Photochemistry_of_Organic

2

A Crash Course in Photophysics and a ClassiÞcation of Primary Photoreactions

2.1 Photophysical Processes

An authoritative compilation of deÞnitions and concepts is available in the Glossary of Terms Used in Photochemistry, 3rd edition,22 which can be downloaded from the IUPAC webpages or from those of the photochemical societies (http://pages.unibas.ch/epa or http://www.i-aps.org/).

2.1.1State Diagrams

The expression ÔJabłonski diagramÕa,25 is commonly used to designate state diagrams of the kind depicted in Figure 2.1. The French community prefers to name such diagrams after J. Perrin.26 Actually, several authors had used similar state or term diagrams to describe the long-lived emission observed from irradiated solid solutions, but none prior to Lewis, Lipkin and Magel in 194127 had attributed the metastable species to a triplet state of the dissolved organic molecules. Firm evidence that the emitting species were normal organic molecules in a triplet state was soon provided by Terenin in 194328 and by Lewis and Kasha in 1944.23 Irrefutable proof identifying the triplet state of an organic molecule was Þnally obtained in 1958 by Hutchison and Mangum,29 who reported ESR spectra of naphthalene held in a small number of known orientations in a durene crystal.

a Excerpts from a personal communication from Professor M. Kasha, Florida State University, to J. W., 25 July 2008: ÔAlexander Jabłonski intended that his diagram applies to dye molecules. In those the lowest S1ÐT1 separation is relatively small (generally about 1000Ð2000 cm-1). Jabłonski gave a neat kinetic analysis which indicated that thermal excitation from the metastable excited (T) state to the nearby excited (S) state was achievable in general. Note that the T-state designation for the metastable excited state was the result of the 1944 Lewis and Kasha study.23 Jabłonski did not accept this interpretation. A paper I presented at the memorial symposium in Torun, Poland, in honor of Jabłonski,24 was commissioned by the Polish Physical Society and was much admired by the Polish physicists. At the meeting I was informed that in his last year or so Jabłonski Þnally accepted the idea that the ÒmetastableÓ state was really the lowest excited triplet state of the dye.Õ

Photochemistry of Organic Compounds: From Concepts to Practice Petr Klán and Jakob Wirz © 2009 P. Klán and J. Wirz. ISBN: 978-1-405-19088-6

Figure 2.1 State diagram (commonly called Jabłonski diagram) depicting molecular states and photophysical processes. The vertical position of the thick horizontal lines represents the electronic energy minimum. Vibrational energy levels are shown as thin lines. The width of the horizontal lines and their position along the abscissa are chosen merely to avoid congestion in the graphical diagram and have no physical connotation

Molecularb electronic states are represented by thick horizontal lines (—) that are arranged in vertical order to indicate relative energies and are labelled consecutively by increasing energy, beginning with the singlet ground state S0, followed by excited singlet states S1, S2, etc. Vibrational states, represented here by thin lines (Ñ), are usually not shown explicitly. When they are shown as in Figure 2.1, the lowest thin line of each electronic state represents the zero-point energy level (y ¼ 0) of that state. States of a given multiplicity are collected in separate columns.c For molecules with a singlet ground state, the left-hand column collects the electronic singlet states and that to the right the triplet states. The lowest triplet state is labelled T1, followed by T2, T3, and so on.

Photophysical processes are radiative or radiationless transitions by which molecules are promoted from one electronic state to another. No structural change results, although the bond lengths and angles generally differ somewhat in different electronic states. Radiative transitions are associated with the absorption or emission of a photon and are represented as straight arrows (" or #), while radiationless transitions are not associated

bWe use the term molecule to comprise all (meta)stable molecular species including ions and radicals.

cHorizontal displacement does not indicate a change in structure. The energy levels represent minima on the potential energy surfaces of a given electronic state. The corresponding structures will be somewhat different for each state. Such a representation is not meaningful if an excited state has no energy minimum near the structure of the ground state.

