Second-order degeneracy

Equation (2.3) describes line positions correctly for spectra with small hyperfine coupling to two or more nuclei provided that the nuclei are not magnetically equivalent. When two or more nuclei are completely equivalent, i.e., both instantaneously equivalent and equivalent over a time average, then the nuclear spins should be described in terms of the total nuclear spin quantum numbers I and mT rather than the individual /, and mn. In this coupled representation , the degeneracies of some multiplet lines are lifted when second-order shifts are included. This can lead to extra lines and/or asymmetric line shapes. The effect was first observed in the spectrum of the methyl radical, CH3, produced by... 

Figure 4.4 Variation of soft-mode frequency, a>, with temperature in (a) second-order transitions and (b) first-order transitions. The numbers next to the curves represent degeneracies.

Spin-orbit coupling in second order (Section 6.4.4.6) raises the degeneracy of orbitally nondegenerate ground terms by amounts of up to a few cm-1, occasionally more. For temperatures of the order of D and E, the magnetic susceptibility is a complicated function of temperature and varies strongly with direction, as the various levels for equation (65) become occupied to different extents.169 At temperatures much smaller than D and E, the susceptibility usually becomes independent of temperature but highly dependent on direction. 

The operator M so couples states of different spin and space symmetries in second order, independent of spatial degeneracies. 

If the degeneracy is lifted completely in first-order or if at least second-order effects do not introduce an additional splitting of degenerate levels, the second-order energy can be expressed as... 

XH. .. Y stretching vibrations respectively, v2 the bending vibration and disregard the degeneracy of the latter. Then we obtain for a vibrational term, to the second order ... 

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