Basis for a representation

The matrix Rij,kl = Rik Rjl represents the effeet of R on the orbital produets in the same way Rik represents the effeet of R on the orbitals. One says that the orbital produets also form a basis for a representation of the point group. The eharaeter (i.e., the traee) of the representation matrix Rij,id appropriate to the orbital produet basis is seen to equal the produet of the eharaeters of the matrix Rik appropriate to the orbital basis Xe (R) = Xe(R)Xe(R) whieh is, of eourse, why the term "direet produet" is used to deseribe this relationship. 

The space spanned by all the j>, /> forms the basis for a representation of which we call the representation induced by y( ). To see the form of the matrices of this representation, consider an element s e applied to an arbitrary basis vector /> ... 

To an energy eigenvalue or state of 3C, there will generally correspond several independent eigenvectors or state functions i,. . . , n is the degeneracy of the state. These functions must form a basis for a representation of the group G if is invariant under G. If i2 is an element of G... 

Such a set of eigenfunctions must form the basis for a representation of the symmetry group of the Hamiltonian, because for every symmetry operation S, Tipi = pi implies that H Spi) = Sp>i) and hence that the transformed wave function Spi must be a linear combination of the basic set of eigenfunctions (/ ,... Pn-... 

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

The distribution of the molecular orbitals can be derived from the patterns of symmetry of the atomic orbitals from which the molecular orbitals are constructed. The orbitals occupied by valence electrons form a basis for a representation of the symmetry group of the molecule. Linear combination of these basis orbitals into molecular orbitals of definite symmetry species is equivalent to reduction of this representation. Therefore analysis of the character vector of the valence-orbital representation reveals the numbers of molecular orbitals... 

We have seen in the previous section that the definition of a set of OjS is intimately bound up with some choice of function space. The reader is cautioned, however, that not all function spaces can be used to define OjjS appropriate for a given point group. For example, the functions cosXi, sinx cosx and sinx, do not forma basis for a representation of the (symmetric tripod) point group xl and xt are the coordinates introduced before (see Fig. 5-2.2). 

CJonsequently, this function space does not provide a basis for a representation of the sv point group. 

The point of changing from Cartesian displacement coordinates to normal coordinates is that it brings about a great simplification of the vibrational equation. Furthermore, we will see that the normal coordinates provide a basis for a representation of the point group to which molecule belongs. 

Next consider the it orbitals of benzene. These MOs will be constructed using the six 2pz carbon AOs as the minimal basis set. These six AOs form the basis for a representation of 6D6A, which we shall call Tw. The characters are readily found by the methods discussed above for example, each 2pz AO goes into its negative upon reflection in the ah plane, so that X (A)= — 6. We find for the characters of Tw (see Fig. 9.4 for the conventions used)... 

Let the functions F ...,FW form a basis for a representation of some point group. Since a symmetry operation R amounts to a rotation (and possibly a reflection) of coordinates, it cannot change the value of a definite integral over all space we have... 

Since the functions (9.144) are transformed into linear combinations of one another by the symmetry operators of the group, they form a basis for a representation VK of dimension + 1) this representation is called the symmetric direct product of TF with itself. We write... 

Any set of algebraic functions or vectors may serve as the basis for a representation of a group. In order to use them for a basis, we consider them to be the components of a vector and then determine the matrices which show how that vector is transformed by each symmetry operation. The resulting matrices, naturally, constitute a representation of the group. We have previously used the coordinates jc, y, and z as a basis for representations of groups C2r (page 78) and T (page 74). In the present case it will be easily seen that the matrices for one operation in each of the three classes are as follows ... 

A thorough treatment of how the transformation properties of the rotations are determined would be an unnecessary digression from this discussion. In simple cases we can obtain the answer in a semipictorial way by letting a curved arrow about the axis stand for a rotation. Thus such an arrow around the 2 axis is transformed into itself by , it is transformed into itself by C3, and its direction is reversed by a . Thus it is the basis for a representation with the characters 1, 1, -1, and so we see that R transforms as A2. 

We use the four H Is orbitals as the basis for a representation we chose the coordinates and label the a functions as shown below. It is necessary to... 

For a two-dimensional representation, we require two orthogonal functions, which jointly form a basis for the representation. We have one, but we require its partner. To obtain it, we recall that any member of a set of functions forming the basis for a representation must be affected by the symmetry operations of the group in one of two ways ... 

Use the set of atomic orbitals as the basis for a representation of the group, and reduce this representation to its irreducible components. 

The naphthalene molecule belongs to the point group Dlh. The set of 10 pn orbitals may be used as the basis for a representation, Tn, of this group. 

As pointed out in Section 6.3, for the (CH)3 case, all the essential symmetry properties of the LCAOs we seek are determined by the operations of the uniaxial rotational subgroup, Cft. When the set of six pn orbitals is used as the basis for a representation of the group C6, the following results are obtained ... 

There is a very important feature of this situation, which we can turn to advantage. It will be observed that the set of four methylene carbon atoms, numbers 1, 4, 5, and 8, possess D4h symmetry by themselves and that the set of four carbon atoms in the ring, numbers 2, 3, 6, and 7, also by themselves constitute a set having D4ll symmetry. Furthermore, the atoms in one set are not equivalent symmetrically to any of those in the other. None of the outer atoms is ever interchanged with any of the inner atoms by any symmetry operation. Thus each of these sets can be used separately as the basis for a representation of the group, and if this is done we obtain from each set a representation, r x, which reduces as follows ... 

Figure 7.46 shows a numbered set of six pn atomic orbitals which will be used to construct the n MOs. Using these AOs as a basis for a representation of the group D3A, we obtain the following results ... 

To determine how to form a set of trigonally directed hybrid orbitals, we begin in exactly the same way as we did in the MO treatment. We use the three a bonds as a basis for a representation, reduce this representation and obtain the results on page 219. However, we now employ these results differently. We conclude that the s orbital may be combined with two of the p orbitals to form three equivalent lobes projecting from the central atom A toward the B atoms. We find the algebraic expressions for those combinations by the following procedure. 

We may use the set of five d wave functions as a basis for a representation of the point group of a particular environment and thus determine the manner in which the set of d orbitals is split by this environment. Let us choose an octahedral environment for our first illustration. In order to determine the representation for which the set of d wave functions forms a basis, we must first find the elements of the matrices which express the effect upon the set of wave functions of each of the symmetry operations in the group the characters of these matrices will then be the characters of the representation we are seeking. 

As shown in Table 9.3, the d orbitals form a basis for a representation of the group D4h that contains the irreducible representations A lR, Bljt, B2g, and r By referring to the character table for D4/ we can obtain the more specific information that the d orbitals correspond with these representations in the following way ... 

To find the contributions of the internal coordinates, C—H bond lengths and HCH angles, to these vibrational modes we first use the set of four C—H bond lengths as the basis for a representation, obtaining TCH shown below. 

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