The irreducible representation

In these irregular cases the negative sign applies in eq. (18) and 

Thus for improper point groups that are formed by the DP of a proper point group with C the character is a class property which is zero for all irregular classes, namely those formed from rotations about proper or improper BB axes. All other improper point groups are isomorphous ( ) with a proper point group and have the same characters and representations as that proper point group. For example C2v D2 D2d C4V D4. 

We now have all the necessary machinery for working out the matrix elements l in the MRs of the proper rotations R in any point group for any required value of The l are given in terms of the Cayley-Klein parameters a, b and their CCs by eq. (11.8.43). The parameters a, b may be evaluated from the quaternion parameters X, A for R, using 

Because b = 0, the only non-vanishing matrix elements 1 are those for which k=0 and 

Exercise 12.8-1 Justify the remark above eq. (4) that, in order for the matrix element to remain finite, the exponent of a must vanish when a is zero. 

---

Having done this we solve the Scln-ddinger equation for the molecule by diagonalizing the Hamiltonian matrix in a complete set of known basis fiinctions. We choose the basis functions so that they transfonn according to the irreducible representations of the synnnetry group. 

As a result the eigenstates of // can be labelled by the irreducible representations of the synnnetry group and these irreducible representations can be used as good quantum numbers for understanding interactions and transitions. 

The characters of the irreducible representations of a synnnetry group are collected together into a character table and the character table of the group 3 is given in table A1.4.3. The construction of character tables for finite groups is treated in section 4.4 of [2] and section 3-4 of [3]. 

In applications of group theory we often obtain a reducible representation, and we then need to reduce it to its irreducible components. The way that a given representation of a group is reduced to its irreducible components depends only on the characters of the matrices in the representation and on the characters of the matrices in the irreducible representations of the group. Suppose that the reducible representation is F and that the group involved... 

The irreducible representations of a symmetry group of a molecule are used to label its energy levels. The way we label the energy levels follows from an examination of the effect of a synnnetry operation on the molecular Sclnodinger equation. 

The rotation-vibration-electronic energy levels of the PH3 molecule (neglecting nuclear spin) can be labelled with the irreducible representation labels of the group The character table of this group is given in table Al.4.10. 

It is recommended that the reader become familiar with the point-group symmetry tools developed in Appendix E before proceeding with this section. In particular, it is important to know how to label atomic orbitals as well as the various hybrids that can be formed from them according to the irreducible representations of the molecule s point group and how to construct symmetry adapted combinations of atomic, hybrid, and molecular orbitals using projection operator methods. If additional material on group theory is needed. Cotton s book on this subject is very good and provides many excellent chemical applications. 

More generally, it is possible to combine sets of Cartesian displacement coordinates qk into so-called symmetry adapted coordinates Qrj, where the index F labels the irreducible representation and j labels the particular combination of that symmetry. These symmetry adapted coordinates can be formed by applying the point group projection operators to the individual Cartesian displacement coordinates. 

The basic idea of symmetry analysis is that any basis of orbitals, displacements, rotations, etc. transforms either as one of the irreducible representations or as a direct sum (reducible) representation. Symmetry tools are used to first determine how the basis transforms under action of the symmetry operations. They are then used to decompose the resultant representations into their irreducible components. 

The functions put into the determinant do not need to be individual GTO functions, called Gaussian primitives. They can be a weighted sum of basis functions on the same atom or different atoms. Sums of functions on the same atom are often used to make the calculation run faster, as discussed in Chapter 10. Sums of basis functions on different atoms are used to give the orbital a particular symmetry. For example, a water molecule with symmetry will have orbitals that transform as A, A2, B, B2, which are the irreducible representations of the C2t point group. The resulting orbitals that use functions from multiple atoms are called molecular orbitals. This is done to make the calculation run much faster. Any overlap integral over orbitals of different symmetry does not need to be computed because it is zero by symmetry. 

Most ah initio calculations use symmetry-adapted molecular orbitals. Under this scheme, the Hamiltonian matrix is block diagonal. This means that every molecular orbital will have the symmetry properties of one of the irreducible representations of the point group. No orbitals will be described by mixing dilferent irreducible representations. 

R = (i/ r) require translations t in addition to rotations j/. The irreducible representations for all Abelian groups have a phase factor c, consistent with the requirement that all h symmetry elements of the symmetry group commute. These symmetry elements of the Abelian group are obtained by multiplication of the symmetry element./ = (i/ lr) by itself an appropriate number of times, since R = E, where E is the identity element, and h is the number of elements in the Abelian group. We note that N, the number of hexagons in the ID unit cell of the nanotube, is not always equal h, particularly when d 1 and dfi d. 

Similarly, it can be shown that the nanotube modes at the T-point obtained from the zone-folding eqn by setting Ai = 1), where 0 < ri < N/2, transform according to the , irreducible representation of the symmetry group e. Thus, the vibrational modes at the T-point of a chiral nanotube can be decomposed according to the following eqn... 

The proof of this theorem follows from theorem A A four-by-four matrix that commutes with the y commuted with their products and hence with an arbitrary matrix. However, the only matrices that commute with every matrix are constant multiples of the identity. Theorem B is valid only in four dimensions, i.e., when N = 4. In other words the irreducible representations of (9-254) are fourdimensional. 

Case (c).- a0< produces a set of functions , which is independent of the set la, which forms a basis for the irreducible representation Ay(u) of H, which is inequivalent to A (u), but which has the same dimension. Df corresponds to two inequivalent irreducible representations of H, A (u), and A (u), such that in this case the anti-unitary operators cause A (u) and A (u) to become degenerate. 

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... 

Table 12-4 gives the characters, basis functions, and the case a, b, or c to which the irreducible representation of Hu and belong for the point T. The degeneracy in and is the usual Kramers spin degeneracy, which is removed in cases (2) and (4) because of the absence of 6 in these symmetry groups. 

Spin Warns.—In Application to Point Groups, above, we considered the irreducible representations of magnetic point groups. These would be useful in obtaining the symmetry properties of localized states in magnetic crystals, as impurity or single ion states in the tight... 

---