Selection rules vibration-rotation transitions

Figure 6.7(a) illustrates the rotational energy levels associated with two vibrational levels u (upper) and il (lower) between which a vibrational transition is allowed by the Au = 1 selection rule. The rotational selection rule governing transitions between the two stacks of levels is... 

Beyond such electronic symmetry analysis, it is also possible to derive vibrational and rotational selection rules for electronic transitions that are El allowed. As was done in the vibrational spectroscopy case, it is conventional to expand i j (R) in a power series about the equilibrium geometry of the initial electronic state (since this geometry is more characteristic of the molecular structure prior to photon absorption) ... 

The selection rules for electronic transitions are not as clear-cut as in the case of vibration and rotation. In the case of molecules consisting of relatively light nuclei, which is the case for many molecules of tropospheric interest, the selection rules... 

Upon absorption of light of an appropriate wavelength, a diatomic molecule can undergo an electronic transition, along with simultaneous vibrational and rotational transitions. In this case, there is no restriction on Au. That is, the selection rule Av = +1 valid for purely vibrational and vibrational-rotational transitions no longer applies thus numerous vibrational transitions can occur. If the molecule is at room temperature, it will normally be in its lower state, v" = 0 hence transitions corresponding to v" = 0 to v = 0,... 

To investigate the spectra of diatomic molecules, we need the selection rules for radiative transitions. We now investigate the electric-dipole selection rules for transitions between vibration-rotation levels belonging to the same 2 electronic state. (Transitions in which the electronic state changes will be considered in Chapter 7.)... 

We now consider radiative transitions foi which both v and J change, but the electronic state does not these transitions give the vibration-rotation spectra of diatomic molecules. The selection rules for these transitions were found in Section 4.4 to be ( 2 states only)... 

A spherical top is a special case of a symmetric top the rotational energy depends only on 7, and the J selection rule for the vibration-rotation transitions is the same as for symmetric tops ... 

Thus a transition between two given electronic states shows many bands, each such band corresponding to a different pair of initial and final vibrational states under high resolution, each band shows many closely spaced lines, each such line corresponding to a different pair of initial and final rotational states. (The electronic spectra of molecules are called band spectra, whereas the electronic spectra of atoms are called line spectra.) Consider the selection rules for electronic transitions. The electric di-... 

The component Mz belongs to the species 4" in the Dah group because fiz is not changed by pure permutations and it changes sign by permutation—inversion operations (Section 4.1). The overall symmetry selection rule therefore allows transitions only between vibration—inversion-rotation states with opposite parity with respect to the operation of inversion (cf. Fig. 6). 

The very different spectra of iodine obtained under continuum and discrete resonance-Raman conditions are illustrated in Fig. 11 for resonance with the B state, whose dissociation limit is 20,162 cm . In the case illustrated of discrete resonance-Raman scattering, Xl =514.5 nm, and specific re-emission results from an initial transition from the v" = 1 vibrational, J" = 99 rotational level of the X state to the v = 58, J = 100 level of the B state, i.e. the transition is 58 - l" R(99). Owing to the rotational selection rule for dipole radiation, AJ = 1, a pattern of doublets appears in the emission. Clearly, the continuum resonance-Raman spectrum of iodine (Xl = 488.0 nm) is very different from the discrete case spectrum. The structure, which arises from the 0,Q, and S branches of the multitude of vibration-rotation transitions occurring, can be analysed in terms of a Fortrat diagram, as done for gaseous bromine (67). 

This is integrated over the Q,Q2Q,-space. If the collision pair wave functions never overlap the vibration wave function Xiku(Qi>Q2>Q3 2Zu) of the QTS, there will be zero contribution to the cross section. In this case, the QTS defines the reaction domain. This is quantized by the corresponding vibration-rotation wave function. Therefore, from all possible collisions among the reactants, only those having a non-zero FC factor will contribute to the reaction rate. This is related to the steric factor, P, in elementary chemical kinetics theory. Selection rules for VR-transitions apply. The probability to find the system in one of the product channel states when starting from a QTS is controlled by the FC integral formed by the products of the type... 

The majority of polyatomic molecular rotational spectra arise from J 1 J, / and P branch transitions and for these, even when they follow asymmetric rotor selection rules, the intensities still increase with increasing frequency see for example SO2 (Figure 1.1). A conspicuous exception to this general rule however occurs when g-branch J J vibration-rotation transitions are present in the spectral range under investigation. 

