Symmetry of Molecules

What is symmetry In this book we are interested in two uses of this word. First, we are interested in the symmetiy of a molecule. When looking at objects, we invariably have some feel as to whether they are highly symmetric or alternatively not very symmetric. This needs to be quantified in some way. Second, we need to classify, in terms of some symmetry description, the molecular orbitals of molecules and fragments. It is just this aspect that tremendously simplifies the construction of molecular orbitals rather than blindly, mechanically solving the secular determinant and equations of Chapter I. Once this symmetry classification has been done, with the use of a few mathematical tools, we will be in a good position to understand how symmetry controls the orbital structure in molecules. 

Geometrical objects (including molecules, if they are regarded as being made up of balls and spokes) possess an associated set of symmetry elements or operations,  

Thomas A. Albright, Jeremy K. Burden, and Myung-Hwan Whangbo. 

The null operation, the identity A rotation around an axis by 360% 

A rotation around an axis by x(360%) x sequential C operations A vertical mirror plane containing the principal rotation axis (the one of highest order) A horizontal mirror plane, one perpendicular to the principal rotation axis A dihedral mirror plane, one which bisects two cTy or operations An improper rotation axis rotation around an axis by 360% followed by reflection in a plane perpendicular to this axis X sequential S operations 

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Systems can possess different extents of complexity. To measure complexity, the information content of the system can be used. Application of information theory is increasingly finitful for modeling biological activities with regard to the symmetry of molecules. 

The metal-free porphyrazine, H Pz (3), was prepared by treatment of MgPz with minimum amount of triflouraaceticacid. When metal ion is removed from the center of core system, two N atoms in pyrrole ring are protonated. That was proved by both IR and UVWIS spectra of the compound. In the IR spectram, the peak around 3,240 cm shows the presence N-H bonds in H Pz. There is splitting in the Q band absorptions of H Pz, because of the symmetry of molecule. The Q band... 

The symmetries of molecules usually change during the vibrations, mostly their symmetries become lower (Figure 2), sometimes they don t change, and in some special cases they may become higher. In most of our discussion, the effects of motion on molecular symmetry will be ignored for... 

A mathematical group consists of a set of elements which are related to each other according to certain rules, outlined later in the chapter. The particular kind of elements which are relevant to the symmetries of molecules are symmetry elements. With each symmetry element there is an associated symmetry operation. The necessary rules are referred to where appropriate. 

The conception of the electrostatic bond has been found to be most valuable in the field of inorganic chemistry and has helped to clarify a great many phenomena. It has frequently made it possible to predict properties both of unknown compounds and of those which have been little investigated. Nevertheless, a number of difficulties have been encountered. For example, in Section 43, it was shown that the theory did not really provide a satisfactory explanation of the low symmetry of molecules. Again, the theory failed to explain the volatility and small dipole moment of the compounds CO, NO, C120, etc., and lastly, it threw no light whatever on the reason for the existence of molecules of elements such as H2, Cl2, Oa, N2, etc., especially as dipole measurements show that the formulae H+H, Ci+ci are excluded. 

The above remarks refer to the symmetries of molecules in crystals. It is very important to remember that the symmetry of a molecule in its crystal setting is not necessarily the full symmetry of an isolated molecule, since, as we have seen, the full symmetries of molecules are not always utilized in forming crystalline arrangements. Suppose, for example, that X-ray and other evidence leads to the definite conclusion that certain molecules in their crystal setting have no symmetry. It does not follow that these molecules in isolation are asymmetric it may be that in isolation they would have axes of symmetry or planes of... 

A symmetry operation is a transformation of a body that sends it to a position physically indistinguishable from its original position, and that preserves the distances between all pairs of points in the body. A symmetry element is a point, line, or plane with respect to which a symmetry operation is carried out. In discussing the symmetry of molecules, we consider the symmetry of the nuclear framework. 

A flowchart for assigning point symmetry. The symmetry elements, and the rules and procedures for their use in determining the symmetry of molecules, can be formalized in a flow chart such as that shown in Fig. 3.16. It contains all of the point groups discussed above (enclosed in square boxes) as well as a few others not commonly encountered. In addition, the symmetries assigned above by inspection may be derived in a more systematic way by the use of this diagram. 

The vehement opposition generated by van t Hoff s proposals was evidently directed against the notion that atoms possessed non-spherical three-dimensional structures. The equivalent, more careful, formulation of Le Bel referred explicitly to the symmetry of molecules, such as methane, and is free of this criticism. It is unfortunate that it was the van t Hoff picture which became established, first in terms of Sommerfeld s elliptic orbits and later on in Pauling s hybrid orbitals. 

I. Rips, Taking into Account the Symmetry of Molecules in Quantum Chemical Calculations, Zinatne, Riga, Latvia, 1978. 

A mathematical group is a very general idea. It is a collection (set) of symbols or objects together with a rule telling us how to combine them. A simple example is a set of two numbers and addition for the rule. The theory of groups has a wide range of applications far beyond pure mathematics especially in physics and chemistry. Symmetry and group theory are inherently related to each other. When the symmetries of molecules are characterized by Schoenflies... 

We are, of course, concerned with the symmetry aspects of the MOs and their construction. As was discussed before, the degeneracy of atomic orbitals is determined by mi. Thus, all p orbitals are threefold degenerate, and all d orbitals are fivefold degenerate. The spherical symmetry of the atomic subshells, however, necessarily changes when the atoms enter the molecule, since the symmetry of molecules is nonspherical. The degeneracy of atomic orbitals will, accordingly, decrease the extent of decrease will depend upon molecular symmetry. 

It should be noted that this book, both in its structure and in its content, is in many respects logically related to our earlier booklet I. S. Dmitriev, Symmetry in the World of Molecules, Mir Publishers, Moscow, 1979. This relation is not accidental since both topology and symmetry of molecules furnish a valuable qualitative and semi-quantitative complementary information about theif structure and properties. 

The symmetry of many molecules and especially of crystals is immediately obvious. Benzene has a six-fold symmetry axis and is planar, buckminsterfullerene (or just fullerene or footballene) contains 60 carbon atoms, regularly arranged in six- and five-membered rings with the same symmetry (point group //,) as that of the Platonic bodies pentagon dodecahedron and icosahedron (Fig. 2.7-1). Most crystals exhibit macroscopically visible symmetry axes and planes. In order to utilize the symmetry of molecules and crystals for vibrational spectroscopy, the symmetry properties have to be defined conveniently. 

In the language of classical dynamics and electrodynamics (which of course give the same results as the quantum mechanics in the case of slow vibrations) this means that any determination of the symmetry of molecules which form part of a collection of molecules arranged at random will in general give the symmetry of the normal state of the molecule, provided that we confine ourselves to sufficiently low temperatures. 

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