Molecular point group

The phrase symmetry adapted basis functions refers to those linear combinations of basis functions (on several atoms) that transform like the particular irreducible representation of the appropriate point group. Molecular symmetry is used at various points in these calculations twenty years ago I would have had to write several chapters on molecular symmetry, point groups, constructing symmetry-adapted combinations of basis functions, factoring a Hamiltonian matrix using symmetry and related topics. The point is that twenty... 

Crystal system (bravais lattices) Crystallographic point groups (molecular point groups )... 

Point groups, molecular, 3-6 Polarization functions, 25, 233 Polyethylene, 193 Polymerization... 

Point groups, molecular, 3-6 Polarization functions, 25, 233 Polyethylene, 193 Polymerization Kaminsky, 193 Ziegler-Natta, 192 Population analysis Mulliken, 91, 236 SHMO, 91-92... 

We now turn to electronic selection rules for syimnetrical nonlinear molecules. The procedure here is to examme the structure of a molecule to detennine what synnnetry operations exist which will leave the molecular framework in an equivalent configuration. Then one looks at the various possible point groups to see what group would consist of those particular operations. The character table for that group will then pennit one to classify electronic states by symmetry and to work out the selection rules. Character tables for all relevant groups can be found in many books on spectroscopy or group theory. Ftere we will only pick one very sunple point group called 2 and look at some simple examples to illustrate the method. 

This is the central Jahn-Teller [4,5] result. Three important riders should be noted. First, Fg = 0 for spin-degenerate systems, because F, x F = Fo. This is a particular example of the fact that Kramer s degeneracies, aiising from spin alone can only be broken by magnetic fields, in the presence of which H and T no longer commute. Second, a detailed study of the molecular point groups reveals that all degenerate nonlinear polyatomics, except those with Kramer s... 

An extensive series of studies for the prediction of aqueous solubility has been reported in the literature, as summarized by Lipinski et al. [15] and jorgensen and Duffy [16]. These methods can be categorized into three types 1 correlation of solubility with experimentally determined physicochemical properties such as melting point and molecular volume 2) estimation of solubility by group contribution methods and 3) correlation of solubility with descriptors derived from the molecular structure by computational methods. The third approach has been proven to be particularly successful for the prediction of solubility because it does not need experimental descriptors and can therefore be applied to collections of virtual compounds also. 

It is recommended that the reader become familiar with the point-group symmetry tools developed in Appendix E before proceeding with this section. In particular, it is important to know how to label atomic orbitals as well as the various hybrids that can be formed from them according to the irreducible representations of the molecule s point group and how to construct symmetry adapted combinations of atomic, hybrid, and molecular orbitals using projection operator methods. If additional material on group theory is needed. Cotton s book on this subject is very good and provides many excellent chemical applications. 

That no degenerate molecular orbitals arose in the above examples is a result of the fact that the C2v point group to which H2O and the allyl system belong (and certainly the... 

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 functions put into the determinant do not need to be individual GTO functions, called Gaussian primitives. They can be a weighted sum of basis functions on the same atom or different atoms. Sums of functions on the same atom are often used to make the calculation run faster, as discussed in Chapter 10. Sums of basis functions on different atoms are used to give the orbital a particular symmetry. For example, a water molecule with symmetry will have orbitals that transform as A, A2, B, B2, which are the irreducible representations of the C2t point group. The resulting orbitals that use functions from multiple atoms are called molecular orbitals. This is done to make the calculation run much faster. Any overlap integral over orbitals of different symmetry does not need to be computed because it is zero by symmetry. 

Most ah initio calculations use symmetry-adapted molecular orbitals. Under this scheme, the Hamiltonian matrix is block diagonal. This means that every molecular orbital will have the symmetry properties of one of the irreducible representations of the point group. No orbitals will be described by mixing dilferent irreducible representations. 

The point groups discussed here are all those that one is likely to use, but there are a few very uncommon ones that have not been included descriptions of these are to be found in the books on molecular symmetry mentioned in the bibliography at the end of this chapter. 

Pyrazine belongs to the D2h point group, and Table A.32 in Appendix A shows that since B u = r(Tj) the Oq band is polarized along the x axis, which is perpendicular to the molecular plane. 

Table 3 (3) shows the influence of branching of the alkyl group on volatility and complexity, usiag titanium and zirconium amyl oxides as examples. Table 3. Boiling Points and Molecular Complexities of Amyloxides of Titanium and of Zirconium...

Mol. ratio epichlorohydrin bis-phenol A Mol. ratio NaOHl epichlorohydrin Softening point rc) Molecular weight Epoxide equivalent Epoxy groups per molecule... 

At higher frequencies (above 200 cm ) the vibrational spectra for fullerenes and their cry.stalline solids are dominated by the intramolecular modes. Because of the high symmetry of the Cgo molecule (icosahedral point group Ih), there are only 46 distinct molecular mode frequencies corresponding to the 180 6 = 174 degrees of freedom for the isolated Cgo molecule, and of these only 4 are infrared-active (all with Ti symmetry) and 10 are Raman-active (2 with Ag symmetry and 8 with Hg symmetry). The remaining 32 eigcnfrequencies correspond to silent modes, i.e., they are not optically active in first order. 

We will find an excitation which goes from a totally symmetric representation into a different one as a shortcut for determining the symmetry of each excited state. For benzene s point group, this totally symmetric representation is Ajg. We ll use the wavefunction coefficients section of the excited state output, along with the listing of the molecular orbitals from the population analysis ... 

Figure 13.17 Molecular structure of some sulfides of arsenic, stressing the relationship to the AS4 tetrahedron (point group symmetry in parentheses).

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