الرئيسية The Chemical Bond : Chemical Bonding Across the Periodic Table
The Chemical Bond : Chemical Bonding Across the Periodic TableFrenking, Gernot, Shaik, Sason S
This is the perfect complement to ''Chemical Bonding - Across the Periodic Table'' by the same editors, who are two of the top scientists working on this topic, each with extensive experience and important connections within the community.
The resulting book is a unique overview of the different approaches used for describing a chemical bond, including molecular-orbital based, valence-bond based, ELF, AIM and density-functional based methods. It takes into account the many developments that have taken place in the field over the past few decades due to the rapid advances in quantum chemical models and faster computers.
The resulting book is a unique overview of the different approaches used for describing a chemical bond, including molecular-orbital based, valence-bond based, ELF, AIM and density-functional based methods. It takes into account the many developments that have taken place in the field over the past few decades due to the rapid advances in quantum chemical models and faster computers.
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Edited by Gernot Frenking and Sason Shaik The Chemical Bond Related Titles Frenking, G., Shaik, S. (eds.) Fleming, I. The Chemical Bond Molecular Orbitals and Organic Chemical Reactions – Student Edition Fundamental Aspects of Chemical Bonding 2014 ISBN: 978-3-527-33314-1; also available in digital formats Reiher, M., Wolf, A. Relativistic Quantum Chemistry The Fundamental Theory of Molecular Science Second Edition 2014 Print ISBN: 978-3-527-33415-5; also available in digital formats 2009 Print ISBN: 978-0-470-74659-2; also available in digital formats Matta, C.F., Boyd, R.J. (eds.) The Quantum Theory of Atoms in Molecules From Solid State to DNA and Drug Design 2007 Print ISBN: 978-3-527-30748-7; also available in digital formats Edited by Gernot Frenking and Sason Shaik The Chemical Bond Chemical Bonding Across the Periodic Table Editors Prof. Dr. Gernot Frenking Philipps-Universität Marburg FB Chemie Hans-Meerwein-Strasse 35032 Marburg Germany Prof. Dr. Sason Shaik Hebrew University Institut of Chemistry Givat Ram Campus 91904 Jerusalem Israel All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: applied for British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliograﬁe; detailed bibliographic data are available on the Internet at <http://dnb.d-nb.de>. c 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microﬁlm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not speciﬁcally marked as such, are not to be considered unprotected by law. Print ISBN: 978-3-527-33315-8 ePDF ISBN: 978-3-527-66468-9 ePub ISBN: 978-3-527-66467-2 mobi ISBN: 978-3-527-66466-5 oBook ISBN: 978-3-527-66465-8 Cover-Design Adam-Design, Weinheim Typesetting Laserwords Private Limited, Chennai, India Printing and Binding betz-druck GmbH, Darmstadt Printed on acid-free paper V Contents Preface XV List of Contributors 1 1.1 1.2 1.2.1 220.127.116.11 1.2.2 1.2.3 1.2.4 1.2.5 1.3 1.4 1.5 1.6 2 2.1 2.2 2.3 2.4 XIX Chemical Bonding of Main-Group Elements 1 Martin Kaupp Introduction and Deﬁnitions 1 The Lack of Radial Nodes of the 2p Shell Accounts for Most of the Peculiarities of the Chemistry of the 2p-Elements 2 High Electronegativity and Small Size of the 2p-Elements 4 Hybridization Defects 4 The Inert-Pair Effect and its Dependence on Partial Charge of the Central Atom 7 Stereo-Chemically Active versus Inactive Lone Pairs 10 The Multiple-Bond Paradigm and the Question of Bond Strengths 13 Inﬂuence of Hybridization Defects on Magnetic-Resonance Parameters 14 The Role of the Outer d-Orbitals in Bonding 15 Secondary Periodicities: Incomplete-Screening and Relativistic Effects 17 ‘‘Honorary d-Elements’’: the Peculiarities of Structure and Bonding of the Heavy Group 2 Elements 19 Concluding Remarks 21 References 21 Multiple Bonding of Heavy Main-Group Atoms 25 Gernot Frenking Introduction 25 Bonding Analysis of Diatomic Molecules E2 (E = N – Bi) 27 Comparative Bonding Analysis of N2 and P2 with N4 and P4 29 Bonding Analysis of the Tetrylynes HEEH (E = C – Pb) 32 VI Contents 2.5 2.6 2.7 3 3.1 3.2 3.3 3.4 3.4.1 3.4.2 3.4.3 3.4.4 3.4.5 3.4.6 3.4.7 3.5 3.5.1 3.5.2 3.5.3 3.6 4 4.1 4.2 4.2.1 4.2.2 4.2.3 4.3 4.3.1 4.3.2 4.4 Explaining the Different Structures of the Tetrylynes HEEH (E = C – Pb) 34 Energy Decomposition Analysis of the Tetrylynes HEEH (E = C – Pb) 41 Conclusion 46 Acknowledgment 47 References 47 The Role of Recoupled Pair Bonding in Hypervalent Molecules David E. Woon and Thom H. Dunning Jr. Introduction 49 Multireference Wavefunction Treatment of Bonding 50 Low-Lying States of SF and OF 53 Low-Lying States of SF2 and OF2 (and Beyond) 58 SF2 (X1 A1 ) 58 SF2 (a3 B1 ) 59 SF2 (b3 A2 ) 61 OF2 (X1 A1 ) 62 Triplet states of OF2 62 SF3 and SF4 63 SF5 and SF6 64 Comparison to Other Models 64 Rundle–Pimentel 3c-4e Model 64 Diabatic States Model 66 Democracy Principle 67 Concluding Remarks 67 References 68 49 Donor–Acceptor Complexes of Main-Group Elements 71 Gernot Frenking and Ralf Tonner Introduction 71 Single-Center Complexes EL2 73 Carbones CL2 73 Isoelectronic Group 15 and Group 13 Homologues (N+ )L2 and (BH)L2 82 Donor–Acceptor Bonding in Heavier Tetrylenes ER2 and Tetrylones EL2 (E = Si – Pb) 88 Two-Center Complexes E2 L2 94 Two-Center Group 14 Complexes Si2 L2 –Pb2 L2 (L = NHC) 95 Two-Center Group 13 and Group 15 Complexes B2 L2 and N2 L2 101 Summary and Conclusion 110 References 110 Contents 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.7.1 5.7.2 5.8 5.9 5.9.1 5.9.2 5.9.3 5.9.4 5.10 6 6.1 6.2 6.2.1 6.2.2 6.3 6.3.1 6.4 6.4.1 6.4.2 6.5 6.5.1 