Symmetry operation simple

In Section 4.3.f it was shown that there are 3N — 5 normal vibrations in a linear molecule and 3N — 6 in a non-linear molecule, where N is the number of atoms in the molecule. There is a set of fairly simple rules for determining the number of vibrations belonging to each of the symmetry species of the point group to which the molecule belongs. These rules involve the concept of sets of equivalent nuclei. Nuclei form a set if they can be transformed into one another by any of the symmetry operations of the point group. For example, in the C2 point group there can be, as illustrated in Figure 6.18, four kinds of set ... 

Thus, the planes of the lattice are found to be important and can be defined by moving along one or more of the lattice directions of the unitcell to define them. Also important are the symmetry operations that can be performed within the unit-cell, as we have illustrated in the preceding diagram. These give rise to a total of 14 different lattices as we will show below. But first, let us confine our discussion to just the simple cubic lattice. 

The hfs (or quadrupole) tensors of geometrically (chemically) equivalent nuclei can be transformed into each other by symmetry operations of the point group of the paramagnetic metal complex. For an arbitrary orientation of B0 these nuclei may be considered as nonequivalent and the ENDOR spectra are described by the simple expressions in (B 4). If B0 is oriented in such a way that the corresponding symmetry group of the spin Hamiltonian is not the trivial one (Q symmetry), symmetry adapted base functions have to be used in the second order treatment for an accurate description of ENDOR spectra. We discuss the C2v and D4h covering symmetry in more detail. 

A fundamental characteristic of isotactic polymers is the presence of translational symmetry with periodicity equal to a single monomer unit In the representation 4 and 5 successive monomer units can be superimposed by simple translation (30-32). In a syndiotactic structure this superimposition is not possible for two successive groups. The corresponding symmetry operator, if one... 

An Abelian group has each element in a class by itself. The converse of this theorem is also true A group with each element in a class by itself is Abelian. The proof is simple By hypothesis C 1AC=A for all elements A and C left multiplication by C gives AC—CA. Q.E.D. The number of symmetry operations in each class of a point group is indicated by an integer in front of the symbol for the symmetry operation, and it is therefore easy to see whether a group is Abelian by looking at the top line of the character table. 

We can now show that the eigenfunctions for a molecule are bases for irreducible representations of the symmetry group to which the molecule belongs. Let us take first the simple case of nondegenerate eigenvalues. If we take the wave equation for the molecule and carry out a symmetry operation, / , upon each side, then, from 5.1-1 and 5.1-2 we have... 

Finally, in class 7 we have four types of symmetry operation (1) simple (unit) translation (2) transverse reflection (3) twofold rotation and (4) glide reflections. As in class 5, not all of these symmetry operations are independent. If we begin with class 1 and introduce explicitly only the glide reflection and one transverse reflection, all the other operations will arise as products of these. Again, this is analogous to the way point groups behave. 

The Mirror Plane, cr Most flowers, cut gems, pairs of gloves and shoes, and simple molecules have a plane of symmetry. A single hand, a quartz crystal, an optically active molecule, and certain cats at certain times4 do not possess such a plane. The symmetry element is a mirror plane, and the symmetry operation is the reflection of the molecule in the mirror plane. Some examples of molecules with and without mirror planes are shown in Fig. 3.1... 

For low order diagrams the determination of ao is quite simple. With a little experience one can easily identify by inspection those symmetry operations leaving the diagram invariant. For example diagram 5.2a ctd = 1 diagrams 5.3a, b Go = 2 diagram 5.3c = 2 2 = 4. (operations ... 

Table 4.1 shows that the MRs T(T) of the symmetry operators T e C3v for the basis (ei e2 e3 all have the same block-diagonal structure so that T = Tj T3. We shall soon deduce a simple rule for deciding whether or not a given representation is reducible, and we shall see then that T3 is in fact irreducible. 

There is more LC extended over two n.n. double-bonds within the (+) electron-hole symmetry class - and, of course, there is much more LC more extended - but we believe the two selected ones are the more important [37]. For instance, we saw that the S-LC is in the (-)class of symmetry, however, with the simple product form of the electron-hole symmetry operator, a LC with two S-LC - or more generally with an even number of S-LC - is in the (+) class of symmetry and should be considered here [37], However, this kind ofLC are sufficiently high in energy to be reasonably neglected. 

An example of I is the overlap integral with different atom. Integral I will vanish unless the integrand is invariant under all symmetry operations of the point group to which the molecule belongs. This condition is a generalization of the simple case of... 

There are three symmetry operations each involving a symmetry element rotation about a simple axis of symmetry (C ), reflection through a plane of symmetry (a), and inversion through a center of symmetry (i). More rigorously, symmetry operations may be described under two headings Cn and Sn. The latter is rotation... 

There are two types of molecular symmetry that cause chemical-shift equivalence. Nuclei or groups of nuclei that are interchangeable by a symmetry operation involving a simple n-fold axis of symmetry (Cn) have been termed equivalent, and are isochronous in chiral and... 

A simple, two-fold axis of symmetry (C2), and, hence, equivalence of the protons interchanged by this symmetry operation, is quite common in specific conformations of inositols and their derivatives, for example, the half-chair conformation (60) of myo-inosose-2 phe-nylosotriazole derivatives,187 and in derivatives of alditols having an even number of chain carbon atoms,188 for example, 2,3 4,5-dianhydro-D-iditol and its 1,6-diacetate and -benzoate189 (61). The 2,5-O-meth-... 

Since a representation—reducible or irreducible—is a set of matrices corresponding to all symmetry operations of a group, the representation can be described by the set of characters of all these matrices. For the simple basis of A/t and Ar2 used before for the HNNH molecule in the C2h point group, the representation consisted of four 2x2 matrices ... 

As mentioned before, the symmetry properties of the one-electron wave function are shown by the simple plot of the angular wave function. But, what are the symmetry properties of an orbital and how can they be described We can examine the behavior of an orbital under the different symmetry operations of a point group. This will be illustrated below via the inversion operation. 

The next step is to determine how these group orbitals transform in the D(,h point group. The D6h character table is given in Table 6-7. Since most of the AOs in the suggested group orbitals are transformed into another AO by most of the symmetry operations, the representations will be quite simple, though still reducible ... 

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