Irreducible representation notation

Familiarity is also assumed with the concepts of representation and irreducible representation (IR). A representation r of dimension n associates to each group element s an n X n matrix D(s), with matrix elements D(s)y, in such a way that for every s, t, D(s)D(t) =D(st), with the product formed by ordinary matrix multiplication. We will sometimes use the bra-ket notation... 

In the so-calledMu/Z/ten notation, representations A and B (which does not appear in Table 7.2) are mono-dimensional, E representations are bi-dimensional, and T representations are three-dimensional. Other irreducible representations of higher order are G (three-dimensional) and H (tetra-dimensional). We will also state now that the dimension of a representation gives the degeneracy of its associated energy level. 

In the MuUiken notation, the subscripts u (ungerade = odd) and g (gerade = even) indicate whether an irreducible representation is symmetric (g) or anti-symmetric(M), in respect to the inversion operation (/). 

Another common notation for the irreducible representations is the so-called Bethe notation, in which the representations are denoted by E, symbols (i = 1,2,...), where the subscript i denotes the dimension. It is not simple to establish an equivalence between these two types of notation, since it depends on the symmetry group. For the moment, we will just mention both notations so that readers will be famihar with any character table. 

Let us consider e.g. a two-orbital two-electron model system with the orbitals a and b which can be understood as notation for one-dimensional irreducible representations of the point group of a TMC. In this case it is easy to see that the corresponding singlet and triplet states and (T = B, S = 0,1) are given correspondingly by ... 

The symbols formulated by R, S. Mulliken are used to distinguish the irreducible representations of the various point groups. In this section we will outline the general points of the notation and the reader is referred to Mulliken reportf for the details. 

A reducible representation is said to be the direct sum of the irreducible representations of which it is made up. In the standard notation (Section 9.5) for point-group irreducible representations, the 03v representation (9.29) is called A ( and the representation (9.28) is called E. If we denote the reducible representation (9.25) by T, then... 

Here x(C4), for instance, means the character for the covering operation consisting of rotation about one of the fourfold or principal cubic axes (normals to cube faces) by 2a-/4. Any rotation about such an axis leaves two atoms invariant, and hence x(Cs) = x(C4) = 2- On the other hand, x((Y)=x(Q)—0 since no atoms are left invariant under rotations about the twofold or secondary cubic axes (surface diagonals) or about the threefold axes (body diagonals). Inversion in the center of symmetry is denoted by I. By using tables of characters for the group Oh, one finds that the irreducible representations contained in the character scheme (2) are, in MuIIiken s notation,4... 

In Table I we give the irreducible representations, in Mulliken s notation, contained in the central and attached orbitals for compounds of other types of symmetry. When 3 or 4 atoms are trigonally or tetragonally attached, we have supposed that the plane of these atoms is a plane of symmetry, as in (N03) or Ni(CN)4 When there is no such symmetry plane, as in NHg, the distinctions between u and g, or between primes and double primes, are to be aholished,9 and the symmetries degenerate to Csv, Civ instead of DSh, D h- When 6 atoms are attached in the scheme Z>3, or 8 in % they are arranged respectively at the corners of a trigonal and a square prism. 

It can therefore be inferred that 0(3) electrodynamics is a theory of Rieman-nian curved spacetime, as is the homomorphic SU(2) theory of Barrett [50], Both 0(3) and SU(2) electrodynamics are substructures of general relativity as represented by the irreducible representations of the Einstein group, a continuous Lie group [117]. The Ba> field in vector notation is defined in curved spacetime by... 

The crystal-field multiplets are classified according to the irreducible representations of the respective double-group where both, the Bethe and the Mulliken (in parentheses) notations are written. DsH means the spin-Hamiltonian D-value accounting for all excitations AmIi - the lowest energy levels difference using the model-Hamiltonian in the first iteration Affl - the... 

Pi irreducible representation of the double group (Bethe notations)... 

Before concluding the discussion on the notation of the irreducible representations, we use C2v point group as an example to repeat what we mentioned previously since this point group has only four symmetry species, A, A2, B, and B2, the electronic or vibrational wavefunctions of all C2V molecules (such as H2O, H2S) must have the symmetry of one of these four representations. In addition, since this group has only one-dimensional representations, we will discuss degenerate representations such as E and T in subsequent examples. 

In these equations, Hw(qa, q, 0) and jls(qa, qfo, 0) are the dipole moment operators at initial time belonging, respectively, to the irreducible representations B and A of the C2 symmetry group that transforms themselves, the first one, according to x and v, and the last one, according to x2 and y2 (allowed Raman transition), where x and y and the Cartesian coordinates that are perpendicular to the C2 symmetry axis. Here, we prefer the notations g and u in place of A and B of group theory. 

First, let us consider the customary notations. Assume that a hypothetical ground state molecule of the C2v point group has four electrons, two in an A symmetry and two in a B symmetry orbital. In short notation this can be written as a b. An electron occupying an A, symmetry orbital is represented by al3 the lower-case letter indicating that this is the symmetry of an orbital and not of an electronic state. If two electrons occupy an orbital, the notation is a. The symmetry of a state is represented by capital letters, just as are the irreducible representations. 

Notice that in this table we retained the Longuet-Higgins notation for the irreducible representations An in accordance to that of ethane [4]. 

Figure 2.7-6 A Assignment of the Cartesian coordinate axes and the symmetry operations of a planar molecule of point group C2,.. B Character table, 1 symbol of the point group after Schoen-flies 2 international notation of the point group 3 symmetry species (irreducible representations) 4 symmetry operations 5 characters of the symmetry operations in the symmetry species +1 means symmetric, -1 antisymmetric 6 x, y, z assignment of the normal coordinates of the translations, direction of the change of the dipole moment by the infrared active vibrations, R, Ry, and R stand for rotations about the axes specified in the subscript 7 x, xy,. .. assign the Raman active species by the change of the components of the tensor of polarizability, aw, (Xxy,. ...

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