6.6 6.7 6.8 6.A 6.B Electron-Counting Rules in Cluster Bonding – Polyhedral Boranes, Elemental Boron, and Boron-Rich Solids 113 Chakkingal P. Priyakumari and Eluvathingal D. Jemmis Introduction 113 Wade’s Rule 114 Localized Bonding Schemes for Bonding in Polyhedral Boranes 119 4n + 2 Interstitial Electron Rule and Ring-Cap Orbital Overlap Compatibility 122 Capping Principle 125 Electronic Requirement of Condensed Polyhedral Boranes – mno Rule 126 Factors Affecting the Stability of Condensed Polyhedral Clusters 134 Exo-polyhedral Interactions 134 Orbital Compatibility 135 Hypoelectronic Metallaboranes 136 Electronic Structure of Elemental Boron and Boron-Rich Metal Borides – Application of Electron-Counting Rules 139 α-Rhombohedral Boron 139 β-Rhombohedral Boron 140 Alkali Metal-Indium Clusters 142 Electronic Structure of Mg∼5 B44 143 Conclusion 144 References 145 Bound Triplet Pairs in the Highest Spin States of Monovalent Metal Clusters 149 David Danovich and Sason Shaik Introduction 149 Can Triplet Pairs Be Bonded? 150 A Prototypical Bound Triplet Pair in 3 Li2 150 The NPFM Bonded Series of n+1 Lin (n = 2−10) 152 Origins of NPFM Bonding in n+1 Lin Clusters 152 Orbital Cartoons for the NPFM Bonding of the 3 Σ+u State of Li2 154 Generalization of NPFM Bonding in n+1 Lin Clusters 156 VB Mixing Diagram Representation of the Bonding in 3 Li2 156 VB Modeling of n+1 Lin Patterns 158 NPFM Bonding in Coinage Metal Clusters 161 Structures and Bonding of Coinage Metal NPFM Clusters 161 Valence Bond Modeling of the Bonding in NPFM Clusters of the Coinage Metals 163 NPFM Bonding: Resonating Bound Triplet Pairs 167 Concluding Remarks: Bound Triplet Pairs 168 Appendix 170 Methods and Some Details of Calculations 170 Symmetry Assignment of the VB Wave Function 170 VII VIII Contents 6.C The VB Conﬁguration Count and the Expressions for De for NPFM Clusters 171 References 172 7 Chemical Bonding in Transition Metal Compounds 175 Gernot Frenking Introduction 175 Valence Orbitals and Hybridization in Electron-Sharing Bonds of Transition Metals 177 q Carbonyl Complexes TM(CO)6 (TMq = Hf2− , Ta− , W, Re+ , Os2+ , Ir3+ ) 181 Phosphane Complexes (CO)5 TM-PR3 and N-Heterocyclic Carbene Complexes (CO)5 TM-NHC (TM = Cr, Mo, W) 187 Ethylene and Acetylene Complexes (CO)5 TM-C2 Hn and Cl4 TM-C2 Hn (TM = Cr, Mo, W) 190 Group-13 Diyl Complexes (CO)4 Fe-ER (E = B – Tl; R = Ph, Cp) 195 Ferrocene Fe(η5 -Cp)2 and Bis(benzene)chromium Cr(η6 -Bz)2 199 Cluster, Complex, or Electron-Sharing Compound? Chemical Bonding in Mo(EH)12 and Pd(EH)8 (E = Zn, Cd, Hg) 203 Metal–Metal Multiple Bonding 211 Summary 214 Acknowledgment 214 References 214 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 8 8.1 8.2 8.2.1 8.2.2 18.104.22.168 22.214.171.124 8.2.3 8.2.4 8.2.5 8.3 8.3.1 8.3.2 8.3.3 8.3.4 8.3.5 8.4 8.4.1 8.4.2 Chemical Bonding in Open-Shell Transition-Metal Complexes 219 Katharina Boguslawski and Markus Reiher Introduction 219 Theoretical Foundations 220 Deﬁnition of Open-Shell Electronic Structures 221 The Conﬁguration Interaction Ansatz 222 The Truncation Procedure 222 Density Matrices 222 Ab Initio Single-Reference Approaches 223 Ab Initio Multireference Approaches 224 Density Functional Theory for Open-Shell Molecules 229 Qualitative Interpretation 230 Local Spin 230 Broken Spin Symmetry 233 Analysis of Bond Orders 235 Atoms in Molecules 237 Entanglement Measures for Single- and Multireference Correlation Effects 239 Spin Density Distributions—A Case Study 243 A One-Determinant Picture 243 A Multiconﬁgurational Study 245 Contents 8.5 9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 10 10.1 10.2 10.3 10.3.1 10.3.2 10.3.3 10.3.4 10.3.5 10.4 10.4.1 10.4.2 10.5 10.6 10.6.1 10.6.2 10.6.2.1 10.6.2.2 10.6.2.3 10.7 Summary 246 Acknowledgments References 247 247 Modeling Metal–Metal Multiple Bonds with Multireference Quantum Chemical Methods 253 Laura Gagliardi Introduction 253 Multireference Methods and Effective Bond Orders 253 The Multiple Bond in Re2 Cl82− 254 Homonuclear Diatomic Molecules: Cr2 , Mo2 , and W2 255 Cr2 , Mo2 , and W2 Containing Complexes 259 Fe2 Complexes 264 Concluding Remarks 265 Acknowledgment 266 References 266 The Quantum Chemistry of Transition Metal Surface Bonding and Reactivity 269 Rutger A. van Santen and Ivo A. W. Filot Introduction 269 The Elementary Quantum-Chemical Model of the Surface Chemical Bond 272 Quantum Chemistry of the Surface Chemical Bond 276 Adatom Adsorption Energy Dependence on Coordinative Unsaturation of Surface Atoms 276 Adatom Adsorption Energy as a Function of Metal Position in the Periodic System 284 Molecular Adsorption; Adsorption of CO 286 Surface Group Orbitals 296 Adsorbate Coordination in Relation to Adsorbate Valence 301 Metal Particle Composition and Size Dependence 303 Alloying: Coordinative Unsaturation versus Increased Overlap Energies 303 Particle Size Dependence 305 Lateral Interactions; Reconstruction 310 Adsorbate Bond Activation and Formation 317 The Reactivity of Different Metal Surfaces 317 The Quantum-Chemical View of Bond Activation 321 Activation of the Molecular π Bond (Particle Shape Dependence) 321 The Uniqueness of the (100) Surface 323 Activation of the Molecular σ Bond; CH4 and NH3 325 Transition State Analysis: A Summary 328 References 333 IX X Contents 11 11.1 11.2 11.3 11.3.1 11.3.2 11.4 11.5 11.6 11.6.1 11.6.2 11.6.3 11.7 12 12.1 12.2 12.3 12.3.1 12.3.2 12.4 12.4.1 12.4.2 12.5 12.5.1 12.5.2 12.5.3 12.6 13 13.1 13.2 Chemical Bonding of Lanthanides and Actinides 337 Nikolas Kaltsoyannis and Andrew Kerridge Introduction 337 Technical Issues 338 The Energy Decomposition Approach to the Bonding in f Block Compounds 338 A Comparison of U–N and U–O Bonding in Uranyl(VI) Complexes 339 Toward a 32-Electron Rule 340 f Block Applications of the Electron Localization Function 341 Does Covalency Increase or Decrease across the Actinide Series? 