By Hiroshi Suzuki
Digital absorption spectra and geometry of natural molecules.
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Extra resources for Electronic Absorption Spectra and Geometry of Organic Molecules. An application of molecular orbital theory
It is to be noted here that the AO's have a nodal plane coinciding with the molecular plane, that is, they change sign under the reflection in the molecular plane, that is, operation ah. , respectively. , and changes its sign by application of operations 48 3. GROUP THEORY C2', C2, /, iC2 ( = (TA), etc. Thus, the transformation properties of this MO are specified by the one-dimensional representation corresponding to the symmetry species A2u of point group D 6 / l . That is, we can say that this MO belongs to the symmetry species A2u , and also that this MO forms a basis for the corresponding nondegenerate representation.
Thus the symmetry of an object can be indicated by specifying the symmetry group to which that object belongs. Symmetry operations and corresponding symmetry elements are denoted by symbols. Some of symbols for operations of interest to us are given below. E (or / ) Cn on σν ad i Identity operation. Rotation about an axis of symmetry by an angle 2π/η. The axis is called an «-fold symmetry (or rotational) axis. The symmetry axis with the largest value of n in a figure is called the principal axis of symmetry of the figure and is usually taken as the z coordinate axis.
Therefore, we have R(R<£J = R(a0J = a^n = Φη · (3-7) 42 3. GROUP THEORY It follows that a2 = 1. This means that the value of a must be either + 1 or — 1. This conclusion is the same as that deduced above from the nondegeneracy of the molecular wavefunction. This means that in molecules belonging to point groups containing no rotational symmetry higher than twofold all the molecular wavefunctions must be nondegenerate, except for the rare case of accidental degeneracy, where two or more eigenfunctions have the same eigenvalue but behave differently under symmetry operations.