Molecular symmetry, and point

Molecular symmetry, and point groups, 53-59 Molecules reactivity of. 203 stereochemically nonrigid, 723-730... 

T2.6 The planar from of NHj molecule would have a trigonal planar molecular geometiy with the lone pair residing in nitrogen s p orbital that is perpendicular to the plane of the molecule. This structure belongs to a Djh point group (for discussion of molecular symmetry and point groups refer to Chapter 6). Now we can consult Resource Section... 

Molecular symmetry and point groups Complexes electronic spectra and magnetism (C3) (H8) ... 

Figure 4-11 The Potential Energy Form for Ethane. The midpoint of the range of oj is m =0° and the end points are 180°. The end points and the minima are identical by molecular symmetry and correspond to the stable staggered form.

The starting points for many conventions in spectroscopy are the paper by R. S. Mulliken in the Journal of Chemical Physics (23, 1997, 1955) and the books of G. Herzberg. Apart from straightforward recommendations of symbols for physical quantities, which are generally adhered to, there are rather more contentious recommendations. These include the labelling of cartesian axes in discussions of molecular symmetry and the numbering of vibrations in a polyatomic molecule, which are often, but not always, used. In such cases it is important that any author make it clear what convention is being used. 

As was discussed earlier in Section 1.2.8 a complication arises in that two of these properties (solubility and vapor pressure) are dependent on whether the solute is in the liquid or solid state. Solid solutes have lower solubilities and vapor pressures than they would have if they had been liquids. The ratio of the (actual) solid to the (hypothetical supercooled) liquid solubility or vapor pressure is termed the fugacity ratio F and can be estimated from the melting point and the entropy of fusion. This correction eliminates the effect of melting point, which depends on the stability of the solid crystalline phase, which in turn is a function of molecular symmetry and other factors. For solid solutes, the correct property to plot is the calculated or extrapolated supercooled liquid solubility. This is calculated in this handbook using where possible a measured entropy of fusion, or in the absence of such data the Walden s Rule relationship suggested by Yalkowsky (1979) which implies an entropy of fusion of 56 J/mol-K or 13.5 cal/mol-K (e.u.)... 

Molecules that are not superimposable on their mirror images are termed dissymmetric. This term is used rather than asymmetric, since the latter means, literally, have no symmetry that is, it is applicable only to a molecule belonging to point group Cx. All asymmetric molecules are dissymmetric, but the converse is not true. Dissymmetric molecules can and often do possess some symmetry. It is possible to give a very simple, compact rule expressing the relation between molecular symmetry and dissymmetric character ... 

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. 

The assignment of orbital phases must take appropriate account of molecular symmetry, and although this is easy for open-chain systems, it is much less straightforward for cyclic ones. You usually will be able to avoid this problem by always trying to set up the orbitals so that the transition state will have no nodes, or just one node at a point where a bond is being made or broken. 

When correlating the number of infrared-active CO-stretching modes with molecular structure, the phase in which measurements are made has some bearing on the effective molecular symmetry and must therefore be considered. Measurement of the infrared spectra of compounds in the gas phase is ideal because mtermolecular interactions can be neglected the selection rules which determine the number and activity of the CO-stretching modes are those associated with the point group of the isolated molecule. Because of the limited volatility of many carbonyl complexes and their tendency to decompose at higher temperatures, gas-phase measurements have been limited chiefly to the binary carbonyls (44). It... 

Usually in the study of the electric properties of molecules, it is assumed that the electron and nuclear dynamics cannot destroy the molecular symmetry, and hence the individual characteristics of molecules are conceptually determined by the symmetry of the nuclear framework only. This statement is based on the assumption that the charge distribution (either classical or quantic) is totally symmetric (with respect to the nuclear configuration) i.e., it transforms into itself under the operations of the point symmetry group of the molecule. In cases in which the elec-... 

