Spinors symmetry transformation

We are not going to review here the transformation properties of spatial wave functions under the symmetry operations of molecular point groups. To prepare the discussion of the transformation properties of spinors, we shall put some effort, however, in discussing the symmetry operations of 0(3)+, the group of proper rotations in 3D coordinate space (i.e., orthogonal transformations with determinant + 1). Reflections and improper rotations (orthogonal transformations with determinant -1) will be dealt with later. 

The operators V and S operate on the spatial and spin degrees of freedom respectively and transform like pseudovectors under the symmetry operations. Now, taking a general unitary transformation on a fixed set of spinors,... 

A different approach is chosen when the screening of nuclear potential due to the electrons is incorporated in /z . Transformation to the eigenspinor basis is then only possible after the DHF equation is solved which makes it more difficult to isolate the spin-orbit coupling parts of the Hamiltonian. Still, it is also in this case possible to define a scalar relativistic formalism if the so-called restricted kinetic balance scheme is used to relate the upper and lower component expansion sets. The modified Dirac formalism of Dyall [24] formalizes this procedure and makes it possible to identify and eliminate the spin-orbit coupling terms in the selfconsistent field calculations. The resulting 4-spinors remain complex functions, but the matrix elements of the DCB Hamiltonian exhibit the non-relativistic symmetry and algebra. 

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. 

Figure 2. Symmetry adapted graphical representation of a (6 spinor, 3 electron) space. The totally symmetric irrep is F,. The spinors transform according to the fermion irreps T4 through Fft.

As a case in point, the JT Hamiltonian for orbital systems is limited to the symmetrized square, whereas the Zeeman Hamiltonian only arises if the antisymmetrized square contains the symmetries of an axial field. For systems with an odd number of electrons, which therefore transform according to spinor representations, the selection rules are exactly opposite. 

The fictitious spin operator indeed transforms as a Ti operator and has the tensorial rank of a p-orbital. However, as we have shown, the full Hamiltonian also includes a Jf part, which involves an /-like operator. To mimic this part by a spin Hamiltonian, one thus will need a symmetrized triple product of the fictitious spin, which will embody an /-tensor, transforming in the octahedral symmetry as the T irrep. These /-functions can be found in Table 7.1 and are of type z 5z - ir ). But beware To find the corresponding spin operator, it is not sufficient simply to substitute the Cartesian variables by the corresponding spinor components, i.e., z by 5j , etc. indeed, while products of x, y, and z are commutative, the products of the corresponding operators are not. Hence, when constmcting the octupolar product... 

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