Symmetry in molecules

Hollas, J. M. (1972) Symmetry in Molecules, Chapman and Hall, London. 

Excited states formed by light absorption are governed by (dipole) selection rules. Two selection rules derive from parity and spin considerations. Atoms and molecules with a center of symmetry must have wavefunctions that are either symmetric (g) or antisymmetric (u). Since the dipole moment operator is of odd parity, allowed transitions must relate states of different parity thus, u—g is allowed, but not u—u or g—g. Similarly, allowed transitions must connect states of the same multiplicity—that is, singlet—singlet, triplet-triplet, and so on. The parity selection rule is strictly obeyed for atoms and molecules of high symmetry. In molecules of low symmetry, it tends to break down gradually however,... 

FIGURE 1.12 Axes of symmetry in molecules (a) twofold axis in OF2, (b)... 

It may be noted that, when a center of inversion exists, restrictions are placed on the numbers of all atoms, or all but one atom, in the molecule. Since the center is a point, only one atom may be at the center. If there is an atom at the center, that atom is unique, since it is the only one in the molecule that is not shifted when the inversion is performed. All other atoms must occur in pairs, since each must have a twin with which it is exchanged when the inversion is performed. From this it follows that we need not bother to look for a center of symmetry in molecules that contain an odd number of more than one species of atom. 

Ladd, M. F. C. Symmetry in Molecules and Crystals Wiley New York, 1989. Hyde. B. G. Andersson, S. Inorganic Crystal Structures Wiley New York, 1989. 

The analogy between transition moment vectors and spin-orbit coupling vectors extends to the effects of symmetry. In molecules that belong to point groups with symmetry oper-... 

Meso Compounds. Symmetry in Molecules with Stereogenic Centers... 

Contents B. T Sutcliffe The Concept of Molecular Structure. - 0. E. Polansky Topology and Properties of Molecules. -J.RDahl Symmetry in Molecules. -L D. Barron Chirality of Molecular Structures - Basic Principles and their Consequences. - J. E. Boggs Interplay of Experiment and Theory in Determining Molecular Geometries. A. The Experiments. -... 

You may be wondering at this point why so much attention is given to symmetry in molecules when, after all, most molecules are not symmetric. Lets begin with perhaps the most fundamental reason chemists use the symmetry properties of diatomic molecules to describe individual chemical bonds in any molecule. In essence, we apply a local bond model to the chemical bond, allowing us to consider—just as a first approximation—that the bond depends only on the two atoms that it connects. In this section we see how to label the MOs of diatomic molecules, and this will inform many of our conclusions about bonding in polyatomic molecules. 

Symmetry in molecules (a) octahedral SFe, (b) trigonal-hipyramidal PCI5, (c) SiCfiPh and (d) a square-planar platinum complex. 

We must therefore consider first how to determine the symmetry of molecular vibrations and, secondly, what the consequences of these symmetry selection rules might be. If you are not already familiar with the use of symmetry in molecules, please read Section 2.3 before continuing. 

Usually it is not difficult to see most elements of symmetry in molecules, the exception being improper axes, which tend to be a little tricky. The six-fold improper rotation in staggered ethane is not too hard to envision when it is applied once. If we rotate clockwise by 60° and reflect. Hi replaces H5, H5 replaces H2, etc. It is a little... 

We will return to the application of symmetry to wavefunctions in a later section. Symmetry in molecules has several immediate consequences. For example, the dipole moment of a molecule depends in part on how the atoms in the molecule are arranged. It can be shown that any molecule whose structure has a point group symmetry of C, C , or C , with n > 1, is polar, and molecules that do not have such symmetry are nonpolar. Further, it can also be shown that a three-dimensional molecule that does not have an S axis is chiral. An S axis may not be explicitly listed in the point group, so care must be used with this requirement. Chirality is an important issue in organic chemistry and is the basis of stereochemistry. 

The symmetry in molecules helps to distinguish chiral structures from achiral ones... 

We now detail a second approach to the evaluation of the degree of symmetry deficiency, from a different point of view, namely the fuzzy sets approach. Fuzzy set methods are especially suitable to reflect the Heisenberg uncertainty relation and other aspects of quantum chemistry " and computational chemistry. Symmetry in molecules relies on the concepts of distance and metric. For the approximate symmetries of fuzzy electron densities, fuzzy set methods, fuzzy distance or fuzzy metric, and fuzzy symmetry are of importance. In Section 3.3 below, we compare and generalize the two approaches. 

Hollas, J. Symmetry in Molecules-, CRC Press Boca Raton, FL, 1972. 

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