Rotational Fine Structure in Electronic Band Spectra

6 ROTATIONAL FINE STRUCTURE IN ELECTRONIC BAND SPECTRA 

In Section 4.4 we worked out the El electronic and vibrational selection rules for electronic band spectra, and it remains for us to determine the selection rules that govern the rotational fine structure. We have seen that no symmetry selection rule exists for Ay, but that the vibrational band intensities are proportional to Franck-Condon factors in the Born-Oppenheimer approximation. To understand the selection rules for simultaneous changes in electronic and rotational state, we must find how l eiZrot) = transforms under 

In heteronuclear diatomics li eiXrot) can be classified according to its behavior under molecule-fixed reflection t . We already know that the molecule-fixed states ej obey 

In homonuelear diatomics an additional symmetry element is needed to classify state symmetries in the D f, point group, and we can use i (molecule-fixed) for this purpose. A new complication, peculiar to homonuelear molecules 

Since the vibrational state lxvib depends only on R = R, it is unaffected by exchange of the nuclei. To decide the effect of on we use the identity 

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For rotational fine structure in electronic band spectra, the P- and R-branch line positions are still given by Eqs. 4.76, except that Vq now becomes Tg -I- G v ) — G"(y"). The important physical difference here is that B and B" are often grossly different in transitions between different electronic states. For example, B and B are 0.029 and 0.037 cm S respectively, for the <- transition in Ij (Fig. 4.7). The rotational line positions in the P and R branches are no longer even approximately equally spaced. For that matter, Vp and Vr both vary as (B — B")J for large J—since they are then dominated by terms quadratic in J—and hence they run in the same direction as functions of J. In contrast, Vp and Vr run in opposite directions as functions of J for small J, where the linear terms dominate. Hence, either the P or the R branch turns around as a function of J (depending on whether B < B" or B > B") at the bandhead as shown in Fig. 4.27. The approximate value of J at which one of these branches turns around can be found by differentiating the... 

In this chapter, we extend our treatment of rotation in diatomic molecules to nonlinear polyatomic molecules. A traditional motivation for treating polyatomic rotations quantum mechanically is that they form a basis for experimental determination for bond lengths and bond angles in gas-phase molecules. Microwave spectroscopy, a prolific area in chemical physics since 1946, has provided the most accurate available equilibrium geometries for many polar molecules. A background in polyatomic rotations is also a prerequisite for understanding rotational fine structure in polyatomic vibrational spectra (Chapter 6). The shapes of rotational contours (i.e., unresolved rotational fine structure) in polyatomic electronic band spectra are sensitive to the relative orientations of the principal rotational axes and the electronic transition moment (Chapter 7). Rotational contour analysis has thus provided an invaluable means of assigning symmetries to the electronic states involved in such spectra. 

Analysis of the rotational fine structure of electronic bands is also possible, and the rotation constant for a molecule can in principle be determined for each of many vibrational states of each electronic state involved in the transition. In practice, smdies have largely been limited to diatomic molecules, partly because of the high resolution needed to observe rotational fine structure of an electronic band, and partly because the excited state can have a very short lifetime, with a consequent uncertainty in the energy levels because of the Uncertainty Principle, leading to broadening of the lines. More information about obtaining rotational information from electronic spectra is available in the on-line supplement for Chapter 7. 

As in diatomics, vibrational transitions in polyatomic molecules are inevitably accompanied by rotational fine structure. In linear molecules, the vibrational and rotational selection rules in vibration-rotation spectra are closely analogous to the electronic and rotational selection rules, respectively, in diatomic electronic band spectra. When applied to a molecule, the general symmetry arguments of the previous Section lead to the El selection rules... 

Many of the ideas that are essential to understanding polyatomic electronic spectra have already been developed in the three preceding chapters. As in diatomics, the Born-Oppenheimer separation between electronic and nuclear motions is a useful organizing principle for treating electronic transitions in polyatomics. Vibrational band intensities in polyatomic electronic spectra are frequently (but not always) governed by Franck-Condon factors in the vibrational modes. The rotational fine structure in gas-phase electronic transitions parallels that in polyatomic vibration-rotation spectra (Section 6.6), except that the rotational selection rules in symmetric and asymmetric tops now depend on the relative orientations of the electronic transition moment and the principal axes. Analyses of rotational contours in polyatomic band spectra thus provide valuable clues about the symmetry and assignment of the electronic states involved. 

