Group theory and molecular vibrations

Cotton, F. A., Chemical Applications of Group Theory, Second Edition, Wiley, New York, 1970. A popular, nonrigorous introduction with applications to MO theory and molecular vibrations. 

The next section describes the application of group theory to molecular vibrations and its consequences, e.g. the determination of the number of vibrations of all symmetry species. 

In addition the reader may find tables with selection rules for the Resonance Raman and Hyper Raman Effect in the book of Weidlein et al. (1982). Special discussions about the basics of the application of group theory to molecular vibrations are given in the books of Herzberg (1945), Michl and Thulstrup (1986), Colthup et al. (1990) and Ferraro and Nakamoto (1994). Herzberg (1945) and Brandmiiller and Moser (1962) describe the calculation of thermodynamical functions (see also textbooks of physical chemistry). For the calculation of the rotational contribution of the partition function a symmetry number has to be taken into account. The following tables give this number in Q-... 

Symmetry-forbidden transitions. A transition can be forbidden for symmetry reasons. Detailed considerations of symmetry using group theory, and its consequences on transition probabilities, are beyond the scope of this book. It is important to note that a symmetry-forbidden transition can nevertheless be observed because the molecular vibrations cause some departure from perfect symmetry (vibronic coupling). The molar absorption coefficients of these transitions are very small and the corresponding absorption bands exhibit well-defined vibronic bands. This is the case with most n —> n transitions in solvents that cannot form hydrogen bonds (e 100-1000 L mol-1 cm-1). 

But perhaps the most determining factor happened to be the war. As prisoner for five years, Barriol s only instruments were paper and pencil. The war obliged me to a complete theoretical activity . During this period, besides teaching, he worked on the group theory and its applications to crystalline and molecular vibrations. This topic would eventually constitute his doctoral dissertation in 1946. 

In the next chapter, we will present various chemical applications of group theory, including molecular orbital and hybridization theories, spectroscopic selection rules, and molecular vibrations. Before proceeding to these topics, we first need to introduce the character tables of symmetry groups. It should be emphasized that the following treatment is in no way mathematically rigorous. Rather, the presentation is example- and application-oriented. 

The as yet unknown pentagonal dodecahedrane molecule (5) with its high />, (icosahedral) symmetry and aesthetic allure has recently been studied from group theory, graph theory, and molecular orbital theory viewpoints.434 Its vibrational... 

Infrared radiation is only absorbed by the irradiated molecule at the appropriate frequency if the corresponding vibration results in a change in molecular dipole moment. This means that not all vibrational modes are infrared active. An analysis of which vibrational modes in a polyatomic molecule are active is based on group theory and the symmetry properties of the molecule. More details about this subject may be found in monographs devoted to spectroscopy [G4]. 

Here we will consider two applications of symmetry and group theory, in the realms of chirality and molecular vibrations. In Chapter 5 we will also examine how symmetry can be used to understand chemical bonding, perhaps the most important application of symmetry in chemistry. 

Vibrational frequencies may be extracted from the PES by performing a normal mode analysis. This analysis of the normal vibrations of the molecular configurations is a difficult topic and can be pursued efficiently only with the aid of group theory and advanced matrix algebra. In essence, the 3 translational, 3 rotational and 3N-6 vibrational modes (2 rotational and 3N-5 vibrational modes for linear molecules) may be determined by a coordinate transformation such that all the vibrations separate and become independent normal modes, each performing oscillatory motion at a well defined vibrational frequency. As a more concrete illustration, assume harmonic vibrations and separable rotations. The PES can thus be approximated by a quadratic form in the coordinates... 

D. Willock (2009) Molecular Symmetry, Wiley, Chichester - A student text introducing symmetry and group theory and their applications to vibrational spectroscopy and bonding. 

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. 

A nonlinear molecule of N atoms with 3N degrees of freedom possesses 3N — 6 normal vibrational modes, which not all are active. The prediction of the number of (absorption or emission) bands to be observed in the IR spectrum of a molecule on the basis of its molecular structure, and hence symmetry, is the domain of group theory [82]. Polymer molecules contain a very high number of atoms, yet their IR spectra are relatively simple. This can be explained by the fact that the polymer consists of identical monomeric units (except for the end-groups). 

It is the objective of the present chapter to define matrices and their algebra - and finally to illustrate their direct relationship to certain operators. The operators in question are those which form the basis of the subject of quantum mechanics, as well as those employed in the application of group theory to the analysis of molecular vibrations and the structure of crystals. 

In this chapter we present some results obtained by our group on scar theory in the context of molecular vibrations, and in particular for the LiNC/LiCN molecular system. This kind of (generic) systems exhibits a dynamical behavior in which regular and chaotic motions are mixed (Gutzwiller, 1990), a situation which presents significant differences with respect to the completely chaotic case considered in most references cited above, and are very important in many areas of physics and chemistry. 

I have left group theory as the last chapter. If it is desired to cover group theory earlier in the course, Sections 9.1-9.8 of the group-theory chapter can be studied prior to Chapter 6 on molecular vibrations, and Sections 9.9-9.12 read in conjunction with Chapter 6. 

Just as group theory enables one to find symmetry-adapted orbitals, which simplify the solution of the MO secular equation, group theory enables one to find symmetry-adapted displacement coordinates, which simplify the solution of the vibrational secular equation. We first show that the matrices describing the transformation properties of any set of degenerate normal coordinates form an irreducible representation of the molecular point group. The proof is based on the potential-energy expression for vibration, (6.23) and (6.33) ... 

Tirana afld comparison with (9.189) shows that the group-theory selection rules for electronic transitions are the same as for vibrational transitions, except that we must consider the symmetry species of the electronic wave functions, rather than the vibrational wave functions. One complication is that the molecular geometry may change on electronic excitation in this case, we use the point group of lower symmetry to classify the wave functions and determine the selection rules. 

Microwave spectrometer, 219-221 Microwave spectroscopy, 130, 219-231 compilations of results of, 231 dipole-moment measurements in, 225 experimental procedures in, 219-221 frequency measurements in, 220 and molecular structure, 221-225 and rotational barriers, 226-228 and vibrational frequencies, 225-226 Mid infrared, 261 MINDO method, 71,76 and force constants, 245 and ionization potentials, 318-319 Minimal basis set, 65 Minor, 14 Modal matrix, 106 Molecular orbitals for diatomics, 58 and group theory, 418-427 for polyatomics, 66... 

In applying the methods of group theory to problems related to molecular structure or dynamics, the procedure that is followed usually involves deriving a reducible representation for the phenomenon of interest, such as molecular vibration, and then decomposing it into its irreducible components. (A reducible representation will always be a sum of irreducible ones.) Although the decomposition can sometimes be accomplished by inspection, for the more general case, the following reduction... 

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