In this chapter we have introduced the matrix as a means of handling sets of objects and discussed the key aspects of matrix algebra. A great deal of this chapter has involved a cataloguing of the properties and types of matrices, but we have also tried to emphasize the chemical importance of matrices, in particular in the vital role they play in the classification of molecular symmetry and the development of group theory. The key points discussed include ... 

The thermal stability of the novel compounds DCNDBQT and CNBTPA were compared to two other thiophene molecules widely used in literature P,P -dibutylquaterthiophene (DBQT) and p,P -dibutylsexithiophene (DBST). As a first characterisation we used DSC and TGA. DCNDBQT has a higher melting point compared to DBQT and a lower one than DBST. This can be explained by different steric influences of the a,p-substituted groups and the number of thiophene units to molecular symmetry and steric stracture of the thiophene derivatives. 

As shown in the previous section, molecular symmetry largely affects the spectroscopic features of the molecular materials. The oligothiophenes have a well-defined molecular symmetry and belong to the point group of either C2/, or C2v according to whether the molecule has an even or odd number of thiophene units, respectively (refer to Figure 8.14). This molecular symmetry is retained almost completely in the crystals of both the neutral and doped forms (see Section 3.2). 

This was the first example of classification based on molecular shape and gave some indication of the physical properties of molecules that were classified as symmetric compared with those that were labelled asymmetric. However, chirality is not the only manifestation of molecular symmetry, and so a more complete classification of molecular shape has been developed the system of point groups. To classify the symmetry of a molecule we derive its point group, which carries much more geometric information than Pasteur s symmetric or asymmetric designation. 

In all the above treatment, no explicit use has been made of either molecular point-group symmetry or of the equivalent positions within the unit cell. These will have to take care of themselves in the calculation. The Bloch function approach accounts for translational symmetry only, and therefore the independent part of the whole calculation is the unit cell contents. A combination like 6.13 must be done for each of the Nbs atomic orbitals in the unit cell, and the resulting total energies are referred to the molecular contents of the whole unit cell. Nevertheless, molecular symmetry and equivalent positions within the cell will determine the asymmetric unit of the Brillouin zone, thus reducing the amount of independent k-space to be sampled. 

An interesting point is that infrared absorptions that are symmetry-forbidden and hence that do not appear in the spectrum of the gaseous molecule may appear when that molecule is adsorbed. Thus Sheppard and Yates [74] found that normally forbidden bands could be detected in the case of methane and hydrogen adsorbed on glass this meant that there was a decrease in molecular symmetry. In the case of the methane, it appeared from the band shapes that some reduction in rotational degrees of freedom had occurred. Figure XVII-16 shows the IR spectrum for a physisorbed H2 system, and Refs. 69 and 75 give the IR spectra for adsorbed N2 (on Ni) and O2 (in a zeolite), respectively. 

We hope that by now the reader has it finnly in mind that the way molecular symmetry is defined and used is based on energy invariance and not on considerations of the geometry of molecular equilibrium structures. Synnnetry defined in this way leads to the idea of consenntion. For example, the total angular momentum of an isolated molecule m field-free space is a conserved quantity (like the total energy) since there are no tenns in the Hamiltonian that can mix states having different values of F. This point is discussed fiirther in section Al.4.3.1 and section Al.4.3.2. 

Atoms have complete spherical synnnetry, and the angidar momentum states can be considered as different synnnetry classes of that spherical symmetry. The nuclear framework of a molecule has a much lower synnnetry. Synnnetry operations for the molecule are transfonnations such as rotations about an axis, reflection in a plane, or inversion tlnough a point at the centre of the molecule, which leave the molecule in an equivalent configuration. Every molecule has one such operation, the identity operation, which just leaves the molecule alone. Many molecules have one or more additional operations. The set of operations for a molecule fonn a mathematical group, and the methods of group theory provide a way to classify electronic and vibrational states according to whatever symmetry does exist. That classification leads to selection rules for transitions between those states. A complete discussion of the methods is beyond the scope of this chapter, but we will consider a few illustrative examples. Additional details will also be found in section A 1.4 on molecular symmetry. 

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