Doublets Kramers

As a preliminary step in the construction of the Kramers doublets we shall compute the spin-orbit coupling matrix within the ta-orbitals 5, Y), The e-orbitals do not enter into the discussion since they are presumed to be far removed in energy, as evidenced by the fact that hemoglobin azide is a low spin compound. The pertinent part of the spin-orbit coupling operator is given by 

Finally we may compute the matrix of s within the basis frmctions in O (Eqs. 67). As expected, by analogy with the atomic case, the matrix of / s (Table 12) is diagonal in this basis set and, moreover, has only two distinct values. These are 1 and — associated with the two-fold 

Examination of the matrix in Table 10 reveals that 5+, yj+ and have non-vanishing matrix elements among themselves and similarly for the set 5 , 7)- and C - Matrix elements of orbitals from one set with orbitals from the other vanish. The general structure of the Kramers doublet will therefore be 

Although there appear to be six constants, they are not all independent. According to Kramers theorem (27, 26), must satisfy the relation 

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More general situations have also been considered. For example. Mead [21] considers cases involving degeneracy between two Kramers doublets involving four electronic components a), a ), P), and P ). Equations (4) and (5), coupled with antisymmetry under lead to the following identities between the various matrix elements... 

If the electric quadrupole splitting of the 7 = 3/2 nuclear state of Fe is larger than the magnetic perturbation, as shown in Fig. 4.13, the nij = l/2) and 3/2) states can be treated as independent doublets and their Zeeman splitting can be described independently by effective nuclear g factors and two effective spins 7 = 1/2, one for each doublet [67]. The approach corresponds exactly to the spin-Hamiltonian concept for electronic spins (see Sect. 4.7.1). The nuclear spin Hamiltonian for each of the two Kramers doublets of the Fe nucleus is ... 

The crystal field interaction gives rise to an energy splitting into a number of Kramers doublets. In the case of high-spin Fe " with spin S = 5/2, there are three Kramers doublets, each of which give rise to separate contributions in the Mossbauer spectra of samples with slow paramagnetic relaxation. For 1 = 0 and a = 0, they can be labeled l/2), 3/2) and 5/2). 

Any set of energetically well-isolated levels can be described by an effective spin Hamiltonian operator by choosing S to match the corresponding number of levels. This can be just one isolated Kramers doublet of a high-spin multiplet if the... 

These studies provide the first advanced EPR studies of a non-Kramers doublet spin system, and they suggest the possibility of investigating similar signals from other proteins in this family. EXAFS spectroscopy does not exhibit Fe- -Fe backscattering in Hred (38). This observation could reflect thermal motion, or it may suggest that, as a consequence of the reduction of Hox to Hred, the Fe-Fe distance has... 

When spin-orbit coupling is introduced the symmetry states in the double group CJ are found from the direct products of the orbital and spin components. Linear combinations of the C"V eigenfunctions are then taken which transform correctly in C when spin is explicitly included, and the space-spin combinations are formed according to Ballhausen (39) so as to be diagonal under the rotation operation Cf. For an odd-electron system the Kramers doublets transform as e ( /2)a, n =1, 3, 5,... whilst for even electron systems the degenerate levels transform as e na, n = 1, 2, 3,. For d1 systems the first term in H naturally vanishes and the orbital functions are at once invested with spin to construct the C functions. 

Of these there are no established examples of the first two cases, but for the d3 and ds systems the metallocenes V(Cp)2 and Ni(Cp)2 respectively have been shown to have the appropriate ground states, and in the latter case a number of bis-arene species are also known. For d5 complexes Mn(Cp)2 is known to be able to exist in the high-spin 62+ form, but the bis-arenes all possess the low-spin 2E+(a S4) ground state which yields little information from susceptibility measurements since it consists only of a single Kramers doublet. 

Hagen, W.R. 1982b. EPR of non-Kramers doublets in biological systems characterization of an S = 2 system in oxidized cytochrome c oxidase. Biochimica et Biophysica Acta 707 82-98. 

In lanthanide complexes, the ligand field created by the surrounding ligands will split the atomic /-multiplets into several components. The latter are doubly degenerate (Kramers doublets (KDs)) for systems with odd number of electrons and non-degenerate (in the absence of symmetry) for systems with even number of electrons. 

Table 6.2 Energies of the lowest spin-free states originating from the SH multiplet and the energies of the low-lying Kramers doublets of the DyZn3 complex.

Atomic multiplet Spin free states J multiplet Kramers Doublets... 

Main values of the g tensor in the ground Kramers doublet ... 

Figure 6.2 (a) Orientation of the main magnetic axis in the ground Kramers doublet of [Cp ErCOT], [62]. Colour scheme Er, purple C, yellow H, white, (b) Polar plot of the angular dependence of the single-crystal magnetic susceptibility recorded in the a-b plane, comparison between the results of the... 

Table 6.3 Energies (per centimetre) and main values of the g tensors of the lowest Kramers doublets at the Dy3 site of the dysprosium triangle (from [37]).

II electronic states, 638-640 vibronic coupling, 628-631 triatomic molecules, 594-598 Hamiltonian equations, 612-615 pragmatic models, 620-621 Kramers doublets, geometric phase theory linear Jahn-Teller effect, 20-22 spin-orbit coupling, 20-22 Kramers-Kronig reciprocity, wave function analycity, 201 -205 Kramers theorem ... 

The effects of spin-orbit coupling on geometric phase may be illustrated by imagining the vibronic coupling between the two Kramers doublets arising from a 2E state, spin-orbit coupled to one of symmetry 2A. The formulation given below follows Stone [24]. The four 2E components are denoted by e, a), e a), e+ 3), c p), and those of 2A by coa), cop). The spin-orbit coupling operator has nonzero matrix elements... 

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