Basis of an irreducible representation

The application of group theory to the construction of molecular orbitals leads us to study the way in which atomic orbitals (or linear combinations of these orbitals) transform when the operations of the point group are applied. Several definitions, which are also appropriate for other types of function besides orbitals, will be presented and illustrated by some simple examples. 

If a set of functions/ = fi,fz, , fi, , fn is such that any symmetry operation, Ru, of the group G transform one of the functions,/, into a linear combination of the various functions of the set/, the set is said to be globally stable and to constitute a basis for the representation of the group G. As the symmetry operations maintain the positions of the atoms or interchange the positions of equivalent atoms, it can be shown that the set of atomic orbitals (AO) of a molecule constitute a basis for the representation of the point-group symmetry of the molecule. In what follows, we shall adopt the usual notation in group theory, and indicate a basis for a representation by T. 

Suppose that a basis F, of dimension n, can be decomposed into several bases F , whose dimensions are smaller (n,), each of which is globally stable with respect to all the symmetry operations of the group. Suppose also that it is not possible to decompose any of the representations Fi into representations whose dimensions are smaller than n . The reducible representation F is said to have been decomposed into a sum of irreducible representations F,-, which is written  

To illustrate this point, we shall consider the set of valence AO of the atoms in the water molecule Ishi on the hydrogen atoms and 

and 2pz on the oxygen atom. The water molecule possesses a two-fold rotation axis (z), and two planes of symmetry, xz andyz (6-14) its point group is therefore Czv The set of atomic orbitals constitutes a basis (F) for the representation of this point group. 

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As the starting geometries for iterative calculation, we take all the possible structures in which bond lengths are distorted so that the set of displacement vectors may form a basis of an irreducible representation of the full symmetry group of a molecule. For example, with pentalene (I), there are 3, 2, 2 and 2 distinct bond distortions belonging respectively to a, b2 and representations of point group D21,. 

Equation (30j is called the reduction spectrum of the tensor T n). Each irreducible tensor of rank n and weight J has 3" components labeled by n indices, and its 3" components form the basis of an irreducible representation of the three-dimensional rotation group of degree 27 -h 1. Elence, each irreducible tensor with weight J has U 4- 1 independent components. The superscript y distinguishes between different components of identical weight. For example, the reduction spectrum of the hyperpolarizability tensor f3 (rank three) is ... 

They form a basis of an irreducible representation of the symmetry group of the moleeule, or in other words, they transform aeeording to this irredueible representation. 

In particular, if Q and Qi correspond to the same classical frequency Vk = vi, then the two corresponding wave functions of the type given in (2) will correspond to the same energy and will form the basis of an irreducible representation of the point group of the molecule, just as do Qk and Qi themselves. In fact, the representation formed by the s will be identical with that formed by the Q s. 

Irreducible Co-Representations of Nonunitary Croups.—To determine the co-representation matrices D , let us select a set of l functions, which forms a basis for an irreducible representation A (u) of the unitary subgroup H. That is... 

For G2v, each of the functions x,y, and z taken alone forms a basis for an irreducible representation for example,... 

Suppose that, instead of writing the secular determinant from an n x n array of atomic orbitals, we use an n x n array of n orthonormal, linear combinations of the basis set orbitals. Suppose, furthermore—and this is the key—we require these linear combinations to be SALCs, that is, each one is required to be a function which forms a basis for an irreducible representation of the point group of the molecule. Then, as shown in Chapter 5, all integrals of the types... 

Consider next the water molecule. As we have seen, it has a dipole moment, so we expect at least one IR-active mode. We have also seen that it has CIt, symmetry, and we may use this fact to help sort out the vibrational modes. Each normal mode of iibratbn wiff form a basis for an irreducible representation of the point group of the molecule.13 A vibration will be infrared active if its normal mode belongs to one of the irreducible representation corresponding to the x, y and z vectors. The C2 character table lists four irreducible representations A, Ait Bx, and B2. If we examine the three normal vibrational modes for HzO, we see that both the symmetrical stretch and the bending mode are symmetrical not only with respect to tbe C2 axis, but also with respect to the mirror planes (Fig. 3.21). They therefore have A, symmetry and since z transforms as A, they are fR active. The third mode is not symmetrical with respect to the C2 axis, nor is it symmetrical with respect to the ojxz) plane, so it has B2 symmetry. Because y transforms as Bt, this mode is also (R active. The three vibrations absorb at 3652 cm-1, 1545 cm-1, and 3756 cm-, respectively. 

