Molecular Orbital Theory; an Introductory Lecture Note and Reprint Volume

Registered Works Database (Author Search) 1. Registration Number: RE-642-366 Title: Molecular orbital theory; an introductory lecture note and reprint volume. By acCarl J. Ballhausen and Harry B. Gray. Claimant: Carl J. Ballhausen (A) Effective Registration Date: 1Nov93 Original Registration Date: 7Dec64; Original Registration Number: A357882. Original Class: A Permission granted for scanning and deposition in Caltech CODA by telephone call from Carl J. Ballhausen, June 1, 2006 per Dana L. Roth. FRONTIERS IN CHEMISTRY Ronald Breslow and Martin Karplus, Editors Columbia University CONTRIBUTIONS TO THE THEORY OF CHEMICAL KINETICS T. A. Bak MOLECULAR ORBITAL THEORY C. J. Ballhausen K0benhavns Universitet H. B. Gray Columbia University THERMODYNAMICS OF SMALL SYSTEMS : Parts I and II T. L. Hill University of Oregon LECTURES ON QUANTUM THEORY OF MOLECULAR ELECTRONIC STRUCTURE R. G. Parr The Johns Hopkins University THE BIOSYNTHESIS OF STEROIDS, TERPENES, AND ACETOGENINS J. H. Richards California Institute of Technology J. B. Hendrickson Brandeis University OXIDATION MECHANISMS : Applications to Organic Chemistry R. Stewart University of British Columbia COMPUTER PROGRAMMING FOR ORGANIC CHEMISTS K. J. Wiberg Yale University MOLECULAR ORBITAL THEORY An Introductory Lecture Note and Reprint Volume C.J. BALEHAUSEN i.1 F Kmbenhavns Universitet and HARRY B. GRAY Columbia University W. A. BENJAMIN, INC. 1965 New York Amsterdam MOLECULAR ORBITAL THEORY: An Introductory Lecture Note and Reprint Volume Copyright @ 1964 by W. A. Benjamin, Inc. All rights reserved Library of Congress Catalog Number 65-12062 Manufactured in the United States of America This manuscript was put into production on May 7,1964; this volume was published on December 7,1964; second printing with corrections September 20,1965 W. A. BENJAMIN, INC. New York, New York 10016 Preface These notes are based on lectures on molecular orbital theory that we have presented at the University of Copenhagen and Columbia University. They were designed primarily for advanced-undergraduate and first-year graduate students as an introduction to molecular orbital theory. It is apparent that the molecular orbital theory is a very useful method of classifying the ground and excited states of small molecules. The transition metal complexes occupy a special place here, and the last chapter is devoted entirely to this subject. We believe that modern inorganic chemists should be acquainted with the methods of the theory, and that they will find approximate one-electron calculations as helpful as the organic chemists have found simple Hiickel calculations. For this reason, we have included a calculation of the permanganate ion in Chapter 8. On the other hand, we have not considered conjugated pi systems because they are excellently discussed in a number of books. Our intuitive approach in the use of symmetry methods is admittedly nonrigorous and therefore will be unsatisfactory to purists, but we believe this is the best way to introduce symmetry ideas to the majority of students. Once the student has learned how to use symmetry methods, it will be easier for him to appreciate more formal and rigorous treatments. Several reprints of papers on molecular orbital theory are included in the back of the book. The papers treat a substantial number of the important molecular geometries. The reader should be able to follow the discus- sions after reading through the lecture notes. We thank our colleagues in New York and Copenhagen for help with the manuscript. We gratefully acknowledge the help of Dr. Arlen Viste and Mr. Harold Basch in preparing Appendix 8-B. Finally, it is a plea- sure to acknowledge the expert assistance of Mrs. Diane Celeste in preparing the final manuscript. C. J. BALLHAUSEN,KGbenhavn HARRYB. GRAY,New York October 1964 Contents Preface 1 Atomic Orbitals 1-1 The Schrodinger Equation 1-2 "Hydrogen-Like" Orbitals 1-3 The Pauli Exclusion Principle 2 Diatomic Molecules 2-1 Molecular Orbitals for Diatomic Molecules 2-2 Symmetry Considerations 2-3 Diatomic Molecules with Different Atomic Nuclei 3 Electronic States of Molecules 4 Hybridization 5 Band Intensities 6 Triatomic Molecules 6-1 The C02Molecule 6-2 The H20Molecule 6-3 NO2 6-4 0:(and SO2 7 Selected Molecules with Four or More Atoms 7-1 Hz02 7-2 Formaldehyde, H2C0 7-3 The Boron Hydride BzHG vii viii CONTENTS 8 Molecular Orbitals Involving d Valence Orbitals 8- 1 General Considerations 8-2 Molecular Orbitals for an Octahedral Molecule 8-3 Ligand-Orbital Representations 8-4 Group Overlap of Metal and Ligand Orbitals 8-5 Energy Calculations 8-6 Electronic Spectra of Metal Complexes Appendix 8A Evaluation of G[eR(cr)] Appendix 8B Example Calculations 107 8-7 Basis Functions 107 8-8 Normalization Including Ligand-Ligand Overlap 107 8-9 Evaluation of a Group Overlap Integral: GT2(d, crs) 111 8-1 0 Radial Functions 117 8-1 1 Bond Distances 117 8-1 2 Overlap Integrals 117 8-1 3 Coulomb Integrals (Hii) 118 8-14 Exchange Integrals (Hij) 118 8-15 Overlap Correction for Hiiof Ligands 118 8- 16 Calculation of MnO; 123 8- 17 Calculation of CrFi- 128 References 132 Problems 133 Suggested Reading 138 Reprint Collection General Papers 1. C. A. Coulson and I. Fischer, "Notes on the Molecular Orbital Treatment of the Hydrogen Molecule," Phil. Mag., 40,386-393 (1949). 