Symmetry operations product

H2O Models for Identifying the Results of Symmetry Operation Products... 

Whenever a fiinction can be written as a product of two or more fiinctions, each of which belongs to one of the synnnetry classes, the symmetry of the product fiinction is the direct product of the syimnetries of its constituents. This direct product is obtained in non-degenerate cases by taking the product of the characters for each symmetry operation. For example, the fiinction xy will have a symmetry given by the direct product of the syimnetries of v and ofy this direct product is obtained by taking the product of the characters for each synnnetry operation. In this example it may be seen that, for each operation, the product of the characters for Bj and B2 irreducible representations gives the character of the representation, so xy transfonns as A2. 

Except for the multiplication of by we follow the rules for forming direct products used in non-degenerate point groups the characters under the various symmetry operations are obtained by multiplying the characters of the species being multiplied, giving... 

Applying the above symmetry formulation to armchair (n = m) and zigzag (m = 0) nanotubes, we find that such nanotubes have a symmetry group given by the product of the cyclic group and Cj , where 2n consists of only two symmetry operations the identity, and a rotation by 2ir/2n about the tube axis followed by a translation by T/2. Armchair and zig-... 

It is noted that two successive symmetry transformations of a system leave that system invariant. The product of the two operations is therefore also a symmetry operation of the system. The set of symmetry transformations is therefore closed under the law of successive transformations. An identity transformation that leaves the system unchanged clearly belongs to the set. It is not difficult to see that any given symmetry transformation has an inverse that also belongs to the set. Since successive transformations of the set obey the associative law it finally follows that the set constitutes a group. 

In order to apply the direct product representation to the derivation of selection rules, recognize that a matrix element of the form ipi, O lpj) will be equal to zero for symmetry reasons if there is even one symmetry operation that takes the integrand into its negative. The argument follows exactly the course of that of section 10.2. Thus the matrix element will vanish unless the direct product representation is totally symmetric (Ai), or contains A upon reduction. 

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. 

We now return to the symmetry analysis of orbital products. Such knowledge is important because one is routinely faced with constructing symmetry-adapted N-electron configurations that consist of products of N individual orbitals. A point-group symmetry operator S, when acting on such a product of orbitals, gives the product of S acting on each of the individual orbitals... 

If the molecule is rotated around the z axis by 120° (360°/3), an equivalent configuration of the molecule is produced. The boron atom does not change its position, and the fluorine atoms exchange places depending upon the direction of the rotation. The rotation described is the symmetry operation associated with the C3 axis of symmetry, and the demonstration of its production of an equivalent configuration of the BF3 molecule is what is required to indicate that the C3 proper axis of symmetry is possessed by that molecule. 

Finally, a word of warning because the order of operations in a product is important (the one to the right always being carried out first), one must be careful when manipulating equations involving symmetry operations, for example, if PQ = R then we can write TPQ = TR (combining T on the left of both sides) or PQT = RT... 

Before showing further applications of direct-product representations to quantum mechanics, we quote without proof a theorem we will need. Let rij a and rkip be the elements of the matrices corresponding to the symmetry operation R in the two different nonequivalent irreducible representations Ta and T it can be shown that... 

The mn product functions (9.127) are thus transformed into linear combinations of one another by the symmetry operators of the group they therefore form a basis for some mn-dimensional representation Ty of the group. The representation T, is called the direct product (or Kronecker product) of the representations and TG this is symbolized by... 

This is the desired result The character of any symmetry operation in the direct-product representation TC is the product of its characters in the representations TF and Tc. (The direct product of matrices is not, in general, commutative however, A<8>B and B A have equal traces, and thus the corresponding direct-product representations are equivalent to each other.)... 

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 ... 

---