Irreducible representation Direct 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. 

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. 

The group (E, J) has only two one-dimensional irreducible representations. The representations of 0/(3) can therefore be obtained from those of 0(3) as direct products. The group 0/(3) is called the three-dimensional rotation-inversion group. It is isomorphic with the crystallographic space group Pi. 

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... 

Since Qh is a direct product group, its irreducible representations are also direct products. We denote a representation of <3A by a Young diagram giving the representation of S , together with a letter g or u according as the representation is even or odd with respect to to-... 

Here we find a new concept, the direct product between irreducible representations of a symmetry group. This direct product is related to the product of their corresponding space functions. For our purposes, we will only mention that the direct product between two, Pj and A, (or more) irreducible representations of a group is a new... 

The next example is provided to clarify how to work with direct products between irreducible representations. Below, in Section 7.6, we will apply the selection rule given by Equation (7.10) to relevant examples. 

EXAMPLE 7.4 The direct product between irreducible representations of group O. 

Similarly, we could perform other direct products between irreducible representations of the 0 group, and then decompose them into irreducible representations of this group. We would obtain ... 

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-... 

Distribution of the two additional electrons to 8 required for dianion formation among the degenerate LUMO orbitals of E, symmetry gives rise to four new states, since, within the 5v symmetry group, the direct product Ej Ej may be reduced to a sum of Aj, and 2 irreducible representations. The A2 state represents a triplet, while Aj and 2 are singlet states. 

These matrix elements are nonzero by spatial symmetry only if the direct products r, (8)rj and share a common irreducible representation [58]. 

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. 

Problem 10-5. In a homonuclear diatomic molecule, taking the molecular axis as z, the pair of LCAO-MO s tpi = 2p A + PxB and tp2 = 2 PyA + 2 PyB forms a basis for a degenerate irreducible representation of D h, as does the pair 3 = 2pxA PxB and 4 = PxA — PxB Identify the symmetry species of these wave functions. Write down the four-by-four matrices for the direct product representation by examining the effect of the group elements on the products 0i 03, 0i 04, V 2 03) and 02 04- Verify that the characters of the direct product representation are the products of the characters of the individual representations. 

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. 

This reducible representation (the occupancy of two e orbitals in the anion gives rise to more than one state, so the direct product e x e contains more than one symmetry component) can be decomposed into pure symmetry components (labels T are used to denote the irreducible symmetries) by using the decomposition formula given in Appendix E ... 

In general, the direct product representations are reducible and using the formulae of 7-4 we have, if T are irreducible representations... 

To what irreducible representations can the following direct product representations be reduced for the specified point group ... 

If we consider in the direct product representation rH P then since Hitf belong to P, rH P = P and therefore TH — P. Hence, any operator which commutes with all 0M of a point group can be said to belong to the totally symmetric irreducible representation P. 

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... 

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