Kernel subgroup

We may see that all A2" characters of D h which are in the E, C3 and CTv headed columns (the corresponding operations constitute the C v group) are equal to -1-1. It means that these symmetry operations are conserved during any nuclear displacement (such as vibrations) described by IR A2" of Dsh group. The subgroup formed in this way is called kernel or kernel subgroup K(G, A) where G is the parent group and A denotes the IR of this displacement [11], In our case this relation is described by the formula... 

Cs(ah) denotes the kernel subgroup with Oh mirror plane of Dsh being preserved). 

Since character-by-character multiplication is commutative, and the direct product of any irrep of D2/1 with Ag leaves it unchanged, it follows that 7(field) = 61 the field must be set up parallel to the 2 axis. We also know that such a field will reduce the symmetry to because - as a glance at Table 2.2 will show - it is the kernel subgroup of... 

Desymmetrization to the co-kernel subgroups will become important in connection with the mechanism of reactions involving transition metal complexes as well as of highly symmetric organic molecules, such as cubane or tetrahedrane. A Table of Kernels and Co-Kernels is, therefore, included as Appendix B, but reference to it will be deferred until the the ideas developed in this chapter are extended beyond single atoms, first to diatomic and then to polyatomic molecules. 

Table 1. Kernel subgroups of one-diniei onal representations ...

The point group of a JT distorted molecule is a subgroup of the point group G of the parent molecule without JT distortion. In the JT active configuration space v, there exists a minimal subgroup which consists of such symmetry operations g in G that leave all allowed distortions invariant. Such a subgroup will be referred to as a kernel of v in G, K(G, v) ... 

Epikernels are intermediate subgroups between the parent group and the kernel... 

Theorem. Let G be an algebraic affine group scheme. Then 7c0(Jc[G]) represents an etale group n0 G, and all maps from G to etale groups factor through the canonical map G - jr0 G. The kernel G° of this map is a connected closed normal subgroup represented by the factor ofk[G] on which s is nonzero. The construction of ic0G and G° commutes with base extension. 

Theorem. Let G be an affine group scheme over a field. Let N be a closed normal subgroup. Then there is a quotient map G - H with kernel precisely N. 

A degenerate (multidimensional) representation describes the symmetry of a set of coordinates (such as vibrations). The elements of this set are called the components of the representation and span a multidimensional distortion space [10]. Because the degeneracy leaves the direction of the distortion unspecified, one has to scan all directions of the distortion space. A minimal subgroup of symmetry elements, which must be conserved in any case, consists of those symmetry operations that leave all distortions invariant and is said to form the kernel K(G, A) of the degenerate representation A. The kernel of the degenerate mode may be obtained from the character tables by collecting all symmetry operations with the same characters as the identity operation [11]. In the case of group for E type IR with all characters equal to -1-2 we obtain... 

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