Framework of light-matter interactions and the origin of photo-

1.1 The foundations of quantum mechanics

2.1.2 Framework of light-matter interactions and the origin of photo-

The most intuitive place to begin in building the working framework of photochemistry is with one of the electronic properties of a molecule, its (molecular) electric dipole moment, µ. This quantity captures the distributions of charges (electrons and nuclei) within a system as µ=µe+µn =−e n X i ri+e N X j ZjRj, (2.1)

where µe and µn are the electronic and nuclear dipoles due to n electrons at positions

ri and N nuclei at Rj with nuclear charge Zj. respectively. The issue is as with HF,

where are the electrons? Since they are not localised, a quantum mechanical framework is most appropriate, utilising our previous discussion of wavefunctions and their linear expansions of basis functions. Furthermore, the subsequent calculation of electronic properties are only accurately determined by post-HF methods.

When radiation of a specific wavelength is incident on the molecule, the molecular dipole moment may resonate with the oscillating electromagnetic field of the radiation, triggering the absorption of that quantum of radiation, and promoting an electron to a higher-lying energy orbital; the molecule goes from being in its molecular ground state, Ψi into a electronically excited state, Ψj. This process is usually adequately captured

by treating this ‘light-matter’ interaction as a weak perturbation. The first-order result

via time-dependent perturbation theory is known as Fermi’s golden rule,1,3,111 _which

models the probability (Pj←i) of the transition from state Ψi to Ψj as

Pj←i =

2π

~ | hΨj|µ·ˆ|Ψii |

2_δ₍_E

j−Ei±~ω), (2.2)

for radiation of frequencyω, with electric field in direction ˆ. Immediately it is obvious that if a change of electronic state occurs, it must be accompanied by the absorption (+~ω) or the emission (−~ω) of radiation equal to the energy gap between the two states involved (Ej −Ei), justifying the usual phenomenological interpretation of absorption

and emission.

This can be taken further by first invoking the BO approximation as before; one can write the wavefunction as a product of electronic, vibrational, and spin wavefunctions,

i.e. Ψ =ψeψvψs. Substituting this into Equation 2.2 yields

Pj←i =

2π

~ | hψj,eψj,vψj,s|µ·ˆ|ψi,eψi,vψi,si |

2_δ₍_E

j−Ei±~ω). (2.3) Usingµ=µe+µn, these integrals can be split up within the BO approximation as

Pj←i =

2π

(2.4)

Since electronic wavefunctions are orthogonal, hψj,e|ψi,ei = 0 and the above expression

reduces to

Pj←i =

2π

2_δ₍_E

j −Ei±~ω). (2.5) Putting the conservation of energy term aside for a moment, and concentrating on the three integrals in this expression, this expression highlights the components of the probability amplitude which determines how likely a transition (j → i) is to occur. A summary of the possible photochemical and photophysical transitions a molecule may undergo is given in a Jablonski diagram (Figure 2.2).110,112 _{The origins of these}

processes are discussed below.

hψj,e|µe·ˆ|ψi,ei. This integral forms the so-called ‘orbital-selection rules’.1 These

place restrictions on the change in quantum numbers (l, ml) when an electronic tran-

sition takes place (cf. Section 1.1.4). This integral is non-zero only when there is a parity change between orbitals ((−1)l_{, for orbital angular momentum quantum number}

l). Similarly, orbital angular momentum must be conserved, since a photon (which is a boson) has l = 1, thus a change of state which involves a photon being absorbed or emitted must be accompanied with an electron changing orbital by ∆l =_±1. Finally, a selection rule involving the magnetic orbital angular moment quantum number (ml)

is deduced from noting photons have the intrinsic property of helicity (σh; the projec-

tion of spin onto linear momentum) which is a conserved physical quantity. Photons have σh =±1, which is conserved if a transition involves a change in magnetic orbital

angular momentum of ∆ml= 0,±1.1 Many of these selection rules which are formally

Ground state, S0 ν0 νm ν1 ν2 ν3 ν4 ν5 ν6 ν7 ν8 ν9 . . . . . . . .. .. .. .. .. .. ... ... ... S1 S2 T1 Absorption Fluorescence Phosphorescence IC IC ISC ISC VR VR VR VR VR _VR IC

Internal Conversion (IC), Si −→Sj.

ISC

Intersystem Crossing (ISC), Si −→Tj.

Vibrational Relaxation, (VR).

