Classifying symmetry operations

A test for recognizing a covering operation is that it would go unnoticed by an observer who was temporarily not looking while the operation was being performed. 

The complete list of covering operations of any object is a precise description of its symmetry. 

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Using the C2 character table (Table A. 11 in Appendix A) the characters of the vibrations under the various symmetry operations can be classified as follows ... 

The cyclobutene-butadiene interconversion can serve as an example of the reasoning employed in construction of an orbital correlation diagram. For this reaction, the four n orbitals of butadiene are converted smoothly into the two n and two a orbitals of the ground state of cyclobutene. The analysis is done as shown in Fig. 11.3. The n orbitals of butadiene are ip2, 3, and ij/. For cyclobutene, the four orbitals are a, iz, a, and n. Each of the orbitals is classified with respect to the symmetiy elements that are maintained in the course of the transformation. The relevant symmetry features depend on the structure of the reacting system. The most common elements of symmetiy to be considered are planes of symmetiy and rotation axes. An orbital is classified as symmetric (5) if it is unchanged by reflection in a plane of symmetiy or by rotation about an axis of symmetiy. If the orbital changes sign (phase) at each lobe as a result of the symmetry operation, it is called antisymmetric (A). Proper MOs must be either symmetric or antisymmetric. If an orbital is not sufficiently symmetric to be either S or A, it must be adapted by eombination with other orbitals to meet this requirement. 

Table 6-1. C2(l molecular poinl group. The electronic stales of the flat T6 molecule are classified according lo the lwo-1 old screw axis (C2). inversion (/). and glide plane reflection (o ) symmetry operations. The A and lt excited slates transform like translations Oi along the molecular axes and are optically allowed. The Ag and Bg stales arc isoniorphous with the polarizability tensor components (u), being therefore one-photon forbidden and Iwo-pholon allowed.

A symmetry operation is one which leaves the framework of a molecule unchanged, such that an observer who has not watched the operation cannot tell that an operation has been carried out on the molecule (of course me presupposes the structure of the molecule from other experimental sources). The geometry of the molecule is governed by the geometry of the orbitals used by the constituent atoms to form the molecule. There are five kinds of symmetry operations which are necessary for classifying a point group. 

The theory of optical activity would be understood in terms of symmetry considerations at the first stage. The elements of symmetry are the geometric elements in relation to which the symmetry operations are carried out, and are classified in the following ... 

Normal modes that have the same behavior with respect to the set of symmetry operations of a molecule are said to belong to the same symmetry species. An example is vx and v2 for H20. (Recall that we classified molecular orbitals and molecular electronic wave functions according to their symmetry species. Further discussion of symmetry species will be found in Chapter 9.)... 

Polyatomic molecules. The same term classifications hold for linear polyatomic molecules as for diatomic molecules. We now consider nonlinear polyatomics. With spin-orbit interaction neglected, the total electronic spin angular momentum operator 5 commutes with //el, and polyatomic-molecule terms are classified according to the multiplicity 25+1. For nonlinear molecules, the electronic orbital angular momentum operators do not commute with HeV The symmetry operators Or, Os,. .. (corresponding to the molecular symmetry operations R, 5,. ..) commute... 

The symmetry group to which an A(B, C,. . . ) molecule belongs is determined by the arrangement of the pendent atoms. The A atom, being unique, must lie on all planes and axes of symmetry. The orbitals that atom A uses in forming the A—(B, C,. . . ) bonds must therefore be discussed and classified in terms of the set of symmetry operations generated by these axes and planes—that is, in terms of the overall symmetry of the molecule. Thus, our first order of business is to examine the wave functions for AOs and consider their transformation (symmetry) properties under the various operations which constitute the point group of the A(B, C,. . . ) molecule. 

The symmetry operations of the water molecule are collected in Table 6-3, together with the transformation properties of the sundry orbitals classified under C2V, the point group of water (see Figure 6-9). 

Now the trace of the new representation should correspond to a character of the irreducible representation of the double group. By inspecting all classes of symmetry operations, the given z-th energy level is unambiguously classified according to the character table of the double group. 

As for the case of the two-leg ladder, we select the Local Configurations (LC) which are the much important for the ground state wave function. This selection of the relevant LC is made by combining energetic and symmetry considerations. For that purpose, we use the electron-hole symmetry operator, J, to classify the LC. On the Fock space of a single site n, the action of this operator are summarized as follows [40]... 

Naphthalene and anthracene are archetypes of the even and odd members of the polyacene series. In each subseries, one can start by classifying the classical Kekule structures by using the symmetry operations i, C2, and point group. Then one can form symmetry-adapted linear combinations of the mutually transformable Kekule structures and deduce their bonding characteristics. Finally, these 1 Ag and 1 B2u symmetry-adapted combinations are allowed to mix and form the states of interest, the ground and first covalent excited states (16). 