342 Multi-conﬁgurational Descriptions of Bonding in f Element Complexes 347 U2 : A Quintuply Bonded Actinide Dimer 347 Bonding in the Actinyls 349 Oxidation State Ambiguity in the f Block Metallocenes 350 Concluding Remarks 353 References 354 Direct Estimate of Conjugation, Hyperconjugation, and Aromaticity with the Energy Decomposition Analysis Method 357 Israel Fernández Introduction 357 The EDA Method 359 Conjugation 361 Conjugation in 1,3-Butadienes, 1,3-Butadiyne, Polyenes, and Enones 361 Correlation with Experimental Data 363 Hyperconjugation 370 Hyperconjugation in Ethane and Ethane-Like Compounds 370 Group 14 β-Effect 371 Aromaticity 372 Aromaticity in Neutral Exocyclic Substituted Cyclopropenes (HC)2 C = X 374 Aromaticity in Group 14 Homologs of the Cyclopropenylium Cation 375 Aromaticity in Metallabenzenes 376 Concluding Remarks 378 References 379 Magnetic Properties of Aromatic Compounds and Aromatic Transition States 383 Rainer Herges Introduction 383 A Short Historical Review of Aromaticity 384 Contents 13.3 13.3.1 13.3.2 13.3.3 13.4 13.4.1 13.4.2 13.4.3 13.4.4 13.4.5 13.4.6 13.4.7 13.4.8 13.5 Magnetic Properties of Molecules 386 Exaltation and Anisotropy of Magnetic Susceptibility 387 Chemical Shifts in NMR 391 Quantum Theoretical Treatment 392 Examples 397 Benzene and Borazine 397 Pyridine, Phosphabenzene, and Silabenzene 398 Fullerenes 400 Hückel and Möbius Structures 401 Homoaromatic Molecules 403 Organometallic Compounds 404 Aromatic Transition States 406 Coarctate Transition States 411 Concluding Remarks 415 References 415 14 Chemical Bonding in Inorganic Aromatic Compounds 421 Ivan A. Popov and Alexander I. Boldyrev Introduction 421 How to Recognize Aromaticity and Antiaromaticity? 422 ‘‘Conventional’’ Aromatic/Antiaromatic Inorganic Molecules 426 Inorganic B3 N3 H6 Borazine and 1,3,2,4-Diazadiboretiidine B2 N2 H4 427 Aromatic P42− , P5− , P6 and Their Analogs 428 ‘‘Unconventional’’ Aromatic/Antiaromatic Inorganic Molecules 430 σ-Aromatic and σ-Antiaromatic Species 431 σ-/π-Aromatic, σ-/π-Antiaromatic, and Species with σ-/π-Conﬂicting Aromaticity 432 σ-/π-/δ-Aromatic, σ-/π-/δ-Antiaromatic, and Species with σ-/π-/δ-Conﬂicting Aromaticity 436 Summary and Perspectives 440 Acknowledgments 441 References 441 14.1 14.2 14.3 14.3.1 14.3.2 14.4 14.4.1 14.4.2 14.4.3 14.5 15 15.1 15.2 15.2.1 15.2.2 15.2.3 15.2.4 15.2.5 15.3 Chemical Bonding in Solids 445 Pere Alemany and Enric Canadell Introduction 445 Electronic Structure of Solids: Basic Notions 447 Bloch Orbitals, Crystal Orbitals, and Band Structure 447 Fermi Level and Electron Counting 449 Peierls Distortions 451 Density of States and its Analysis 453 Electronic Localization 456 Bonding in Solids: Some Illustrative Cases 458 XI XII Contents 15.3.1 15.3.2 15.3.3 15.3.4 15.4 16 16.1 16.2 16.3 16.3.1 16.3.2 16.3.3 16.3.4 16.4 16.4.1 16.4.2 16.4.3 16.5 16.6 17 17.1 17.2 17.3 17.4 17.5 17.5.1 17.5.2 17.5.3 17.5.4 17.5.5 17.5.6 Covalent Bonds in Polar Metallic Solids: A3 Bi2 and A4 Bi5 (A = K, Rb, Cs) 459 Electronic Localization: Magnetic versus Metallic Behavior in K4 P3 462 Crystal versus Electronic Structure: Are There Really Polyacetylene-Like Gallium Chains in Li2 Ga? 466 Ba7 Ga4 Sb9 : Do the Different Cations in Metallic Zintl Phases Play the Same Role? 470 Concluding Remarks 473 Acknowledgments 474 References 474 Dispersion Interaction and Chemical Bonding 477 Stefan Grimme Introduction 477 A Short Survey of the Theory of the London Dispersion Energy 480 Theoretical Methods to Compute the Dispersion Energy 485 General 486 Supermolecular Wave Function Theory (WFT) 486 Supermolecular Density Functional Theory (DFT) 488 Symmetry-Adapted Perturbation Theory (SAPT) 490 Selected Examples 492 Substituted Ethenes 492 Steric Crowding Can Stabilize a Labile Molecule: Hexamethylethane Derivatives 493 Overcoming Coulomb Repulsion in a Transition Metal Complex 494 Conclusion 495 Computational Details 496 References 496 Hydrogen Bonding 501 Hajime Hirao and Xiaoqing Wang Introduction 501 Fundamental Properties of Hydrogen Bonds 502 Hydrogen Bonds with Varying Strengths 504 Hydrogen Bonds in Biological Molecules 506 Theoretical Description of Hydrogen Bonding 508 Valence Bond Description of the Hydrogen Bond 508 Electrostatic and Orbital Interactions in H Bonds 509 Ab Initio and Density Functional Theory Calculations of Water Dimer 510 Energy Decomposition Analysis 511 Electron Density Distribution Analysis 513 Topological Analysis of the Electron Density and the Electron Localization Function 514 Contents 17.5.7 17.5.8 17.6 Resonance-Assisted Hydrogen Bonding 515 Improper, Blueshifting Hydrogen Bonds 516 Summary 517 Acknowledgment 517 References 517 18 Directional Electrostatic Bonding 523 Timothy Clark Introduction 523 Anisotropic Molecular Electrostatic Potential Distribution Around Atoms 524 Electrostatic Anisotropy, Donor–Acceptor Interactions and Polarization 528 Purely Electrostatic Models 530 Difference-Density Techniques 531 Directional Noncovalent Interactions 533 Conclusions 534 Acknowledgments 534 References 534 18.1 18.2 18.3 18.4 18.5 18.6 18.7 Index 537 XIII xv Preface The chemical bond is the backbone of chemistry. It deﬁnes chemistry as the science of understanding and transformation of the physical world in terms of interatomic interactions, which are considered as chemical bonds. Thus, chemistry appears right at the beginning as a fuzzy discipline because the distinction between chemical and nonchemical interatomic interactions is not exactly deﬁned, which creates ongoing controversial debates. But the fuzziness of chemical bonding