The optical transitions between the coupled vibrational—electronic levels of a
molecule can be understood by invoking the Franck-Condon principle. This
says that the electronic transitions take place so rapidly that the nuclei do not move significantly during the transition. The Franck—Condon principle follows from the Born—Oppenheimer approximation, and is a consequence of the fact that electrons are much lighter than the nuclei.
The steps that take place when photons are absorbed and re-emitted by a molecule according to the Franck—Condon principle are illustrated schemati- cally in Fig. 8.6. The molecule starts in the ground state with a mean nuclear separation of r;. The absorption of the photon promotes an electron to the excited state without altering r. The transition thus leaves the molecule in the excited state with a mean nuclear separation of r; instead of the equilibrium separation r2. The separation of the nuclei rapidly relaxes to rz before re- emitting a photon. This leaves the molecule in the ground state with a mean nuclear separation of r2. Further rapid relaxation processes occur to complete the cycle and bring the molecule back to its equilibrium separation in the ground state. These four steps correspond to the four processes indicated in the simplified energy level diagram shown in Fig. 8.4.
The ‘rapid relaxation’ processes that accompany the optical transitions need some clarification. If we think of the nuclear vibrations as analogous to those of a spring, we can see that the transition leaves the molecular spring in a compressed or extended stationary state at time f = 0. We know that in this
8.3. Optical spectra of molecules 173
relaxation
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absorption UOISSIUIo
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ry J2
Fig. 8.6 Schematic representation of the pro- cesses that occur during the absorption and emission of photons by vibronic transitions in a molecule. The Franck—Condon principle states that the nuclei do not move during the optical transitions. r; and r2 are the equilibrium separations of the nuclei in the ground and excited states respectively. One of the atoms has a larger radius in the excited molecule because the atom itself is in an excited state.
174 Molecular materials
Fig. 8.7 Configuration diagram for two elec- tronic states in a molecule. Vibrational—
electronic transitions are indicated by the vertical arrows, together with a schematic representation of the absorption and emission spectrum. The probability amplitudes for the relevant vibrational levels for the absorption transition are shown in Fig. 8.8.
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excited
State
emission
ground
State
situation the spring will immediately begin to vibrate at its own natural fre- quency fort > 0. This is equivalent to instigating the oscillations of a spe- cific vibrational mode in one particular molecule. However, the molecule may have other vibrational modes, and it can also interact with the other molecules that surround it. The relaxation of the vibrational energy created during the transition thus involves the spreading of the localized vibrational energy of a particular molecule to the other modes of the molecule and to the other molecules. This is a more technical way of saying that the excess energy ends up as heat. The vibrational relaxation typically occurs in less than 1 ps in a solid, which is much faster than the ~ 1 ns taken to re-emit a photon.
The Franck—Condon principle implies that we represent the optical transi- tions by vertical arrows in configuration diagrams, as shown in Fig. 8.5. The absorption of a photon puts the molecule in an excited vibrational state as well as an excited electronic state. The excess vibrational energy is lost very rapidly through non-radiative relaxation processes, as indicated by the dotted lines in Fig. 8.5. The frequencies of the photons absorbed and emitted are given by eqns 8.3 and 8.5. These describe a series of sharp lines with equal energy spacing.
In more complicated molecules with many degrees of freedom, the vibra- tional motion is described in terms of the normal modes of the coupled system.
These vibrational modes are usually represented by a generalized coordinate Q, which has the dimensions of length. The Born—Oppenheimer approxima- tion allows us to produce configuration diagrams in which we plot the elec- tronic energy as a function of Q. Figure 8.7 is an example of such a configura- tion diagram. In general, the ground state and excited state have approximately parabolic minima at different values of the configuration coordinate. The op- tical transitions are indicated by vertical arrows, as prescribed by the Franck—
Condon principle. The absorption and emission spectra consist of a series of lines with frequencies given by eqns 8.3 and 8.5, as shown in the right-hand side of the figure.