Þrst-order rate law. When several processes contribute to the decay of a large number of molecules in an excited state M that was generated by absorption of a short light pulse at time t ¼ 0, the decay of the initial concentration cM (0) will be given by Equation 2.2:

cM*ðtÞ ¼ cM*ð0Þe ðSkÞt

Equation 2.2 First-order rate law

The lifetime of M , t(M ), is deÞned as the time in which the initial concentration cM (0) is reduced to cM (0)e 1 0.368cM (0), that is, t(M ) ¼ 1/Sk, where Sk is the sum of the rate constants of all processes contributing to the decay of M . More detailed rules for estimating the rate constants of photophysical processes for individual molecules will be given in the following sections.

The difference between photophysical and photochemical processes is not always clear cut. For example, the E–Z isomerization of alkenes represents a photochemical process by the above deÞnition, yet the same process might be seen as photophysical in terminal alkenes, where geometric isomers are indistinguishable in the absence of isotopic labels.

The term luminescence denotes the spontaneous emission of radiation from an electronically excited species and comprises ßuorescence and phosphorescence, and also chemiluminescence (Section 5.6).

2.1.2Beer–Lambert Law

The absorbance A(l) of a sample is deÞned by Equation 2.3:g,10,22

AðlÞ ¼ logðP0l=Ptrl Þ ¼ log½TðlÞ&; TðlÞ ¼ 10 AðlÞ

Equation 2.3

where P0l and Ptrl are the incident and transmitted spectral radiant power, respectively, and TðlÞ ¼ Ptrl =P0l is the internal transmittance of the sample at the given wavelength l. For

g Radiant power, P, is the radiant energy, Q, emitted, transferred or received at all wavelengths per unit time, P ¼ dQ/dt. If the radiant energy Q is constant over the time interval considered, P ¼ Q/t. The SI unit of radiant power is watt, [P] ¼ W. The spectral radiant power is a differential, Pl ¼ dP(l)/dl, representing the radiant power P per wavelength interval. Spectral radiant power

expressed in wavenumbers, Pn~ ¼ dPðn~Þ=dn~, is equal to Pl/n~2, because |dl/dn~| ¼ 1/n~2. The factor 1/n~2 cancels in the ratio T(l) ¼ Pl0/Pltr ¼ Pn~ 0/Pn~tr. The units of spectral radiant power are [Pl] ¼ W m 1 and [Pn~ ] ¼ W m. The intensity or irradiance,

symbol I, is the radiant power received at a surface, [I] ¼ W m 2. The ratios P0/Ptr and I0/Itr are equal. The expression ÔintensityÕ has traditionally been used indiscriminately for photon ßux, irradiance or radiant power. When considering an object exposed to radiation, it should therefore be used only for qualitative descriptions.

The frequency distribution rn ¼ dr/dn of a black-body radiator is given by Planck s law (Equation 2.6):

8phn~3

rn ¼ ehn=kBT 1

Equation 2.6 Planck distribution of a black-body radiator

where T is the temperature of the black body.

Once the gas has reached equilibrium with the radiation Þeld, the populations of its energy levels m and n, Nm and Nn, must be constant, dNm/dt ¼ dNn/dt ¼ 0. With the Einstein coefÞcients deÞned in Figure 2.3 one obtains Equation 2.7:

dNm=dt ¼ BnmNnrn Nm½Amn þ Bmnrn& ¼ 0

Equation 2.7

Solving for rn yields Equation 2.8:

Amn=Bnm

rn ¼ Nn=Nm Bmn=Bnm

Equation 2.8

The ratio of equilibrium populations Nn/Nm in the denominator of Equation 2.8 must obey Boltzmann s equation (Equation 2.9):

Nn=Nm ¼ eDE=kBT ¼ ehn=kBT

Equation 2.9 Boltzmann s law

where DE ¼ hn is the energy difference between two molecular states m and n. Equation 2.8 has the same form as Equation 2.6 if and only if Bmn ¼ Bnm ¼ B, i.e.