posed no obstacle for the advancement of chemical research from an esoteric playground to a highly sophisticated academic discipline, which became the basis for industrial growth and economic wealth. In the absence of a physical understanding of chemical bonding, chemists used their imagination and creativity for designing models that proved to be very useful for rationalizing experimental observations. The development of chemical bonding models, which is an integral part of the progress in experimental chemistry, is a fascinating chapter in the history of mankind. It goes beyond the mere realm of natural science and is part of the evolution of human culture. Chemists analyzed and synthesized in the past centuries a steadily increasing number of new compounds, which required a systematic ordering system to become comprehensible. In order to understand the enormous diversity of molecules and solids, which constitute the chemical universe, chemists developed bonding models that served two purposes. One purpose was to provide an understanding for the observed species, which were classiﬁed according to well-deﬁned rules. The second purpose was to establish a guideline for new experiments; a goal that needed a scientiﬁc hypothesis in order to distinguish research from random activity. Originally, chemists suggested bonding models that appeared as simple sketches of connections between atoms that were ﬁnally recognized as the elementary building blocks of molecules and solids. The heuristically developed models were continuously reﬁned like a neural network where experimental observations and hypotheses served as guidelines for improving the patterns and archetypes that were used to rationalize new ﬁndings. Already in the nineteenth century, chemists realized that atoms might possess different valencies for the formation of chemical bonds without knowing their physical meaning. It is amazing how much progress was made in experimental chemical research without actually knowing much about the constitution of atoms and the nature of the interatomic interactions. The xvi Preface bonding models became the language by which chemists circulate information about molecular structures and reactivities using simple terms that were reﬁned along with the progress in chemical research. An important step for bridging the gap between chemistry and physics was made in 1916 when G.N. Lewis suggested that the bond line which so far was only used as pictorial representation for a chemical bond without physical meaning, should be identiﬁed with a pair of electrons. Lewis knew that his suggestion did not explain the nature of the chemical bond in terms of basic laws of physics, because the classical expression for electrostatic interactions did not agree at all with experimental data. Lewis proposed his model despite the deﬁance of the physical laws, because of his ﬁrm belief that the immense body of chemical facts supported his idea (e.g., the preponderance of molecules with an even number of electrons). Thus, to justify his belief, he postulated that in the atomic world there might be different forces than in the macroscopic world. This intuitively derived model of electron-pair bonding is still the most commonly used archetype for a chemical bond. He could not have foreseen the revolution, which followed from quantum theory that was introduced by Schrödinger and Heisenberg in 1926. One year later, Heitler and London published their landmark paper where they applied quantum theory to describe the interactions between two hydrogen atoms in the bonding and antibonding state of H2 . It was the birth of quantum chemistry, which provided the ﬁrst physically sound description of the covalent chemical bond. It is remarkable that the work by Heitler and London that outlined for the ﬁrst time a physically correct description of the chemical bond did not replace the Lewis picture of electron-pair bonding that was based on intuition rather than on elementary physics. One reason is the dramatically different appeal of the two approaches for human imagination of the chemical bond. The Lewis picture is simple to use and it proved as extremely powerful ordering scheme for molecular structures and reactivities. Chemists are generally happy with such models. The quantum theoretical description of interatomic interactions introduced the wave function Ψ as the central term for chemical bonding, which is in contrast an elusive object for human imagination, as evidenced by the intensive discussions about the meaning and the interpretation of Ψ mainly in the physics community. An important step for building a bridge between the Lewis picture of chemical bonding and quantum theory was made by Pauling in his seminal book The Nature of the Chemical Bond, which was published in 1939. Pauling’s work showed that the heuristically derived electron-pair paradigm of Lewis could be dressed by quantum theory keeping the notion of localized bonds in terms of the valence bond (VB) approach. It is thus not surprising that the VB model was well accepted by the chemical community, which quickly adapted the VB notions such as resonance and hybridization for discussing molecular structures and reactivities. The alternative approach of molecular orbital (MO) theory that was developed by Mulliken and Hund was initially met with scepticism by most chemists, because the picture of a localized chemical bond did not seem to be contained in the delocalized MOs. The resistance against MO theory did not change by the fact that phenomena Preface such as the stability of aromatic compounds and spectroscopic data could easily be explained with MOs. The situation gradually changed during the 1950s till the 1970s, when