Selection rules rotational fine structure

We now consider the rotational fine structure of gas-phase IR bands, beginning with linear molecules. For nearly all known linear polyatomic molecules, the ground electronic state is a 2 state and we do not have to worry about the interaction of rotational and electronic angular momenta. A linear molecule in a 2 electronic state is a symmetric top with 7 =0 the selection rules are [(6.76) and (6.77)]... 

As is the case for diatomic molecules, rotational fine structure of electronic spectra of polyatomic molecules is very similar, in principle, to that of their infrared vibrational spectra. For linear, symmetric rotor, spherical rotor and asymmetric rotor molecules the selection rules are the same as those discussed in Sections 6.2.4.1 to 6.2.4.4. The major difference, in practice, is that, as for diatomics, there is likely to be a much larger change of geometry, and therefore of rotational constants, from one electronic state to another than from one vibrational state to another. 

Both the rotational fine structure and the contours of bands are helpful in assigning vibrations of gaseous compounds with a known structure. It is well known that vibrational transitions of diatomic AS molecules without changes in the rotational state ( -branch, AJ = 0) are not allowed in the IR spectrum. Likewise, selection rules exist for linear molecules, spherical tops, symmetric and asymmetric tops. With different irreducible representations, these lead to characteristic band contours, which are discussed in detail in another part of this book (Sec. 4.3). 

In accordance with these considerations, the pure-rotational Raman spectrum (selection rule AJ= 2) of has every second line missing, whereas that of Na has all lines present, but those arising from even-J states axe more intense than those arising from. odd-J states (2). Yoshino and Bernstein (ll) have observed intensity alternations having statistical origins in both the pure-rotational Raman spectrum of Ha, and in the rotational fine-structure (selection rules AJ=0, 2) of the vibrational band in the Raman spectra of both and Da. 

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... 

This chapter begins with a classical treatment of vibrational motion, because most of the important concepts that are specific to vibrations in polyatomics carry over naturally from the classical to the quantum mechanical description. In molecules with harmonic potential energy functions, vibrational motion occurs in normal modes that are mutually uncoupled. Coupling between vibrational modes inevitably occurs in the presence of anharmonic potentials (potentials exhibiting cubic and/or higher order terms in the nuclear coordinates). In molecules with sufficient symmetry, the use of group theory simplifies the procedure of obtaining the normal mode frequencies and coordinates. We obtain El selection rules for vibrational transitions in polyatomics, and consider the rotational fine structure of vibrational bands. We finally treat breakdown of the normal mode approximation in real molecules, and discuss the local mode formulation of vibrational motion in polyatomics. 

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... 

This variety in rotational selection rules, coupled with our natural endowment of molecules with diverse rotational constants, leads to wide variations in the rotational fine structure exhibited by symmetric and near-symmetric tops. For definiteness, we consider a prolate symmetric top whose rotational energy levels are given in Eq. 5.26. Rotational lines will be found at the frequencies... 

These spectra serve to illustrate the sensitivity of rotational fine structure to the transition moment orientations and rotational constants. In practice, individual rotational lines cannot be resolved in most infrared vibration-rotation spectra, because the rotational constants are too small. In spectra such as that in Fig. 6.14, the bunched groups of Q-branch lines frequently materialize as single intense bands, while the more sparse P and R branches form weak continua. Rotational structures are frequently analyzed by comparing them with computer-generated spectra derived from assumed rotational constants and selection rules. By weighting the rotational line intensities with appropriate Boltzmann factors (cf. Eq. 3.28) and assigning each rotational line a frequency width commensurate with the known instrument resolution, realistic simulations of experimental spectra are possible if the rotational constants and selection rules are properly adjusted. 

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. 

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 normal frequencies are the observed result of normal vibrations (hereinafter abbreviated n.v. in this chapter) corresponding to transitions from the vibrational ground state to the first excited one, and are usually observed at room temperature. Generally the selection rule Jv = +1 is obeyed. By infrared radiation rotational motions of the molecules are also excited. Their absorptions are superimposed on those of the vibrational modes. However, no example is known where rotational fine structure has been resolved for a n complex. 

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. 

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. 

Figure 9.31. FIR laser magnetic resonance spectrum of CO in the a 3n state, observed using the 393.6 pm line from formic acid [62]. This spectrum arises from the J = 7 — 6 rotational transition in the Q = 2 fine-structure state, and the transitions obey the selection rule A Mj = +1. The lower Mj states are indicated in the diagram.

