M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 34
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The coefficient (the normalization factor) in thesymmetry-adapted linear combinations can be determined at a laterstage by normalization. In an actual calculation this is necessary,whereas here we are interested only in the symmetry aspects, whichare well represented by the relative values. In fact, the normalizationfactors will be ignored throughout our discussions.2124 Helpful Mathematical ToolsApplication of the projection operator will also be demonstrated pictorially in forthcoming chapters.
These representations willemphasize the results of summation of symmetry-sensitive propertieswhile the absolute magnitudes will not be treated rigorously. Thus,for example, the directions of vectors will be summed in describingvibrations, and the signs of the angular components of the electronicwave functions will be summed in describing the electronic structure.4.10. Dynamic PropertiesMolecular properties can be of either static or dynamic nature. A staticproperty remains unchanged by every symmetry operation carried outon the molecule. The geometry of the nuclear arrangement in themolecule is such a property: a symmetry operation transforms thenuclear arrangement into another which will be indistinguishable from∗the initial. The mass and the energy of a molecule are also staticproperties.Dynamic properties, on the other hand, may change undersymmetry operations.
Molecular motion itself is a most commondynamic property. In our previous discussions of molecular structure,the molecules were mostly assumed to be motionless, and only thesymmetry of their nuclear arrangement was considered. However, realmolecules are never motionless, and their chemical behavior is influenced by their motion to a great extent.In order to appreciate the effects of symmetry operations on motion,an example from our macroscopic world is invoked here, followingan idea of Orchin and Jaffe [27].
Suppose there exists a long wallof mirror and one walks alongside this mirror (Figure 4-19 left). Ourmirror image will be walking with us with the same speed and in thesame direction (its velocity will be the same as ours). If we walk nowfrom a distance towards the mirror perpendicularly to it, our mirrorimage will have a different velocity from ours: the speed will be thesame again but the direction will be just the opposite. Both we andour mirror image will be walking towards the plane of the mirror, andif we do not stop in time, we shall collide in that plane (Figure 4-19right).∗Unless, of course, identical atoms are distinguished by labels as, e.g., in Figs.
4-2and 4-3.4.11. Where Is Group Theory Applied?213Figure 4-19. Symmetric (left) and antisymmetric (right) consequencies of the“mirror operation” for two movements. Drawing courtesy of the late György Doczi.The consequences of the mirror operation were different for the twomovements. One was symmetric, and the other was antisymmetric.There are analogous phenomena for all kinds of molecular motionwhich may be symmetric and antisymmetric with respect to thevarious symmetry operations of the molecular point group. The twomain kinds of motion in a molecule are nuclear and electronic.The nuclear motion may be translational, rotational and vibrational(Chapter 5).
The electronic motion is basically the changes in the electron density distribution (Chapter 6).4.11. Where Is Group Theory Applied?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 grouptheory. On the other hand, this theory need not be used to determinethe point group symmetry of the nuclear arrangement of a molecule,as has been shown before (cf.
Figure 3-5).The first step in the symmetry determination of the dynamic properties is the selection of the appropriate basis. Appropriate here meansthe correct representation of the changes in the properties examined.In the investigation of molecular vibrations (Chapter 5), either Cartesian displacement vectors or internal coordinate vectors are used.In the description of the molecular electronic structure (Chapter 6),the angular components of the atomic orbitals are frequently used2144 Helpful Mathematical Toolsbases. Since the angular wave function changes its “sign” undercertain symmetry operations, its behavior will be characteristic ofthe spatial symmetry of a particular orbital. Molecular orbitals canalso be used as basis of representation.
The simple scheme belowshows some important areas in chemistry where group theory is indispensable, and the most convenient basis functions are also indicated:AreaConstruction of molecularorbitalsConstruction of hybrid orbitalsPredicting the decrease ofdegeneracies of d orbitalsunder a ligand fieldPredicting the allowedness ofchemical reactionsDetermining the number andsymmetries of molecularvibrationsNormal coordinate analysis(symmetry coordinates)Basis functionsAtomic orbitalsPosition vectors pointing towardthe ligandsd Atomic orbitalsMolecular orbitalsCartesian displacement vectorsInternal coordinatedisplacementsGroup theory is also used prior to calculations to determine whethera quantum-mechanical integral of the type i ôp.
j d is differentfrom zero or not. This is important in such areas as selection rules forelectronic transitions, chemical reactions, infrared and Raman spectroscopy, and other spectroscopies.References1. A. L. Mackay, A Dictionary of Scientific Quotations, Adam Hilger, Bristol,1991, p. 12/123.2. J.
R. Newman (ed.) The World of Mathematics, Dover Publications, Inc.,New York, 2003.3. F. A. Cotton, Chemical Applications of Group Theory, Third Edition, WileyInterscience, New York, 1990.4. A. Nussbaum, Applied Group Theory for Chemists, Physicists and Engineers,Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1971.References2155. L. H. Hall, Group Theory and Symmetry in Chemistry, McGraw-Hill Book Co.,New York, 1969.6. G. Burns, Introduction to Group Theory with Applications, Material ScienceSeries, A. M. Alper, A.
