Coupling, of momenta

Let us notice that antisymmetrization involves summations over permutations of electron coordinates, whereas the coupling of momenta and their projections involves summations over functions with different values of the magnetic quantum numbers. Since these are independent finite summations, the order in which they are performed is interchangeable we can, in principle, either antisymmetrize, first, and couple, second, or vice versa. In practice, the Pauli principle restricts the possible values of the projections and as a result of these complications it proves to be more convenient to couple first, and antisymmetrize second, with the antisymmetrization not always being done by means of coordinate permutations. 

The wave function of the two-shell configuration (17.42) corresponds to the representation of uncoupled quasispin momenta of individual shells. The eigenfunction of the square of the operator of total quasispin and its z-projection can be written as follows in the scheme of the vectorial coupling of momenta in quasispin space ... 

Both isospin operators and quasispin operators of various pairing states commute with each other, therefore we can use the vectorial coupling of momenta of individual pairing states and introduce the isospin and quasispin of the JV-particle state. 

Thus, with the mlNln2lNl configuration we can do without the wave functions derived using the vectorial coupling of momenta of individual shells and use the functions characterized by eigenvalues of the commuting operators N, T2, Tz, L2, Lz, S2, Sz... 

Wave functions in an isospin basis can be represented as a linear combination of appropriate quantities obtained by vectorial coupling of momenta of individual shells... 

We have used the shortened forms NiN2N N4 and Ni — IN2N2 + IJV4) for the wave function of the configurations in (26.16) and have specified the form of coupling scheme A. Here N = Ni + IV3 and the last multiplier in (26.17) for the pair and successive couplings of momenta is equal to... 

In the more general case S 0 and the molecular angular momenta can be coupled in various ways. It is of primary importance to ascertain to what extent the interaction of the spin momentum S with the orbital momentum L is comparable to the rotation of the molecule, as well as to the interaction of each of the momenta L and S with the internuclear axis. An attempt to establish a hierarchy of interactions yields a number of possible, certainly idealized, coupling cases between angular momenta, first considered by Hund and known as Hund s coupling cases. Here we will discuss the three basic (out of five) cases of coupling of momenta in a linear molecule. 

The case of intermediate coupling of momenta (between Hund s cases (a) and (6)), as well as that of breaking weak field approximation axe discussed in [294]. The molecular (/-factors for Hund s case (c) coupling are discussed in [92, 364]. 

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