the advantages of implementing the MO theory became more and more apparent. The development of computer codes by Dewar and Pople, ﬁrst for semiempirical approaches and then for ab initio methods, paved the way for the acceptance of MO methods. MO theory could much easier become coded into computer programs, which along with the dramatic development of computer hardware produced numerical results with increasing accuracy. Ruedenberg showed that the delocalized MOs could be converted into localized orbitals via unitary transformations, which recovered the Lewis picture even from MO calculations. The breakthrough came with the big success of MO theory of not only explaining but also predicting the reaction course and the stereochemistry of pericyclic reactions. The MO-based frontier orbital theory and Woodward–Hoffmann rules became a standard model for chemical bonding and reactivity, which culminated in awarding the 1981 Nobel Prize to Kenichi Fukui and Roald Hoffmann. While VB theory nearly disappeared during that time from the horizon of quantum chemistry, it remained remarkably alive in the description of molecular structures and bonding by experimental chemists. In spite of the sweeping success of MO-based quantum chemical methods, which were increasingly used by the general chemical community, the qualitative models that chemists continued to use for sketching molecules and chemical reactions rested often on the picture of localized two-electron bonds. This is because of the unsurpassed simplicity and usefulness of the Lewis bonding model and the associated rules. What can be observed in 2014 is a complementary coexistence of VB and MO models, where the choice of a chemist for answering a question depends on the particular problem and his preference for a speciﬁc approach. Also, there has been a remarkable renaissance of VB methods in recent years, which provide an arsenal of bonding models that have been very helpful for explaining molecular structures and reactivities. The development of quantum chemical methods focussed for a long time on more accurate techniques and efﬁcient algorithms for obtaining numerical results for increasingly larger molecules and for the calculation of reaction pathways. The famous request by Charles Coulson Give us insight, not numbers seemed to have been buried under the quest for more reliable data and little attention was paid to the interpretation of the calculated numbers. The situation has clearly changed in the past two decades. Numerous methods were developed, which aimed at building a bridge between the wealth of numerical data and conceptual models that convert the calculated results into a qualitative understanding of molecular structures and reactivities. Quantum theoretical results can often be presented in terms of ﬁgures and schemes rather than merely by presenting tables with numbers, which appeals to the aptitude of most experimental chemists for sensory perception. This does not mean that traditional models such as the Lewis electron pair for chemical bonding have to be abandoned. On the contrary, well-deﬁned partitioning schemes have been introduced, which make it possible to assign calculated numbers that come from accurate quantum chemical calculations to classical concepts replacing xvii xviii Preface handwaving arguments. Although such numbers are not measurable values, they can often be interpreted in terms of physically meaningful expressions, which provide a well-deﬁned ordering scheme for molecular structures and reactivities. Volume 1 of this book has 11 chapters that present and discuss the physical understanding of the chemical bond and introduce the most important methods that are presently available for the interpretation of molecular structures and reactivities. On the other hand, Volume 2 has 18 chapters, which describe the application of modern theoretical models to chemical bonding in molecules containing main elements, transition metals, and lanthanides and actinides, including clusters, solids, and surfaces across the periodic table. An important section of Volume 2 is dedicated to the weak interactions, such as dispersion, halogen bonding, and hydrogen bonding. There is some overlap between some chapters, which is intended. Chemical bonding can be described with several models and it is sometimes useful to consider it from different perspectives. In judging the performance of different methods one should consider the device that bonding models are not right or wrong, they are more or less useful. It was the goal of the editors to give a comprehensive account of the present knowledge about the chemical bond and about the most important quantum chemical methods, which are available for describing chemical bonding. The two volumes of The Chemical Bond are intended to be an authoritative overview of the state-of-the-art of chemical bonding. Marburg, Germany Jerusalem, Israel Gernot Frenking Sason Shaik XIX List of Contributors Pere Alemany Universitat de Barcelona Institut de Quı́mica Teòrica i Computacional (IQTCUB) and Departament de Quı́mica Fı́sica Diagonal 647 08028 Barcelona Spain Katharina Boguslawski ETH Zurich Laboratorium für Physikalische Chemie Vladimir-Prelog-Weg 2 8093 Zurich Switzerland and McMaster University Department of Chemistry and Chemical Biology 1280 Main Street West Hamilton Ontario L8S 4M1 Canada Alexander I. Boldyrev Utah State University Department of Chemistry and Biochemistry 0300 Old Main Hill Logan, UT 84322-0300 USA Enric Canadell Institut de Ciència de Materials de Barcelona (ICMAB-CSIC) Campus de la UAB 08193 Bellaterra Spain Timothy Clark Computer-Chemie-Centrum der Friedrich-Alexander-Universitaet Erlangen-Nürnberg Nägelsbachstrasse 25 