Irreducible representations benzene molecules

For nonlinear molecules, each electronic wave function is classified according to the irreducible representation (symmetry species) to which it belongs the symmetry properties of i cl follow accordingly. For example, for the equilibrium nuclear configuration of benzene (symmetry ), the... 

Note first that y(E) = 6 while all other characters are zero. The reason is that the operation E transforms each into itself while every rotation operation necessarily shifts every 0, to a different place. Clearly this kind of result will be obtained for any /z-membered ring in a pure rotation group C . Second, note that the only way to add up characters of irreducible representations so as to obtain y = 6 for E and y - 0 for every operation other than E is to sum each column of the character table. From the basic properties of the irreducible representations of the uniaxial pure rotation groups (see Section 4.5), this is a general property for all C groups. Thus, the results just obtained for the benzene molecule merely illustrate the following general rule ... 

The use of symmetry—at least the translational subgroup—is essential to modem first-principles calculations on crystalline solids. Group theory is simplest for Abelian groups such as the translational subgroup of a crystal or the six-fold-rotational subgroup of the benzene molecule. For such simple cyclic groups, the irreducible representations are characterized by a phase, exp(ifc), associated with each step in a direction of periodicity. For one-dimensional (or cyclic) periodicity,... 

Table 1 Double-ionizations of the benzene molecule to singlet dication states, predicted in the standard enhanced ADC(2) approximation [5] and in the diagonal approximation described in the text. In this and subsequent tables, Term indicates the term symbol for a transition. Character the sum of the squares of the normalised transition eigenvector coefficients associated with the dominant basis configuration, and AE the small symmetry-breaking energy splitting in degenerate irreducible representations introduced by the diagonal approximation (see text)...

Derivative, of Lower Symmetry. In order to apply the product rule in the cases in which isotopic substitution lowers the symmetry, it is necessary to make use of the correlation tables between the irreducible repro-sentations of the group of the parent molecule and the subgroup which expresses the symmetry of the derivative. As an example, the case of p-dideuterobenzene, 3C = 1), will be considered. If the z axis of benzene is correlated with the 2 axis of p-dideuterobenzene, the relation between the irreducible representations is as indicated in Table 10-6. 

The transition vector necessarily possesses all the symmetry elements of the nuclear configuration of a transition state, i.e., corresponds to one of the irreducible representations of the symmetry point group of this state. Figure 1.10 shows, for example, that the transition vector for the transformation of Dewar benzene into benzene corresponds to the fully symmetrical a -type vibration of the C2v transition state structure, i.e., the transition vector is symmetrical with respect to the reflection in each of the two symmetry planes of the molecule and to the rotation about the C2 axis. 

As a first step it is advisable to classify all atoms of a molecule in equivalent sets. Such a set includes all nuclei, which go over into themselves by a symmetry operation. In the benzene molecule, for example, there are two equivalent sets—considering all the C atoms and all the H atoms—since they can be brought into identical position by rotation around the sixfold axis of symmetry. The lower the symmetry is, the more equivalent sets of atoms must be present in a molecule and the fewer irreducible representations belong to the point group. In general, every equivalent set or every bond between equivalent sets can contribute no more than one stretching vibration to each representation. 

By means of group theory, we can find directly the molecular orbitals which form bases for irreducible representations of the symmetry group of the benzene molecule. The procedure is identical with that followed in finding the proper linear combinations of bond eigenfimctions. In place of the set of five bond eigenfunctions we use the set of six atomic orbitals as the basis for the reducible representation. The character of this representation is... 

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