Epikernel principle

According to the epikernel principle, the symmetry of the stable minima of the corannulene ion is Cs. If corannulene monoanion is a static JT system, the 11 peaks of the hyperhne structure with intensity ratio 1 10 45 120 210 252 210 120 45 10 1 due to the 10 equivalent hydrogens can be split into 3s = 243 peaks. 

Since the ab initio calculation on coronene monoanion indicates that the stable configuration has C2h symmetry, we must take higher-order anharmonic terms up to sixth order into consideration which is not considered in the derivation of the epikernel principle to obtain a JT surface with the adequate structure. We will discuss this point later. 

The permutational proof also elucidates two important general properties of JT potentials the epikernel principle, and the symmetry of the dynamic ground state. 

Since kernel K(G,A) is a subgroup of epikernel E(G,A), kernel extrema (if they exist) will be more numerous than epikernel extrema of a given type. In order to be stationary at all these equivalent points, the JT PES must be of considerable complexity. Only higher order term in the perturbation expansion (7) are able to generate non-symmetrical extrema. However - from a perturbational point of view -the dominance of higher order terms over the first (and second) order contributions is (extremely) unlikely. This rationalizes the epikernel principle as well. 

Thus the B3LYP obtained Cav structures are epikernels of the parent group in agreement with the epikernel principle. 

It must be mentioned that C2 structures have been obtained using MP2/cc-pVDZ treatment which are not predicted by epikernel principle for the e type JT coordinate (these are possible for the e" one only). Nevertheless, further theoretical studies using larger basis sets and more exact methods are desirable. 

It has been mentioned [25-27] that the optimized structure of the monoanionic state of coronene does not have (epikernel) D2h symmetry expected from the epikernel principle but has (kernel) C2h symmetry. Hence the JT distortion of coronene monoanion is an exception for the epikernel principle. This is because the epikernel... 

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... 

DshCE or E") C2v( Ai) or C2v("B2) This is in agreement with the epikernel principle since... 

The method of epikernel principle seems to be incomplete due to its restriction to the 1st order perturbation theory and linear extension of the perturbation potential. Using more complete perturbation may produce the results comparable with the other method on account of higher elaborateness. The JT caused loss of planarity or of symmetry center in JT systems can be explained by pseudo-JT mechanisms only. Another problem is the applicability to the groups with complex characters (C , S , and C h for n > 2, T and Th). 

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). 

An epikernel is an intermediate subgroup in the decomposition scheme of a given point group. The epikernel principle states that the preferred distortions of Jahn-Teller unstable molecules are directed towards the maximal allowed epikernels of the undistorted parent group. The group theoretical foundations of this principle are explained, and a wide variety of applications in different areas of chemistry is discussed. 

Previously we presented a detailed discussion of both antithetical aspects of the JT effect, on the basis of a case study of a tetrahedral instability [3]. The present article provides a generalized treatment, which focuses on the common structure of all types of JT instabilities. The treatment is followed by an exhaustive overview of the applications of the epikernel principle in all point group symmetries, except for the fourfold and fivefold degenerate representations in icosahedral symmetry. The purely mathematical aspects of this treatment have been published in a separate paper [4]. 

In this survey, we will not be concerned with the dynamic aspects of the JT theorem. In fact the concepts of symmetry destruction and symmetry conservation, to which the epikernel principle refers, are properties of the adiabatic potential energy surface only. When the kinetic energy of the nuclei is included, the JT coupling does not destroy the initial symmetry, but just replaces the electronic degeneracy by a vibronic degeneracy, due to dynamic tunneling between the different equivalent minima on the JT surface. 

Before proceeding with an exposition of the epikernel principle, it is well to stress that our approach is not limited to structural predictions. More generally, we are interested in the global properties of potential surfaces and the possibilities of chemical (isomerization or exchange) reactions as specific paths on these surfaces. Obviously, in the neighborhood of a degeneracy, the Jahn-Teller theorem and the epikernel principle will strongly influence the structure of the surfaces and the orientation of the preferred reaction paths. 

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