Representations of point groups

Consider the effects of the symmetry operations of the Cji, point group on the set of x, y, and z coordinates. [The set of p orbitals p, Py, Pz) behaves the same way, so this is a useful exercise.] The water molecule is an example of a molecule having C2, symmetry. It has a C2 axis through the oxygen and in the plane of the molecule, no perpendicular C2 axes, and no horizontal mirror plane, but it does have two vertical mirror planes. 

Each symmetry operation may be expressed as a transformation matrix as follows  

As examples, consider how transformation matrices can be used to represent the symmetry operations of the C2J, point group  

Cl- Rotate a point having coordinates (x, y, z) about the C2(z) axis. The new coordinates are given by 

The transformation matrices for the four symmetry operations of the group are 

Verify the transformation matrices for the E and a- iyz) operations of the C2v 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,. 

In the anion radical of XXIII, the starting distorted structures belonging to the Og, bjg and b2u irreducible representations of point group D2k all converge into the unique set of bond lengths corresponding to... 

Given the IRs T of H, all the irreducible co-representations F of G can be determined from eqs. (40)-(42). Although the equivalence of T, T and the sign of c(Z) provide a criterion for the classification of the co-representations of point groups with antiunitary operators, this will be more useftd in the form of a character test. 

The following tables show how irreducible vector representations of point groups are re-labeled or reduced when the symmetry of the point group is lowered. The tables are in the reverse order to that given at the beginning of Appendix A3. For groups with pairs of complex conjugate representations, E means the direct sum 1H 2E, and similarly for Eg and E . 

Hurley, A. C. (1966) Ray representations of point groups and the irreducible representations of space groups and double space groups. Phil. Trans. Roy. Soc. (London) A260, 1-36. 

Table 68 Reduction of the (2/ + 1) states of R3 and R 3 to irreducible representations of point groups...

Table 7.5 Interrelations between Irreducible Representations of Point Groups 2, and Da,...

By incorporating these symmetries in the 4-spinor basis functions, as we have done in our BERTHA code [50-54], we can make substantial computational economies in computing interaction integrals. The angular stracture of Dirac 4-spinors described here is also exploited by the major computer package TSYM, which utilizes projection operators to construct relativistic molecular symmetry orbitals for double valued representations of point groups [77-79]. 

E, identity transformation, 3, 18, 51 Eigenfunctions, as a basis for representation of point groups, 112 classification of, 3, 14ff degeneracy of, 9... 

Ceulemans, A., Beyens, D. Monomial representations of point-group symmetries. Phys. Rev. A 27,621 (1983)... 

Small Representations of a Little Group. Projective Representations of Point Groups... 

Projective representations were first introduced by Shur [32], who developed a general theory of projective representations and worked out methods for constructing projective representations of finite groups. The connection between projective representations of point groups and representations of space group was demonstrated by Lyubarskii, Kovalev, Bir [27,31,33]. To find with the factor qrstem (3.40)... 

This short description of projective representations of point groups allows us to understand information given in different tables and on the site [16]. When using different existing tables for small representations of little groups one has to remember that the projective representations of point groups can be ordered in different ways and may appear to be p equivalent to each other. 

The first and third columns in Table 3.6 contain the symbols of the point subgroups C2v and Civ and their irreps. The rest of the columns give the symbols of the induced representations of point group Td decomposed over the irreps of this group. 

Table 3.11. Space group 0 correspondence between small representations of little groups Gr, Gr, Gx and Gm and irreducible representations of point groups Oh and Dih...

For the symmetry directions in the BriUouin zone A(FX), A FL), E (see Fig. 3.2) the small representations of both space groups arep-equivalent to ordinary irreducible representations of point groups C4 ,C3 and C v The notations of these representations are taken from [17]. For symmetry directions on the surface of the BriUouin zone Z XW),S smaU representations of space group are p-equivalent to ordinary irreducible representations of point group C v, for the symmetry direction - to ordinary irreducible representations of group G. For the nonsymmorphic space group... 

A FIGURE 6.9 Symmetry representations of point group Cj. A function with A symmetry has the same sign and magnitude on both sides of the mirror plane, whereas a function with A" symmetry has the same magnitude on both sides but opposite sign. 

Ceulemans A, Beyens D (1983) Monomial representation of point-group symmetries. Phys Rev A 27 621-631... 

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