Determination of molecular point groups

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

Elow Ghart for Determination of Molecular Point Groups... 

FLOW CHART FOR DETERMINATION OF MOLECULAR POINT GROUPS... 

The set of molecular data required for statistical thermodynamic calculations includes molecular mass, structural parameters for determination of a point group, a symmetry number a and calculation of a product of principal moments of inertia IA IB Ic, as well as 3n - 6 frequencies of normal vibrations for an n-atomic molecule. 

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. 

The preceding paragraph illustrates two points about Raman and IR spectroscopies. First, the two techniques are complementary—information obtained by one or the other alone is generally incomplete. For example, for a molecule that is centrosymmetric (possesses an inversion center, like CO2) IR-active normal modes are forbidden in the Raman, and vice versa. Second, it is implicit that molecular symmetry matters—if CO2 were a bent molecule instead of linear, the conclusions about IR and Raman activity of the modes (indeed, the number of modes) would not be the same. The determination of molecular symmetry groups and how molecular symmetry in turn determines spectroscopic and... 

Using the flowchart in order to determine the molecular point group of NH3, there are no rotational axes (a axis will only occur in a linear molecule). The ammonia molecule does, however, have a principal proper rotational axis, C3. It does not have any Cj or C4 axes and there is only one (not four) C3 axes. Thus, the molecule belongs to one of the point groups in the lower box of Figure 8.9. There are no C2 axes in NH3 that are perpendicular to the principal axis and there are no improper rotational axes. There are also no horizontal mirror planes, but there are n = 3 vertical mirror planes. Thus, the molecular point group for NH3 is 3. ... 

Example 8-6. Determine the molecular point group in the Schoenflies notation for each of the following molecules (a) H2O, (b) PPh3, and (c) BH3. 

Determine the molecular point group for each of the following molecules or ions ... 

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... 

Coordinates such as these, which have the symmetry properties of the point group are known as symmetry coordinates. As they transform in the same manner as the IRs when used as basis coordinates, they factor the secular determinant into block-diagonal form. Thus, while normal coordinates most be found to diagonalize the secular determinant, the factorization resulting horn the use of symmetry coordinates often provides considerable simplification of the vibrational problem. Furthermore, symmetry coordinates can be chosen a priori by a simple analysis of the molecular structure. 

Further, we will find in this chapter that wavefunctions (nuclear or electronic) must be functions which form bases for the irreducible representations of the point group to which the molecule belongs. With this knowledge we are able to determine which integrals over molecular wavefunctions are necessarily zero and this in turn (next chapter) leads to well known spectroscopic selection rules. 

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 two characteristic features of normal modes of vibration that have been stated and discussed above lead directly to a simple and straightforward method of determining how many of the normal modes of vibration of any molecule will belong to each of the irreducible representations of the point group of the molecule. This information may be obtained entirely from knowledge of the molecular symmetry and does not require any knowledge, or by itself provide any information, concerning the frequencies or detailed forms of the normal modes. 

Symmetry Notation.—A state is described in terms of the behavior of the electronic wave function under the symmetry operations of the point group to which the molecule belongs. The characters of the one-electron orbitals are determined by inspection of the character table the product of the characters of the singly occupied orbitals gives the character of the molecular wave function. A superscript is added on the left side of the principal symbol to show the multiplicity of the state. Where appropriate, the subscript letters g (gerade) and u (ungerade) are added to the symbol to show whether or not the molecular wave function is symmetric with respect to inversion through a center of symmetry. 

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