139 2. G. E. Kimball, "'Directed Valence," J. Chem. Phys., 8, 188-198 (1940). 147 CONTENTS ix 3. R. S. Mulliken, C. A. Rieke, D. Orloff, and H. Orloff, "For- mulas and Numerical Tables for Overlap Integrals," J. Chem. Phys., 17, 1248-1267 (1949). 158 4. J. A. Pople, "The Molecular-Orbital and Equivalent-Orbital Approach to Molecular Structure," Quart Revs., 11, 273-290 (1957). 178 5. A. D. Walsh, "The Electronic Orbitals, Shapes, and Spectra of Polyatomic Molecules. Part I. AH2 Molecules," pp. 2260- 2265; "Part 11. Non-hydride AB2 and BAC Molecules," pp. 2266-2289, J. Chem. Soc. (1953). 196 Transition Metal Complexes 1. C. J. Ballhausen and H. B. Gray, "The Electronic Structure of the Vanadyl Ion," Inorg. Chem., 1, 111-122 (1962). 226 2. H. B. Gray and C. J. Ballhausen, "A Molecular Orbital Theory for Square Planar Metal Complexes," J. Am. Chem. Soc., 85, 260-265 (1963). 238 3. H. B. Gray and N. A. Beach, "The Electronic Structures of Octahedral Complexes. 1. Metal Hexacarbonyls and Hexacy- anides," J. Am. Chem. Soc., 85,2922-2927 (1963). 244 4. W. Moffitt, "The Electronic Structure of Bis-cyclopentadienyl Compounds," J. Am. Chem. Soc., 76, 3386-3392 (1954). 250 5. J. H. Van Vleck, "The Group Relation Between the Mulliken and Slater-Pauling Theories of Valence," J. Chem. Phys., 3, 803-806 (1935). 257 6. M. Wolfsberg and L. Helmholz, "The Electronic Structure of MnOc, Cr0z;and ClO4," J. Chem. Phys., 20,837-834 (1952). 261 Corrections Index The publisher wishes to thank the American Institute of Physics, the American Chemical Society, Taylor and Francis, and the Chemical Society (London) for permission to reprint material from their journals. MOLECULAR ORBITAL THEORY /~tornicOrbitals One of the most important questions confronting the chemist is the problem of describing how atoms are held together in molecules. Such a description can only be obtained by using quantum mechanical methods. We are in general interested in accounting for the properties of excited states, as well as the ground state. A simple and useful method of describing the electronic structures of molecules starts with wave functions which are localized on individual atoms in the molecule, and then proceeds to combine these functions in various trial combinations. The combinations are called molecular orbitals and are used as approximate solutions to the molecular SchrGdinger equation. We test their "goodness" as wave functions for a molecule by calculating observable quantities with them and comparing the results with experiment. In these notes we shall treat the fundamental principles governing the construction and properties of such molecular orbitals. We must realize, however, that such a description of a molecule involves drastic approximations; thus only approximate numerical results can be obtained. It is p~ssibleby performing elaborate numerical calculations to obtain better and better approximations for the molecular wave functions. Here we shall be interested only in semiquantitative approximate schemes which allow us to place the low-lying electronic states of molecules. 2 MOLECULAR ORBITAL THEORY 1-1 THE SCHR~DINGEREQUATION The Schrijdinger equation must be understood in the same way as Newton's laws, and, just like these, it cannot be proved by reasoning. The presence of Planck's constant, h, shows that the equation cannot be derived from classical mechanics. We know that all observable quantities which contain this constant are discontinuous and, consequently, not classical. The proof of its validity lies, therefore, in its applicability. Our calculations yield results that can be tested experimentally. If we do our calculations as accurately as possible, it turns out that our calculated results agree with experience. It is, however, obvious that the more we apply approximate solutions to the Schrijdinger equation, the less we can expect to obtain results that "fit perfectly." When we, therefore, in many cases are happy just to "under- stand" what happens, it is natural that we must rely on the validity of the Schrijdinger equation. There is nothing in our experience which indicates that the equation should not be true, provided the particles under consideration do not have such large velocities that relativistic effects become important. Let us first review the different atomic orbitals. These orbitals ~ are obtained by solving the Schriidinger equation for a particle of mass m, the electron, in a central field produced by the nucleus: where W is the total energy (a constant), V is the potential energy as a function of the position (x,y,z) of m, and A is Planck's constant h divided by 2n.

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