Angular Momentum Coupling Cases

Hand s case (a) describes the majority of molecules with A 0 that exhibit small spin-orbit coupling. Since the electronic orbital and spin angular momenta L and S are not mutually strongly coupled, they precess independently about the quantization axis (the molecular axis) established by the magnetic fields arising from electronic motion. The projections Ah and Zh of h and S respectively along the molecular axis are then conserved, as is their sum Qh. In contrast, the parts of L and S normal to the molecular axis oscillate rapidly they are denoted Lj and (Fig. 4.17). 

For the total angular momentum J, the quantity Jp = J J + l)h is necessarily a constant of motion. (J was previously used for rotational angular momentum in state molecules in Chapter 3 it is now reserved for total 

We finally compute the rotational energy = N /2/ for a case (a) molecule. Since the portion of J which lies in a plane perpendicular to the molecular axis is 

The quantum analog of this expression for the rotational energy in cm is (in the absence of centrifugal distortion) 

For atoms with Russell-Saunders coupling (those in which can be treated as a perturbation) the spin-orbital correction to the energy was seen to behave 

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See [15,16] for the alternative diabatic representations which correspond to the alternative Himd s angular momentum coupling cases in the spectroscopy and dynamics of a diatomic system. 

Franck-Condon factors. The most common angular momentum coupling cases are discussed, and rotational fine structure in electronic transitions (cf. Fig. 4.3) is rationalized for heteronuclear and homonuclear diatomics using Herzberg diagrams. 

Even though angular momentum coupling considerations would allow L = L (because coupling two angular momenta with j = 1 and j = L should give L +l, L, and L -l), the 3-j symbol vanishes for the L = L case since 3-j symbols have the following... 

First we recall the details of the angular momentum coupling scheme for a 2n molecule described as Hund s case (b). It is as follows ... 

We have already shown the importance of the Zeeman effect, both in identifying the J quantum numbers involved in each line, and in providing effective g-factors for the levels. These g-factors serve as additional labels for each level, and provide information concerning the best angular momentum coupling scheme. We now develop the theory of the Zeeman effect in Hund s case (c). 

The mechanism by which spin-orbit coupling can impact molecular bonding becomes more obvious if the wave function is reformulated in terms of molecular spinors (Hund s case c coupling). One should keep in mind, however, that even for the heaviest elements the angular momentum coupling is generally intermediate between case a and case c, and the following picture is therefore an oversimplification. 

We now have to treat a system of 9 degrees of freedom. It is possible to use techniques of angular momentum coupling that are analogous to those employed for the two-body case. We define the angular momentum operators... 

In case of excited molecules the angular momentum coupling scheme is often not known, in particular if hyperfine structure or perturbations between different electronic states affect the coupling of the various angular momenta. The total angular... 

More generally, further complications arise for angular momentum couplings when the fragments show electronic angular momentum, but the fundamental symmetry rules are not affected by this. Early examples for detailed correlations include triatomic systems, and another more recent example is the correlation for atom + linear molecule. While for such simple cases, and planar molecules in general, parity is... 

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