Symmetry examples

Because many physical systems possess certain types of symmetry, its adaptation has become an important issue in theoretical studies of molecules. For example, symmetry facilitates the assignment of energy levels and determines selection rules in optical transitions. In direct diagonalization, symmetry adaptation, often performed on a symmetrized basis, significantly reduces the numerical costs in diagonalizing the Hamiltonian matrix because the resulting block-diagonal structure of the Hamiltonian matrix allows for the separate... 

For example, the graph of cosx is identical to the graph of cos(x+2 7t), and also to the graph of cos(—x). Shifting the x value by 2tt and multiplying x by — 1 are, in that example, symmetry operators. For any given function, there may be many symmetry operators, and the symmetry does not have to be restricted to the spatial symmetry of Cartesian coordinates like x, y, z. 

Material Examples Symmetry (structure) site (symmetry)... 

To illustrate further how operations can fall into the same class, we will look at the 31i point group, for which cyclopropane (Figure 5.5) serves as an example molecule. Figure 5.5 also shows some example symmetry elements from Dsh. The principal C3 axis is perpendicular to the plane of the three C atoms and passes through the centre of the triangle... 

The symmetry elements for Ot, were discussed in Section 3.9. To set up the reference frame for the complex it is usual to align the axis system with the metal ligand bonds so that the ligands can be thought of as lying at the corners of the standard octahedral shape. This arrangement is also used in the three-dimensional model from Appendix 4, which also has the same set of example symmetry elements drawn on the paper octahedron. 

Table 5.14 The transformations of the p- and d-orbital functions under example symmetry operations from each class in Oh. The symmetry elements that were used are those shown on the paper model from Appendix 4. The normalization factor for the d i orbital has been omitted for the sake of clarity. 

A simplified basis of just the a-donor orbitals from H2O is shown in Figure 7.38, along with some example symmetry axes from the Ot, rotational classes. Appendix 12 gives the rotational subgroup given for 0 as 0. 

Symmetry breaking in wavefunctions occurs because insufficient correlation between electrons of opposite spin is included. 47 example, symmetry... 

As an example, symmetry coordinates for the U3 and U4 species are derived from irreducible representations for the U point (see Table V.2),... 

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