Thiophosgene possesses the planar molecular structure of Figure 2.1 in the ground electronic state and can be classified by the symmetry operations of... 

Along with the primitive translations, and glide-reflections when appropriate, there axe other symmetry operations belonging to the space group. In bipartite systems, it is relevant to classify any symmetry operation according to whether it leaves each sublattice invariant or transforms one into each other (see, for instance, Fig. 2). 

The inversion operation i which leads to the g/u classification of the electronic states is not a true symmetry operation because it does not commute with the Fermi contact hyperfine Hamiltonian. The operator i acts within the molecule-fixed axis system on electron orbital and vibrational coordinates only. It does not affect electron or nuclear spin coordinates and therefore cannot be used to classify the total wave function of the molecule. Since g and u are not exact labels, it was realised by Bunker and Moss [265] that electric dipole pure rotational transitions were possible in ll], the g/u symmetry breaking (and simultaneous ortho-para mixing) being relatively large for levels very close to the dissociation asymptote. The electric dipole transition moment for the 19,1 19,0 rotational transition in the ground electronic state was calculated... 

We are now in a position to describe and classify the various molecular point groups. In this designation bold face symbols represent the point groups and the non-bold symbols are used to denote symmetry operations. The bold face symbols are used for the point groups are based for the most part on the principal symmetry elements of the group and that subscripts added to some of the point group symbols further tie in the symbol with elements in the group. 

The symmetry groups to which real molecules may belong are very numerous. However they may be systematically classified by considering how to build them up using increasingly more elaborate combinations of symmetry operations. The outline that follows, though neither unique in its approach nor rigorous in its procedure, affords a practical scheme for use by most chemists. 

Regarding the property symmetric , antisymmetric , or degenerate with respect to all. symmetry operations, vibrations can be classified according to symmetry species. Each symmetiy species possesses certain spectroscopic characteristics, like forbidden in IR and Raman spectra , or IR-active with dipole moment change in. v-direction , or modulates the xy component of the polarizability tensor . They are given in character tables (Figure 2.7-6), Sec. 7. 

Every atom of a molecule has three degrees of freedom, i.e., it can move independently along each of the axes of a Cartesian coordinate system. A molecule containing n atoms therefore has 3 motional degrees of freedom. In order to classify these, g symmetry operators of the point group of the molecule have to be applied to them. 

Since pericyclic reactions proceed with a cyclic reorganization of bonding electron pairs, it is necessary to evaluate changes in the associated MOs that take place in going from reactants to products. These orbitals may be classified by two independent symmetry operations a mirror plane (m) perpendicular to the functional plane and bisecting the molecule, and a twofold axis of rotation (C2). 

Various MOs of reactants and products are classified according to two independent symmetry operations, either a plane of symmetry (m) or a twofold axis of symmetry (C2). For... 

The symmetry operations for objects in ordinary three-dimensional space can be classified into four fundamental types, each of which is defined by a synunetry element around which the synunetry operation takes place. The four fundamental types of synunetry operations and their corresponding synunetry elements are listed in Table 1. 

Symmetry concepts can be extremely useful in chemistry. By analyzing the symmetry of molecules, we can predict infrared spectra, describe the types of orbitals used in bonding, predict optical activity, interpret electronic spectra, and study a number of additional molecular properties. In this chapter, we first define symmetry very specifically in terms of five fundamental symmetry operations. We then describe how molecules can be classified on the basis of the types of symmetry they possess. We conclude with examples of how symmetry can be used to predict optical activity of molecules and to determine the number and types of infrared-active stretching vibrations. 

Symmetry operations apply not only to the unit cells shown in Fig. 2-3, considered merely as geometric shapes, but also to the point lattices associated with them. The latter condition rules out the possibility that the cubic system, for example, could include a base-centered point lattice, since such an array of points would not have the minimum set of symmetry elements required by the cubic system, namely four 3-fold rotation axes. Such a lattice would be classified in the tetragonal system, which has no 3-fold axes and in which accidental equality of the a and c axes is allowed. 

In addition, as far as electron-electron interaction is neglected, the tt electrons are subject to a potential with the full spatial symmetry of the CNT topology. The electronic wavefunctions can then be classified according to their transformation properties under the symmetry operations leaving the CNT invariant. As far as dipole approximation holds, both the linear and nonlinear optical response of a CNT are governed by matrix elements of the dipole operator between two electronic wavefunctions. Being the selection rules of dipole matrix elements essentially determined by the wavefunctions symmetry, it turns out that the optical properties of CNT are essentially rooted in their topology. 

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