91052 Erlangen Germany and University of Portsmouth Centre for Molecular Design King Henry Building King Henry I Street Portsmouth PO1 2DY United Kingdom XX List of Contributors David Danovich The Hebrew University Institute of Chemistry and the Lise-Meitner Minerva Center for Computational Quantum Chemistry Edmond J. Safra Campus 91904 Jerusalem Israel Thom H. Dunning Jr. University of Illinois Department of Chemistry 600 S. Mathews Avenue Urbana, IL 61801 USA Israel Fernández Universidad Complutense de Madrid Departamento de Quı́mica Orgánica Facultad de Ciencias Quı́micas Avda. Complutense s/n 28040 Madrid Spain Ivo A. W. Filot Eindhoven University of Technology Institute for Complex Molecular Systems and Laboratory for Inorganic Materials Chemistry 5600 MB Eindhoven The Netherlands Gernot Frenking Philipps-Universität Marburg Fachbereich Chemie Hans-Meerwein-Strasse 35032 Marburg Germany Laura Gagliardi University of Minnesota Department of Chemistry Supercomputing Institute and Chemical Theory Center Minneapolis MI 55455 USA Stefan Grimme Mulliken Center for Theoretical Chemistry Institut für Physikalische und Theoretische Chemie der Universität Bonn Beringstr.4 53115 Bonn Germany Rainer Herges University of Kiel Otto Diels-Institute for Organic Chemistry 24118 Kiel Germany Hajime Hirao Nanyang Technological University Division of Chemistry and Biological Chemistry School of Physical and Mathematical Sciences 21 Nanyang Link 637371 Singapore Eluvathingal D. Jemmis Indian Institute of Science Department of Inorganic and Physical Chemistry Bangalore 560012 India List of Contributors Nikolas Kaltsoyannis University College London Christopher Ingold Laboratories Department of Chemistry 20 Gordon Street London WC1H 0AJ UK Martin Kaupp Technische Universität Berlin Institut für Chemie Theoretische Chemie/Quantenchemie Sekr. C7 Strasse des 17. Juni 115 10623 Berlin Germany Andrew Kerridge University College London Christopher Ingold Laboratories Department of Chemistry 20 Gordon Street London WC1H 0AJ UK Ivan A. Popov Utah State University Department of Chemistry and Biochemistry 0300 Old Main Hill Logan, UT 84322-0300 USA Chakkingal P. Priyakumari Indian Institute of Science Education and Research Thiruvananthapuram School of Chemistry CET Campus Thiruvananthapuram Kerala 695 016 India Markus Reiher McMaster University Department of Chemistry and Chemical Biology 1280 Main Street West Hamilton Ontario L8S 4M1 Canada Rutger A. van Santen Eindhoven University of Technology Institute for complex molecular sciences and Laboratory for Inorganic Materials Chemistry 5600 MB Eindhoven The Netherlands Sason Shaik The Hebrew University Institute of Chemistry and the Lise-Meitner Minerva Center for Computational Quantum Chemistry Edmond J. Safra Campus 91904 Jerusalem Israel Ralf Tonner Philipps-Universität Marburg Fachbereich Chemie Hans-Meerwein-Strasse 35032 Marburg Germany XXI XXII List of Contributors Xiaoqing Wang Nanyang Technological University Division of Chemistry and Biological Chemistry School of Physical and Mathematical Sciences 21 Nanyang Link Singapore 637371 David E. Woon University of Illinois Department of Chemistry 600 S. Mathews Avenue Urbana, IL 61801 USA 1 1 Chemical Bonding of Main-Group Elements Martin Kaupp 1.1 Introduction and Deﬁnitions Prior to any meaningful discussion of bonding in main-group chemistry, we have to provide a reasonably accurate deﬁnition of what a main-group element is. In general, we assume that main-group elements are those that essentially use only their valence s- and p-orbitals for chemical bonding. This leads to a number of borderline cases that require closer inspection. Assuming that the outer d-orbitals, that is, those with principal quantum number n equal to the period in question, are not true valence orbitals (see discussion of outer d-orbital participation in bonding later in the text), we may safely deﬁne groups 13–18 as main-group elements. Group 1 is also reasonably assigned to the main groups, albeit under extreme hydrostatic pressures it appears that the elements K, Rb, and Cs turn from ns1 metals into transition metals and use predominantly their inner (n−1) d-orbitals for bonding . However, sufﬁciently high pressures may change fundamental bonding in many elements and compounds . We disregard here such extreme pressure conditions and count group 1 in the main groups. Matters are less straightforward for group 2: whereas Be and Mg utilize only their s- and p-orbitals, Ca, Sr, Ba, and Ra use their inner (n−1) d-orbitals predominantly in covalent bonding contributions when sufﬁciently positively charged, as we discuss in the following text . This leads to a number of peculiar structural features that bring these elements into the realm of ‘‘non-VSEPR d0 systems’’ that encompass early transition elements and even lanthanides, and they have also been termed ‘‘honorary d-elements’’ . The heavy group 2 elements are nevertheless usually placed with the main groups, and it seems appropriate to include the discussion of these interesting features in the last section of this chapter. It thus remains to discuss the inclusion of groups 11 and 12. The group 11 elements Cu, Ag, Au, and Rg clearly have a too pronounced involvement of their (n −1) d-orbitals in bonding, even in their lower oxidation states, to be safely considered main-group elements. The group 12 elements Zn, Cd, Hg, and Cn are usually considered to be main-group or ‘‘post-transition’’ elements. Yet recently quantum-chemical predictions  of oxidation-state Hg(+IV) in the form of the The Chemical Bond: Chemical Bonding Across the Periodic Table, First Edition. Edited by Gernot Frenking, Sason Shaik. c 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA. 