Rotational structure

If the experunental technique has sufficient resolution, and if the molecule is fairly light, the vibronic bands discussed above will be found to have a fine structure due to transitions among rotational levels in the two states. Even when the individual rotational lines caimot be resolved, the overall shape of the vibronic band will be related to the rotational structure and its analysis may help in identifying the vibronic symmetry. The analysis of the band appearance depends on calculation of the rotational energy levels and on the selection rules and relative intensity of different rotational transitions. These both come from the fonn of the rotational wavefunctions and are treated by angnlar momentum theory. It is not possible to do more than mention a simple example here. 

A-B relative or external motion undergo free-free transitions (E., E. + dE.) (Ej Ej+ dE within the translational continuum, while the structured particles undergo bound-bound (excitation, de-excitation, excitation transfer) or bound-free (ionization, dissociation) transitions = (a, 3) ->/= (a, (3 ) in their internal electronic, vibrational or rotational structure. The transition frequency (s ) for this collision is... 

Figure 2-88. The permutation matrices of the fragments of the rotated trans Isomers. The rotated structure (right-hand side) has two descriptors of (-1) whereas the initial structure (left-hand side) had two values of (-r 1). The overall descriptor of both sides is obtained by multiplication (+1)(+1) = (41) and (-1)(-1) = (41),...

Each vibrational peak within an electronic transition can also display rotational structure (depending on the spacing of the rotational lines, the resolution of the spectrometer, and the presence or absence of substantial line broadening effects such as... 

Figure 6.9 The 1-0 Stokes vibrational Raman spectrum of CO showing the 0-, Q-, and 5-branch rotational structure...

Figure 9.46 Rotational structure of the Ojj bands in the fluorescence excitation spectra of s-tetrazine dimers at about 552 run. Bottom Ojj band of planar dimer. Middle Ojj band of T-shaped dimer with transition in monomer unit in stem of T. Top Ojj band of T-shaped dimer with transition in monomer unit in top of T. (Reproduced, with permission, from Haynam, C. A., Brumbaugh, D. V and Levy, D. H., J. Chem. Phys., 79, f58f, f983)...

Fig. 0.2. (a) The comb spectrum of N2 considered as a quantum rotator. The envelope of the rotational structure of the Q-branch slightly split by the rotovibra-tional interaction is shaded, (b) The depolarized rotovibrational spectrum of N2 at corpuscular density n = 92 amagat, T = 296 K and pressure p = 100 atm. The central peak, reported in a reduced (x30) scale is due to a polarized component [5] (V) experimental (—) best fit. 

The quantum theory of spectral collapse presented in Chapter 4 aims at even lower gas densities where the Stark or Zeeman multiplets of atomic spectra as well as the rotational structure of all the branches of absorption or Raman spectra are well resolved. The evolution of basic ideas of line broadening and interference (spectral exchange) is reviewed. Adiabatic and non-adiabatic spectral broadening are described in the frame of binary non-Markovian theory and compared with the impact approximation. The conditions for spectral collapse and subsequent narrowing of the spectra are analysed for the simplest examples, which model typical situations in atomic and molecular spectroscopy. Special attention is paid to collapse of the isotropic Raman spectrum. Quantum theory, based on first principles, attempts to predict the. /-dependence of the widths of the rotational component as well as the envelope of the unresolved and then collapsed spectrum (Fig. 0.4). 

The fluctuating cage model presented in Chapter 7 is an alternative. The idea came from comparison of the different kinds of absorption spectra of HC1 found in liquid solutions (Fig. 0.5). In SFg as a solvent the rotational structure of the infrared absorption spectrum of HC1 is well resolved [15, 16], while in liquid He it is not resolved but has... 

The origin of the rotational structure of the isotropic Q-branch (Av = 0, Aj = 0) is connected with the dependence of the vibrational transition frequency shift on rotational quantum number j [121, 126]... 

