Kramers symmetry

The Kramers-restricted form of the Hamiltonian that was used in Cl theory is not suitable for Coupled Cluster theory because it mixes excitation and deexcitation operators. One possibility is to define another set of excitation operators that keep the Kramers pairing and use these in the exponential parametrization of the wavefunction. This would automatically give Kramers-restricted CC equations upon rederivation of the energy and amplitude equations. A more pedestrian but simpler alternative is to start from the spin-orbital formulation and inspect the relations that follow from the Kramers relation of the two-electron integrals. This method does also readily give the relations between the Kramers symmetry-related amplitudes. We will briefly discuss the basic steps in this approach, a detailed description of a possible algorithm is given in reference [47],... 

This concludes our introduction to relativistic symmetry. Our aim has been to relate closely to features that should be familiar to the practicing quantum chemist. In particular, we have put some emphasis on the double groups, which represent a rather straightforward extension of the methods and concepts of nonrelativistic symmetry. We have also provided a more general discussion that shows how the double group symmetry arises as the direct product of the underlying symmetries in the two separate physical spaces considered—spin space and the four-space spanned by the Lorentz transformations. In the chapters to follow, we will repeatedly exploit both SU 2) (g) G, G 0(3) symmetry and Kramers symmetry to develop and simplify methods for quantum chemical calculations on relativistic systems. 

From our experience to date with the transformations, we can immediately foresee a problem. If the transformation is some complicated function of the momentum, we might not be able to separate out the perturbation from the zeroth-order Hamiltonian. This would be unfortunate, because magnetic operators break Kramers symmetry and we would be forced to perform calculations without spin (or time-reversal) symmetry. We might also be forced to perform finite-field calculations. We will address this problem as it arises. 

Table 12-4 gives the characters, basis functions, and the case a, b, or c to which the irreducible representation of Hu and belong for the point T. The degeneracy in and is the usual Kramers spin degeneracy, which is removed in cases (2) and (4) because of the absence of 6 in these symmetry groups. 

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. 

FIGURE 5.9 Effective g-values for half-integer spin systems in axial symmetry. The scheme gives the values for all transitions within the Kramer s doublets of S = nl2 systems assuming gmal = 2.00 and S S S B. 

FIGURE 7.2 Zero-field manifold for 5 = 2. The energy levels on the left hand are for axial symmetry (E = 0 and a = 0), that is, two non-Kramer s doublets and a singulet. The degeneracy of the doublets is lifted by addition of an E-term and subsequent addition of an a-term. 

For higher integer spins the number of allowed zero-field interaction terms further increases, and so does the convolution of comparable effects, except once more for a unique term that directly splits the highest non-Kramer s doublet. For S > 3 we have the addition, valid in cubic (and, therefore, in tetragonal, rhombic, and triclinic) symmetry ... 

If the Ln3+ centre is a Kramers ion, the spectra can be interpreted in terms of a doublet with largely anisotropic effective -values. If one neglects the admixture of higher lying/ multiplets and considers an axial symmetry, the effective g values will be... 

Finally, we note that, to the best of our knowledge, only one report exists about EPR spectra of non-Kramers lanthanide ions in molecular magnets. In 2012, Hill and coworkers [51] performed a multifrequency study on powder and single crystal samples of NagHofWgOj H20, in both the pure form and when doped into the isostructural Y3+ derivative. While crystallizing in a triclinic unit cell, the symmetry of the lanthanide ion in this family is very close to Did. For this reason, susceptibility data had been previously fitted by a purely axial Hamiltonian [50]. 

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. 

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... 

However, as mentioned above, T c)3) will be orthogonal to all the k states, and T ) is nonzero. This implies that the number of total states of the same eigenvalue E is (k + 1), which contradicts our initial hypothesis. Thus, we conclude that k must be even, and hence proved the generalized Kramers theorem for total angular momentum. The implication is that we can use double groups as a powerful means to study the molecular systems including the rotational spectra of molecules. In analyses of the symmetry of the rotational wave function for molecules, the three-dimensional (3D) rotation group SO(3) will be used. 

It should be pointed out that a somewhat different expression has been given for the Knight shift [32] and used in the analysis of PbTe data that in addition to the g factor contains a factor A. The factor A corresponds to the I PF(0) I2 probability above except that it can be either positive or negative, depending upon which component of the Kramers-doublet wave function has s-character, as determined by the symmetry of the relevant states and the mixing of wavefunctions due to spin-orbit coupling. 

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