Group theory identity

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. 

All molecules possess the identity element of symmetry, for which the symbol is / (some authors use E, but this may cause confusion with the E symmetry species see Section 4.3.2). The symmetry operation / consists of doing nothing to the molecule, so that it may seem too trivial to be of importance but it is a necessary element required by the mles of group theory. Since the C operation is a rotation by 2n radians, Ci = I and the symbol is not used. 

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). 

If the identity is represented by the symbol , condition (ii) can be written E such that EX = XE = X, V2f. Clearly, E plays the role of unity in group theory and it is not by chance that the symbol is the same as thatl ed for the unit matrix (see Section 7.3). 

Although not listed among the symmetry elements for a structure, there is also the identity operation, E. This operation leaves the orientation of the molecule unchanged from the original. This operation is essential when considering the properties that are associated with group theory. When a Cn operation is carried out n times, it returns the structure to its original orientation. Therefore, we can write... 

First, it is possible to simplify the secular equation (2) by means of symmetry. It can be shown by group theory (140) that, in general, the integrals Hi and Si are nonzero only if the orbitals < , and j have the same transformation properties under all the symmetry elements of the molecule. As a simple example, the interaction between an s and a pn orbital which have different properties with respect to the nodal plane of the pn orbital is clearly zero. Interaction above the symmetry plane is cancelled exactly by interaction below the plane (Fig. 13). It is thus possible to split the secular determinant into a set of diagonal blocks with all integrals outside these blocks identically zero. Expansion of the determinant is then simply the product of those lower-order determinants, and so the magnitude of the... 

For nonlinear molecules with identical nuclei that are interchangeable by rotation, the derivation of the nuclear statistical weights is not as simple as for linear molecules. The problem is most efficiently dealt with using group theory. We will not attempt a complete discussion, but will only point out some features of the results.16... 

Many of the properties of IRs that are used in applications of group theory in chemistry and physics follow from one fundamental theorem called the orthogonality theorem (OT). If F, F are two irreducible unitary representations of G which are inequivalent if i -/ j and identical if i = j, then... 

The symbol E comes from the German word Einheit meaning unity. This element generates an operation E which leaves the molecule unchanged. All molecules possess such an element or operation. The need for this seemingly trivial do nothing or leave it alone operation arises from the mathematical requirements of group theory, as we shall see in Chapter 7. Note that in some books the symbol I (for identity) is used in place of E. 

The concept of symmetry is of ancient vintage and in many ways almost identical with the equally elusive concepts of beauty and harmony, i.e. beauty of form arising from balanced proportions. Although symmetry can be described in mathematically precise terms, symmetry in the physical world, like beauty1, never absolutely obeys the mathematical requirements of group theory even the most perfect crystal has a surface that spoils the symmetry. 

The Group Theory for Non-Rigid Molecules considers isoenergetic isomers, and the interconversion motions between them. Because of the discernability between the identical nuclei, each isomer possesses a different electronic Hamiltonian operator in (3), with different eigenfunctions, but the same eigenvalue. In contrast, a non-rigid molecule has an unique effective nuclear Hamiltonian operator (7). 

State, A5 in the notation of group theory, and may here be taken simply as a band label. The Bloch state of angular form zy gives an identical energy symmetry requires the two bands to have the same energy for k in this direction. 

Force constant calculations are facilitated by applying symmetry concepts. Group theory is used to find the appropriate linear combination of internal coordinates to symmetry-adapted coordinates (symmetry coordinates). Based on these coordinates, the G matrix and the F matrix are factorized, which makes it possible to carry out separate calculations for each irreducible representation (c.f. Secs. 2.133 and 5.2). The main problem in calculating force constants is the choice of the potential function. Up until now, it has not been possible to apply a potential function in which the number of force constants corresponds to the number of frequencies. The number of remaining constants is only identical with the number of internal coordinates (simple valence force field SVFF) if the interaction force constants are neglected. If this force field is applied to symmetric molecules, there are often more frequencies than force constants. However, the values are not the same in different irreducible representations, a fact which demonstrates the deficiencies of this force field (Becher, 1968). 

Decomposition of the tensor product space of equation 3.2, which is the foundation for many of the group theory calculations that can be performed using the files on the CD-ROM, leads to the identity... 

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