2 1 Chemical Bonding of Main-Group Elements molecular tetra-ﬂuoride have been conﬁrmed by low-temperature matrix-isolation IR spectroscopy . This molecule is clearly a low-spin square-planar d8 complex and thus a transition-metal species. Other, less stable Hg(+IV) compounds have been examined computationally , and calculations predict that the tetra-ﬂuoride should be yet more stable for Cn (eka-Hg, element 112) . However, in the largest part of the chemistry of these two elements, and in all of the accessible chemistry of Zn  and Cd, d-orbital participation in bonding is minor. We note in passing that Jensen  has vehemently opposed assignment of mercury to the transition metals based on the ‘‘extreme conditions’’ of the low-temperature matrix study of HgF4 . The present author disagrees with this argument, as the role of the matrix is only to separate the HgF4 molecules from each other and to thus prevent aggregation and stabilization of HgF2 . The low temperature is admittedly needed for entropic reasons. It is large relativistic effects that render the borderlines between the later d-elements and the main groups fuzzy in the sixth and seventh period. We nevertheless agree that the oxidation state +IV of Hg or Cn is an exception rather than the rule in group 12 chemistry, and thus discussion of the more ‘‘regular’’ bonding in group 12 belongs to this chapter. If d-orbital participation in bonding is absent or at least an exception, main-group bonding may be considered simpler than that for transition-metal and f-element species. Although this is true from a general viewpoint, the variety of unusual bonding situations in main-group chemistry is nevertheless fascinating, ranging from the more common localized situations encountered in organic chemistry to delocalized bonding in, for example, electron-poor borane clusters or electron-rich noble-gas compounds to situations with even more complicated bonding-electron counts, for example, for radical-ion species or the bonding situations that may be envisioned for amorphous carbon . We observe that the relative sizes of the valence s- and p-orbitals crucially inﬂuence periodic trends, and that hybridization is a more complicated matter than usually considered. In this chapter we focus mostly on general aspects and periodic anomalies, providing a basis for more speciﬁc discussions in some of the other chapters of this book. 1.2 The Lack of Radial Nodes of the 2p Shell Accounts for Most of the Peculiarities of the Chemistry of the 2p-Elements The eigenfunctions of a Hermitean operator form a (complete) orthonormal set. This seemingly abstract mathematical condition is fundamental for the presence of nodes in wave functions. The nodes are needed to ensure orthogonality of the, exact or approximate, solutions of the Schrödinger equation. From the simplest examples like a particle in a box (closely related to the nodal characteristics of valence orbitals of extended π-systems), it is a short way to the radial and angular eigenfunctions of the hydrogen atom, which also deﬁne qualitatively the nodal structure of the atomic orbitals (AOs) of many-electron atoms. Orthogonality of AOs may be ensured either via the angular part (determined by the quantum numbers l and ml ) or via 1.2 Importance of Lack of Radial Nodes of the 2p Shell the radial part (determined by quantum number n). AOs with different angular momentum are generally orthogonal via their angular part. But valence (and outer core) AOs with increasing principal quantum number n develop nodes in their radial part to stay orthogonal to inner orbitals (with lower n) of the same angular momentum. Although this is exactly true only for the isolated atom, where n, l, and ml are good quantum numbers, it also carries over approximately to atoms in molecules, with strong consequences for chemistry. The introduction of radial nodes by this ‘‘primogenic repulsion’’  (a term indicating the necessity of staying orthogonal to the inner shells) moves the outer maximum of a radial wave function successively outwards, which makes it more diffuse. At the same time, we have to recall that the radial solutions of the Schrödinger equation for the hydrogen atom may also be written as an effective differential equation of a particle in an l-dependent potential. This effective potential contains a repulsive term due to centrifugal forces for l ≥ 1 but not for l = 0. As a consequence, s-orbitals have ﬁnite amplitude at the nucleus, whereas p-, d- or f-orbitals vanish at the nucleus (within a nonrelativistic framework). However, most importantly for the present discussion, the repulsive centrifugal term moves the outer maximum of the p-orbitals to larger radii. Therefore, np-orbitals tend to be larger and more diffuse than the corresponding ns-orbitals, with the major exception of n = 2 – because of the lack of a radial node, the 2p-orbital is actually similar in size to the 2s-orbital, which has one radial node to stay orthogonal to the 1s-shell . The approximate relative sizes of the valence s- and p-orbitals in period 2 and 3 many-electron atoms from relativistic Hartree–Fock calculations are shown in Figure 1.1. On average, the 2p-orbitals are less than 10% larger than the 2s-orbitals, whereas the 3p-orbitals Na 4 3.5 Li Al Mg <r> in a.u. 3 2.5 Si P Be S CI 2 Ar B 3p 1.5 C 3s N 1 O 2p 2s F Ne Figure 1.1 Radial expectation values for the valence s- and p-orbitals in periods 2 and 3 of the periodic table (approximate numerical Dirac-Hartree-Fock values from Ref. ). Figure adapted from Ref. . 