Hund coupling scheme

Here and below we follow the usual spectroscopic notations for molecular terms [8, 9], In the case of the Hund coupling scheme a the notation a +1Aw refers to a molecular term with total electron spin S, the projection A (A > 0) of electronic orbital angular momentum on the molecular axis, and parity w. 

Under this condition a collision practically does not induce intramultiplet mixing, but during a collision there is an appreciable distortion of atomic functions. The mere fact of the existence of the molecular term A2ni/2 described by the Hund coupling scheme a means that the electronic spin follows rotation of the molecular axis. To this approximation there will be depolarization of the P1/2 state although the relevant molecular term... 

We now write down the possible interaction potential in the quasimolcculc under consideration. In contrast to the Hund coupling scheme (Table 1), we separate various interactions of cui ion and atom at Uu ge distances in the form... 

Let us summarize the Hund coupling scheme [5, 6, 7] that is given in Table 1 together with the quantum numbers of the quasimolecule for each case of Hund coupling. We denote by L the total electron angular momentum of the molecule, S is the total electron spin, J is the total electron momentum of the molecule, n is the unit vector along the molecular axis, K is the rotation momentum of nuclei, A is the projection of the angular momentum of electrons onto the molecular axis, H is the projection of the total electron momentum J onto the molecular axis, 5 is the projection of the electron spin onto the molecular axis, Lyv, Si, Jn are projections of these momenta onto the direction of the nuclear rotation momentum N. Below we will take this scheme as a basis. 

A partly filled shel > exhibits a number of states of different energies which arise as a result of the interactions or couplings of the electrons in the shell. These states can be determined using the Russell-Saunders coupling scheme (Hund s rules) (Figgis, 1966). A characteristic property of a state is the spin multiplicity which is related to the number of unpaired electrons in a shell. A singlet state has a spin multiplicity of one (two electrons of opposite spin), a doublet state has a multiplicity of two and... 

In the presence of Coulomb correlation only, the wave function is characterized by the total spin S = SSj and the total angular momentum L = 2,1 of the 5 f electrons, and the total momentum J is given by Hund s rule (J = L S). Important spin orbit coupling will mix LS multiplets and only J remains a good quantum number. The Russell-Saunders coupling scheme is no longer valid and an intermediate coupling scheme is more appropriate. 

FIGURE 26 (a) MP2/6-31G(d) geometry of the cyclopropyl radical 40 according to Reference 246. (b) Spin-coupling scheme of the cyclopropyl radical according to the intra-atomic Hund rule and spin coupling within bonds. Hyperfine splitting aN at nucleus N will be > 0 ( < 0) if valence electrons at N possess oc (/ ) spin... 

Let the diatomic (or linear) molecule satisfy the conditions for the Hund s case (a) coupling scheme (see Section 1.2, Fig. 1.3(a)), where the electronic orbital and spin angular momenta are coupled with the internuclear axis. The magnetic moment which is directed along the internuclear axis and corresponds to the projection Cl of the total angular momentum J upon the internuclear axis, has the value... 

Fig. 4.24. Vector scheme for the interaction of momenta of a diatomic molecule in a magnetic field in Hund s case (a) coupling scheme. The frequency u)q of nutation of fl and (j,q around J considerably exceeds the frequency u>j of the precession of J around B.

With the introduction of electronic angular momentum, we have to consider how the spin might be coupled to the rotational motion of the molecule. This question becomes even more important when electronic orbital angular momentum is involved. The various coupling schemes give rise to what are known as Hund s coupling cases they are discussed in detail in chapter 6, and many practical examples will be encountered elsewhere in this book. If only electron spin is involved, the important question is whether it is quantised in a space-fixed axis system, or molecule-fixed. In this section we confine ourselves to space quantisation, which corresponds to Hund s case (b). 

In the Born-Oppenheimer approximation the basis set for 3Q,i would consist of products of electronic space and spin functions. Transformation to the gyrating axis system may involve transformation of both space and spin variables, leading to a Hamiltonian in which the spin is quantised in the molecule-fixed axis system (as, for example, in a Hund s case (a) coupling scheme) or transformation of spatial variables only, in which case spatially quantised spin is implied (for example, Hund s case (b)). We will deal in detail with the former transformation and subsequently summarise the results appropriate to spatially quantised spin. 

To take a specific example, let us consider P = S, the electron spin angular momentum for a diatomic molecule in a Hund s case (a) coupling scheme where the basis functions are simple products of orbital, rotational and spin functions. Using standard... 

Symmetric top eigenfunctions can be expressed in terms of rotational matrices as follows for a Hund s case (a) coupling scheme ... 

As Lefebvre-Brion and Field [61] point out, the only coupling cases for which the electronic and nuclear motions can be separated are cases (a) and (c) consequently only in these cases can potential curves be defined unambiguously and accurately. However, as we have already pointed out, Hund s coupling cases are idealised descriptions and for most molecules the actual coupling corresponds to an intermediate situation. Moreover, the best description of the vector coupling often changes as the molecular rotation increases. In this section we consider the nature of the intermediate coupling schemes in more detail some of these will appear elsewhere in this book in connection with the observed spectra of specific molecules. 

When considering the effect of Pi2 on a Hund s case (a) or case (b) wave function, we must also take the nuclear spin wave function into account. The nuclear spin is usually very weakly coupled to the other angular momenta and so can be described by a separate factor (i/ ns). We shall discuss the detailed form of this function shortly but for the moment, we simply need to recognise that there are two types of functions which can arise from the coupling scheme... 

The coupling scheme for N2 in this state is Hund s case (b). Therefore we have... 

The ratio B0/A0 for the OH radical in its ground 2 n state is -0.1333, large compared with, for example, the ratio for CIO which is —0.0022. Consequently its coupling scheme shows a considerable departure from the Hund s case (a) limit towards case... 

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). 

In a Hund s case (b) coupling scheme, the Zeeman interaction is described by the following matrix element ... 

Hund mles apply if the Russell-Saunders coupling scheme is valid. Sometimes the first rule is apphed to molecules. 

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