Irreducible R-matrix

A category of reactions with a characteristic irreducible R-matrix is a set of basis reactions. The basis reactions correspond to the traditional classification of organic reactions . A basis reaction is best characterized in graph theoretical terms (ref. 13). The educts and the products of a basis reaction are expressed by a graph (see Fig. 7.2) whose nodes correspond to the reactive centers and whose lines indicate the bond orders of the covalent bonds that are directly affected by the reaction. The... 

An irreducible R-matrix is one that does not have a row/column containing all zeroes. Any two chemical reactions which are represented by the same irreducible R-matrix belong to the same R-category. The Brandt categories are characterized by R-matri-ces which may have up to three off-diagonal pairs of positive and negative entries. 

The elimination of all rows and columns containing only zeros from an R-matrix yields the corresponding irreducible R-matrix. 

Any two chemical reactions which are represented by the same irreducible R-matrix belong to the same R-category. 

W, the irreducible R-matrix the intact BE-matrix a, its associated atom vector... 

It is evident that the matrix elements of the Hamiltonian and overlap are independent of the index r of BT in Eq. (13) and only the first diagonal element of the irreducible representation matrix, D P), is required, which has been well discussed [31,33,42,43], and is easily determined. It is worthwhile to emphasize that Eqs. (21) and (22) are the unique formulas of the matrix elements in the spin-free approach, even though one can take some other forms of VB functions. For example, it is possible to construct VB functions by Young operator [2], but the forms of the matrix elements are identical to Eqs. (21) and (22) [44],... 

The beauty of this simple concept lies in the fact that an R-matrix may be applied to any ensemble of molecules (B) to show the products (E) characteristic of the reaction (R). Therefore, the basic irreducible R-matrices (the reaction core ) constitute the highest level of a hierarchy. These R-matrices are also equivalent to mathematical formulations of Arens operators placed on the matrices of the cyclic atoms, in order to be applied to the computer using matrix algebra. 

Familiarity is also assumed with the concepts of representation and irreducible representation (IR). A representation r of dimension n associates to each group element s an n X n matrix D(s), with matrix elements D(s)y, in such a way that for every s, t, D(s)D(t) =D(st), with the product formed by ordinary matrix multiplication. We will sometimes use the bra-ket notation... 

The matrix D R) is the matrix representative of R. Furthermore, the representation D R) is irreducible, so that to every energy state of 3C, an irreducible representation of G can be assigned. 

These matrix elements are nonzero by spatial symmetry only if the direct products r, (8)rj and share a common irreducible representation [58]. 

In addition, group theory can be used to assess when transition dipole moments must be zero. The product of the irreducible representations of the two wave functions and the dipole moment operator within the molecular point group symmetry must contain the totally symmetric representation for the matrix element to be non-zero (note that, if the molecule does not contain an inversion center, the operator r does not belong to any single irrep, except for the trivial case of Ci symmetry see Appendix B for more details). A consequence of this consideration is that, for instance, electronic transitions between states of the same symmetry are forbidden in molecules possessing inversion centers. 

Another correspondence between finite subgroups of SU(2) and Dynkin diagrams was given by McKay [55]. Let R0,Ri,---, Rn be the irreducible representations of T with R the trivial representation. Let Q be the 2-dimensional representation given by the inclusion T C SU(2). Let us decompose Q Rk into irreducibles, Q Rk = dkiRu where aki is the multiplicity. Then the matrix 27 — (aki)ki is an affine Cartan matrix of a simply-laced extended Dynkin diagram, An Dn E or... 

This theorem states that if T and Ty are two non-equivalent irreducible representations with matrices D iR) and Dv(fl) (of dimensions and nv, respectively) for each operation R of the group <3 then the matrix elements are related by the equation... 

For a given unitary irreducible representation r there will be nj matrix elements corresponding to each R. If the operations are R Rv Rtf. .. Rr, then the g matrix elements of a chosen i and j value,... 

This theorem states that if for some group there are two different irreducible representations T and TT with matrices DH(R) and Dy(/t) of dimension and nv respectively and if a rectangular matrix A exists such that... 

Let Rao be the matrix representing the symmetry operation R in the representation rAG. Equation (9.64) means that there is a similarity transformation that transforms RAO to RAO, where RAO is in block-diagonal form, the blocks being the matrices R],R2,...,Rfc of the irreducible representations ri,r2,...,ri. Let rAO denote the block-diagonal representation equivalent to TAO and let A be the matrix of the similarity transformation that converts the matrices of I AO to TAO Rao = A, RaoA. We form the following linear combinations of the AOs ... 

Since any operator can be written as the sum of Hermitian and anti-Hermitian operators, we can restrict our discussion to these two types only. Further, any operator can be written as a linear combination of irreducible symmetry operators, so we can restrict ourselves to irreducible tensor operators. An operator matrix 0(r, K) that transforms according to the symmetry (T, K) obeys the relationship... 

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