Symmetry, axes operation

For an Abelian group, each element is in a class by itself, since X 1AX = AX IX = A. Since rotations about the same axis commute with each other, the group e is Abelian and has n classes, each class consisting of one symmetry operation. 

Consider first the Ax SALC. It must have the same symmetry as the s orbital on atom A. This requires that it be everywhere positive and unchanged by all symmetry operations, and thus it must be gx + g2 + negative amplitude where it is negative. To match the p orbitals, on a 1 1 basis, the combinations clearly must be... 

In order to apply group theory to the physical properties of crystals, we need to study the transformation of tensor components under the symmetry operations of the crystal point group. These tensor components form bases for the irreducible repsensentations (IRs) of the point group, for example x, x2 x3 for 7(1) and the set of infinitesimal rotations Rx Ry Rz for 7(1 )ax. (It should be remarked that although there is no unique way of decomposing a finite rotation R( o n) into the product of three rotations about the coordinate axes, infinitesimal rotations do commute and the vector o n can be resolved uniquely... 

In many applications, the operator L is, besides being skew symmetric, also a Poisson operator, which means that (AX,LBX) is a Poisson bracket denoted hereafter by the symbol A,B A and B are real-valued functions (sufficiently regular) of x. We recall that the Poisson bracket, in addition to satisfying the skew symmetry A, B = — B,A, satisfies also the Jacobi identity A, B,C + B, C,A ... 

Many prochiral groups, however, cannot be interchanged by any type of symmetry operation, because of the presence of one or more chiral centers in the molecule. They are diastereotopic and anisochronous, and they constitute AB, AX, etc., systems. For this reason, when encountering prochiral groups, chemists should expect them to be diastereotopic and should be, perhaps, pleasantly surprised when they are not. 

The behavior of the matrix elements of the primitive functions with respect to symmetry operations of the helical chain is important in determining die character of the matrix elements in the first Brillouin zone. The -matrix element is completely symmetric, of course. Consider next, however, the matrix element, eq (F13). Gx-Ax can be rewritten as... 

It is important to note that no symmetry operation exchanges an axial ligands with an equatorial one. As these two types of ligands are therefore non-equivalent, both chemically and according to group theory, they can he considered separately. The characters of the representations Per (eq) and P (ax) are given in Table 6.22. From the reduction formula (6.5), we find ... 

The kinetic energy of a molecule possesses similar properties, except that it is a function of the velocities instead of the positions of the particles. At any instant the kinetic energy is determined b - the values of d Ax )/dt, d Aya)/dt, d Aza)/dt. These quantities may be thought of as components of velocity vectors, one for each atom. Fhe effect of a symmetry operation is then to transform these velocity vectors in just the same way as the displacement vectors are transformed. Therefore, the kinetic energy will have the. same numerical value in any two states of motion which are symmetrically related. 

Figure 3.7. Left the symmetry operations of the two-dimensional square lattice the thick straight lines indicate the reflections (labeled ax,ay,ai,as) and the curves with arrows the rotations (labeled C4, C2, C4). Right the Irreducible Brillouin Zone for the two-dimensional square lattice with full symmetry the special points are labeled F, S, A, M, Z, X.

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