The total parity of a given class of levels (F fine structure component for E-states, upper versus lower A-doublet component for II-states) is found to alternate with 7. The second type of label, often loosely called the e// symmetry, factors out this (—l) 7 or (—l)-7-1/2 7-dependence (Brown et al., 1975) and becomes a rotation-independent label. (Note that e/f is not the parity of the symmetrized nonrotating molecule ASE) basis function. In fact, for half-integer S, it is not possible to construct eigenfunctions of crv in the form [ A, S, E) —A, S, — E)], because, for half-integer S, vice versa.) The third type of parity label arises when crv is allowed to operate only on the spatial coordinates of all electrons, resulting in a classification of A = 0 states according to their intrinsic E+ or E- symmetry. Only A = 0) basis functions have an intrinsic parity of this last type because, unlike A > 0) functions, they cannot be put into [ A) — A)] symmetrized form. The peculiarity of this E symmetry is underlined by the fact that the selection rule for spin-orbit perturbations (see Section 3.4.1) is E+ <-> E, whereas for all types of electronic states and all... 

Figure 6.22 displays the rotational plus electronic fine structure of the NO 15/ <— A2E+(v = 1) transition from the N = 3, Ms = —1 /2 Zeeman component of the intermediate level (Guizard, et at, 1991). The parity selection rule permits transitions from the — parity TV = 3 initial level of the A2E+ state to the three + parity N+ = 1,3,5 rotational clusters (separated by 10B+ and 18B+) of an nf complex. The structure of an nf <— A2E+IV = 3, Ms = +1/2 transition (not shown) is identical to that originating from the Ms = —1/2 Zeeman component. The electric dipole transition operator operates exclusively on the spatial coordinates of the electron, thus AMs = 0 is a rigorous selection rule. Since the d-character of the A2E+ state is exclusively responsible for making the nf <— A2E+ transition allowed, one expects Z-polarized transitions that terminate on mi = —2, —1,0, +l,+2 Zeeman components in each N+ cluster. The observed intensity patterns in Fig. 6.22b are in excellent agreement with those calculated for an uncoupled case (d) <— case (b) pure / d transition (Guizard, et al., 1991). 

The propensity rules for collision-induced transitions between electronic states and among the fine-structure components of non-1E+ states depend on the identity of the leading term in the multipole expansion of the molecule/collision-partner interaction potential. Alexander (1982a) has considered the dipole-dipole term, which included both permanent and transition dipole contributions. In the limit that first-order perturbation theory applies (not the usual circumstance for thermal molecular collisions), the following collisional propensity rules for the permanent dipole term can be enumerated from the selection rules for both perturbations and pure rotational transitions... 

The rotational selection rule for electronic transitions between states at the same Hund s case (see Section 3.2.1) limit is A J = AN = 1,0. If either electronic state is not at a Hund s limiting case or if the limiting cases are not identical in the upper and lower electronic states, less restrictive rotational selection rules apply. However, the number of rotational eigenstate J or N components present in a (to) created by a typical spectroscopically realizable pluck (i.e., a single, non-chirped, not-saturating excitation pulse) is small, typically 2 or 3. The possibility that each of these rotational components is split by spin fine structure (see Section 3.4) is neglected in the following discussion. 

Examples of selection rules have already been mentioned in the condition that the atomic quantum number I must change by one unit at a time in the requirement that the rotational quanta responsible for the fine structure of molecular band spectra change by one unit or zero and, in a more general way, in the prohibition of transfers from symmetrical to antisymmetrical states. 

In addition to the rotational structure 16.20, the inversion spectrum has a hyper-fine structure. For the main nitrogen isotope thehyperfine structure is dominated by the electric quadrupole interaction ( 1 MHz) [69]. Because of the dipole selection rule, AAl = 0 and the levels with 7 = AT are metastable. In beam experiments, the width of the corresponding inversion lines is usually determined by collisional broadening. In astrophysical observations, the lines with 7 = AT are also narrower and stronger than others, but the hyperfine structure of the spectra with high redshifts is unresolved. 

Molecular vibrational spectra exhibit fine structure in gases, because rotational transitions can occur simultaneously with vibrational transitions. In diatomics with small vibration-rotation coupling, the selection rules on Av and AJ are exactly as in the cases of pure vibrational and pure rotational spectroscopy, respectively ... 

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