Linear symmetry groups

THE APPLICATION OF THE ONE-DIMENSIONAL OR LINEAR SYMMETRY GROUPS TO THE SYSTEMATIZATION OF LINEAR STRUCTURE SERIES AND THEIR REPRESENTATIVES... 

Figure 1. Examples of zigzag lines, numeric codes and linear symmetry groups for structures with f2 T (left) and Cl = 2 (right) a) Example of a homogeneous structure b - d) Inhomogeneous structures of different construction.

Appendix Short review of linear symmetry groups... 

If the symmetry operators change the geometric coordinates and the colour of the frieze parts, the two coloured linear symmetry groups are usefui for describing such objects. The colour changing operators are notated in the linear symmetry group symbol by means of an apostrophe. In the case of there being only two colours there are 31... 

We therefore conclude that, for a combination of model, numerical and conceptual reasons the OHAO basis is well-adapted to a theory of valence. The hybrid orbital basis (for simple molecules) has a distinctive symmetry property it carries a permutation representation of the molecular symmetry group the equivalent orbitals are always sent into each other, never into linear combinations of each other. This simple fact enables the hybrid orbital basis to be studied in a way which is physically more transparent than the conventional AO basis. 

A consequence of the symmetry of the molecule is that states must transform according to representations of the appropriate symmetry group. In terms of coordinates, this implies that one must form internal symmetry coordinates. These are linear combinations of the internal coordinates. For example, denoting in Fig. 6.1 by sx, s2, s3,, v4, j5, s6 the stretching coordinates of the six C-H bonds, the internal symmetry coordinates are linear combinations... 

As an example, to construct the character table for the Oh symmetry group we could apply the symmetry operations of the ABg center over a particularly suitable set of basis functions the orbital wavefunctions s, p, d,... of atom (ion) A. These orbitals are real functions (linear combinations of the imaginary atomic functions) and the electron density probability can be spatially represented. In such a way, it is easy to understand the effect of symmetry transformations over these atomic functions. 

Ih ,12(75,12(71,20C 3,15C 2, fl2S io,12S o, 2056,15(7 in. Continuous groups symmetry groups of linear molecules some viruses regular icosahedron... 

For the symmetry groups of nonlinear molecules, it should be expected that for any eigenvalue , there are only a small finite number of linearly... 

Such a set of eigenfunctions must form the basis for a representation of the symmetry group of the Hamiltonian, because for every symmetry operation S, Tipi = pi implies that H Spi) = Sp>i) and hence that the transformed wave function Spi must be a linear combination of the basic set of eigenfunctions (/ ,... Pn-... 

The distribution of the molecular orbitals can be derived from the patterns of symmetry of the atomic orbitals from which the molecular orbitals are constructed. The orbitals occupied by valence electrons form a basis for a representation of the symmetry group of the molecule. Linear combination of these basis orbitals into molecular orbitals of definite symmetry species is equivalent to reduction of this representation. Therefore analysis of the character vector of the valence-orbital representation reveals the numbers of molecular orbitals... 

The notion of a group is a natural mathematical abstraction of physical symmetry. Because quantum mechanical state spaces are linear, symmetries in quantum mechanics have the additional structure of group representations. Formally, a group is a set with a binary operation that satisfies certain criteria, and a representation is a natural function from a group to a set of linear operators. 

Other than linear molecules. If molecules of symmetry other than axial are considered, it is not possible to describe their orientation by an azimuthal and polar angle, Euler angles, Q = a pi, y and Wigner rotation matrices are then needed as Eq. 4.8 suggests. In that case, besides the set of parameters X, X2, A, L that has been used for linear molecules, two new parameters, u, with i = 1,2, occur that enter through the rotation matrices. These must be chosen so that the dipole moment is invariant under any rotation belonging to the molecular symmetry group. The rotation matrix is expressed as a linear combination of such... 

The dipole moment p must be invariant under any rotation of the molecular symmetry group applied to any one of the two molecules [166, 374]. When one of these rotations is applied, the rotation matrix D V(Q) is transformed into a linear combination of the D v, Cl) matrices with different o. The proper linear combinations of the D%V(C1) are invariant under the rotational symmetry group. Such linear combinations are obtained from group-theoretical arguments. For example, for the case of methane pairs in the ground vibrational state, for Ai = 3, 4 and 6, we have the combinations... 

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