Field Symmetries

Ceulemans paper of 1984 generalizes the work of Griffith [25,26] on particle-hole conjugation for the specific case of d" electrons split by an octahedral symmetry field. He relies on the use of matrices and determinants, in particular Laplace s expansion of the determinant in terms of complementary minors, for the analysis. He bases his selection mle analysis on the properties of a novel particle-hole conjugation operator... 

The authors have interpreted their data using a statistical model, which reveals that the energy gap between the 1 Aj ground state and the lowest of the spin-orbitally split ST2 levels is temperature dependent the form of this temperature dependence changes with x. The model includes spin-orbit coupling, low-symmetry field distortion, covalency, and dynamic as well as static effects of local ligand vibrations. 

Here Sg and are the tetragonal and orthorhombic components of the low symmetry field, sometimes called strain terms. The signs of Sg, are chosen to conform with the signs for the two coordinates Qg and respectively. All parameters in Eqs. (1) and (2) can have units of energy if dimensionless units are used for Qg and [2,8]. 

The lowest energy, spin-allowed transition in a low-spin cobalt(III) complex is Ti Ai. In Cav or Dah symmetry, the Ti orbital degeneracy is partially lifted to give and 2 levels. The second band in an octahedral complex is the T2 Ai transition, but this does not seem to be so susceptible to lower symmetry fields. Transitions are expected to take place with the following energies 24, 4 ) ... 

We shall make use of the effective Hamiltonian formalism [14] that enables us to isolate effects of interest from irrelevant complications. We divide the electronic Hamiltonian into a strong part H° and a weak part H, and we shall suppose that H° is simple enough to be solved exactly. The Hamiltonian including the cubic field and interelectronic repulsion only is the usual choice for H in the case of the 3d group ions. Then H should include all other interactions (spin-orbit coupling, lower symmetry fields, electron-phonon interaction, external fields, strain etc). The most important assumption is that the perturbations, described by the H Hamiltonian (in particular the JT interaction) must be smaller relative to the initial splitting due to H°. In the case of the 3d metal ions the assumption is usually well justified. 

In the discussion of terms in low symmetry fields it is useful to adopt a ket notation of the form... 

Actually, the above argument was used in the reverse order to determine the orientation of the histidine on the basis of ESR data before x-ray determinations were available. Nevertheless, even though subsequent x-ray studies did not invalidate the conclusions based on ESR data, there is some doubt in attibuting the low symmetry field to the position of the histidine (37). The magnitude of the effect does not appear to be sufficient. 

Next, it is necessary to estimate A Ex, A Ey and A Ez. It is recalled that Ti arises from the electronic configuration e. If the ta-orbitals are not degenerate, as would be the case in a low symmetry field, the three spatial components of Ti will have different energies corresponding to the different ways in which four electrons can distribute themselves among the three ta-orbitals. These arrangements are C and... 

From Fig. 20 it is seen that the ground state in a cubic field for a high spin ferrous complex is tj e T. It will, of course, be necessary to apply spin-orbit coupling to obtain departures from the spin-only value. It appears that a cubic field with spin-orbit coupling is sufficient without the necessity of invoking lower symmetry fields (38). 

Spin-free Fe(diimine)zX2 complexes show a weak broad band at 8.5—12.5 AK with two more or less clearly resolved components separated by ca. 2 AK. Kdnig et al. (33,34,40) associate this band with the 5X2 - transition, the excited E state being split by low symmetry field components. For spin-free l e(bipy)2(NCS)2 and Fe(phen)2(NCS)2, 10 Dj is estimated at 11.2 and 11.8AK, respectively. 

Do - Fi transitions implies a low symmetry field at the europium site (23). 

The calculation of the matrix elements of the lower symmetry fields and of the spin-orbit interaction is not so easy. The general procedure is given by Tanabe and Kamimura and the tables for this purpose are also given in their papers. . ... 

Because they can, and usually do, remove orbital degeneracies—and thus reduce the orbital contribution to the magnetism—low-symmetry ligand fields cannot be ignored in a study of the magnetic properties of transition metal complexes. Unfortunately, it is no easy matter to determine the splitting effects of low symmetry fields usually it is necessary to work with two... 

---