Symmetry descent

A free atom belongs to the continuous rotation group R3. The irreducible representations of group R3 are labelled by the quantum number /. The spherical harmonic functions Yl m form the basis of the irreducible representation of R3 with the dimension 2/ + 1. 

The operation of rotation through an angle a about the z-axis yields 

Such a rotation has a representation expressed through the 21 + 1 dimensional matrix 

Therefore the character of this operation (a trace of the transformation matrix) is the sum of the geometric series, i.e. 

Then the standard form of the character table is arranged in Table 8.13. 

---

The 6x6 secular equation within the ground2 T2g term on the symmetry descent (Table 13) is fully factored and yields analytical formulae for the energy levels (Table 30) also in the case of the anisotropic orbital reduction factors. Then the energy gap S67 = / (/ 7)a - (/, ) is < 67 = A.[- 2v + 3kz - (Av2 + 4kzv + 8/c2 + k2 )1/2 ] /4 and disappears upon recovery of the octahedral geometry. 

On tetragonal distortion two different reference situations occur as displayed in Fig. 60. For the compressed tetragonal bipyramid the ground CFT is3 g. The magnetism could be modeled with the Figgis anisotropic Hamiltonian (Eq. 196) on the symmetry descent. Much more precise, however, are the calculations in the complete d4 space spanned by 210 functions. These results are displayed in Fig. 61. The magnetic anisotropy is visible far above room temperature. 

Scheme 1 JT symmetry descent paths of Oh parent group and its subgroups (upper lines in rectangles) for IRs (bottom lines in rectangles) Eg (a), Tig and Tgg (b). Analogous schemes may be obtained for ungerade IRs ( , Tiu, T2u) replacing subscripts g by the u ones where appropriate. For continuation see Schemes 2 Td and T groups), 4 Sn group) and 6 (Du and Dg groups)...

Scheme 2 JT symmetry descent paths of Td parent group and its subgroups (upper lines in rectangles) for IRs (bottom lines in rectangles) E (a), Tj and T2 (b)...

We performed MP2/cc-pVTZ geometry optimization of cyclo-CsHs radical using Gaussian03 software [24], We have found Cs stable structure ( A electronic state) and PES saddle points of C2v ( Bi and A2 electronic states) and Cs i A" electronic state) symmetries (see Table 4 and Fig. 1) in agreement with step-by-step descent method (because the original Oh plane of the parent Dsh group is not conserved in the non-planar cyclopropenyl radical). Two symmetry descent paths of Scheme 6b may be employed ... 

The remaining groups preserve double electron degeneracy and are subjects to further JT symmetry descent (moreover, the symmetry group is not feasible for coronene). There are two possible ways to JT stable C2h group in Bg electronic state ... 

Finally, an absolute agreement between the results of epikernel principle and step-by-step symmetry descent method may be concluded. 

Possible symmetry groups originating in JT symmetry descent of parent group with triple electron degeneracy ( Ti or electronic state for B4+ cluster) in (36) formally agree with the epikernel principle but it does not hold for the... 

Thus the planar C2v structure (C model) in A electronic state cannot be explained by JT symmetry descent from parent Td group. 

It is evident that electronic state of D2h symmetry group cannot arise due to JT effect (by splitting Eg electronic state of D4h group) because it is not allowed by the symmetry descent path (see Scheme 5c)... 

Planar C2v structure with Ai electronic state (C model) cannot be explained by JT symmetry descent from parent group and must be explained by JT symmetry descent of parent D h symmetry group (see Fig. 4) with double electron degeneracy by the symmetry descent path (Scheme 6b)... 

The method of step-by-step symmetry descent does not explain the mechanisms that are responsible for JT distortions. Some opponents argue that its predictions are far too wide on account of selectivity ( all is possible ). On the other hand, this treatment is based exclusively on group theory and does not account for any approximations used in the recent solutions of Schrddinger equation. Chemical thermodynamics does not solve the problems of chemical kinetics but nobody demands to do it as well. Thus we cannot demand this theory to solve also the mechanistic problems despite the epikernel principle solves it. The problem of too wide predictions can be reduced by minimizing the numbers and lengths of symmetry descent paths (see the applications in this study). 

In Td point group, a single electron occupies e orbital. The electronic ground state is E. After the symmetry descent to Dxd the later splits into Ai and In order to obtain JT parameters calculation recipe discussed in Sect. 3 is applied. Calculation method is summarized in Fig. 2 and results are given in Table 2. 

Let us suppose that within an initial group G0 the state vectors corresponding to the degenerate energy level E are transformed according to the irreducible representation 7 When the symmetry descent occurs to a subgroup G, the degenerate level E is split into the levels Et, E2, each corre-... 

Another problem originates in the fact that a multimode vibronic coupling should be considered in higher orders of the theory at least the mode is to be included beginning with the second order. Therefore the analytic forms of the adiabatic potential surfaces need a rederivation. Finally, some structural transitions should be either included or deleted from the Jahn-Teller mechanism, and the symmetry descent concept is useful along these lines. 

In other words, the Jahn-Teller distortion causes a descent in the symmetry G — G and the original electronic state Sd(G°) is split yielding the nondegenerate term Sn(G ). The symbol G° marks the symmetry point group of the reference system. A symmetry descent generates the n-th level subgroup G". For the first-level subgroup G <= G ... 

The stable (equilibrium) geometry corresponds to the nondegenerate electronic state, Sn(G ), described by the one-dimensional irreducible representation T(Sn). Otherwise the system continues in symmetry descent. 

In some cases two or more sets of cartesian components (say - y, xy or xz, yz in D3) belong to the same multidimensional irreducible representation so that a linear combination of them should be considered. Hence, the symmetry descent technique is applicable. 

To determine the other terms multiplicities of the oilier two direct products, the general method, due to Bethe, of the symmetry descent is applied it is based on the observation according which such operation of decreasing symmetry in the crystal field (or of ligands) does not affect the spin, and therefore neither its multiplicity. 

---