. It is also based on the Fermi Golden Rule.

Electron transfer is also a Fermi Golden Rule reaction. V term has the distance dependence, and FC the isoenergetic dependence of the reaction. As we see in 3P3, $V_{eltr}=exp(-\beta r)$, where β is the tunnelling coefficient, the measure of the electrons ability to tunnel through the medium. It's strongly distance dependent. The FC factor is dependent on the 'reorganisation energy', the energy change required for state 1 to take up the eq. geometry of state 2. $FC\propto exp(\frac{\mathrm{\Delta }{G}^{‡}}{k_{B}T}=exp(\frac{-(\mathrm{\Delta }{G}^0+\lambda {)}^2}{4\lambda k_{B}T}$