Direct product of representations

The values of (x, x2) are instead obtained from the direct product of representations, discussed in Appendix B. They are given by... 

Many quantum-mechanical applications of group theory involve the direct product of representations, which we now discuss. 

The direct product of representations is basic to the determination of selection rules. We shall consider only transitions between vibrational levels of the same electronic state of a molecule, but the theory is applicable to electronic and rotational transitions as well. 

The direct products of representations of double groups can be taken in the usual way and reduced to sums of irreducible representations. 

The reprcscntalion for the overtone may be obtained by squaring the irreducible representation for fl, X 82, - <4 ). The result, /4 , transforms as a binary product and therefore is Raman active. For a discussion of direct products of representations as applied to overtones, see Footnote 24. 

Take the direct product of representations for any electron that is in a partially-filled orbital. If all the orbitals are filled, the result is the totally symmetric representation. The term state is labeled by the capital letter for the result of this direct product. For example, in a C2v molecule with all its orbitals filled, such as water, the term is labeled Aj. For ground state H2, with MO configuration Itr, the electronic state must also be labeled by the totally symmetric representation, in this case Xg. 

In order to investigate the symmetry properties of the product we can start by investigating the symmetry properties of the product and then comparing these to the symmetry properties of the dipole moment p. (These quantities commute.) In order to do this it is necessary to define the character of the direct product of representations. This is most easily done with an example where the direct product of the characters for the species (totally symmetric ground state) and the J 2 species (one type of excited state) for the C2V group is obtained as shown below. 

Equation [24] is also valid for the half-integral values / so that direct products of representations needed for the coupling of arbitrary angular momenta can be calculated. However, one should be aware that for half-integral values / the spherical harmonics due to their definition cannot serve as basis for the corresponding SU(2)-representations. 

A vibrational transitioa is allowed if at least one of these integrals differs from zero. The symmetry selection rule states that die integral ( n I c ( I m ) is not zero if the direct product of representations for the i[Pg.206]

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. 

If the system contains symmetry, there are additional Cl matrix elements which become zero. The symmetry of a determinant is given as the direct product of the symmetries of the MOs. The Hamilton operator always belongs to the totally symmetric representation, thus if two determinants belong to different irreducible representations, the Cl matrix element is zero. This is again fairly obvious if the interest is in a state of a specific symmetry, only those determinants which have the correct symmetry can contribute. 

As described above, the ground state vibrational wavefunction is totally symmetric for most common molecules. Therefore, the product, -(1)0 must at least contain a totally symmetric component. The direct product of two irreducible representations contains the totally symmetric representation only if the two irreducible representations are identical. Therefore, transitions can occur from a symmetrical initial state only to those states that have the same symmetry properties as the transition operator, 0. 

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. 

Suppose now that A) and B) belong to an electronic representation I ,. Since H is totally symmetric, Eq. (6) implies that the matrix elements (A II TB) belong to the representation of symmetrized or anti-symmetrized products of the bras (A with the kets 7 A). However, the set TA) is, however, simply a reordering of the set ( A). Hence, the symmetry of the matrix elements in the even- and odd-electron cases is given, respectively, by the symmetrized [Ye x Te] and antisymmetrized Ff x I parts of the direct product of I , with itself. A final consideration is that coordinates belonging to the totally symmetric representation, To, cannot break any symmetry determined degeneracy. The symmetries of the Jahn-Teller active modes are therefore given by... 

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. 

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... 

The calculated state energies, the transition moments, and the symmetry classification are given in Table 3. The symmetry species of the triplet functions is obtained by taking the direct product of irreducible representation of the space and the spin functions Fx, Fy, Fz, which transform as the rotations Rx, Ry, and Rz-... 

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. 

Table VII. Display of the decomposition of products of representations within the direct product group S3-DP-S4.

We denote the tensor of such elements as Dp, which is the tensor representation of the kernel Dp in a basis of p-electron direct products of the spin orbitals 4>j) [46]. The convention introduced in Eq. (8), that the number of indices implicitly specifies the tensor rank, is followed wherever tensors are used in this chapter. 

The set of products fkgt, forms a basis for a representation called the direct product of the representations F/ and F. ... 

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. 

The techniques used earlier for linear molecules extend easily to non-linear molecules. One begins with those states that can be straightforwardly identified as unique entries within the box diagram. For polyatomic molecules with no degenerate representations, the spatial symmetry of each box entry is identical and is given as the direct product of the open-shell orbitals. For the formaldehyde example considered earlier, the spatial symmetries of the n7t and %% states were A2 and Aj, respectively. 

---