Finally it should be noted that if the molecular system has any spatial symmetry, i.e. if there is a point group 0 whose operations St all commute with H, then each function sk in equation (7) must be replaced by a set which forms a basis for an irreducible representation A of 0 ... 

The (2 / + 1) eigenfunctions /. m) for a given value of j and for m values ranging from j in integer steps down to — j are transformed among themselves and with no other functions by the operators Jz and. J, and hence by rotations in general. They thus form the basis for an irreducible representation of dimension (2j + 1) which must be an integer. From this we see that j can take the possible values 0,1 /2,1,3/2,2, ... 

The close relationship between symmetry and vibration is expressed by the following rule Each normal mode of vibration forms a basis for an irreducible representation of the point group of the molecule. 

The vibrational wave function, as any wave function, must form a basis for an irreducible representation of the molecular point group [3],... 

As far as electron-electron interaction is neglected, the Hamiltonian // a tt electron in a CNT commutes with all the element of G making, according to the basic theory of group representation [22], the electronic eigenfunctions a set of basis functions for the Irreducible representations of G. In fact, the basis functions = , of an irreducible representation of dimension I are characterized by the property... 

The d>s functions form a basis for an irreducible representation of the group of permutations of N objects ( the symmetric group ). The particular representation is specified by S and N the total spin of the system S and the total number of electron N. The matrices of the particular representation are denoted by They are of dimension fs, where... 

It can be shown also from Schur s lemma that a 3-F symbol can be defined by means of the coupling coefficients as in Eq. (5). Therefore the reduction of the irreducible product of the two sets ai A and a2 A, each forming a basis for an irreducible representation, can be written in terms of the 3-F symbol,... 

It is possible to be more general than this and state that every eigenfunction of Eq. 5.39 must form a basis for an irreducible representation of Civ if the operator X is only invariant under these four symmetry transformations. The phrase basis for an irreducible representation means that the functions in Eq. 5.40 generate the matrices of an irreducible representation under the point group linear operators. In the same sense a vector along the z axis of a Car system (with satisfies the relations... 

The canonical spin orbitals, which are a solution to this equation, will generally be delocalized and form a basis for an irreducible representation of the point group of the molecule, i.e., they will have certain symmetry properties characteristic of the symmetry of the molecule or, equivalently, of the Fock operator. Once the canonical spin orbitals have been obtained it would be possible to obtain an infinite number of equivalent sets by a unitary transformation of the canonical set. In particular, there are various criteria (see Further Reading) for choosing a unitary transformation so that the transformed set of spin orbitals is in some sense localized, more in line with our intuitive feeling for chemical bonds. 

Fig. C.4. Each energy level corresponds to an irredueible representation of the symmetry group of the Hamiltonian. Its linearly independent eigenfunctions corresponding to a given level form the basis of the irreducible representation, or in other words, transform according to this representation. The number of the basis functions is equal to the degeneracy of the level.

Let us now take a set of variables x[ - x which form a basis for an irreducible representation r / of a group. We also take a set of variables yi Vm which form a basis for an irreducible representation Py/ of the same group. We have then... 

Within the Bom-Oppenheimer approximation, the exact stationary states form a basis for an irreducible representation of the molecular point group. We may enforce the same spatial symmetry on the approximate state by expanding the wave function in determinants constmcted from a set of symmetry-adapted orbitals. For atoms, in particular, the use of point-group... 

This is an immediate consequence of the lowering of the symmetry as, even in the regular octahedral geometry, group theory tells us that the highest dimension of the irreducible representation is three. This is the basis of Crystal Field Theory, whose deeply symmetry-based formalism was developed by Bethe in 1929 [16]. 

Each set of four numbers ( 1) constitutes an irreducible representation (i.r.) of the symmetry group, on the basis of either a coordinate axis or an axial rotation. According to a well-known theorem of group theory [2.7.4(v)], the number of i.r. s is equal to the number of classes of that group. The four different i.r. s obtained above therefore cover all possibilities for C2V. The theorem thus implies that any representation of the symmetry operators of the group, on whatever basis, can be reduced to one of these four. In summary, the i.r. s of C2 are given by Table 1. 

If there is a similarity transformation (or, what is the same thing, a change of basis) such that all the matrices in some representation T are brought into identical block form, then T is said to be a reducible representation. If there is not, then T is said to be an irreducible representation. 

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