Spinors spin-orbital representation

Factorization of the Cl space is difficult in the relativistic case. The first problem is the increase in number of possible interactions due to the spin-orbit coupling. The second problem is the rather arbitrary distinction in barred and unbarred spinors that should be used to mimic alpha and beta-spinorbitals. Unlike the non-relativistic case the spinors can not be made eigenfunctions of a generally applicable hermitian operator that commutes with the Hamiltonian. If the system under consideration possesses spatial symmetry the functions may be constrained to transform according to the representations of the appropriate double group but even in this case the precise distinction may depend on arbitrary criteria like the choice of the main rotation axis. 

It should be remembered, of course, that Lie groups have an independent existence apart from their role in the theory of f electrons in the lanthanides. Some of the bizarre properties that turn up in the f shell might well derive from isoscalar factors that receive a ready explanation in another context. An example of this is provided by the vanishing of the spin-orbit interaction when it is set between F and G states belonging to the irreducible representation (21) of G2. In the f shell, (21) is merely a 64-dimensional representation of no special interest. However, for mixed configurations of p and h electrons, it fits exactly into the spinor representations (iiiHi 2) of SO(14) with dimensions 2 (Judd 1970). These are the analogs of the spinor representations of eq. (130), and (21) describes the quasiparticle basis of the configurations (p-t-h)". The spinor representations of SO(3) and (Hiii) of SO(ll) provide the quasi-particle bases for the p and h shells respectively and their SO(3) structures, namely S 1,2 and 512+ 912+ is/2> when coupled, must yield the L structure of (21) of G2, namely D-l-F-l-G-l-H-t-K-t-L. In this context, the F and G terms of (21) are associated with the different irreducible representations Sj,2 and S9/2. It is this property,... 

The discussion above applies to uncontracted basis sets. Contracted basis sets present a few further problems. To properly represent the spin-orbit splitting, the two spin-orbit components should be contracted separately. The contraction is now j -dependent, rather than f-dependent, and can only be represented directly in a 2-spinor basis. The problem is not now confined to the small component. If the large-component scalar basis set includes contractions for both spin-orbit components, the product of the contracted basis functions for each spin-orbit component with the spin functions generates a representation for both spin-orbit components. Thus there is a duplication of the basis set that is close to linearly dependent, and some kind of scheme to project out linearly dependent components, either numerically or by conversion to a 2-spinor basis, is mandatory. The same applies to the small component. For example, the contracted p sets for the large-component and d sets both span the same space, but because of the contraction the (i-generated set cannot be made a subset of the -generated set, even if a dual family basis set is used. 

In the previous discussion of the symmetry of the Dirac equation (chapter 6), it was shown that the Dirac equation was symmetric under time reversal, and that the fermion functions occur in Kramers pairs where the two members are related by time reversal. We will have to deal with a variety of operators, and in most cases the methodologies will be developed in the absence of an external magnetic field, or with the magnetic field considered as a perturbation. Consequently, we can make the developments in terms of a basis of Kramers pairs, which are the natural representation of the wave function in a system that is symmetric under time reversal. The development here is primarily the development of a second-quantized formalism. We will use the term Kramers-restricted to cover techniques and methods based on spinors that in some well-defined way appear as Kramers pairs. The analogous nonrelativistic situation is the spin-restricted formalism, which requires that orbitals appear as pairs with the same spatial part but with a and spins respectively. Spin restriction thus appears as a special case of Kramers restriction, because of the time-reversal connection between a and spin functions. 

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