Zero-order vibrational levels

Thus, there is a condition relating triplets of beat frequencies (i.e., cotJ, coJK, ai,K) in Eq. (3.12). One can show that each beat frequency in Iy(t) is a member of (N — 2) such triplets, and that present in Iy(t) are N(N — 1 )(N — 2)/6 distinct triplets obeying Eq. (3.15). Finally, one notes that y can take on any one of N values, corresponding to each of the zero-order vibrational levels. 

In this equation we have chosen to consider a near prolate symmetric top (like anthracene). The symbols J and Ka have their usual meanings as the rotational quantum numbers of such a molecule and refer in the equation to rotational levels in the manifold of the a> zero-order vibrational level. As for the other factors in Eq. (3.36), T is the rotational temperature of the sample, W(J, Ka, T) is a weighting factor for each ro-vibrational level a,J,Ka, and Iy(J,Ka,t) is the y-type fluorescence decay which arises from the coupling of a,J, Ka) with the same rotational levels of the other zero-order vibrational states. Iy(J, Ka, t) is just given by Iy(t) of Eq. (3.12). [Note that in deriving Eq. (3.36) we have neglected the possibility of any coherence effects arising from the coherent preparation of rotational levels within the same vibrational state. This possibility is the subject of Section III D 2.]... 

The combination of low-resolution and spectral unzipping into noninteracting polyads enables systematic, model-free surveys of deperturbed Franck-Condon factors, deperturbed zero-order energy levels, and trends in intramolecular vibrational redistribution (IVR) rates and pathways [3]. The H[ res,/i polyad model permits extraction of the most important resonance strengths directly from fits to a few polyads [6-8]. Once these anharmonic... 

Their system consists of a dilute crystal such that solute-solute interactions are negligible. Interactions between solvent and solute are important only insofar as the lattice vibrations are coupled with the zero-order nonstationary levels of the final state. The initial state is considered to be an equilibrated vibronic state, which implies v = 0 state for temperatures around 77°K. 

By monitoring excitation spectra with a time-resolved detection of the emission, briefly called time-resolved excitation spectroscopy , it is possible, to identify specific relaxation paths. Although, these occur on a ps time scale, only measurements with a ps time resolution are required. It is shown that the relaxation from an excited vibrational state of an individual triplet sublevel takes place by a fast process of intra-system relaxation (on the order of 1 ps) within the same potential surface to its zero-point vibrational level. Only subsequently, a relatively slow crossing to a different sublevel is possible. This latter process is determined by the slow spin-lattice relaxation. A crossing at the energy of an excited vibrational/phonon level from this potential hypersurface to the one of a different substate does not occur (Fig. 24, Ref. [60]). This method of time-resolved excitation spectroscopy, applied for the first time to transition metal complexes, can also be utilized to resolve spectrally overlapping excited state vibrational satellites and to assign these to their triplet substates. 

To calculate Iy(t) from Eq. (3.36) it is very convenient to make several approximations. The first has been mentioned already the molecule is taken to be an approximate symmetric top. Thus, the rotational energy of rotational level JK in the manifold of the zero-order vibrational state y> is58... 

The population probabilities Pn t) defined in Eqs. (8)-(13) should not be confused with the population probabilities which have been considered in the extensive earlier literature on radiationless transitions in polyatomic molecules, see Refs. 28 and 29 for reviews. There the population of a single bright (i.e. optically accessible from the electronic ground state) zero-order Born-Oppenheimer (BO) level is considered. Here, in contrast, we define the electronic population as the sum of all vibrational level populations within a given (diabatic or adiabatic) electronic state. These different definitions are adapted to different regimes of time scales of the system dynamics. If nonadiabatic interactions are relatively weak, and radiationless transitions relatively slow, the concept of zero-order BO levels is useful the populations of these levels can be prepared and probed using suitable laser pulses (typically of nanosecond duration). If the nonadiabatic transitions occur on femtosecond time scales, the preparation of individual zero-order BO levels is no longer possible. The total population of an electronic state then becomes the appropriate concept for the interpretation of time-resolved experiments. ° ... 

Equation 9.24 looks essentially the same as Eq. 9.4 for the rotational energy of a linear molecule, but the non-linearity of the molecule causes the degeneracy of a particular value of / to be higher. A given J value can correspond to any of a number of possible combinations of rotations about the a and c axes. Their spectra turn out to be more complicated than implied by this zero-order energy level expression, because molecular vibrations partially break the rotational state degeneracy. 

We now discuss the lifetime of an excited electronic state of a molecule. To simplify the discussion we will consider a molecule in a high-pressure gas or in solution where vibrational relaxation occurs rapidly, we will assume that the molecule is in the lowest vibrational level of the upper electronic state, level uO, and we will fiirther assume that we need only consider the zero-order tenn of equation (BE 1.7). A number of radiative transitions are possible, ending on the various vibrational levels a of the lower state, usually the ground state. The total rate constant for radiative decay, which we will call, is the sum of the rate constants,... 

Fig. 29. Group-theoretical predictions of the polarizations of the vibronic transitions, allowed to second order, from the individual zero-field levels of the lowest triplet state of 2,3-dichIoro-quinoxaline to vibrational levels of the ground electronic state. Solid line transitions gain intensity by spin-orbit mixing between states which differ in the electronic type of one electron e.g., S n and T . The dashed line transitions require the mixing to occur between states of the same electronic type (e.g., S and T n ) and is expected to be weaker. The dash-dotted transition could involve the favorable mixing between states that differ in the electronic type of one electron, but a spin-vibronic perturbation is needed. (From Tinti and El-Sayed, Ref. ))...

The excited vibrational states can be considered as quasi-eigenstates [41]. As can be seen in the simplified scheme of Figure 2.2, these states are a result of the relatively strong coupling between a zero-order bright state (ZOBS), namely i >, with several zero-order dark states (ZODS), l > [48], that are further weakly coupled to the bath states that include a dense manifold of nearly equally coupled levels with a finite decay rate. 

The rate is smallest for 1B2u,v vibrational levels far below the zero-order crossing ... 

The rate is slower for a deuterated species in a vibrational level 1B2u,v not near the zero-order crossing than for a nondeuterated species in a vibrational level 1B2u,v of the same energy. 

Freed and Jortner226 have reworked the formal theory of radiationless transitions described in this paper. They carefully account for the difference between distinguishable and indistinguishable levels, and allow for variable coupling of the sparse system to the dense system of states. Of course, only certain vibrational modes in the dense manifold have the appropriate symmetries to couple to the sparse manifold and thereby contribute to the radiationless transition. Freed and Jortner take this into account in the fashion in which the zero-order manifolds of the molecule are classified. 

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