Irreps dimensions

The most powerful theorem in group theory, for our purposes, is the great orthogonality theorem (GOT) which states that for irreps D and D, of respective dimensions na and n, ... 

This theorem provides a connection between a degenerate irrep of dimension n — 1 and the existence of an orbit of n equivalent and equidistant sites. We will... 

The definitions of the perturbation corrections to the interaction energy, as given by Eqs. (8), (9), and (11), involve nonsymmetric operators, like H0, V, and Rq. These operators do not include all electrons in a fully symmetric way, so H0, V, and Rq must act in a larger space them the dimer Hilbert space Hab adapted to a specific irrep of 4. To use these operators we have to consider the space Ha Hb, the tensor product of Hilbert spaces Ha and Hb for the monomers A and B. For the interaction of two closed-shell two-electron systems this tensor product space should be adapted to the irreps of the 2 2 group. In most of the quantum chemical applications the Hilbert spaces Hx, X = A and B, are constructed from one-electron spaces of finite dimension Cx - In the particular case of two-electron systems in singlet states we have Hx — x x, where the symbol denotes the symmetrized tensor product. We define the one-electron space as the space spanned by the union of two atomic bases associated with the monomers in the dimer. The basis of the space includes functions centered on all atoms in the dimer and, consequently, will be referred to as the dimer-centered basis. We assume that the same one-electron space is used to construct the Hilbert spaces Hx 1 X = A and B, i.e., Hx = In such case Ha )Hb can be represented as a direct sum of Hilbert spaces H adapted to the irreps entering the induced product [2] [2] 4, i.e., Ha ( Hb = H[2i] f[3i] f[4]- This means that every function from H , v = [22], [31], or [4], can be expanded in terms of functions from Ha Hb-... 

Foldy representation for finite mass So far we have ignored the discrete operators P and T which generate the other 3 classes of It proves necessary to double the dimension of each irrep, so that for a representation (m,y) for mass m and spin y the dimension is 2(2y-l-1). Foldy [57] gives a canonical form in terms of a representation space of vector-valued functions P(ct,x) with inner product... 

Hence, every irrep occurs as many times as its own dimension. The sum of all these irreducible blocks must yield the regular representation. Thus one has... 

Theorem 6 Let L2 and T2 be two irreducible representations of a group G, and consider vectors formed by taking elements [i j] and kl from the respective representation matrices for every element of the group. Then these vectors are orthogonal to each other, and their squared norm is equal to the order of the group, divided by the dimension of the irrep ... 

Hence, if the multiplicity is greater than one, an additional label preceding the irrep label has to be introduced in order to distinguish SALCs with the same symmetry, and, by varying the / index of the projector, aU these can be projected out. Note that the maximal invariance space of a symmetry group is bound to be the regular representation hence, multiplicities of an invariant function space cannot exceed the dimensions of the irreps and thus will always be covered by the variation of index 1. If the multiplicity is smaller than dim(f2), variation of I wiU give rise to redundancies. 

Here, we have made use of the fact that the sum over all characters multiplied by the dimension of the irrep is the character of the regular representation, and this vanishes for all R except for the unit element, where it is equal to G (see Eq. (4.41)). In Dirac terminology this reads... 

The 19-dimensional Hiickel matrix thus will be resolved into five blocks, one of dimension 5, two identical blocks of dimension 4, and three identical blocks of dimension 2. In Table 4.9 we display the blocks for each irrep and the corresponding SALCs for one component. The corresponding secular equations are ... 

In this case the electronic degeneracy is equal to the dimension of an irrep of the point group. 

In a cyclic model of a crystal with N primitive cells in the main region a band rep is an Np dimensional reducible rep of a space group. An induced rep is a particular case of a band rep as it satisfies both properties 1 and 2 with p = nqU/ n is the dimension of the site-symmetry group irrep for a point q belonging to the set of n, points in the unit cell). 

The great orthogonality theorem must still hold for the double group, as must the various orthogonalities between rows and columns of the character table. Also, the sum of squares of the dimensions of the irreducible representations (irreps) must equal the order of the group. If the nonrelativistic group has ki irreps with dimensions... 

The double group has the additional operations E, C2x = CixE, C y = C2yE, Ciz = C2zE, and has order 8. Construction of the character table can be done from general group theoretical considerations. The additional irreps must be either four irreps of dimension 1 or one irrep of dimension 2. In the first case the total number of irreps is 8, there must be 8 classes, every element is in a class by itself, the group is Abelian, and the elements must commute. For any of these twofold rotations we can show that... 

What consequences does this have for the relation of the Kramers pairs to the representations of the (double) point group Let us initially restrict our discussion to groups with irreps of dimension 1 or 2, which covers all groups except the cubic and icosahedral groups. There are three cases to consider ... 

For a finite group, the number of classes equals the number of irreps. Furthermore, the sum of the squares of the dimensions di) of the irreps must equal the order ( ) of the group... 

---