Here te, tc are the correlation times of rotational and vibrational frequency shifts. The isotropic scattering spectrum corresponding to Eq. (3.15) is the Lorentzian line of width Acoi/2 = w0 + ydp- Its maximum is shifted from the vibrational transition frequency by the quantity coq due to the collapse of rotational structure and by the quantity A due to the displacement of the vibrational levels in a medium. 

Condition (3.19) is usually satisfied in processes of vibrational dephasing [41, 130, 131, 132], Because of this condition the dephasing is weak and the effect of rotational structure narrowing is pronounced. A much more important constraint is imposed by inequality (3.18). It shows that perturbation theory must be applied to a rather dense medium and even then only the central part of the spectrum (at Aa> < 1/t , 1/tc) is Lorentzian. 

Formulae (3.15)—(3.17) are quite general because the method is indifferent to how the perturbation changes in time. It may be a sequence of collisions or random continuous noise. Thus, the results are valid when the gas is condensed into a liquid. Motional narrowing of rotational structure progresses with increase of density as long as... 

The quasi-classical description of the Q-branch becomes valid as soon as its rotational structure is washed out. There is no doubt that at this point its contour is close to a static one, and, consequently, asymmetric to a large extent. It is also established [136] that after narrowing of the contour its shape in the liquid is Lorentzian even in the far wings where the intensity is four orders less than in the centre (see Fig. 3.3). In this case it is more convenient to compare observed contours with calculated ones by their characteristic parameters. These are the half width at half height Aa)i/2 and the shift of the spectrum maximum ftW—< > = 5a>+A, which is usually assumed to be a sum of the rotational shift 5larger scale A determined by vibrational dephasing. 

It should be noted that there is a considerable difference between rotational structure narrowing caused by pressure and that caused by motional averaging of an adiabatically broadened spectrum [158, 159]. In the limiting case of fast motion, both of them are described by perturbation theory, thus, both widths in Eq. (3.16) and Eq (3.17) are expressed as a product of the frequency dispersion and the correlation time. However, the dispersion of the rotational structure (3.7) defined by intramolecular interaction is independent of the medium density, while the dispersion of the vibrational frequency shift (5 12) in (3.21) is linear in gas density. In principle, correlation times of the frequency modulation are also different. In the first case, it is the free rotation time te that is reduced as the medium density increases, and in the second case, it is the time of collision tc p/ v) that remains unchanged. As the density increases, the rotational contribution to the width decreases due to the reduction of t , while the vibrational contribution increases due to the dispersion growth. In nitrogen, they are of comparable magnitude after the initial (static) spectrum has become ten times narrower. At 77 K the rotational relaxation contribution is no less than 20% of the observed Q-branch width. If the rest of the contribution is entirely determined by... 

Fig. 3.12. The room temperature CARS spectra of CH4 obtained in [162] at the following densities (1) 0.1 amagat of pure CH4 (2) 5 amagat CH4 (3) 5 amagat CH4 + 35 amagat Ar (4) 5 amagat CH4 + 85 amagat Ar. The position of the vibration frequency wv is indicated as well as the centre of gravity of the Q0i branch rotational structure wv + coq.

The quasi-classical theory of spectral shape is justified for sufficiently high pressures, when the rotational structure is not resolved. For isotropic Raman spectra the corresponding criterion is given by inequality (3.2). At lower pressures the well-resolved rotational components are related to the quantum number j of quantized angular momentum. At very low pressure each of the components may be considered separately and its broadening is qualitatively the same as of any other isolated line in molecular or atomic spectroscopy. 

We will show below when and how the line interference and its special case, spectral exchange , appear in spectral doublets considered as an example of the simplest system. It will be done in the frame of conventional impact theory as well as in its modern non-Markovian generalization. Subsequently we will concentrate on the impact theory of rotational structure broadening and collapse with special attention to the shape of a narrowed Q-branch. 

