Symmetry classification, of molecules

Figure 3.10 A five-stage procedure for the symmetry classification of molecules.

The symmetry classification of wavefunctions is based on the symmetry properties of molecules. Most small molecules possess certain symmetry elements such as a plane (a), or an w-fold axis (OJ, or a centre of symmetry (i), or perhaps a variety of these elements in combination. In order to be as definite as possible we shall develop the argument in terms of a specific example. The ground state of... 

The effects of all the PI group operations on the Euler angles 0, rotational levels in the ground state of the NH3 molecule. 

Point group (symmetry point group) The symmetry classification of a molecule based on its symmetry elements (axes, planes, etc.). 

As the algebraic treatment of molecular properties outlined in Section II.A is based upon a classification of molecules according to the symmetry of the skeleton and the number of ligands attached to the skeleton, it is immediately obvious that the molar rotations of pentatetraenes should be described by an approximation ansatz corresponding to that of Equation 516. Referring to pentatetraenes with two identical ligands, such as 68, their molar rotations should be given by Equation 55 ([ Id — x). 

Classification of molecules according to their rotational symmetry is a key point in rotational spectroscopy as the expression of the rotational Hamiltonian and the solution of the corresponding eigenvalue equation vary noticeably upon symmetry. 

The description of the vibrations of polyatomic molecules only becomes mathematically tractable by treating the system as a set of coupled harmonic oscillators. Thus a set of 3N - 6 (3N - 5 for linear molecules) normal modes of vibrations can be described in which aU the nuclei in the molecule move in phase in a simple harmonic motion with the same frequency, normal-mode frequencies are solved, the normal coordinates for the vibrations can be determined, and how the nuclei move in each of the normal modes of vibration can be shown. There are two important points that follow from this. First, each normal mode can be classified in terms of the irreducible representations of the point group describing the overall symmetry of the molecule [7, 8]. This symmetry classification of the... 

These simple examples serve to show that instinctive ideas about symmetry are not going to get us very far. We must put symmetry classification on a much firmer footing if it is to be useful. In order to do this we need to define only five types of elements of symmetry - and one of these is almost trivial. In discussing these we refer only to the free molecule, realized in the gas phase at low pressure, and not, for example, to crystals which have additional elements of symmetry relating the positions of different molecules within the unit cell. We shall use, therefore, the Schdnflies notation rather than the Hermann-Mauguin notation favoured in crystallography. 

Since the presence of a plane of symmetry in a molecule ensures that it will be achiral, one a q)ro h to classification of stereoisomers as chiral or achiral is to examine the molecule for symmetry elements. There are other elements of symmetry in addition to planes of symmetry that ensure that a molecule will be superimposable on its mirror image. The trans,cis,cis and tmns,trans,cis stereoisomers of l,3-dibromo-rranj-2,4-dimethylcyclobutaijte are illustrative. This molecule does not possess a plane of symmetry, but the mirror images are superimposable, as illustrated below. This molecule possesses a center of symmetry. A center of symmetry is a point from which any line drawn through the molecule encouniters an identical environment in either direction fiom the center of ixnimetry. 

A necessary prelude to determining the combinations of AOs which give a hybrid orbital of correct symmetry is the classification of the AOs of the central atom A in terms of the irreducible representations of the point group to which the molecule belongs. This is discussed in 11-2. In 11-4 we consider 77--bonding systems and in the final section we discuss the relationship between localized and non-localized MO theory. 

A molecule that has a mirror image is also said to be dissymmetric while one that docs not (an achiral molecule) have an enantiomer is noiidissyiinnetric. The classification of a given structure as dissymmetric or nondissymmetric is based upon the presence (or lack) of symmetry elements (axes, planes) in the structure. 

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