M. Hargittai, I. Hargittai - Symmetry through the Eyes of a Chemist (793765), страница 27
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Oberhammer,“The Chlorofluorophosphoranes PCln F5–n (n = 1–4). Gas-Phase Structuresand Vibrational Analyses.” Inorg. Chem. 1986, 25, 2828–2835.100. I. D. Brown, “Topology and Chemistry.” Struct. Chem. 2002, 13, 339–355.101. R.
Hoffmann, B. F. Beier, E. L. Muetterties, A. Rossi, “7-Coordination –Molecular-Orbital Exploration of Structure, Stereochemistry, and ReactionDynamics.” Inorg. Chem. 1977, 16, 511–522.102. Gillespie, Hargittai, The VSEPR Model of Molecular Geometry.103. See, e.g., W. B. Jensen, “Abegg, Lewis, Langmuir, and the Octet Rule.”J. Chem. Educ. 1984, 61, 191–200.104. G. N.
Lewis, “The Atom and the Molecule.” J. Am. Chem. Soc. 1916, 38,762–785; G. N. Lewis, Valence and the Structure of Atoms and Molecules,Chemical Catalog Co., New York, 1923.105. Ibid.106. N. V. Sidgwick, H. M. Powell, “Bakerian Lecture. Stereochemical Types andValency Groups.” Proc. R. Soc. London, Ser A 1940, 176, 153–180.107. R. J. Gillespie, R. S. Nyholm, “Inorganic Stereochemistry.” Quart. Rev. Chem.Soc. 1957, 11, 339–380.108. R. F. W. Bader, P. J. MacDougall, C.
D. H. Lau, “Bonded and NonbondedCharge Concentrations and Their Relation to Molecular-Geometry and Reactivity.” J. Am. Chem. Soc. 1984, 106, 1594–1605; R. F. W. Bader, Atoms andMolecules: A Quantum Theory, Oxford University Press, Oxford, U. K., 1990.109. See, e.g., R. S. Berry, in Quantum Dynamics of Molecules. The New Experimental Challenge to Theorists, R.
G. Wooley, ed., Plenum Press, New Yorkand London, 1980.110. See, e.g., M. Hargittai, I. Hargittai, “Linear, bent, and quasilinear molecules.”In Structures and Conformations of Non-rigid Molecules, J. Laane,M. Dakkouri, B. van der Veken, and H. Oberhammer, eds., NATO ASI SeriesC.: Mathematical and Physical Sciences, Vol. 410, pp. 465–489, KluwerAcademic Publishers, Dordrecht, Boston, London, 1993.111. Ibid.112. W. v. E. Doering, W. R. Roth, “A Rapidly Reversible Degenerate CopeRearrangement.
Bicyclo[5.1.0]octa-2,5-diene. A rapidly reversible degenerate Cope rearrangement.” Tetrahedron 1963, 19, 715–737 Tetrahedron 1963, 19, 715–737; G. Schroeder, “Preparation and Properties ofTricyclo[3,3,2,04,6]deca-2,7,9-triene (Bullvalene).” Angew. Chem. Int. Ed.Engl. 1963, 2, 481–482; M. Saunders, “Measurement of the Rate of Rearrangement of Bullvalene.” Tetrahedron Lett. 1963, 4, 1699–1702.113. Ibid.114. J. S. McKennis, L. Brener, J.
S. Ward, R. Pettit, “The Degenerate Cope Rearrangements in Hypostrophene, A Novel C10 H10 Hydrocarbon.” J. Am. Chem.Soc. 1971, 93, 4957–4958.115. Ibid.116. R. S. Berry, “Correlation of Rates of Intramolecular Tunneling Processes,with Application to Some Group V Compounds.” J. Chem. Phys. 1960, 32,933–938.References167117. G. M. Whitesides, H. L. Mitchell, “Pseudorotation in (CH3 )2 NPF4 .” J. Am.Chem. Soc. 1969, 91, 5384–5386.118. L.
S. Bartell, M. J. Rothman, A. Gavezzotti, “Pseudopotential SCF-MOStudies of Hypervalent Compounds .4. Structure, Vibrational Assignments,and Intramolecular Forces in IF7 .” J. Chem. Phys. 1982, 76, 4136–4143 andreferences therein; K. O. Christe, E. C. Curtis, D. A. Dixon, “On the Problemof Heptacoordination – Vibrational-Spectra, Structure, and Fluxionality ofIodine Heptafluoride.” J. Am. Chem.
Soc. 1993, 115, 1520–1526.119. E. L. Muetterties, W. H. Knoth, Polyhedral Boranes, Marcel Dekker,New York, 1968.120. W. N. Lipscomb, “Framework Rearrangement in Boranes and Carboranes.”Science 1966, 153, 373–378.121. Ibid., and see also, D. M. P. Mingos, D. J. Wales, in Electron Deficient Boronand Carbon Clusters, G. A. Olah, K. Wade, R. E. Williams, eds., Wiley,New York, 1991; D.
