Group theoretical method

Sokolov, A. V., and Shirokovski, V. P., Group Theoretical Methods in the Quantum Physics of Solids (Spatial Symmetry), Soviet Physics—Uspekhi, 3, 651 (1961). 

The vibrations of the free molecule can be correlated with the vibrations of the crystal by group theoretical methods. Starting with the point group of the molecule Did)> the irreducible representations (the symmetry classes) have to be correlated with those of the site symmetry (C2) in the crystal and, as a second step, the representations of the site have to be correlated with those of the crystal factor group (D2h) [89, 90]. Since the C2 point group is not a direct subgroup of of the molecule and of D211 of the crystal, the correlation has to be carried out in successive steps, for example ... 

S. Kim (1998) Group Theoretical Methods and Applications to Molecules and Crystals (Cambridge University Press, Cambridge). 

In 1937 Jahn and Teller applied group-theoretical methods to derive a remarkable theorem nonlinear molecules in orbitally degenerate states are intrinsically unstable with respect to distortions that lower the symmetry and remove the orbital degeneracy.37 Although Jahn-Teller theory can predict neither the degree of distortion nor the final symmetry, it is widely applied in transition-metal chemistry to rationalize observed distortions from an expected high-symmetry structure.38 In this section we briefly illustrate the application of Jahn-Teller theory and describe how a localized-bond viewpoint can provide a complementary alternative picture of transition-metal coordination geometries. 

MSN.96. A. P. Grecos and 1. Prigogine, Irreversible processes in quanmm theory, in Group Theoretical Methods in Physics, Lecture Notes in Physics, Vol. 94, Springer, 1979, pp. 229-237. 

This phenomenon of vibronic coupling can be treated very effectively by using group theoretical methods. As will be shown in Chapter 10, the vibrational wave function of a molecule can be written as the product of wave functions for individual modes of vibration called normal modes, of which there will be 3n - 6 for a nonlinear, /i-atomic molecule. That is, we can... 

Unit tensors are especially important for group-theoretical methods of studying the lN configuration. We can express the infinitesimal operators of the groups [10, 24, 98], the parameters of irreducible representations of which are applied to achieve an additional classification of states of a shell of equivalent electrons, in terms of them. 

Group-theoretical methods of classification of the states of a shell of equivalent electrons. Casimir operators... 

In Chapter 14 we have already discussed the group-theoretical method of classification of the states of a shell of equivalent electrons. Remembering that second-quantization operators in isospin space have an additional degree of freedom, we can approach the classification of states in isospin basis in exactly the same way. 

The group-theoretic method to find nontrivial maps S3 —> S2 is based on the isomorphism between S3 and the group manifold SU(2). Every point g C SU(2) can be written as... 

Kim, S. K. (1999) Group Theoretical Methods. Cambridge Cambridge University Press. 

T.H. Seligman, Group Theoretical Methods in Physics, Nijmegen, Netherlands, 1973. 

In order to determine into how many Stark terms a given energy level splits when put into a ligand field without making a detailed calculation of the values, the group-theoretical methods of Bethe (66) are convenient. In this method it is noted that the spherical harmonics transform according to the Ith irreducible repre-... 

This is one of the reasons for the power and generality of group theoretical methods in discussing the properties of molecules for although the number of different imaginable molecules is unbounded, this is not true of their possible systems of axes and planes of symmetry. These are severely restricted by geometrical considerations and it is possible to write down a list of all the molecular symmetry groups that can exist and to discuss... 

A few additional things need to be mentioned before embarking on chemical applications of group theoretical methods. For detailed descriptions and proofs we refer to References [21-23],... 

It is primarily the description of the dynamic properties that is facilitated by group-theoretical methods. This is, in fact, an understatement. The dynamic properties cannot be fully discussed without group theory. On the other hand, this theory need not be used to determine the point group symmetry of the nuclear arrangement of a molecule, as has been shown before (cf. Figure 3-5). 

The structure of our book has not changed. Following the Introduction (Chapter 1), Chapter 2 presents the simplest symmetries using chemical and non-chemical examples. Molecular geometry is discussed in Chapter 3. The next four chapters present group-theoretical methods (Chapter 4) and, based on them, discussions of molecular vibrations (Chapter 5), electronic structures (Chapter 6), and chemical reactions (Chapter 7). For the last two chapters we return to a qualitative treatment and introduce space-group symmetries (Chapter 8), concluding with crystal structures (Chapter 9). 

The PI group operations are defined by their effect on the space-fixed coordinates of the atomic nuclei and electrons. Since our molecular wavefunctions are written in terms of the vibrational coordinates, the Euler angles and the angle p, we must first determine the effect of the PI group operations on these variables. In the case of inversion this can lead to certain problems both in the understanding of the concepts of molecular symmetry and in the proper use of group theoretical methods in the classification of the states of ammonia. 

Whereas framework groups are more informative than point symmetry groups, they are able to describe only a rather restricted aspect of molecular shape. Alternative group theoretical methods, notably, the Shape Group Methods (SGM) of molecular topology, are more suitable for a detailed shape characterization. The shape groups will be discussed in Chapter 5 of this book. 

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