Molecular point group determination

Number of species for molecular point group determined from Appendix2. 

Molecular point-group symmetry can often be used to determine whether a particular transition s dipole matrix element will vanish and, as a result, the electronic transition will be "forbidden" and thus predicted to have zero intensity. If the direct product of the symmetries of the initial and final electronic states /ei and /ef do not match the symmetry of the electric dipole operator (which has the symmetry of its x, y, and z components these symmetries can be read off the right most column of the character tables given in Appendix E), the matrix element will vanish. 

The molecular point groups of (CF3)2S02, ( 03)2802 and (CBr3)2S02 can be either C2V or C2 according to electron diffraction and vibrational spectroscopic data. The molecular model and projection formula for ( 13)2802 are shown in Figure 11. The molecular geometry of the bromine derivative has not been determined but its vibrational... 

In 11-2 we showed how all AOs can be classified according to the irreducible representations of the different molecular point groups. Therefore, if we consult Table 11-2.2, we can determine the splitting of the energy level of a single electron for any particular perturbing environment. 

The most important and frequent use for projection operators is to determine the proper way to combine atomic wave functions on individual atoms in a molecule into MOs that correspond to the molecular symmetry. As pointed out in Chapter 5, it is essential that valid MOs form bases for irreducible representations of the molecular point group, we encounter the problem of writing SALCs when we deal with molecules having sets of symmetry-equiv-... 

The MO (molecular orbitals) of a polyatomic system are one-electron wave-function k which can be used as a (more or less successful) result for constructing the many-electron k as an anti-symmetrized Slater determinant. However, at the same time the k (usually) forms a preponderant configuration, and it is an important fact67 that the relevant symmetry for the MO may not always be the point-group determined by the equilibrium nuclear positions but may be a higher symmetry. For many years, it was felt that the mathematical result (that a closed-shell Slater determinant contains k which can be arranged in fairly arbitrary new linear combinations by a unitary transformation without modifying k) removed the individual subsistence... 

Representation theory of molecular point groups tells us how a rotation or a reflection of a molecule can be represented as an orthogonal transformation in 3D coordinate space. We can therefore easily determine the irreducible representation for the spatial part of the wave function. By contrast, a spin eigenfunction is not a function of the spatial coordinates. If we want to study the transformation properties of the spinors... 

We are not going to review here the transformation properties of spatial wave functions under the symmetry operations of molecular point groups. To prepare the discussion of the transformation properties of spinors, we shall put some effort, however, in discussing the symmetry operations of 0(3)+, the group of proper rotations in 3D coordinate space (i.e., orthogonal transformations with determinant + 1). Reflections and improper rotations (orthogonal transformations with determinant -1) will be dealt with later. 

We exemplify the procedure of determining the spinor transformation properties under molecular point group operations for the Czv double group. Other double groups can be treated analogously. The character tables of the 32 molecular double groups may be found, e.g., in Ref. 68. 

Scheme for determining molecular point groups (Schonflies notation). 

The representation of a product function can be determined by forming the direct product of the original functions. The representation of a direct product will contain the totally symmetric representation only if the original functions whose product is formed belong to the same irreducible representation of the molecular point group. This follows directly from rules 2 and 3 in Section 4.5. 

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. 

Character tables, which can be found in several vibrational spectroscopy books, allow the determination, for any molecular point group, of the species (or irreducible representations) in relation to the symmetry elements typical of that group. As further cited below, the classification in terms of a particular symmetry species determines the activity (IR activity, Raman activity, both IR and Raman activity or inactivity) of any mode. 

Some properties of groups 3-6. Classification of point groups 3-7. Determination of molecular point groups A. 3-1. The Rearrangement Theorem Problems... 

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