Selection Rules in Atoms

Note that, by convention, a linear polarized hght wave is often termed n-polarization, and right-hand and left-hand circular-polarized waves are termed and O -polarization respectively. These are of great importance when considering photon transition probabihties and selection rules in atoms and molecules. 

Figure Bl.4.9. Top rotation-tunnelling hyperfine structure in one of the flipping inodes of (020)3 near 3 THz. The small splittings seen in the Q-branch transitions are induced by the bound-free hydrogen atom tiiimelling by the water monomers. Bottom the low-frequency torsional mode structure of the water duner spectrum, includmg a detailed comparison of theoretical calculations of the dynamics with those observed experimentally [ ]. The symbols next to the arrows depict the parallel (A k= 0) versus perpendicular (A = 1) nature of the selection rules in the pseudorotation manifold.

The theory of molecular symmetry provides a satisfying and unifying thread which extends throughout spectroscopy and valence theory. Although it is possible to understand atoms and diatomic molecules without this theory, when it comes to understanding, say, spectroscopic selection rules in polyatomic molecules, molecular symmetry presents a small barrier which must be surmounted. However, for those not needing to progress so far this chapter may be bypassed without too much hindrance. 

The strength or intensity of absorption is related to the dipole strength of transition D or square of the transition moment integral M m , and is pressed in terms of oscillator strength / or integrated molar extinction jfe Jv. A transition with /= 1, is known as totally allowed transition. But the transitions between all the electronic, vibrational or rotational states are not equally permitted. Some are forbidden which can become allowed under certain conditions and then appear as weak absorption bands. The rules which govern such transitions are known as selection rules. For atomic energy levels, these selection rules have been empirically obtained from a comparison between the number of lines theoretically... 

Platt has discussed the application of selection rules to the spectra of conjugated organic molecules, in particular condensed ring systems (8). He cites the various kinds of forbiddenness and then compares these with the corresponding selection rules for atomic systems. His treatment is useful and the following statements are taken directly from his article ... 

Insufficient experimental data is available to demonstrate either the occurrence or lack of selection rules. In Table 15, some of the fastest reactions do correspond to transitions allowed for both atoms. Singlet helium transfers to Ne at a rate consistent with the data in Fig. 26, whereas triplet helium transfers comparatively slowly neither of the He transitions is allowed. However, there appear to be other cases, discussed earlier, where a forbidden transition is preferred to an allowed transition. Stepp and Anderson146 have suggested that there is partial conservation of electronic angular momentum accompanying energy transfer between atoms, and interpreted experiments on mercury fluorescence by means of the steps... 

Relaxation of the rules can occur, especially since the selection rules apply strongly only to atoms that have pure Russell-Saunders (I-S) coupling. In heavy atoms such as lanthanides, the Russell-Saunders coupling is not entirely valid as there is the effect of the spin-orbit interactions, or so called j mixing, which will cause a breakdown of the spin selection rule. In lanthanides, the f-f transitions, which are parity-forbidden, can become weakly allowed as electric dipole transitions by admixture of configurations of opposite parity, for example d states, or charge transfer. These f-f transitions become parity-allowed in two-photon absorptions that are g g and u u. These even-parity transitions are forbidden for one photon but not for two photons, and vice versa for g u transitions [46],... 

This represents a formidable practical problem, as one is very unlikely to find isolated atoms with two nonorthogonal dipole moments and quantum states close in energy. Consider, for example, a V-type atom with the upper states 11), 3) and the ground state 2). The evaluation of the dipole matrix elements produces the following selection rules in terms of the angular momentum quantum numbers J — J2 = 1,0, J3 — J2 = 1,0, and Mi — M2 = M3 — M2 = 1,0. Since Mi / M3, in many atomic systems, p12 is perpendicular to p32 and the atomic transitions are independent. Xia et al. [62] have found transitions with parallel and antiparallel dipole moments in sodium molecules (dimers) and have demonstrated experimentally the effect of quantum interference on the fluorescence intensity. We discuss the experiment in more details in the next section. Here, we point out that the transitions with parallel and antiparallel dipole moments in the sodium dimers result from a mixing of the molecular states due to the spin-orbit coupling. 

One very important aspect of two-photon absorption is that the selection rules for atoms or S5munetrical molecules are different from one-photon selection rules. In particular, for molecules with a centre of symmetry, two-photon absorption is allowed only for gog or uou transitions, while one-photon absorption requires g-o-u transitions. Therefore, a whole different set of electronic states becomes allowed for two-photon spectroscopy. The group-theoretical selection rules for two-photon spectra are obtained Ifom the symmetries... 

As already mentioned, the phenomenon of magnetic circular dichroism in photoemission originates from spin-orbit and exchange interactions in combination with the dipole selection rules. In the atomic model picture, the splitting of the 3p level (into sublevels with orbital momentum m) is caused by the electrostatic interaction of the core level with the magnetically polarized valence electrons [57]. The observed intensity differences and the respective asymmetry values in photoemission from the Fe 3p levels are small (typically 3%) compared to the large MCDAD and MLDAD asymmetries (up to about 12%) observed in valence band photoemission [27]. 

The Bohr-Sommerfeld theory provided a reasonably satisfactory explanation of the spectra of atoms having only one valence electron. With two or more electrons discrepancies occur and certain arbitrary selection rules for atomic transitions were required. In an attempt to solve this problem, several persons, including de Broglie, Schrodinger, Heisenberg, and Born, combined quantum mechanical and wave mechanical concepts. A detailed account of these efforts is beyond the scope of this book. However, some qualitative understanding of these concepts will be helpful and thus a brief account is included. 

Since each of the hydrogenlike levels with / 9 0 is now split into doublets with = / + 5, it becomes necessary to augment the El selection rules on An, A/, and Aw with El selection rules on Aj. It turns out that these are Aj = 0, 1. We will demonstrate this selection rule in the case of rYSi/2 - n Pj transitions, where j can be i or. The coupled (total) angular momentum states lsjmy = jm in atoms can be expressed as a superposition of uncoupled states /w,sWs> weighted by Qebsch-Gordan coefficients [1,3],... 

Compared with the early models of Black and Dalgarno (1977), there have been a few developments in the molecular data relevant to the excitation calculations. Although the collisional excitation rates for H + H2 and H2 + H2 are still uncertain, more accurate values for the latter system have been provided by Schaefer (1985). The rotational excitation rates of CO by H2 have recently been recomputed independently by Schinke et af. (1985) and Flower and Launay (1985), and show good agreement. Monteiro and Flower (1987) have reminded us of a neglected selection rule for atomic line structure excitation, which suggests that the J=0— 1 excitation of C and O by H2 is in first order forbidden. Finally, Chambaud et af. (1988) have computed rotational excitation cross sections for C2 by H2. 

Raman spectroscopy also has selection rules. The gross selection rule for a Raman-active vibration is related to the polarizability of the molecule. Polarizability is a measure of how easily an electric field can induce a dipole moment on an atom or molecule. Vibrations that are Raman-active have a changing polarizability during the course of the vibration. Thus, a changing polarizability is what makes a vibration Raman-active. The quantum-mechanical selection rule, in terms of the change in the vibrational quantum number, is based on a transition moment that is similar to the form of M in equation 14.2. For allowed Raman transitions, the transition moment [a] is written in terms of the polarizability a of the molecule ... 

Specific selection rules for atoms and molecules can also be determined using group-theoretical analyses of the functions in equation 15.1, exactly as we did in the previous chapter for allowed IR and Raman vibrational transitions. 

---