Apart from molecular vibrations, also rotational states bear a significant influence on the appearance of vibrational spectra. Similar to electronic transitions that are influenced by the vibrational states of the molecules (e.g. fluorescence, Figure 3-f), vibrational transitions involve the rotational state of a molecule. In the gas phase the rotational states may superimpose a rotational fine structure on the (mid-)IR bands, like the multitude of narrow water vapour absorption bands. In condensed phases, intermolecular interactions blur the rotational states, resulting in band broadening and band shifting effects rather than isolated bands. 

Sharp absorption bands are typically not observed in UV and visible absorption spectra of liquid samples. This is the consequence of the presence of the vibrational and rotational fine structure that become superimposed on the potential energy surfaces of the electronic transitions. Fine structure in UV/vis absorption spectra can be detected for samples in vapor phase or in nonpolar solvents. 

In some cases, the UV /VIS spectra will show the different energies associated with the vibrational sublevels. For example, simple molecules in the gas phase often show the vibrational levels superimposed on the electronic transitions, as seen in Fig. 5.11, the gas phase spectrum for benzene. The sharp peaks on top of the broad bands are called vibrational fine structure . This fine structure is usually lost at high temperatures in the gas phase due to population of higher vibrational energy levels in the ground state electronic level, with the result that many more lines are seen. Molecules in solution (such as the spectrum shown in Fig. 5.1) usually do not exhibit vibrational structure due to interactions between the solvent and the solute molecules. Compare the gas phase spectrum of benzene (Fig. 5.11) to the solution spectrum for benzene (Fig. 5.12) and note the loss of much of the fine structure in solution. Fine structure due to rotational sublevels is never observed in routine UV/VIS spectra the resolution of commercial instrumentation is not high enough to separate these lines. 

For solids and liquids, electronic absorption bands are usually broad and essentially featureless, but more information is obtainable from electronic spectra of gas-phase molecules. Transitions between two levels with long lifetimes are the most informative. Such an electronic transition for a gas-phase sample has various possible changes in vibrational and rotational quantum numbers associated with it, so that the spectrum, however it is obtained, consists of a number of vibration bands, each with rotational fine structure, together forming an electronic system of bands. The selection rules governing the changes in vibrational and rotational quantum numbers depend on the nature of the electronic transition, and they can be ascertained by analyzing the pattern and structures of the bands. 

The empirical material on molecular spectra is interpreted in principle for the rotation or vibration of the nuclear framework of the molecules. But as soon as the electronic motion comes into play we are far away from understanding. The fine structure of the bands originates here, as well as the ordering of the electronic jumps. As far as the fine structure is concerned, certain types have been found and a close relation to the atomic spectra has been more or less established. For some molecules recently some electron jumps have been characterised by terms. ... 

Gas phase electronic spectra of polyatomic molecules are more complicated than the spectra of diatomics. The number of vibrational modes and the possibility of combination bands usually lead to numerous vibrational bands, and these may be overlapping. Also, the rotational fine structure tends to be more complicated, as we might expect from the differences between diatomic and polyatomic infrar (IR) spectra. Conventional absorption spectra can prove to be a difficult means of measuring and assigning transitions, and so numerous experimental methods have been devised to select molecules in specific initial states and to probe the absorption or the emission spectrum with narrow frequency range lasers. 

Electronic spectra are almost always treated within the framework of the Bom-Oppenlieimer approxunation [8] which states that the total wavefiinction of a molecule can be expressed as a product of electronic, vibrational, and rotational wavefiinctions (plus, of course, the translation of the centre of mass which can always be treated separately from the internal coordinates). The physical reason for the separation is that the nuclei are much heavier than the electrons and move much more slowly, so the electron cloud nonnally follows the instantaneous position of the nuclei quite well. The integral of equation (BE 1.1) is over all internal coordinates, both electronic and nuclear. Integration over the rotational wavefiinctions gives rotational selection rules which detemiine the fine structure and band shapes of electronic transitions in gaseous molecules. Rotational selection rules will be discussed below. For molecules in condensed phases the rotational motion is suppressed and replaced by oscillatory and diflfiisional motions. 

The formation of vibrationally excited products is nearly always energetically possible in an exothermic reaction, and these products can be detected by observing either an electronic banded system in absorption or the vibration-rotation bands in emission. In principle, rotational level distributions may be determined by resolving the fine structure of these spectra, but rotational energy is redistributed at almost every collision, so that any non-Boltzmann distribution is rapidly destroyed and difficult to observe. In contrast, simple, vibrationally excited species are much more stable to gas-phase deactivation and the effects of relaxation are less difficult to eliminate or allow for. 

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