Fig. 4.1. The La line of the H atom and its structure in the constant electric field (a) and the rotational structure of the vibrational transition (b). Wavy arrows show collision-induced transitions, thick horizontal arrows indicate the optical transitions that mutually interfere.

The quantum theory must describe not only the shape of a resolved rotational structure of the Q-branch but its transformation with increase of pressure to a collapsed and well-narrowed spectrum as well. A good example of such a transformation is shown in Fig. 4.6. The limiting cases of very low and very high pressures are relatively easy to treat as they relate to slow modulation and fast modulation limits of frequency exchange. 

The best resolution of Q-branch rotational structure in a N2-Ar mixture was achieved by means of coherent anti-Stokes/Stokes Raman spectroscopy (CARS/CSRS) at very low pressures and temperatures (Fig. 0.4). A few components of such spectra obtained in [227] are shown in Fig. 5.9. A composition of well-resolved Lorentzian lines was compared in [227] with theoretical description of the spectrum based on the secular simplification. The line widths (5.55) are presented as... 

At higher pressures only Raman spectroscopy data are available. Because the rotational structure is smoothed, either quantum theory or classical theory may be used. At a mixture pressure above 10 atm the spectra of CO and N2 obtained in [230] were well described classically (Fig. 5.11). For the lowest densities (10-15 amagat) the band contours have a characteristic asymmetric shape. The asymmetry disappears at higher pressures when the contour is sufficiently narrowed. The decrease of width with 1/tj measured in [230] by NMR is closer to the strong collision model in the case of CO and to the weak collision model in the case of N2. This conclusion was confirmed in [215] by presenting the results in universal coordinates of Fig. 5.12. It is also seen that both systems are still far away from the fast modulation (perturbation theory) limit where the upper and lower borders established by alternative models merge into a universal curve independent of collision strength. 

The pressure being higher, all features of rotational structure disappear and the difference between spectra of various spin modifications becomes so smooth that any of them practically reproduces the whole contour shape. In Fig. 5.14 the theoretical contours are shown calculated with and without adiabatic correction of the impact operator for an ideal nitrogen solution in Ar. They are compared with the experimental one related to the same value of... 

As can be seen from the above, the shape of the resolved rotational structure is well described when the parameters of the fitting law were chosen from the best fit to experiment. The values of estimated from the rotational width of the collapsed Q-branch qZE. Therefore the models giving the same high-density limits. One may hope to discriminate between them only in the intermediate range of densities where the spectrum is unresolved but has not yet collapsed. The spectral shape in this range may be calculated only numerically from Eq. (4.86) with impact operator Tj, linear in n. Of course, it implies that binary theory is still valid and that vibrational dephasing is not yet... 

Fig. 6.1. A spectral exchange scheme between components of the rotational structure of an anisotropic Raman spectrum of linear molecules. The adiabatic part of the spectrum is shadowed. For the remaining part the various spectral exchange channels are shown ( - — ) between branches (<— ) within branches.

Let us demonstrate that the tendency to narrowing never manifests itself before the whole spectrum collapses, i.e. that the broadening of its central part is monotonic until Eq. (6.13) becomes valid. Let us consider quantity x j, denoting the orientational relaxation time at ( = 2. If rovibrational interaction is taken into account when calculating Kf(t) it is necessary to make the definition of xg/ given in Chapter 2 more precise. Collapse of the Q-branch rotational structure at T = I/ojqXj 1 shifts the centre of the whole spectrum to frequency cog. It must be eliminated by the definition... 

Since its solution is rather complex, let us restrict ourselves to consideration of a collapsed spectrum at T 1, when it is already symmetrical with a centre shifted to frequency coq=0. As we are interested only in its broadening, we may neglect the rotational structure of the Q-branch in Eq. (6.27) assuming... 

We need only one simplification neglect of the Q-branch rotational structure. It is conditioned by Eq. (6.29). This assumption restricts the buffer gas densities, which should not be too low ... 

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