Direct product, of groups

The original formulation of de Bruijn s theorem was for a quite general problem of this type, with a broad definition of the "weight" of a mapping. We assume that R is the union of a finite number of pairwise disjoint sets R- (i = 1,. .., k and that // is a direct product of groups //j, where //j acts on / j. For each there is a weight function where n is the number of elements of D that are... 

The complete nuclear pemnitation inversion (CNPI) group of the PH molecule is the direct product of the complete nuclear pemnitation (CNP) group 3 (see (equation Al.4.19)) and the inversion group P= E, E ]. This is a group of 12 elements that we call... 

Each such nonual mode can be assigned a synuuetry in the point group of the molecule. The wavefrmctions for non-degenerate modes have the following simple synuuetry properties the wavefrmctions with an odd vibrational quantum number v. have the same synuuetry as their nonual mode 2the ones with an even v. are totally symmetric. The synuuetry of the total vibrational wavefrmction (Q) is tlien the direct product of the synuuetries of its constituent nonual coordinate frmctions (p, (2,). In particular, the lowest vibrational state. 

Molecular point-group symmetry can often be used to determine whether a particular transition s dipole matrix element will vanish and, as a result, the electronic transition will be "forbidden" and thus predicted to have zero intensity. If the direct product of the symmetries of the initial and final electronic states /ei and /ef do not match the symmetry of the electric dipole operator (which has the symmetry of its x, y, and z components these symmetries can be read off the right most column of the character tables given in Appendix E), the matrix element will vanish. 

Clearly, the above procedure can be continued (in principle) as many times as required. Thus, if the wave function includes n = —4 3 paths, we have simply to dehne the function I 4((t)) = —+ 8ti), and then map onto the (j) = 0 16ti cover space, which will unwind the function completely. In general, if there are h homotopy classes of Feynman paths that contribute to the Kernel, then one can unwind ihG by computing the unsymmetrised wave function ih in the 0 2hn cover space. The symmetry group of the latter will be a direct product of the symmetry group in the single space and the group... 

The arrangement of die elements in the direct-product matrix follows certain conventions. They are illustrated in the following chapter, where the direct product of matrices is employed in the theory of groups. 

As indicated in Section 3.4, the integral of an odd function, taken between symmetric limits, is equal to zero. More generally, the integral of a function that is not symmetric with respect to all operations of the appropriate point group will vanish. Thus, if the integrand is composed of a product of functions, each of which belongs to a particular irreducible representation, the overall symmetry is given by the direct product of these irreducible representations. 

When spin-orbit coupling is introduced the symmetry states in the double group CJ are found from the direct products of the orbital and spin components. Linear combinations of the C"V eigenfunctions are then taken which transform correctly in C when spin is explicitly included, and the space-spin combinations are formed according to Ballhausen (39) so as to be diagonal under the rotation operation Cf. For an odd-electron system the Kramers doublets transform as e ( /2)a, n =1, 3, 5,... whilst for even electron systems the degenerate levels transform as e na, n = 1, 2, 3,. For d1 systems the first term in H naturally vanishes and the orbital functions are at once invested with spin to construct the C functions. 

Vector spaces which occur in physical applications are often direct products of smaller vector spaces that correspond to different degrees of freedom of the physical system (e.g. translations and rotations of a rigid body, or orbital and spin motion of a particle such as an electron). The characterization of such a situation depends on the relationship between the representations of a symmetry group realized on the product space and those defined on the component spaces. 

The group (E, J) has only two one-dimensional irreducible representations. The representations of 0/(3) can therefore be obtained from those of 0(3) as direct products. The group 0/(3) is called the three-dimensional rotation-inversion group. It is isomorphic with the crystallographic space group Pi. 

A tableau may be used to define certain subgroups of which are themselves direct products of smaller permutation groups the symmetrizing and antisymmetrizing operators for these subgroups lead, as we shall see, to projection operators on irreducible representations of... 

Given a tableau t, we first define the subgroup 1, consisting of all permutations q among sites in the same column of t. Evidently, 1 is just the direct product of the permutation groups for the sites in the... 

The examples used above to illustrate the features of the software were kept deliberately simple. The utility of the symbolic software becomes appreciated when larger problems are attacked. For example, the direct product of S3 (order 6) and S4 (isomorphic to the tetrahedral point group) is of order 144, and has 15 classes and representations. The list of classes and the character table each require nearly a full page of lineprinter printout. When asked for, the correlation tables and decomposition of products of representations are evaluated and displayed on the screen within one or two seconds. Table VII shows the results of decomposing the products of two pairs of representations in this product group. 

The direct product representation is usually reducible, unless both component representations are one-dimensional. For instance, in a group such as Dsh, in which no irreducible representation has dimension higher than two, the direct product of Ei and E2 will be four-dimensional, and thus it must be reducible. 

In this section we shall first treat the simple molecular orbital description of pyridine. Each molecular energy level corresponds to a configuration, specified by the occupancy of individual molecular orbitals. Each molecular orbital has the symmetry species of an irreducible representation of the symmetry group, C2v The spatial symmetry of the overall molecular wave function is the direct product of the symmetry species of the occupied orbitals. 

In contrast, the n ==> Jt transition has a ground-excited state direct product of B2 x Bj = A2 symmetry. The C2V s point group character table clearly shows that the electric dipole operator (i.e., its x, y, and z components in the molecule-fixed frame) has no component of A2 symmetry thus, light of no electric field orientation can induce this n ==> Jt transition. We thus say that the n ==> 7t transition is El forbidden (although it is Ml allowed). 

An apparent order in nitrite of 3 or more would also be consistent with a-amino nitrite fragmentation mechanisms if one assumes that nitrite is preferentially consumed in redox or nitrosation reactions elsewhere in the molecule which compete with nitrosation of the dimethylamino group. One such possibility was suggested by Dr. R.N. Loeppky (private communication), as shown in Fig. 6. This mechanism, which postulates the intermediacy of two different o-amino nitrites, le and If, should obey third order kinetics, since dimethylnitrosamine is produced only after aminopyrine reacts with the third mole of nitrite. Moreover, this pathway offers a mechanistic explanation for the direct production of nitrosohydrazide V, which has also been reported to be a product of aminopyrine nitrosation (12.17). 

The deacetyl compound (26-6) is now used for the direct production of cephalexin (25-3) as well as several other related agents that incorporate similar amide side chains. For example, reaction of the protected 7-ADCA derivative (27-2) with the fert-BOC amide from D- flra-hydroxyphenylglyeine (27-1) by the mixed anhydride method gives the amide (27-3). Serial scission of the tert-BOC group and the silyl ester affords the antibiotic cefadroxyl (27-4) [32]. Exactly the same sequence starting... 

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