Molecular point groups establishing

Figure 3-5 shows a possible scheme for establishing the molecular point group that has been widely used to reliably establish molecular symmetries [9],... 

Figure 3-5. Scheme for establishing the molecular point groups (cf. Refs. [3-4] and [3-5]). 

A diagram of the geometry and axis conventions for ketene is shown in Fig. 36. The molecule has a planar structure with symmetry. The molecular orbitals in the ketene molecule are not directly analogous to those of formaldehyde, as was recognized in early ab initio studies Ketene is a 22 electron system with 16 valence electrons. The molecular orbitals can be classified according to their behavior under the symmetry operations of the molecular point group, C, . Detailed MO calculations have established that the ground state orbital occupancy is ... 

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],... 

These observations establish our approach to the role of point group orbits in the characterization of molecular geometry and in point group theory. The specific point symmetry of an orbit is limited by its environment, the surrounding molecular skeleton. Thus, we find triangles of atoms, squares of atoms and so on, identified as distinct orbits in different point... 

Table 1 shows the partition of W (r) for the oxaziridine (VI) at three characteristic points very near the three minima of W (r) found for this molecule (see Fig. 10). Intuitively, each point was associated with one of the lone pairs (of N or 0) present in the molecule. The first column of Table, 1 which refers to a point near the N atom, indicates that the contribution of the N lone pair localized group (two electrons and two unit positive charges on N) is by far the largest (—258,6 kcal/mole). Its contribution turns out to be drastically modified by those of the other groups — the overall value is —79.6 kcal/mole this example shows the importance of the whole molecular framework in establishing the actual value of the potential at a given point. 

The isotropic phase formed by achiral molecules has continuous point group symmetry Kh (spherical). According to the group representations [5], upon cooling, the symmetry Kh lowers, at first, retaining its overall translation symmetry T(3) but reduces the orientational symmetry down to either conical or cylindrical. The cone has a polar symmetry Coov and the cylinder has a quadrupolar one Dooh- The absence of polarity of the nematic phase has been established experimentally. At least, polar nematic phases have not been found yet. In other words, there is a head-to-tail symmetry taken into account by introduction of the director n(r), a unit axial vector coinciding with the preferred direction of molecular axes dependent on coordinate (r is radius-vector). 

In this chapter we will explore further the symmetry operations that are used to describe molecular structure. New operations are introduced to complete the set used in molecular symmetry. Particular sets of operations often recur, with many molecules having the same collection of operations. Once we establish how the properties of a molecule depend on the set of valid operations, this will mean that we can actually infer the properties of many related molecules of the same symmetry. The sets of operations are referred to as point groups, and our main task in this chapter is to introduce the concept of a point group. The formal construction of the point groups most commonly used in chemistry is carried out in Chapter 3. The properties of the point groups, and their application in vibrational spectroscopy and MO theory, are then the subject of the remainder of the book. 

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