Bmn/Bnm ¼ 1. Combining Equation 2.8 with Equation 2.6, we obtain the Þnal relation Equation 2.10, where we have dropped the now superßuous indices m and n:

A ¼ 8phn~3B

Equation 2.10

The probability of stimulated emission by an excited molecule, Brn, is the same as that of the reverse process, absorption by the ground-state molecule. This is a consequence of the law of microscopic reversibility: if the number of excited molecules Nm is equal to the number of ground-state molecules Nn, then the rates of stimulated absorption and emission must be equal. If by some means a population inversion can be produced, Nm > Nn, the net effect of interaction with electromagnetic radiation of frequency n will be stimulated emission (Figure 2.4). This is the operating principle of the laser. LASER is an acronym for light ampliÞcation by stimulated emission of radiation (Section 3.1).

Polyatomic molecules have broad absorption and emission bands due to the contributions of vibronic transitions. Einstein s equation (Equation 2.10) relates the coefÞcients of absorption and emission in a two-level system. Several extensions of

Figure 2.4 Net effects of interaction with electromagnetic radiation

Equation 2.10, which permit the determination of ßuorescence rate constants kf from the broad absorption spectra of molecules, have been derived. The best known is that of Strickler and Berg.31 A simpler version that is easier to use (Equation 2.11)32 is sufÞcient for most practical purposes:

absorption band

Equation 2.11 Determination of fluorescence rate constants from absorption spectra

where n is the refractive index of the solvent, « is the molar absorption coefficient, and n~max is the wavenumber of the absorption band s maximum. However, overlap of absorption bands frequently hampers integration over a single absorption band. It may be helpful in this respect to assume a mirror-image relationship (Section 2.1.9) between the Þrst absorption band and ßuorescence. In general, ßuorescence rate constants kf determined from lifetime measurements (Equation 3.33) are in satisfactory agreement ( 20%) with those calculated from the absorption spectrum (Equation 2.11). Even quick- and-dirty estimates will give the right order of magnitude for kf, which may be quite useful. In some cases, however, experimental values of kf (determined from kf ¼ Ff/tf; see Section 3.9.7, Equation 3.33) are found to be much smaller than those calculated by Equation 2.11. This indicates that the electronic transition to the lowest excited state S1 may be hidden under an absorption band of stronger intensity. Such an observation led to the important identiÞcation of a ÔforbiddenÕ state in linear polyenes (see Section 4.7).

2.1.4Dipole and Transition Moments, Selection Rules

An electric dipole can be represented by two equal but opposite charges, þ q and q, that are held at a distance r. The dipole moment m is a vector pointing from the negative to the positive charge.h The magnitude of m, m ¼ qr, is commonly measured in the non-SI unit debye, 1 D ¼ 3.336 10 30 C m. Thus, for an electron held at a distance of 100 pm from a proton, m ¼ 4.8 D. In quantum mechanics, the dipole moment m of a molecule in the ground state is obtained from the ground-state wavefunction C0 as the expectation value of

h Confusingly, the chemical symbol for a dative bond as in H3N ! BH3 indicates the direction of electron donation and, accordingly, the dipole moment vector points from þ towards in many chemistry textbooks. However, IUPAC deÞnes the direction of the dipole moment as pointing from the negative to the positive charge.

n (υ = 0)

Figure 2.5 The resonance condition for absorption of light with frequency n

the dipole moment operator (Equation 2.12). The vectors ri ¼ (xi, yi, zi) deÞne the positions of the electrons i, rm ¼ (xm, ym, zm) those of the nuclei m, Zm are the charges of the nuclei, and e is the elementary charge. The three components of m, mx, my and mz, are obtained by inserting the respective components of the vectors r.

Equation 2.12

For a quantum mechanical treatment of light absorption, the dual nature of light must be taken into account. Light can be absorbed only if the excitation energy matches the photon energy, DE ¼ hn that is, if the absorption of a photon by the molecule leads to one of the discrete excited states of the molecule (Figure 2.5).