Figure 2.2 | A summary of the important photophysical processes which occur after a

molecule is photoexcited from its electronic ground state, S0, and vibrational ground state,

ν0, to an excited vibronic state, i.e. excited electronic (Sn;n≥1) and vibrational state (νm;

m≥1). Following this, there are a number of radiative (→) and nonradiative (;) processes a molecule may undergo to dissipate its excess energy. Photodissociation, ionisation, and isomerisation are not shown.

can induce a magnetic dipole transition and in turn open up some of these previously forbidden transitions. Such a discussion on the orbital-selection rule is valid for atoms; molecules on the other hand become more complicated since l is no longer a ‘good’ quantum number. In such cases, composite quantum numbers are instead used to pre- dict allowed transitions, the result being the molecular point groups used throughout chemistry and physics.3

hψj,v|ψi,vi. This integral is known as the Franck-Condon (FC) factor.113–115 It is the

overlap integral of the vibrational modes between the electronic states _|i_i and _|j_i. In other words, the probability of an electronic transition is proportional to the degree of overlap between the initial and final vibrational state wavefunctions, perfect overlap (_hψj,v|ψi,vi = 1) results in the most intense (probable) transitions (Figure 2.3). This

ν

j,0

ν

j,1

ν

j,2

ν

j,3

ν

i,0

ν

i,1

ν

i,2

ν

i,3

i

j

Displacement

Energy

ν

i,0

ν

j,2

ν

j,1

ν

j,0

ν

j,3

hψ

j,v

|ψ

i,v

i

h

ψ

j,v

|

ψ

i,v

i

0

1

2

3 ν

j,n

←

ν

i,0

Figure 2.3 | (left): A pictorial analogue of the FC principle for a given transition from elec-

tronic state |ii to electronic state|ji. (top-right): For two representative anharmonic states, the overlap (_hψj,v|ψi,vi) between the ground vibration stateνi,0and the energetically accessible

vibrational states of _|j_i determines how probable each possible transition is. (bottom-right): The result is an absorption spectrum shown as delta functions. The blue arrow between νi,0

and νj,2 depicts a vertical transition (electronic transition with no change in nuclear geom-

etry), which, given the vibrational wavefunctions of these two states have greatest spatial overlap, this transition is most intense. The FC principle can also be applied to transitions between vibrational modes, where the probability of transitioning between two modes depends on how similar the vibrational wavefunctions are (their overlap) and how similar the nuclear geometry is between the modes. As such, the energy difference between the initial vibronic state (depicted asνj,2) andνj,0 means emitted photons are red-shifted compared to absorbed

photons, known as a Stokes shift.

Figure 2.3 depicts vertical transitions, an electronic transition without an accompanying change in nuclear geometry (i.e. a transition within the BO approximation). These transitions are most probable given the nuclear geometry between initial and final states is the same, hence the corresponding vibrational wavefunctions exhibit a large FC factor.

loss of an excited state may occur, a fundamentally important process in photochemistry. Whilst in an excited vibronic state (both electronically and vibrationally excited state), a molecule may release excess vibrational energy as heat, known as Vibrational Energy Transfer (VET). The molecule may distribute excess vibrational energy to the environment, known an Intermolecular Energy Transfer (IET), which, for example, will decrease its vibrational state by conserving energy through transferring heat to a sol- vent bath. Alternatively, or indeed concurrently, it may redistribute vibrational energy throughout the normal mode space available to it, known as Intramolecular Vibrational Energy Transfer (IVR), noting an N-atom polyatomic molecule has 3N - 6 vibrational modes. Such processes are responsible for the Stokes shift in emissions such as with fluorescence. As depicted in the Jablonski diagram, absorption usually results in an excited vibronic state. Once in the emitting electronic state, VET ensues to occupy the ground vibrational state, where fluorescence takes place. Indeed, this sequence of events is more probable than emitting from an excited vibrational state because of the large FC overlap between vibrational states at some nuclear geometry. The result of this is the (often considerable) red-shift of fluorescence profile for a molecule, relative to its absorption spectrum. In fact, fluorescence generally occurs from the first excited state since higher excited states are closer in energy to one another, making non-adiabatic couplings between electronic states more probable. This is known as Kasha’s rule.116

Nuclear vibrations can also initiate electronic transitions. During vibrations, the nuclear geometry may couple (mix) two vibrational modes between two electronic states. The FC factor once again captures this. An electronic state may change if the vibrational wavefunctions between two energy-degenerate vibrational modes, of the two different electronic states, display significant overlap. The probability of this transition is multiplied by the density of states accessible for this transition. This vibrationally initiated change of electronic state is know as Internal Conversion (IC). Importantly, there is no change in vibrational energy.

hψj,s|ψi,si. This is referred to as the spin-selection rule, being an overlap integral of

spin wavefunctions between the two states involved in a transition. As briefly discussed with Hund’s rules for predicting the ground state electronic configuration of a molecular system (Section 1.1.5), the lowest energy configuration for a stable molecular system will have paired electrons in all orbitals (Figure 2.4). The spin multiplicity, M, is defined as M =P_n2ms,n+ 1; for a molecular system with all paired electrons, the sum

of the spin quantum numbers will be zero, and thus M = 1, as such these are referred to as singlet states, Sn (n ≥ 0), where n = 0 denotes the ground state configuration.