The relative intensities of the manifold of vibronic transitions can be calcu- lated in the Franck—Condon approximation. The matrix element for an electric-
dipole transition from an initial state WY; to a final state 2 is given by (c.f.
eqn B.25):
Miz = (ler €oll) = Í dã -:: | đến Weer EW), (8.7) where r is the position vector of the electron, € is the electric field of the light wave, and &;, --- , €y represent the coordinates for all the relevant internal degrees of freedom of the molecule. For the coupled vibrational—electronic states that we are considering here, the total wave function will be a product of
an electronic wave function that depends only on the electron coordinate r, and
a vibrational wave function that depends only on the configuration coordinate Q. We thus write the vibronic wave function for an electronic state 7 and a vibrational level n as:
Win(r, QO) = Wilt) @n(Q — Qo). (8.8)
The vibrational wave function ứn(C — Qo) is just the wave function of a simple harmonic oscillator centred at Qo, the equilibrium configuration for the ith electronic state. On inserting the vibronic wave functions from eqn 8.8 into the matrix element given in eqn 8.7, we find:
Mịy % | | 0] 00ứj;(ể — ỉ0)x Wi@)en(O- Oo)d*rAQ, (89
where we have arbitrarily taken the light to be polarized along the x axis. Note that the equilibrium positions are different because they correspond to different electronic states. The matrix element can be separated into two parts:
Mi x | vÿœ)xwa@ ar x xc ~ Q})~ny(Q — Qo) dO. (8.10)
The first factor is the usual electric-dipole moment for the electronic transition, which we are assuming to be non-zero. The second is the overlap of the initial
and final vibrational wave functions. From Fermi’s golden rule (eqn B.13), we know that the transition rate is proportional to the square of the matrix element.
Hence the intensity of a specific vibronic transition will be proportional to the
Franck-Condon factor:
oo 2
nị,nạ % Í @n„(@ — ểạ) ỉ@n (ỉ — Qo) dQ (8.11)
The Franck—Condon factor basically states that the intensity of the transition is proportional to the overlap of the initial and final vibrational wave functions.
To see how the Franck—Condon factor works in practice, we need to look at the probability amplitudes (i.e. the square of the wave functions) for the vi- brational levels involved in the transition. In the absorption transition shown in Fig. 8.7, the molecule starts in the Ny = 0 level of the ground state, which has its equilibrium position at Qo, and finishes in the Nath level of the excited state, which is centred at Q). The relevant wave functions are shown in Fig. 8.8. We see that that we have a good overlap for several transitions, with the largest
8.3. Optical spectra of molecules 175
In an electronic transition, the x that appears in Mj> is the electron coordinate, because we are specifically considering the interac- tion with the dipole moment of the electron.
The light will, of course, also interact with the dipole moment of the nucleus, but these transitions occur at much lower frequencies in the infrared, and can be ignored in the Franck—Condon approximation.
Harmonic oscillator wave functions peak near the classical turning points at the edge of the potential well as n gets larger. Therefore we can give an approximate rule of thumb that the Franck—Condon factor will be largest for the levels with the edge of their classical potential well close to Qo.
176 Molecular materials
Fig. 8.8 Probability amplitudes for the vibra- tional levels involved in the absorption transi- tion shown in Fig. 8.7. The initial wave func- tion has been shaded. The molecule starts in the n = 0 level of the ground state, and finishes in the Noth level of the excited state.
Qo and Qj are the equilibrium positions for
the ground and excited states respectively.
We do not show spectra for the simplest molecules like Hz here because the first elec- tronic excited state is above the ionization limit of the molecule, and the vibronic lines are not well resolved. Ammonia shows dis- crete vibronic lines because it is a small, rigid molecule with very well defined vibrational frequencies.
Fig. 8.9 Absorption spectrum of ammonia (NH3) in the ultraviolet spectral region. After [1], copyright 1954 American Institute of Physics, reprinted with permission.
Franck—Condon factor for Ng ~ 2. Hence we would expect the intensity to be largest for the No = 2 level, as indicated by the schematic absorption spectrum
shown in Fig. 8.7. |
The reverse argument can be applied to the emission transitions. This leads to the schematic emission spectrum shown on the right of Fig. 8.7. In the sim- ple model, we expect the emission spectrum to be the ‘mirror’ of the absorption spectrum when reflected about the centre frequency fiwp. This is called the
mirror symmetry rule.