Irreducible representations orthogonal

We will not be concerned further with the explicit forms of the co-representation matrices. Instead we need ask only to which of the three cases a specific representation A (u) of the group H belongs when H is considered as a subgroup of O. The co-representation matrices can be written down immediately once this is known. The irreducible representations of H can be obtained by standard means since H is unitary. It, therefore, remains to obtain a method by which one can decide between the three cases given the group 0 and an irreducible representation of H.9 In order to do this we need the fact that the matrices / and A (u) may be assumed to be unitary,6 and that the A((u) matrices satisfy the usual orthogonality relation... 

FIGURE 18. The symmetry-adapted, orthogonal linear combinations of the localized a-orbitals of norbornadiene 75 belonging to the irreducible representations / and 62 of the point group C2v- The A] and B2 combinations are the relay orbitals for through-bond interaction between jra and 7Tb which define, according to equation 56, the orbitals jt and jt... 

This equation (16) is known as the great orthogonality theorem for the irreducible representations of a group and occupies a central position in the theory of group representations. 

Each irreducible representation of a group consists of a set of square matrices of order lt. The set of matrix elements with the same index, grouped together, one from each matrix in the set, constitutes a vector in -dimensional space. The great orthogonality theorem (16) states that all these vectors are mutually orthogonal and that each of them is normalized so that the square of its length is equal to g/li. This interpretation becomes more obvious when (16) is unpacked into separate expressions ... 

A matrix of order l has l2 elements. Each irreducible representation T, must therefore contribute If -dimensional vectors. The orthogonality theorem requires that the total set of Y f vectors must be mutually orthogonal. Since there can be no more than g orthogonal vectors in -dimensional space, the sum Y i cannot exceed g. For a complete set (19) is implied. Since the character of an identity matrix always equals the order of the representation it further follows that... 

For every integral value ol m. there is an irreducible representation of 0(2), given by (23). The orthogonality theorem for characters in this case becomes... 

Here, Cys is the Cl vector in the basis of VB structures, projected such that it transforms according to the irreducible representation phi. Because even the standard CASVB approach involves an expansion of Fvb in terms of structures formed from orthogonal molecular orbitals (the transformation given in Eq. (41)), this implementation is completely straightforward. 

From eqn (7-2.1) (the Great Orthogonality Theorem) we can obtain for the non-equivalent irreducible representations T and Tv ... 

We are now in a position to show that two representations with a one-to-one correspondence in characters for each operation, are necessarily equivalent (see 7-3). If we consider two different nonequivalent irreducible representations then, since the characters are orthogonal (eqn (7-3.4)), there cannot be a one-to-one correspondence. If we consider two different reducible representations T° and Tb then, by eqn (7-4.2), if the characters are the same, the reduction will also be the same, that is the number of times occurs in P (a ) will, by the formula, be the same as the number of times T occurs in Fb. The reduced matrices can therefore be brought to the same form by reordering the basis functions of either Ta or Tb. The reduced matrices are therefore equivalent and necessarily Ta and Tb from whence the reduced matrices came (via a similarity transformation) must also be equivalent. Hence, we have proved our proposition. 

It is convenient if the symmetry orbitals belonging to a degenerate irreducible representation are made orthogonal to each other and this is achieved in the present case by taking combinations which are the sum... 

The factor lj/h, where h is the order of the group and b 18 the dimension of the y th irreducible representation, has been included in (9.67) for convenience. Application of this procedure to the functions / gives us (unnormalized) symmetry-adapted functions g,. This procedure is applicable to generating sets of functions that form bases for irreducible representations from any set of functions that form a basis for a reducible representation. The proof of the procedure (9.67) for one-dimensional representations is outlined in Problem 9.22 we omit its general proof.5 Symmetry-adapted functions produced by (9.67) that belong to the same irreducible representation are not, in general, orthogonal. 

This means that in the set of matrices constituting any one irreducible representation any set of corresponding matrix elements, one from each matrix, behaves as the components of a vector in /i-dimensional space such that all these vectors are mutually orthogonal, and each is normalized so that the square of its length equals hllh This-interpretation of 4.3-1 will perhaps be more obvious if we take 4.3-1 apart into three simpler equations, each of which is contained within it. We shall omit the explicit designation of complex conjugates for simplicity, but it should be remembered that they must be used... 

The vectors whose components are the characters of two different irreducible representations are orthogonal, that is,... 

The simplest way to illustrate physical meaning of these quantities is to consider the perturbations of orthogonally twisted ethylene for which SAB = yAB = <5ab = yab = 0 holds via (1) return to planarity or (2) substitution at one end of the C=C bond. For (1), localized orbitals interact, yAB 0, but their energies are the same, 5AB = 0. Since delocalized orbitals become eventually HOMO and LUMO of planar ethylene, they do not have the same energy, Sab 0, but they do not interact, yab = 0. For (2), orthogonal-substituted ethylene, the situation is different. In the localized basis SAB 0, but the interaction is not present due to the symmetry yAn = 0. (A and 2 S belong to different irreducible representations.) For the delocalized description the energies of these orbitals are the same 5ab = 0 since the orbitals are equally distributed over both carbon atoms. But yab 0, since a and b are not canonical orbitals. 

Two functions that belong to different irreducible representations are necessarily orthogonal to each other. If they belong to the same irreducible representation, they may not be orthogonal, and must be combined to produce a pair of orthogonal linear combinations. Thus the application of group-theoretical principles and the exploitation of molecular symmetry help to fulfil essential quantum-mechanical requirements in the construction of MOs. We now illustrate these principles by looking at the MOs of the H20 molecule. 

The vibration coordinates of interest are displacements in the plane orthogonal to the symmetry axis so they transform to the irreducible representation E. Later on, it will be useful to keep in mind these decompositions... 

Representation theory of molecular point groups tells us how a rotation or a reflection of a molecule can be represented as an orthogonal transformation in 3D coordinate space. We can therefore easily determine the irreducible representation for the spatial part of the wave function. By contrast, a spin eigenfunction is not a function of the spatial coordinates. If we want to study the transformation properties of the spinors... 

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