σ2pz

Singlet, S

0 σ∗ 2pz π2py π∗2py π2px π∗2px σ2pz

Singlet, S

1 σ2∗pz π2py π∗ 2py π2px π∗ 2px σ2pz

Triplet, T

1 σ∗ 2pz π2py π∗2py π2px π∗2px

Figure 2.4 | The ground singlet state (S0), the first excited singlet state (S1), and the first

excited triplet state T1electronic configuration for F2 (n= 1 electrons are omitted for clarity,

cf. Figure 1.4).

singlet states for example the first excited singlet state, S1. Triplet states on the other

hand have a set of unpaired spins, the sum of the spins are 1, and thus M = 3 and hence the name triplet states. These are denoted as Tm (m ≥ 1), where the lowest

energy triplet state is T1. A representative example of this is given for F2 (Figure 2.4).

As such, if there is no change in spin multiplicity (e.g. singlet to singlet or triplet to triplet transition) in a transition, _hψj,s|ψi,si = 1. Thus, a transition which results in

a change in spin multiplicity is formally forbidden. In practice, magnetic interactions involving an electron’s spin can lift this restriction making some transitions possible.

The magnetic interactions (couplings) involving electron spin can occur from: (i) coupling to the orbital angular momentum of the electron known as spin-orbit coupling; (ii) coupling to an external magnetic field, such as an applied laboratory field,

may only occur if one of these sources of magnetic interaction is available to conserve magnetic energy. Such a transition is known as Intersystem Crossing (ISC). In terms of organic photochemistry, spin-orbit coupling is by far the strongest (and therefore dominant) source of magnetic interaction, and remains the subject of this discussion. ISC is commonplace in photochemistry, perhaps most notably through the observa- tion of phosphorescence, an analogue of fluorescence which involves a change of spin multiplicity (Figure 2.2).

Since ISC to the zeroth order is forbidden, the probability of ISC compared to other radiationless relaxation processes (IC and VET) is significantly smaller, and thus is often out-competed. However, within a molecular ensemble, there will be a statisti- cal distribution of molecules undergoing any one competitive pathway, based on their probability. In the case of spin-orbit coupling, increasing the magnetic interaction will increase the probability amplitude of ISC, PISC

PISC =hΨj|Hˆso|Ψii, (2.6)

where ˆHso is the spin-orbit coupling operator, which is approximately the product of

the spin and orbital angular momentum dipole moments, µs and µl respectively,

PISC =hΨj|ηsoµsµl|Ψii, (2.7)

where ηso is known as the spin-orbit coupling constant, which is the effective nuclear

charge coupling to the electron spin. This constant is atom-dependent, where heavy atoms (atomic number Z) contribute more greatly to ηso, thus inducing a higher rate

of ISC. For hydrogenic atoms,110,117_η

so ∼Z4, although for molecules this effect will be

smaller given the electron will observe a screened nuclear charge from the rest of the molecule. However, in both cases, atom or molecule, the closer the electron is to the heavy atom (smalll quantum numbers), the larger the spin-orbit interaction. Finally, a transition which involves a change in orbital angular momentum, analogous to fluorescence, but involves a change in spin multiplicity, may occur, known as phosphorescence. Generally, these undergo a greater Stokes shift than with fluorescence since they will occur from the lowest triplet state, which is lower in energy than the first excited singlet state (Hund’s first rule).

The take-home message from this discussion is that an electronic transition initiated by the interaction of its electric dipole moment with radiation (and indeed radiationless transitions) is determined by a series of probabilistic integrals which depend on the electric dipole moment (orbital selection rules), and are proportional to the spatial overlap

Table 2.1: The typical times for the processes discussed thus far.118

Process Time taken / s

Absorption 10−18 _{– 10}−15 Intersystem Crossing 10−14 _{to 10}−8 Vibrational Relaxation 10−12 _{to 10}−10 Internal Conversion 10−11 _{to 10}−9 Fluorescence (S1 → S0) 10−7 to 10−1 Phosphorescence (T1 → S0) 10−6 to 1

of involved wavefunctions (FC factor), and the overlap of spin states (spin-selection rule). Importantly, many of these are zeroth-order rules, which means some formally forbidden transitions are possible under certain higher order interactions. Perhaps most notably the magnetic field of the radiation, and the magnetic properties of electrons have been ignored since they are not included in the perturbation term which was used to derive Equation 2.2. When considered, transitions between formally forbidden states, such as those of different spin multiplicities, are possible.

In general, a particular transition will have a finite probability of occurring; in essence all allowed relaxation mechanisms are in competition with one another where the ‘winning’ transition is that with the greatest probability amplitude (for an ensemble, as previously stated, a distribution of mechanisms will occur drawn from each probability amplitude). Such amplitudes are generally sufficiently well described (with the given addenda) within Fermi’s golden rule (Equation 2.2). On the macroscale, these probabilities are observed as timescales; an unlikely event will take longer to happen (in terms of sampling phase space) and thus have a longer ‘lifetime’ (Table 2.1), the core measurement of ultrafast photochemistry.

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