When the Bohr resonance condition is met, the probability of light absorption may be calculated from the interaction of the electric vector of the light wave of frequency n with the electrons and nuclei in the molecule. Molecules being much smaller than the wavelengths of UVÐVIS radiation (Figure 1.6), the electromagnetic radiation Þeld acting on a molecule is treated as a linear electric Þeld and magnetic interactions are neglected. We assume that the molecule is initially in the lowest vibrational level (y ¼ 0) of the electronic ground state n and reaches some vibrational level of the excited state m, for example, y0 ¼ 2 as shown in Figure 2.5. Conversely, for emission, one would assume that y0 ¼ 0 and y could take any value. The transition moment for a single vibronic absorption band Mn,0 ! m,y0, is then given by Equation 2.13.

n;0 ! m;y0 e Cn;0 ^ Cm;y0

M ¼ h jMj i

Equation 2.13 Transition moment

Like the dipole moment m, the vector Mn,0 ! m,y0 has a magnitude, measured in debye, and a direction relative to the Cartesian coordinate system of the molecule. Computer programs make use of Equations 2.12 and Equation 2.13 to calculate dipole and transition moments, respectively, from the appropriate wavefunctions Cn,0 and Cm,y0. The comparison of calculated and experimental (Section 3.6) transition moments is an essential test to validate simpliÞed models of electronic excitation (Section 4.4). Note that the dipole and transition moments of a molecule are not related; in fact, they may well be orthogonal to each other.

To calculate the transition moment Mn,0 ! m,y0, we use the approximate wavefunctions that are factored into the product of an electronic function Cel, a vibrational wavefunction

Equation 2.14

The spin functions are orthonormal: The integral hsn|smi equals unity if sn ¼ sm and it vanishes otherwise, so that the transition moment vanishes for singletÐtriplet and tripletÐsinglet transitions (or doubletÐquartet transitions in radicals). This is our Þrst selection rule: Electronic transitions between states of different multiplicity are forbidden

(the transition moment is zero). Note that selection rules, of which we will encounter several more in Chapter 4, do not hold strictly; they are broken like all rules in real life. They hold only within the limited view of some approximations, in the present case the neglect of magnetic interactions. The multiplicity selection rule is a strong one in the absence of heavy atoms, because magnetic interactions in most organic molecules are very weak. As we have seen in Section 2.1.1, absorption coefÞcients of singletÐtriplet transitions and rate constants of phosphorescence, which are directly related to the absorption coefÞcients (see Special Topic 2.1 in Section 2.1.3), are orders of magnitude smaller than those for the corresponding spin-allowed transitions, singletÐsinglet absorption and ßuorescence.

Using the deÞnition of ^ (Equation 2.12), the Þrst integral of Equation 2.14 can be

M

expanded as follows:

Note the factorization of the two terms; the vibrational wavefunctions x can be separated because they do not depend on the electronic positions ri and the electronic wavefunctions can be separated if we disregard their parametric dependence on the precise nuclear positions.i The second term then vanishes due to the orthogonality of the electronic wavefunctions Cel,n and Cel,m, so that the transition dipole moment of a spin-allowed transition becomes

Mn;0 ! m;y0 ¼ ehCel;njSrijCel;mihx0jxy0 i ¼ Mel;n ! mhx0jxy0 i

Equation 2.16

Equation 2.16 represents the transition moment for a single vibronic transition from the zeroth vibrational level of the ground state to a single vibrational level y0 of the electronically excited state. The vibrational overlap integrals hx0jxy0i are called Franck–Condon integrals; they will generally be different from zero, because the two vibrational wavefunctions refer to different electronic potential energy surfaces (vibrational wavefunctions calculated from a single harmonic potential are orthonormal). The transition moment of a single vibronic band is thus determined by the product of two factors, the electronic transition moment (Equation 2.17) and the FranckÐCondon integral

i This approximation must be dropped for forbidden electronic transitions, i.e. when hCel,njSrijCel,mi ¼ 0 (symmetry-forbidden transitions, Section 4.4).