Symmetry lowering

It is unlikely that the ground states of really existing metal complexes are degenerate (T or E terms), for three reasons  

The actual crystal packing may require lower than cubic symmetry. 

The relativistic double group approach allows only doubly degenerate levels and in the case of cubic symmetry the maximum degeneracy of Tg is four consequently no orbital triplets are allowed, only spin-orbital quadruplets. 

Consequently the complex has a lower than cubic symmetry, which is equivalent to the addition of a low-symmetry crystal field potential to the spin Hamiltonian 

Symmetry lowering owing to the Jahn-Teller effect 

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For the d9 configuration, with a 2II(a2 it3 S4) ground state, only one sandwich complex, Co(HMBz)2, is known (108). This was shown to have a moment of 1.86 0.08 B.M. at 83 and 295 K, and a permanent dipole moment of 1.78 D in benzene solution. Although there are no indications of a static distortion to be found from X-ray diffraction studies (7), this latter observation appears to afford clear cut evidence of an effective symmetry lower than Cxv, and the magnetic data were shown (101) to imply A 1100 cm-1 in order to reproduce the essentially temperature independent moment observed. However, two cautions should be observed in assessing this evidence. Firstly the predicted temperature... 

When a lanthanide ion is placed in a ligand environment with symmetry lower than spherical, the energies of its partly filled 4f orbitals are split by the electrostatic field of the ligand. The result is a splitting of the 2/ + 1 degeneracy of the free ion states (see Figure 1.2). 

Strictly speaking, there is one minor qualification to be made to this general conclusion. In the case of CH4, nine independent variables render the optimisation process numerically rather redundant - the strict conclusion is therefore slightly weaker for no GHO basis with a symmetry lower than that of the molecule was an energy found lower than that of a symmetrical basis. 

Measurements of NMR for Ti, Ti [33], and Sr [34,35] were carried out for STO 16 and STO 18-96. Ti and Sr nuclear magnetic resonance spectra provide direct evidence for Ti disorder even in the cubic phase and show that the ferroelectric transition at Tc = 25 K occurs in two steps. Below 70 K, rhomb ohedral polar clusters are formed in the tetragonal matrix. These clusters subsequently grow in concentration, freeze out, and percolate, leading to an inhomogeneous ferroelectric state below Tc. This shows that the elusive ferroelectric transition in STO 18 is indeed connected with local symmetry lowering and impHes the existence of an order-disorder component in addition to the displacive soft mode [33-35]. Rhombohedral clusters, Ti disorder, and a two-component state are found in the so-called quantum paraelectric... 

A prediction of crystal field theory as outlined in the preceding subsections is that the crystal field splitting parameter, A, should be rather critically dependent upon the details of the crystal lattice in which the transition metal ion is found, and that the splittings of the /-orbital energies should become larger and quite complicated in lattices of symmetry lower than cubic. The theory could not be expected to apply, for example, to the spectra of transition metal ions in solution. 

In symmetries lower than cubic the (/-orbitals mix with the donor atom s—p hybrid orbitals to varying extents in molecular orbitals of appropriate symmetry. However, the mixing is believed to be small and the ligand field treatment of the problem proceeds upon the basis that the effective d-orbitals still follow the symmetry requirements as (/-orbitals should. There will be separations between the MOs which can be reproduced using the formal parameters appropriate to free-ion d-orbitals. That is, the separations may be parameterized using the crystal field scheme. Of course, the values that appear for the parameters may be quite different to those expected for a free ion (/-orbital set. Nevertheless, the formalism of the CFT approach can be used. For example, for axially distorted octahedral or tetrahedral complexes we expect to be able to parameterize the energies of the MOs which house the (/-orbitals using the parameter set Dq, Ds and Dt as set out in Section 6.2.1.4 or perhaps one of the schemes defined in equations (11) and (12). 

For complexes of symmetry lower than cubic no general relationships which connect the parameters describing departure from cubic symmetry, Ds and Dt say, with features of M and L or their combination appear. For example, it is difficult to obtain a coherent account even of the signs of the splittings of the cubic field spectral bands of Cr(en)2X2+ ions using the parameters Aoct(Cr(en)33+) and Aoct(CrX63 ). The predictive power of the LFT parameterization scheme does not seem to extend to symmetries lower than cubic. 

An important step in developing the mean-field concept was done by Landau [8, 10]. Without discussing the relation between such fundamental quantities as disorder-order transitions and symmetry lowering, we just want to note here that his theory is based on thermodynamics and the derivation of the temperature dependence of the order parameter via the thermodynamic potential minimization (e.g., the free energy A(r),T)) which is a function of the order parameter. It is assumed that the function A(rj,T) is analytical in the parameter 77 and thus near the phase transition point could be expanded into the series in 77 usually it is a polynomial expansion with temperature-dependent coefficients. Despite the fact that such a thermodynamical approach differs from the original molecular field theory, they are quite similar conceptually. In particular, the r.h.s. of the equation of state for the pressure of gases or liquids and the external field in ferromagnetics, respectively, have the same polynomial form. 

Symmetry lowering case, anisotropic [elongated bipyramid of Ti(III), S = 1/2]... 

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