Point group of a molecule

A flow chart for determining the point group of a molecule... 

Of course, the point group of a molecule is always a subgroup of the point group of its ellipsoid of inertia. For example, asymmetric tops can belong to the point groups... 

Symmetry is a fundamental concept of paramount importance in art, mathematics, and all areas of natural science. In the context of chemistry, once we know the symmetry characteristics (i.e., point group) of a molecule, it is often possible for us to draw qualitative inferences about its electronic structure, its vibrational spectra, as well as other properties such as dipole moment and optical activity. 

To help students to determine the symmetry point group of a molecule, various flow charts have been devised. One such flow chart is shown in Table 6.2.3. However, experience indicates that, once we are familiar with the various operations and with visualizing objects from different orientations, we will dispense with this kind of device. 

A fundamental property of the wave function is that it can be used as basis for irreducible representations of the point group of a molecule [13], This property establishes the connection between the symmetry of a molecule and its wave function. The preceding statement follows from Wigner s theorem, which says that all eigenfunctions of a molecular system belong to one of the symmetry species of the group [14],... 

Various methods (described in Chapter 4) can be used to determine the symmetry of atomic orbitals in the point group of a molecule, i. e., to determine the irreducible representation of the molecular point group to which the atomic orbitals belong. There are two possibilities depending on the position of the atoms in the molecule. For a central atom (like O in H20 or N in NH3), the coordinate system can always be chosen in such a way that the central atom lies at the intersection of all symmetry elements of the group. Consequently, each atomic orbital of this central atom will transform as one or another irreducible representation of the symmetry group. These atomic orbitals will have the same symmetry properties as those basis functions in the third and fourth areas of the character table which are indicated in their subscripts. For all other atoms, so-called group orbitals or symmetry-adapted linear combinations (SALCs) must be formed from like orbitals. Several examples below will illustrate how this is done. 

A systematic procedure for determining the point group of a molecule is outlined in the following flowchart. Follow the path by answering the question in each diamond box until you arrive at one of the group designations. Note that if you answer NO to every question, you will wind up at Ci. 

Many molecular orbital programs determine the symmetry point group of a molecule before using this information to simplify the calculations, but some require you to state the point group in the information that you give the computer. 

Various methods (described in Chapter 4) can be used to determine the symmetry of atomic orbitals in the point group of a molecule, i.e., to determine the irreducible representations of the molecular point group to which the atomic orbitals belong. There are two possibilities depending on the position of the atoms in the molecule. For a central atom (such as O in HjO or N in NH ),... 

The chemist perceives no fewer than seven nontrivial symmetry operations in naphthalene (point group D2h or mmm), the corresponding symmetry elements are three mutually perpendicular mirror planes, three mutually perpendicular twofold rotation axes, and the center of inversion. In addition, the trivial symmetry operation, onefold rotation, is always present. Each of these, however, yields one of the above four permutations. The notion of point group of a molecule will be introduced later. 

FIGURE 13.14 Flowchart for determining the point group of a molecule. It does not include O, T, or Rh(3T Rh( ) is the point group of any single atom or ion. Source Adapted from P. W. Atkins, Physical Chemistry, 5th ed., Freeman, New York, 1994. 

There is an alternative approach to the topic which is of no less validity. This is the study of commuting operators. The statement that two operators commute is equivalent to the statement that you can, in principle at least, make simultaneous measurements of the physical properties associated with each of them. The set of operators that make up the point group of a molecule commute with the Hamiltonian operator and so one can have knowledge of energies which correspond to symmetry-distinct levels. If one can recognize the existence of a valid operator then it can be used in the classification of energy levels there is no requirement that one can implement in some way the associated physical process. The permutation operator and time-reversal operators... 

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