Vibrational thermal equilibrium

This section briefly introduces the generalized coupled master equation within the Born-Oppenheimer adiabatic (BOA) approximation. In this case, the non-adiabatic processes are treated as the vibronic transitions between the vibronic manifolds. Three types of the rate constant are then introduced to specify the nature of the transitions depending on whether the electronically excited molecular system achieves its vibrational thermal equilibrium or not. The radiationless transitions can occur between two... 

Depending on the method of pumping, the population of may be achieved by — Sq or S2 — Sq absorption processes, labelled 1 and 2 in Figure 9.18, or both. Following either process collisional relaxation to the lower vibrational levels of is rapid by process 3 or 4 for example the vibrational-rotational relaxation of process 3 takes of the order of 10 ps. Following relaxation the distribution among the levels of is that corresponding to thermal equilibrium, that is, there is a Boltzmann population (Equation 2.11). 

Equation (9) was obtained using the assumption that the vibrational subsystem is in the state of thermal equilibrium corresponding to the initial electron state. The expression for the effective frequency a>eff has the form5... 

Minima in Ti are usually above the So hypersurface, but in some cases, below it (ground state triplet species). In the latter case, the photochemical process proper is over once relaxation into the minimum occurs, although under most conditions further ground-state chemistry is bound to follow, e.g., intermolecular reactions of triplet carbene. On the other hand, if the molecule ends up in a minimum in Ti which lies above So, radiative or non-radiative return to So occurs similarly as from a minimum in Si. However, both of these modes of return are slowed down considerably in the Ti ->-So process, because of its spin-forbidden nature, at least in molecules containing light atoms, and there will usually be time for vibrational motions to reach thermal equilibrium. One can therefore not expect funnels in the Ti surface, at least not in light-atom molecules. 

Nitrosobenzene was studied by NMR and UV absorption spectra at low temperature146. Nitrosobenzene crystallizes as its dimer in the cis- and fraws-azodioxy forms, but in dilute solution at room temperature it exists only in the monomeric form. At low temperature (—60 °C), the dilute solutions of the dimers could be obtained because the thermal equilibrium favours the dimer. The only photochemistry observed at < — 60 °C is a very efficient photodissociation of dimer to monomer, that takes place with a quantum yield close to unity even at —170 °C. The rotational state distribution of NO produced by dissociation of nitrosobenzene at 225-nm excitation was studied by resonance-enhanced multiphoton ionization. The possible coupling between the parent bending vibration and the fragment rotation was explored. 

An emission of radiation occurring spontaneously from an electronically or vibrationally excited species that is not in thermal equilibrium with its environment. 

If the equilibrium position of the excited state C is located outside the configurational coordinate curve of the ground state, the excited state intersects the ground state in relaxing from B to C, leading to a nonradiative process. As described above, the shape of an optical absorption or emission spectrum is decided by the Franck-Condon factor and also by the electronic population in the vibrational levels at thermal equilibrium. For the special case where both ground and excited states have the same angular frequency, the absorption probability can by calculated with harmonic oscillator wavefunctions in a relatively simple form ... 

This assignment is supported, indirectly, by the measured activities of these experiments. For example, the activities were measured to be (in M/g-catalyst hour) 0.98 (MS-Hz) 0.61 (MS-Dz) 180 (US-Hz) and 190 (US-Dz). Hence, the 250-fold greater activities of the ultrasound systems is consistent with the expected, more rapid, statistical C-H/D dissociation process as compared to the conventional (e.g., stirred/silent) mediated systems. Additional support for this model arises from a study of gas phase cA-2-butene isomerization to fra/rs-2-butene [15] at 291 K. Here the c O extrapolated trans deuterium number of -0.27 is supportive of C3-H/D elimination predicted by tra/jsition-state theory in this system at thermal equilibrium (e.g., vibrational temperature equal to tra/jslational temperature). 

At high temperatures, vibrational states must also be included in the partition sum above. The nuclear weights are gj for hydrogen we have, for example, gj = 1 for even j, and gj = 3 for odd j. However, we mention that in low-temperature laboratory measurements as well as in astrophysical applications, para-H2 and ortho-H2 abundances may actually differ from the proportions characteristic of thermal equilibrium (Eq. 6.53). In such a case, at any fixed temperature T, one may account for non-equilibrium proportions by assuming gj values so that the ratio go/gi reflects the actual para to ortho abundance ratio. Positive frequencies correspond to absorption, but the spectral function g(co T) is also defined for negative frequencies which correspond to emission. We note that the product V g a> T) actually does not depend on V because of the reciprocal F-dependence of Pt, Eq. 6.52. 

In solution, the excess vibrational energy following an FC transition is lost very quickly — there are indications that only a few picoseconds are needed for the complex to come to thermal equilibrium with the medium with respect to vibrational excitation.19 We speak of the thermally equilibrated excited state, or, as an abbreviation, of the thexi state. Photochemical and photophysical processes very often involve thexi states. 

Chemical reactivity of vibrationally excited molecules may be much higher than for the same species in thermal equilibrium. Although direct IR excitation is limited by the choice of lasers available and usually lends itself predominantly to the excitation of the first few vibrational levels and to IR active transitions,2, 3 no such restriction applies to molecular excitation by E-V-R transfer. 

Moreover, the Gj(t) functions also depend on the temperature, cf. Eq. (15), that has its origin from the Boltzmann factor Eq. (11). For simplification, we will now restrict them to low temperatures, i.e., to a region where vibrational modes of the excited state are essentially unoccupied at thermal equilibrium. From Eq. (11) we see that a vibrational level with a fundamental frequency of, e.g., cOj = 200 cm-1, less than 1% will be populated at T = 60 K. In this case, no accepting modes j = a are occupied in the excited state since they usually correspond to stretching vibrations which have larger energy quanta. Gj(t) then has a more convenient expression [59,63]... 

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