J. Wales, “Rearrangement Mechanisms of B12 H12 2– andC2 B10 H12 .” J. Am. Chem. Soc. 1993, 115, 1557–1567.122. Lipscomb, Science, 373–378.123. R. K. Bohn, M. D. Bohn, “Molecular Structures of 1,2-, 1,7-, and1,12-Dicarba-closo-dodecaborane(12), B10 C2 H12 .” Inorg. Chem. 1971, 10,350–355.124. B. F. G. Johnson, R. E. Benfield, “Structures of Binary Carbonyls and RelatedCompounds. 1. New Approach to Fluxional Behavior.” J. Chem. Soc.
DaltonTrans. 1978, 1554–1568.125. B. E. Hanson, M. J. Sullivan, R. J. Davis, “Direct Evidence for BridgeTerminal Carbonyl Exchange in Solid Dicobalt Octacarbonyl by VariableTemperature Magic Angle Spinning C-13 NMR-Spectroscopy.” J. Am. Chem.Soc. 1984, 106, 251–253.126. R. E. Benfield, B. F. G. Johnson, “The Structures and Fluxional Behaviourof the Binary Carbonyls – A New Approach. 2.
Cluster Carbonyls Mm (CO)n(n = 12,13,14,15, or 16).” J. Chem. Soc. Dalton Trans. 1980, 1743–1767.127. Johnson, Benfield, J. Chem. Soc. Dalton Trans. 1554–1568.Chapter 4Helpful Mathematical ToolsWhat we have to learn to do we learn by doing.Aristotle [1]4.1. GroupsSo far our discussion has been non-mathematical. Ignoring mathematics, however, does not make things necessarily easier. Grouptheory is the mathematical apparatus for describing symmetry operations. It facilitates the understanding and the use of symmetries.
Itmay not even be possible to successfully attack some complex problems without the use of group theory. Besides, groups are fascinating.In his book series, the mathematician, James Newman, characterizedgroup theory in the following way: “The Theory of Groups is a branchof mathematics in which one does something to something and thencompares the result with the result obtained from doing the same thingto something else, or something else to the same thing” [2].This introductory chapter gives the reader the tools necessaryto understand the next three chapters in which molecular vibrations, electronic structure, and chemical reactions are discussed.Further reading is recommended for broader knowledge of the subject[3–11].A mathematical group is a very general idea.
It is a collection(set) of symbols or objects together with a rule telling us how tocombine them. A simple example is a set of two numbers and additionfor the rule. The theory of groups has a wide range of applicationsfar 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 SchoenfliesM. 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 4, 1691704 Helpful Mathematical ToolsFigure 4-1. Symmetry operations in the C2v point group.symbols, for example, C 2v , C 3v or C2h , these symbols represent welldefined groups of symmetry operations. Let us consider first the C2vpoint group.
It consists of a twofold rotation, C2 , and two reflectionsthrough mutually perpendicular symmetry planes, v and v , whoseintersection coincides with the rotation axis. All the correspondingelements are shown in Figure 4-1. One more operation can be addedto these, called the identity operation, E. Its application leaves themolecule unchanged. The set of the operations C2 , v , v , and Etogether make a mathematical group.A mathematical group is a set of elements related by certain rules.They will be illustrated on the symmetry operations.1.
The product of any two elements of a group is also an element ofthe group. The product here means consecutive application of theelements rather than common multiplication. Thus, for example,the product v · C2 means that first a twofold rotation is applied to∗an operand and then reflection is applied to the new operand. Letus perform these operations on the atomic positions of a sulfurylchloride molecule as is shown in Figure 4-2a. The same final resultis obtained by simply applying the symmetry plane v , as is alsoshown in Figure 4-2b. Thus∗Shortly, we shall use a wide range of operands related to molecular structure.4.1. Groups171Figure 4-2 (a) Consecutive application of two symmetry operations, C2 and v tothe nuclear positions of the SO2 Cl2 molecule.
(b) Application of v to SO2 Cl2 .v · C2 = vThe products of the elements in a group are generally notcommutative. That means that the result of the consecutive application of the symmetry operations depends on the order in whichthey are applied. This is why it is so important to read the multiplication sign as “preceded by.” Figure 4-3 gives an example forthe ammonia molecule, which belongs to the C3v point group.Depending on whether the C3 operation is applied first and thenthe v or vice versa, the effect is different. There are some groupsfor which multiplication is commutative, they are called Abeliangroups. The C2v point group is an example.
Thus, in Figure 4-2awe could get the same result first applying the v reflection andthen the twofold rotation.2. One element in the group must commute with all other elements inthe group and leave them unchanged. This is the identity element,E. Thus,E·X =X·E=X3.
The products of the elements in a group are always associative.That means that if there is a consecutive application of several1724 Helpful Mathematical ToolsFigure 4-3. Illustration for the non-commutative character of the symmetry operations.symmetry operations, their application may be grouped in any waywithout changing the final result as long as the order of the application remains the same. Thus, for example,C2 · v · v = C2 · (v · v ) = (C2 · v ) · v4. For each element in a group, there is an inverse or reciprocaloperation which is also an element of the group and satisfies thefollowing condition:X · X −1 = X −1 · X = EFor example,C2 · C2−1 = C2−1 · C2 = Eor−1v · −1v = v · v = EThe symmetry operation corresponding to an inverse operation canbe found in group multiplication tables.
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