In the Einstein relation for induced radiation processes the ratio of absorption and emission is proportional to the population density in the two energy levels involved however, we have to account for the degeneracy factor g. This is of particular importance in vibration-rotation transitions with the selection rules AJ=-1 and AJ=+1 for P-branch and R-branch transitions, since there the J-values and hence the degeneracy factors of the two levels involved are different. The relation is... 

The electric dipole allowed vibration-rotation transitions obey the selection rules AN= 1, AG,=0, AG2=0, AF=0, 1. 

The selection rules for vibration-rotation transitions are the same as for separate transitions ... 

As with rotational spectroscopy, there are several ways of stating selection rules for spectral transitions involving vibrational states of molecules. There is a gross selection rule, which generalizes the appearance of absorptions or emissions involving vibrational energy levels. There is also a more specific, quantum-number-based selection rule for allowed transitions. Finally, there is a selection rule that can be based on group-theoretical concerns, which were not considered for rotations. 

As with rotational and vibrational transitions, there is a selection rule for electronic transitions dictating which electronic wavefunctions participate in allowed transitions. Allowed electronic transitions must have a nonzero transition moment as given by the expression... 

The selection rules for Raman transitions are different from those of absorption or emission, and this makes it possible to observe transitions that are forbidden in emission or absorption spectroscopy. The Raman selection rules for rotational and vibrational transitions are ... 

Diatomic Molecules (Spin Neglected), 258. Symmetry Properties of the Wave Functions, 261. Selection Rules for Optical Transitions in Diatomic Molecules, 262. The Influence of Nuclear Spin, 265. The Vibrational and Rotational Energy Levels of Diatomic Molecules, 268. The Vibrational Spectra of Polyatomic Molecules, 273. 

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. 

Atoms have complete spherical synnnetry, and the angidar momentum states can be considered as different synnnetry classes of that spherical symmetry. The nuclear framework of a molecule has a much lower synnnetry. Synnnetry operations for the molecule are transfonnations such as rotations about an axis, reflection in a plane, or inversion tlnough a point at the centre of the molecule, which leave the molecule in an equivalent configuration. Every molecule has one such operation, the identity operation, which just leaves the molecule alone. Many molecules have one or more additional operations. The set of operations for a molecule fonn a mathematical group, and the methods of group theory provide a way to classify electronic and vibrational states according to whatever symmetry does exist. That classification leads to selection rules for transitions between those states. A complete discussion of the methods is beyond the scope of this chapter, but we will consider a few illustrative examples. Additional details will also be found in section A 1.4 on molecular symmetry. 

The result of all of the vibrational modes contributions to la (3 J-/3Ra) is a vector p-trans that is termed the vibrational "transition dipole" moment. This is a vector with components along, in principle, all three of the internal axes of the molecule. For each particular vibrational transition (i.e., each particular X and Xf) its orientation in space depends only on the orientation of the molecule it is thus said to be locked to the molecule s coordinate frame. As such, its orientation relative to the lab-fixed coordinates (which is needed to effect a derivation of rotational selection rules as was done earlier in this Chapter) can be described much as was done above for the vibrationally averaged dipole moment that arises in purely rotational transitions. There are, however, important differences in detail. In particular. 

As before, when pf i(Rg) (or dpfj/dRa) lies along the molecular axis of a linear molecule, the transition is denoted a and k = 0 applies when this vector lies perpendicular to the axis it is called n and k = 1 pertains. The resultant linear-molecule rotational selection rules are the same as in the vibration-rotation case ... 

The rotational selection rule for vibration-rotation Raman transitions in diatomic molecules is... 

Atomic spectra are much simpler than the corresponding molecular spectra, because there are no vibrational and rotational states. Moreover, spectral transitions in absorption or emission are not possible between all the numerous energy levels of an atom, but only according to selection rules. As a result, emission spectra are rather simple, with up to a few hundred lines. For example, absorption and emission spectra for sodium consist of some 40 peaks for elements with several outer electrons, absorption spectra may be much more complex and consist of hundreds of peaks. 

Here a third selection rule applies for linear molecules, transitions corresponding to vibrations along the main axis are allowed if Aj = 1. The A/=0 transition is only allowed for vibrations perpendicular to the main axis. Note that because of this selection rule the purely vibrational transition (called Q branch) appears in the gas phase spectrum of C(X but is absent in that of CO. In both cases, two branches of rotational side bands appear (called P and R branch) (see Fig. 8.3 for gas phase CO). 

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