Quasi-Spin Selection Rules

To examine this exchange behavior for an arbitrary one-electron operator we start by rewriting the general expression of Eq. 6 in a sum over symplectic pairs. In the following and t] are assumed to be compound indices for resp. (mS, mi ) and Also — refers to (—m.v — m,-). The matrix element 

The annihilation operators in this equation must now be represented in their 

The hermitean character of the F operator allows the replacement of F, 4 by the complex conjugate of F 4 r This matrix element can be related to Fin if it is assumed that F either commutes or anticommutes with time reversal. 

In this expression the F operator is broken up into its irreducible quasi-spin parts. The behavior of F under time reversal thereby proves to be the determining factor. Two cases are possible. 

The first term in this expression is proportional to the trace of the F matrix. This is a scalar and therefore has the three quantum numbers Q, S, L equal to zero. The second term is clearly symmetric under exchange of the associated mq partners in the operator part. It thus corresponds to a quasi-spin triplet. 

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The first coordination sphere of acido-pentamine [Cr(NH3)5X]2+ complexes has C4v symmetry and this leads to a tetragonal resolution of the 2Eg state into 2At and 1Bl components. The classical ligand field model predicts that these components will be virtually degenerate. This is based on a combination of pseudo-spherical and quasi-spin selection rules of the shell and will be discussed later on in Sect. 5.2. At present we welcome this example of a pseudo-degeneracy as another opportunity to observe fine details of the interelectronic repulsion interaction which have the proper anisotropy to induce a splitting of the 2Eg term. 

The quasi-spin label is perhaps the most exotic quantum characteristic of these states. It is in any case a true shell characteristic since it extends over several configurations. Most importantly it gives rise to very strong selection rules as we will demonstrate in the next section. 

Knowing the quasi-spin properties of F we can now turn our attention to the associated selection rules. According to the Wigner-Eckart theorem acting in quasi-spin space the selection of an interaction element of an operator K0 ... 

Finally it should also be mentioned that the quasi-spin treatment can be extended to the two-particle Coulomb interaction. This will not be considered here since in the case of the (t2g )3 multiplets the quasi-spin characteristics of the Coulomb operator do not give rise to additional selection rules. 

The s.o.c. operator is a one-electron operator which is even under time reversal, and non-totally symmetric in spin and orbit space. The trace of the spin-orbit coupling matrix for the t2g-shell thus vanishes. As a result the s.o.c. operator is found to transform as the MK = 0 component of a pure quasi-spin triplet (Cf. Eq. 26). Application of the selection rule in Eq. 28 shows that allowed matrix elements must involve a change of one unit in quasi-spin character, i.e. AQ = 1. Since 4S and 2D are both quasi-spin singlets while 2P is a quasispin triplet, s.o.c. interactions will be as follows ... 

Quasi-Spin and Pseudo-Cylindrical Selection Rules... 

Low-symmetry LF operators are time-even one-electron operators that are non-totally symmetric in orbit space. They thus have quasi-spin K = 1, implying that the only allowed matrix elements are between 2P and 2D (Cf. Eq. 28). Interestingly in complexes with a trigonal or tetragonal symmetry axis a further selection rule based on the angular momentum theory of the shell is retained. Indeed in such complexes two -orbitals will remain degenerate. This indicates that the intra-t2g part of the LF hamiltonian has pseudo-cylindrical D h symmetry. As a result the 2S+1L terms are resolved into pseudo-cylindrical 2S+1 A levels (/l = 0,1,..., L ). It is convenient to orient the z axis of quantization along the principal axis of revolution. In this way each A level comprises the ML = A components of the L manifold. In a pseudo-cylindrical field only levels with equal A are allowed to interact, in accordance with the pseudo-cylindrical selection rule ... 

In the perspective of shell-theoretic selection rules the two types of emission clearly expose a difference in quasi-spin characteristics. The -emitting levels... 

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