Permuted cyclically

For the irreducible representation A the symmetrized combination is easily found to be 0-1+02+ <73 The application of Eq. (47) for the representation E yields - 02 - 03. As the shhplMed projectifota operator has beefi employed in this example, the second combination of species E is not given directly. However, it is sufficient in this case to coi tiiict a secOiid litiear cOfffbtiia-don that is compatible with the symmetry Et and ofttiOgohal to the first. A direct method to find the appropriate combination is to permute cycliC iiy the functions obtained above, viz. 

These types of symmetry operation are graphically explained by the diagrams in Fig. 5A-2, where it is shown how an arbitrary point (0) in space is affected in each case. Filled dots represent points above the xy plane and open dots represent points below it. L t us examine first the action of proper rotations, illustrated here by the C4 rotations, that is, rotations by 2ir/4 = 90°. The operation C4 is seen to take the point 0 to the point 1. The application of C4 twice, designated C4, generates point 2. Operation C gives point 3 and, of course, C, which is a rotation by 4 x 2ir/4 = 2ir, regenerates the original point. The set of four points, 0, 1, 2, 3 are permutable, cyclically, by repeated C4 proper rotations and are equivalent points. It will be obvious... 

In the Plackett-Burman designs of 28 experiments for up to 27 factors, a set of three square blocks each of 9 rows and columns is permutated cyclically (table A2.3a). These blocks, labelled A, B, and C, are given in table A2.3b. 

By permuting cyclically h, /, g and summing up the results obtained, we discover that the first sum vanishes because the form a is closed, that is. 

Consider a triatomic system with the three nuclei labeled A, Ap, and Ay. Let the arrangement channel -1- A A be called the X arrangement channel, where Xvk is a cyclic permutation of apy. Let Rx,r be the Jacobi vectors associated with this arrangement channel, where r is the vector from A to and the vector from the center of mass of AyA to A . Let R i, rx be the corresponding mass-scaled Jacobi coordinates defined by... 

With 4) containing a normalization factor and all permutations over the atomic orbital wave functions i (1 = 1,2,... 2n). Likewise, if all electron pairs were exchanged in a cyclic manner, the product wave function, 4>b, has the fonn ... 

This equaciOTi, together with the corresponding results obtained from it by cyclic permutation of the suffixes L, 2, and 3, provides the required explicit form of the flux relations. 

E the cyclic group of order and degree 5, generated by cyclic permutations of s objects ... 

Others follow by cyclic permutations all of them can be symbolized conveniently by the vector relation... 

For this purpose, let us use invariance of the matrix product trace under cyclic permutation of factors and represent (7.11) as... 

The other operator is a cyclic permutation g = (13542) on the five ligands, where (did2d, ...,d ) means replace object by dj, dj by d3,...,d by dj. The use of the standard rules for calculating products of permutations yields ... 

Thus, the trace of the commutator [A, B] = AB - BA is equal to zero. Furthermore, the trace of a continued product of matrices is invariant under a cyclic permutation of... 

The corresponding relations for 62 and b- follow by cyclic permutations of the subscripts (see Chapter 8). 

The second mechanism, due to the permutational properties of the electronic wave function is referred to as the permutational mechanism. It was introduced in Section I for the H4 system, and above for pericyclic reactions and is closely related to the aromaticity of the reaction. Following Evans principle, an aromatic transition state is defined in analogy with the hybrid of the two Kekule structures of benzene. A cyclic transition state in pericyclic reactions is defined as aromatic or antiaromatic according to whether it is more stable or less stable than the open chain analogue, respectively. In [32], it was assumed that the in-phase combination in Eq. (14) lies always the on the ground state potential. As discussed above, it can be shown that the ground state of aromatic systems is always represented by the in-phase combination of Eq. (14), and antiaromatic ones—by the out-of-phase combination. 

The first term in (13), also called the diagonal term (Berry 1985), originates from periodic orbit pairs (p,p ) related through cyclic permutations of the vertex symbol code. There are typically n orbits of that kind and all these orbits have the same amplitude A and phase L. The corresponding periodic orbit pair contributions is (in general) g n - times degenerate where n is the length of the orbit and g is a symmetry factor (g = 2 for time reversal symmetry). 

We consider first the class structure of Sn. To do this, we note that every permutation may be written as the product of a number of independent cyclic permutations. Thus, the permutation s takes the ligand on site 1 to si, that on si to s(si), that on s(si) to s[s(si)], etc. Following this chain, since n is finite, we must eventually reach a site whose ligand is taken to site 1 by s. The closed chain evidently forms a cyclic permutation, which we will denote by writing the sites concerned in order enclosed in parentheses. Thus, (123.../) denotes a permutation s for which si =2, s2=3,. .. s/ i =/, s/= 1. If this first cycle does not include all the sites, we can do the same thing with the lowest-numbered site not appearing in the cycle, and continue until we have broken down the group element completely into a product of cycles.b>... 

The fifth class listed above contains a 1-cycle and a 2-cycle in its permutation part, with an odd number of reflections (in this case, one reflection) on sites involved in the two-cycle. The seventh class has a permutation part consisting of a cyclic permutation of the three sites, with an odd number of reflections (either one or three). 

By definition, a molecule is achiral if it is left invariant by some improper operation (reflection or rotary reflection) of the point group of the skeleton. Writing the permutation s corresponding to a given improper operation in cyclic form,... 

P> A transposition is the exchange of two adjacent elements of an ordered set. A cyclic permutation of n symbols contains n—1 transpositions. 

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