Products of Symmetry Operations

In Sections 3.3-3.6 we have often discussed the question of how we can represent the net effect of applying one symmetry operation after another to a molecule, but only in a limited way. In this section we shall discuss this question with regard to a broader range of possibilities. First, we shall establish a conventional shorthand for stating that operation X is carried out first and then operation Y, giving the same net effect as would the carrying out of the single operation Z. This we express symbolically as 

Note that the order in which the operations are applied is the order in which they are written from right to left, that is, YX means X first and then Y. In general, the order makes a difference although there are cases where it does not. When the result of the sequence XY is the same as the result of the 

One way in which we may approach the problem of finding a single operation which is the product of two others is to consider a general point with coordinates [xj, y, zj. On applying a certain operation, this point will be shifted to a new position with coordinates [x2, v2, z2] if still another operation is applied, it will again be shifted so that its coordinates are now [x3, y3, z3]. The net effect of applying the two operations successively is to shift the point from [x, yi, z,] to [x3, y3, z3]. We now look for a way of accomplishing this in one step. The operation which does so will be the product of the first two. 

whenever C2(x) and C2(y) exist, C2(z) must also exist, because it is their product. 

From these relations we can determine the effect of applying successively a(xz) and then C4(z), namely, 

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The associative law is obviously valid for products of symmetry operations. 

Products of symmetry operators mean carry out the operations specified successively, beginning with the one on the right. Thus, R2R means apply the operator R, first, and then R2. Since the product of two symmetry operators applied to some initial configuration e results in an indistinguishable configuration (r2 in Figure 2.2(d)), it is equivalent to a single symmetry operator R3=R2 R. For example,... 

Products of Symmetry Operations. Symmetry operations are operators that transform three-dimensional space, and (as with any operators) we define the product of two such operators as meaning successive application of the operators, the operator... 

A symmetry operation transforms an object into a position that is physically indistinguishable from the original position and preserves the distances between all pairs of points in the object. A symmetry element is a geometrical entity with respect to which a symmetry operation is performed. For molecules, the four kinds of symmetry elements are an n-fold axis of symmetry (C ), a plane of symmetry (cr), a center of symmetry (i), and an n-fold rotation-reflection axis of symmetry (5 ). The product of symmetry operations means successive performance of them. We have " = , where E is the identity operation also, 5, = o-, and Si = i, where the inversion operation moves a point at x,y, zto -X, -y, -z.Two symmetry operations may or may not commute. 

Fig. 6-4. Example showing that the products of symmetry operations may depend upon the order of application of the operations, cr,"C% CaVr", for the carbonate ion (see Fig. 5-3).

We have seen that the group of symmetry operations for the anunonia molecule leads to a particular group table of product operations (Table 13-7). Let us now see if we can assign a number or matrix to each symmetry operation such that the products of numbers satisfy the same group multiplication table relationships as do the products of symmetry operations. If we can find such a set of numbers or matrices, we say we have a representation for the group. 

That is, the characters for the product 1/ 11/ 2 are equal to the products of the characters for i/ i and 1/ 2- We have demonstrated the rule for one-dimensional representations, but it can be proved for higher-dimensional cases as well. In group theory, the product of two functions, like 1 11 2, is referred to as a direct product to distinguish it from a product of symmetry operations, like cr C. The symbol for a direct product is . 

When the result of a pair of operations depends on the order in this way it is said that the operators do not commute. In the multiplication of real numbers, the product does not matter on the order used, and so multiplication of simple numbers is commutative, whereas the product of symmetry operations may not commute. In the earlier example of H2O we did not see this order dependence of the products, and so there are some pairs of operations that do commute. 

From here and the symmetry condition = rjf it follows that TZl = 7 2-The operator B built into scheme (6) arranges itself as a product of triangle operators ... 

Symmetry operators leave the electronic Hamiltonian H invariant because the potential and kinetic energies are not changed if one applies such an operator R to the coordinates and momenta of all the electrons in the system. Because symmetry operations involve reflections through planes, rotations about axes, or inversions through points, the application of such an operation to a product such as H / gives the product of the operation applied to each term in the original product. Hence, one can write ... 

Here g is the order of the group (the number of symmetry operations in the group- 6 in this case) and %p(R) is the character for the particular symmetry T whose component in the direct product is being calculated. 

Let us first specify what we mean by a complete set of symmetry operations for a particular molecule. A complete set is one in which every possible product of two operations in the set is also an operation in the set. Let us consider as an example the set of operations which may be performed on a planar AB3 molecule. These are E, C3, Cjj, C2, C2, CJ, symmetry operations are possible. If we number the B atoms as indicated, we can systematically work through all binary products for example ... 

The final requirement, that every element of the group have an inverse, is also satisfied. For a group composed of symmetry operations, we may define the inverse of a given operation as that second operation, which will exactly undo what the given operation does. In more sophisticated terms, the reciprocal S of an operation R must be such that RS = SR = E. Let us consider each type of symmetry operation. For <7, reflection in a plane, the inverse is clearly a itself a x a = cr = E. For proper rotation, C , the inverse is C" m, for C x C "m = C" = E. For improper rotation, S , the reciprocal depends on whether m and n are even or odd, but a reciprocal exists in each of the four possible cases. When n is even, the reciprocal of S% is S m whether m is even or odd. When n is odd and m is even, S% — C , the reciprocal of which is Q m. For S" with both n and m odd we may write 5 = C a. The reciprocal would be the product Q ma, which is equal to and which... 

Using matrix methods, verify all the rules concerning the products and the commutation of symmetry operations given on pages 33-34. 

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 set ( bj) therefore closes. The other necessary group properties are readily proved and so G is a group. Direct product (DP) without further qualification implies the outer direct product. Notice that binary composition is defined for each group (e.g. A and B) individually, but that, in general, a multiplication rule between elements of different groups does not necessarily exist unless it is specifically stated to do so. However, if the elements of A and B obey the same multiplication rule (as would be true, for example, if they were both groups of symmetry operators) then the product at bj is defined. Suppose we try to take (a,-, bj) as a, bj. This imposes some additional restrictions on the DP, namely that... 

We already know from the invariance of the scalar product under symmetry operations that spatial symmetry operators are unitary operators, that is they obey the relation R R = R R1 E, where E is the identity operator. It follows from eq. (3.5.7) that the set of function operators / are also unitary operators. 

The A 2 representation of the C2v group can now be explained. The character table has four columns it has four classes of symmetry operations (Property 2 in Table 4-7). It must therefore have four irreducible representations (Property 3). The sum of the products of the characters of any two representations must equal zero (orthogonality. Property 6). Therefore, a product of A and the unknown representation must have 1 for two of the characters and — 1 for the other two. The character for the identity operation of this new representation must be 1 [x(i ) = 1 ] in order to have the sum of the squares... 

The advantage to this formulation is that higher dimensional evolution operators are replaced by a product of lower dimensional evolution operators. This is always a far easier computation. In addition, because products of evolution operators replace the full evolution operator, a variety of mathematical properties are retained, such as unitarity, and thus time reversal symmetry. 

These are indeed spherical tensor operators they are obviously related to step operators by means of = +A/V2. The typical group theoretical route for constructing bilinear forms respecting the SO(3) symmetry, contained in U(4), is to make use of tensor products of boson operators. 

The set of symmetry operations is ordinarily divided into two main classes. The product of the symmetry operations with the identity of the translation group (zero translation) yields the category termed essential space group operations. The other products of the symmetry operations with primitive lattice translations are termed nonessential... 

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