S. Nowich, eds., Academic Press, New York, 1977.7. A. Vincent, Molecular Symmetry and Group Theory: A Programmed Introduction to Chemical Applications, Second Edition, Wiley-Interscience, New York,2001.8. S. F. A. Kettle, Symmetry and Structure: Readable Group Theory for Chemists,John Wiley & Sons, Chichester, 2008.9. H. H. Jaffe, M.
Orchin, Symmetry in Chemistry, Dover Publications, Inc.,New York, 2002.10. P. R. Bunker, P. Jensen, Fundamentals of Molecular Symmetry, Taylor &Francis, London, 2004.11. B. E. Douglas and C.A. Hollingsworth, Symmetry in Bonding and Spectra: AnIntroduction, Academic Press, Orlando, Florida, 1985.12. Cotton, Chemical Applications of Group Theory.13. I. Hargittai, M. Hargittai, “Symmetry of Opposites—Antisymmetry.” Math.Intell.
1994, 16, 60–66.14. A. L. Mackay, “Extensions of Space-Group Theory.” Acta Cryst. 1957, 10,543–548.15. A. V. Shubnikov, Simmetriya i antisimmetriya konechnikh figur, Izd. Akad.Nauk S. S. S. R., Moscow, 1951.16. Ibid.17. Ibid.18. A. Loeb, Color and Symmetry. Wiley-Interscience, New York, 1971.19. A. Loeb, in Patterns of Symmetry, M. Senechal and G. Fleck, eds., Universityof Massachusetts Press, Amherst, Massachusetts, 1977.20.
M. Senechal, Acta Cryst. 1983, A39, 505.21. Cotton, Chemical Applications of Group Theory.22. Nussbaum, Applied Group Theory for Chemists, Physicists and Engineers.23. Hall, Group Theory and Symmetry in Chemistry.24. Cotton, Chemical Applications of Group Theory.25. Ibid.26. Nussbaum, Applied Group Theory for Chemists, Physicists and Engineers.27. M. Orchin, H. H. Jaffe, Symmetry, Orbitals, and Spectra (S.O.S.), WileyInterscience, New York, 1971.Chapter 5Molecular Vibrations...the atoms march in tune.Ralph W. Emerson [1]Vibration is a special kind of motion: the atoms of every moleculeare permanently changing their relative positions at every temperature (even at absolute zero) without changing the position of themolecular center of mass. In terms of the molecular geometry thesevibrations amount to continuously changing bond lengths and bondangles.
Symmetry considerations will be applied to the molecularvibrations in this chapter following primarily References [2–4]. Ourbrief discussion is only an indication of yet another important application of symmetry considerations. The mentioned references and twoother fundamental monographs [5, 6] on vibrational spectroscopy aresuggested for further reading. Our primary concern will be to examinein simple terms the following question: What kind of information canbe deduced about the internal motion of the molecule from the mereknowledge of its point-group symmetry?5.1. Normal ModesThe seemingly random motion of molecular vibrations can alwaysbe decomposed into the sum of relatively simple components, callednormal modes of vibration.
Each of the normal modes is associatedwith a certain frequency. Thus, for a normal mode every atom of themolecule moves with the same frequency and in phase. Three characteristics of normal vibrations will be examined: their number, theirsymmetry, and their type.M. Hargittai, I. Hargittai, Symmetry through the Eyes of a Chemist, 3rd ed.,C Springer Science+Business Media B.V. 2009DOI: 10.1007/978-1-4020-5628-4 5, 2172185 Molecular Vibrations5.1.1.
Their NumberSince vibration is only one of the possible forms of motion, it has tobe separated from the others, translation and rotation. Consider firsta single atom. Its motion can be characterized by the three Cartesiancoordinates of its instantaneous position as shown in Figure 5-1. Inother words, the atom has three degrees of motional freedom. Considernext a diatomic molecule.
It will have 2×3 = 6 degrees of freedom.We might think again that the three Cartesian coordinates of eachatom describe the motion of the molecule in space. However, thisis not quite so. Since the two atoms are not independent from eachother, they must move together in space. This means that three degreesof freedom will account altogether for the translation of a diatomicmolecule (see Figure 5-2) — or of any polyatomic molecule, for thatmatter. Two other degrees of freedom describe the rotation of thediatomic molecule around the center of mass (see Figure 5-3a). Therotation around the z axis (Figure 5-3b) need not be considered as itis the axis of the molecule, and the rotation around it does not changethe position of the molecule.Figure 5-1. Three motional degrees of freedom of an atom.Thus of the six degrees of freedom, five have been accounted for.The sixth will describe the movement of the two atoms relative to eachother without changing the center of the mass.
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