Single vibrational

When one of the cartesian coordinates (i.e. x, y, or z) of a centrosymmetric molecule is inverted through the center of symmetry it is transformed into the negative of itself. On the other hand, a binary product of coordinates (i.e. xx, yy, zz, xz, yz, zx) does not change sign on inversion since each coordinate changes sign separately. Hence for a centrosymmetric molecule every vibration which is infrared active has different symmetry properties with respect to the center of symmetry than does any Raman active mode. Therefore, for a centrosymmetric molecule no single vibration can be active in both the Raman and infrared spectrum. 

Even where the promotion is to a lower vibrational level, one that lies wholly within the 2 curve (such as Vi or V2), the molecule may still cleave. As Figure 7.2 shows, equilibrium distances are greater in excited states than in the ground state. The Franck-Condon principle states that promotion of an electron takes place much faster than a single vibration (the promotion takes... 

When we activated the catalyst system on a large scale, we were unsure of whether the reaction would proceed. The only data for the catalyst activation available to us was in situ IR (React-IR) as shown in Figure 2.3. During activation of the catalyst, a single vibration frequency (-1980 cm"1) of carbon monoxides in Mo(CO)(s became five different frequencies of carbon monoxide in the catalyst solution. This IR data provided us some relief from the risk of running the large scale reaction but did not provide any clues on the structure of the true catalyst. 

The collective modes of vibration of the crystal introduced in the previous paragraph involve all the atoms, and there is no longer a single vibrational frequency, as was the case in the Einstein model. Different modes of vibration have different frequencies, and in general the number of vibrational modes with frequency between v and v + dv are given by... 

A vibronic coupling model for mixed-valence systems has been developed over the last few years (1-5). The model, which is exactly soluble, has been used to calculate intervalence band contours (1, 3, 4, 5), electron transfer rates (4, 5, 6) and Raman spectra (5, 7, 8), and the relation of the model to earlier theoretical work has been discussed in detail (3-5). As formulated to date, the model is "one dimensional (or one-mode). That is, effectively only a single vibrational coordinate is used in discussing the complete ground vibronic manifold of the system. This is a severe limitation which, among other things, prevents an explicit treatment of solvent effects which are... 

To illustrate Equations 5.24a and 5.24b we consider the contribution of a single vibrational mode to VPIE. Comparing CH and CD stretching modes for a typical hydrocarbon at room temperature (300 K) (vCh 3,000 cm-1 in the gas, red shifting 10cm-1 on condensation), we approximate RPFR as... 

Figure 32. Vibronic periodic orbits of a coupled electronic two-state system with a single vibrational mode (Model IVa). All orbits are displayed as a function of the nuclear position x and the electronic population N, where N = Aidia (left) and N = (right), respectively. As a further illustration, the three shortest orbits have been drawn as curves in between the diabatic potentials Vi and V2 (left) as well as in between the corresponding adiabatic potentials Wi and W2 (right). The shaded Gaussians schematically indicate that orbits A and C are responsible for the short-time dynamics following impulsive excitation of V2 at (xo,po) = (3,0), while orbit B and its symmetric partner determine the short-time dynamics after excitation of Vi at (xo,po) = (3, —2.45).

However, for a real crystal at zero temperature it is impossible to group all vibrational motions on the lowest single vibrational mode and, if the crystal behaves as a Debye solid (see later), zero-point energy is expressed as ... 

For simplicity of analysis, we consider just a single vibrational transition, which is characterized by the Raman cross section,... 

This experimental work on the dissociation of excited Nal clearly demonstrated behavior one could describe with the vocabulary and concepts of classical motions.The incoherent ensemble of molecules just before photoexcitation with a femtosecond laser pump pulse was transformed through the excitation into a coherent superposition of states, a wave packet that evolved as though it represented a single vibrationally activated molecule. 

The distinction between the two representations not a rigid one, hinging on the lifetime of the intermediate hydride ion. If the latter is very short (of the order of the time of a single vibration or collision), the two mechanisms clearly become equivalent. 

In this section, we describe the motion of vibronic WPs created in a diatomic molecule that has only one vibrational mode. The influence of other degrees of motions such as rotation and nuclear spins are omitted for simplicity. Since our studies deal with the quantum property of the system in which relaxations can be neglected, the decoherence process is not taken into account in the following formulations. Assuming that the molecule occupies a single vibrational level v = 0 as an initial state, the WP generated by the absorption of a pump laser pulse is given as... 

If we use an ns probe pulse, we can tune its wavelength resonant to one particular vibronic transition. In this case, the LIF signal reflects the population of a single vibrational level involved in the WP. By scanning the wavelength of the probe pulse, we can observe the population distribution of the eigenstates involved in the WP. The peak intensities of the LIF signal are influenced by the Franck-Condon factors and the probe laser intensities, so that the relevant corrections are necessary to obtain the population distribution. 

There are many systems of different complexity ranging from diatomics to biomolecules (the sodium dimer, oxazine dye molecules, the reaction center of purple bacteria, the photoactive yellow protein, etc.) for which coherent oscillatory responses have been observed in the time and frequency gated (TFG) spontaneous emission (SE) spectra (see, e.g., [1] and references therein). In most cases, these oscillations are characterized by a single well-defined vibrational frequency, It is therefore logical to anticipate that a single optically active mode is responsible for these features, so that the description in terms of few-electronic-states-single-vibrational-mode system Hamiltonian may be appropriate. 

The most striking fact about the dipole matrix elements for the H2-H2 overtone band is the large number of components, Tables 4.13 and 4.14. Besides the single vibrational transitions (V2 = 0 — 2 while t i = const, Table 4.13, and vice versa), we now have to consider vibrational double transitions ( i = 0 — 1 and V2 —> 1, Table 4.14). The associated spectra appear at nearly the same frequencies. If one adds to these the various rotational bands, Eq. 4.40, a very large number of spectral components arises that must be accounted for in the computations of the overtone... 

Table 4.13. Analytical form (Eqs. 4.39 and 4.40) of the dipole matrix elements for the H2-H2 first overtone band, single vibrational transitions [284].

Dimers. It is well known that H2 pairs form bound states which are called van der Waals molecules. The discussions above based on the isotropic interaction approximation have shown that for the (H2)2 dimer a single vibrational state, the ground state (n = 0), exists which has two rotational levels f = 0 and 1). If the van der Waals molecule rotates faster ( > 1), centrifugal forces tear the molecule apart so that bound states no longer exist. However, two prominent predissociating states exist which may be considered rotational dimer states in the continuum (/ = 2 and 3). The effect of the anisotropy of the interaction is to split these levels into a number of sublevels. 

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