Symmetry in spin-orbit coupling

Fedorov DG, Gordon MS (2002) Symmetry in spin-orbit coupling. In Hoffmann MR, Dyall KG (eds) Low-lying potential energy surfaces. ACS symposium series, vol 828. American Chemical Society, Washington, pp 276-297... 

K spin-orbit coupling is included, the shape of the APES s may be strongly affected, with the exception of E states in cubic symmetry, where spin-orbit coupling has no influence in first-order. As a general rule, the spin-orbit interaction tends to cancel the JT effect shifting the minima towards the undistorted configuration. ... 

Table 3. Vertical ionization energies / (in eV) of gaseous zinc(II), cadmium(II) and mercury (II) compounds. The lowest / corresponds to a M.O. with symmetry type (and oj = 3/2 in spin-orbit coupling) mainly consisting of halide n pn orbitals (carbon a in the case of dimethylmercury). Further on, the d-like components at high / are given in two groups (separated by a semi-colon) with 2Ds/2 at lower and 2D3/2 at higher /.

An important dilference between these complexes and the examples of crossover already discussed is that the Mossbauer spectra only show evidence for one distinct iron environment in each case. The implication is that the lifetimes of the two spin configurations are small in comparison to the Mossbauer excited-state lifetime of 10 s. In other words the spectrum is time-averaged. In a ligand field of perfect cubic symmetry there is no interaction between the two states and a superposition of the spectra of each of these is predicted. For lower symmetries the spin-orbit coupling causes state mixing of the and via the Ti state, which causes a rapid exchange. 

Indeed, in. some cases it is probable that V2 is not ob.served at all, but that the fine. structure arises from term splitting due to spin-orbit coupling or to distortions from regular octahedral symmetry. 

Fig. 2. Temperature dependence of the HS fraction % according to the Ising model. The employed parameter values are = 150 K, Aj = Aj = 500 cm and X = — 100 em h Here, Aj and Aj are the orbital energy differences between the and levels and between the Bj and levels, respectively, X being the spin-orbit coupling constant. The model parameters A, Aj, and X determine the value of AG. The levels result from the HS iron(II) ground state in orthorhombic symmetry according to Bj -1- B2 + B. The figures on the curves specify the values of...

To examine the effect of covalency on the QS, the values of the QS of some compounds have been calculated with the aid of the Extended Hiickel MO method (i 82, 61). In these calculations the values for the empirical parameters used were obtained from comparison with EPR experiments (see EPR studies). This method is suitable only for molecules with low symmetry, because effectsoof spin-orbit coupling and thermal mixing have been neglected. 

Foyt et al. [137] interpreted the quadrupole-splitting parameters of low-spin ruthenium(II) complexes in terms of a crystal field model in the strong-field approximation with the configuration treated as an equivalent one-electron problem. They have shown that, starting from pure octahedral symmetry with zero quadrupole splitting, A q increases as the ratio of the axial distortion to the spin-orbit coupling increases. 

The AO composition of the SOMO can often be deduced from the dipolar hyperfine matrix, particularly when the radical has enough symmetry to restrict possible hybridization. Thus an axial hyperfine matrix can usually be interpreted in terms of coupling to a SOMO composed of a single p- or d-orbital. A departure from axial symmetry may be due to spin orbit coupling effects, if (for example) /) Az and Ax AyxP(gx gy). If the departure from axial symmetry is larger, it is usually caused by d-orbital hybridization. The procedure is best illustrated by examples. 

Substituting the parameters, we have ab = 0.058. (The upper sign applies if the components are listed in the order x, y, z in Table 4.3, the lower sign if the order is y, x, z.) Finally, we get b2 = 0.660, a2 = 0.005. The dz2 component is not really significant, given the accuracy of the data and the theory, i.e., most of the departure from axial symmetry can be explained by the spin-orbit coupling correction. 

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. 

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