Coov point group

The molecule of hydrogen fluoride, HF, belongs to the Coov point group. The hydrogen atom uses its 1 s atomic orbital to make bonding and antibonding combinations with the 2pz orbital of the fluorine atom, the z... 

Coo signifies the presence of an 00-fold axis of rotation, i.e. that possessed by a linear molecule (Figure 3.7) for the molecular species to belong to the Coov point group, it must also possess an infinite number of planes but no o-ji plane or inversion centre. These criteria are met by asymmetrical diatomics such as HF, CO and [CN] (Figure 3.7a), and linear polyatomics (throughout this book, polyatomic is used to mean a species containing three or more atoms) that do not possess a centre of symmetry, e.g. OCS and HCN. 

At the other end of the second row of the periodic table, fluorine also forms a diatomic molecule with hydrogen. HF also belongs to the Coov point group. The MOs for HF are... 

Hydrogen isocyanide (HNC) is a linear triatomic molecule with Coov point group symmetry. It is a zwitterion and an isomer of hydrogen cyanide (HCN). Both HNC and HCN have large, similar dipole moments, with respectively phnc=3.05 Debye and... 

Polar symmetry (point group Coov or lower) is quite familiar at the molecular level as the symmetry required for the existence of a molecular dipole moment. Molecules possessing higher, nonpolar symmetry, cannot possess a permanent molecular dipole moment even when there are bond dipoles. Until the 1970s, no LC phases were known to possess polar symmetry, in spite of the fact that most mesogens are polar. 

Let us first examine a few special cases that cover most common point groups. A linear molecule, such as HCN (point group Coov) or acetylene (Dxl), will lie along one principal axis, say the z axis, so that the first eigenvalue of the inertial tensor vanishes and the other is doubly degenerate alternatively, by the second case in Eq. 3 x, = v, = 0 for all i, and thus % = 0. 

The symmetry operations E, C, and av (reflection in a plane that contains the axis A-B) are present. All molecules that possess these symmetry properties have the point-group symmetry Coov The orbitals are characterized by symbols similar to those used for a homonuclear diatomic molecule, such as a, n, etc. The character table for CMV is given in Table 2-2. 

Determine whether the molecule belongs to a special group such as Dooh, CoovjTd, Oh or Ih. If the molecule is linear, it will be either Dooh or Coov If the molecule has an infinite number of twofold axes perpendicular to the Coo axis, it will fall into point group Dooh. If not, it is C v-... 

Molecular geometries of the XH diatomic hydrides (X=Cu, Ag, Au) and XFg hexafluorides (X=S, Se, Mo, Ru, Rh, Te, W, Re, Os, Ir, Pt, U, Np, and Pu) molecules were assumed to be of CooV and Oh symmetry with their bond lengths taken from experiments [28-33]. As the spin function is explicidy included in eq.(l), the Cv and Oh point groups reduce to the CooV and Oh double groups, respectively, in the relativistic DV-Xa calculation. Symmetry orbitals... 

Table 4.3 shows the results of such calculations for the CO2 molecule.In order to understand the results, Figure 4.4 shows the normal modes of vibration of CO2. The asymmetric modes, Hu and Su, differ from the symmetric mode. Eg, in several respects. First of all, they destroy some elements of symmetry, changing the point group to and Coov, respectively. 

As said, a free molecule may have a rotation axis of infinite order. The operation C , together with dv generates the point group CooV that characterizes hetero-nuclear diatomic molecules. Homonuclear diatomic molecules are characterized by the point group D h, generated by C , Q and ah, or more conveniently by C , C 2 and i. 

Fig. 3.7 Linear molecular species can be classified according to whether they possess a centre of symmetry (inversion centre) or not. All linear species possess a Coo axis of rotation and an infinite number of a., planes in (a), two such planes are shown and these planes are omitted from (b) for clarity. Diagram (a) shows an as5mmetrical diatomic belonging to the point group Coov, and (b) shows a symmetrical diatomic belonging to the point group Dooh-...

Initially, check whether the molecule adopts one of the four most easily identifiable geometries octahedral (e.g. [FeClg] " (6-4)), tetrahedral (e.g. CH4), linear with an inversion centre (e.g. CO2), or linear without an inversion centre (e.g. HCN). If this is the case, the problem is solved the symbols associated with these four point groups are Ou, Td, Dooh> and Coov respectively. 

For azimuthally isotropic structures, as for instance glass substrates covered by a surfactant, the symmetry properties of the adsorbed LC film are described by the point group Coov The angular distribution of the adsorbed LC molecules in that situation is independent of the azimuthal angle (p and the non-vanishing components of sire ... 

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). 

A cone can be considered as a deformation of a cylinder, in which the poles of the uniaxial direction are no longer equivalent. The symmetry group is reduced accordingly to Coov, where only the vertical symmetry planes remain. Conical symmetry is exemplified by hetero-nuclear diatomic molecules, but it is also the symmetry of a polar vector, such as a translation in a given direction, or a polarized medium or an electric held, etc. Conical molecules have C v symmetries, as was the case for the ammonia model. Again, the smallest trivial member of this series is Civ, which is fully anisotropic. This is the point group of the water molecule. 

Curie understood that under stress, or in the presence of external electric or magnetic fields, the symmetry of a system is changed. The Neumann principle still applies but should no longer be based on the symmetry of the isolated crystal, but on that of the combined system of crystal and external field, as we have considered in Sect. 3.9. In the case of ammonia, application of an electric field has the Coov symmetry of a polar vector. The symmetry that results from the superposition of the field with the molecular point group 3 depends on the orientation (see Appendix B). In the coordinate frame of Fig. 3.1 one has ... 

The symmetry point group of the molecule in the quadrupolar field is thus C2v, which bears the same relation to Coov as D2/1 does to Doo/i- The reduction of symmetry from Doo/i to Coov or from D2/1 to C2V - is a result of a dipolar field along the z axis. The greater electronegativity of oxygen than carbon is the natural equivalent of the artificial dipolar field introduced in Section 2.2.6, the effect of which on an isolated atom was illustrated in Fig. 2.9. 

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