3 4 1 Chemical Bonding of Main-Group Elements exceed the 3s-orbitals by 20–33%. Differences increase further down a given group (modiﬁed by spin-orbit coupling for the heavier elements) . The similarity of the radial extent of 2s- and 2p-shells is a decisive factor that determines the special role of the 2p-elements within the p-block, and the main consequences will be discussed further below. We note in passing that the lack of a radial node of the 3d-shell and the resulting small radial extent is crucial for many properties of the 3d-elements, and the nodelessness and small size of the 4f-shell is important for lanthanide chemistry . Finally, the lack of a radial node of the 1s-AO distinguishes H and He from all other elements of the Periodic Table, with far-reaching consequences (different ones for H and He) for their chemistry. These aspects have been discussed elsewhere . 1.2.1 High Electronegativity and Small Size of the 2p-Elements Because of the small r-expectation value of the nodeless 2p-shell, the valence electrons of the 2p-block elements tend to move on average particularly close to the positive nuclear charge. Shielding by the 1s- and 2s-shells is relatively poor, leading to high ionization potentials and electron afﬁnities and, hence, high electronegativity. Indeed, the electronegativities (ENs) of the p-block elements are known to exhibit a particularly large decrease from the second period (ﬁrst row) to the third period (second row). This holds for all EN scales, with vertical jumps increasing from left to right in the row (from a difference of about 0.4 between B and Al to one of about 0.9–1.0 between F and Cl), consistent with the higher percentage of p-character and the smaller size of the orbitals for the elements further to the right in a given row. The further drop from the third to the lower periods is much less pronounced and nonmonotonous, for reasons discussed later in the text. This makes it already obvious that the nodelessness and small size of the 2p-shell is decisive for rendering the 2p-block elements different from their heavier homologues. The overall small size of the 2p-elements can be appreciated from any tabulation of atomic, ionic, or covalent radii. Again, a jump from the 2p- to the 3p-elements is apparent, with much less pronounced increases (in some cases small decreases from 3p to 4p, see following text) towards the heavier p-block elements. This does of course in turn lead to overall lower coordination-number preferences of the 2p-element, as is well known. The smaller radii and the high electronegativities of the 2p-elements are behind their strict obediance of the octet rule, in contrast to the apparent behavior of their heavier homologues, as will be discussed in more detail later in the text. 126.96.36.199 Hybridization Defects In addition to the absolutely small size of the 2p-shell, the relative similarity of its radial extent to that of the 2s-orbital (see earlier text) is crucial for understanding the differences between the 2p-elements and their heavier congeners. Covalent bonding is often discussed using the tools of valence-bond (VB) theory (see Chapter 5 in Volume 1), and hybridization is a main requirement of VB models. Kutzelnigg, in 1.2 Importance of Lack of Radial Nodes of the 2p Shell his classical article on bonding in higher main-group elements in 1986 , has pointed out that the assumptions of isovalent hybridization are only fulﬁlled for mutually orthogonal hybrids at a given atom. This orthogonality in turn is only achievable to a good approximation when, for example, the valence s- and p-orbitals of a p-block main-group element exhibit comparable overlap with the orbitals of the bonding partner, that is, when they have comparable radial extent. And, as we have seen above, this is the case only in the 2p-series because of the nodelessness of the 2p-shell, whereas for the heavier p-block elements the centrifugal term in the effective potential makes the np-orbitals signiﬁcantly larger than the ns-orbitals (Figure 1.1). Then isovalent hybridization becomes much less favorable. To save the concept of hybridization, it then becomes unavoidable to drop the orthogonality requirement, and Kutzelnigg has introduced the term ‘‘hybridization defects’’ and a corresponding mathematical deﬁnition for the deviations from ideal, orthogonal, isovalent hybrids [13, 17]. However, the use of non-orthogonal hybrids for the heavier main-group elements in turn wipes out the well-known and still widely used relations between hybridization and bond angles and leads to very different p/s hybridization ratios than one might expect at ﬁrst glance from structures. It appears that even more than 25 years after Kutzelnigg’s milestone paper many chemists are not yet aware of these considerations, and inorganic-chemistry text books tend to ignore these aspects (but see Refs [16, 18]). We thus describe the obvious consequences of large hybridization defects for the heavier p-block elements and their relative smallness for the 2p-series in some detail here. Note that the question of orthogonality of hybrids and thus of hybridization defects arises also when considering bonding in transition-metal complexes from a VB point of view [3, 16]. Here it is mainly the (n −1) d- and ns-orbitals that are involved in bonding (see Chapter 7 in this book). It is obvious that their radial extent will not be comparable in all cases, and thus hybridization defects have to be taken into account. Most practical quantum-chemical calculations use some ﬂavor of molecularorbital (MO) theory, in recent years in particular within the framework of Kohn–Sham density functional theory. Within MO theory, hybridization is not needed. But to connect to the widely used qualitative hybridization arguments, we can extract local hybridizations a posteriori by using some kind of population analysis. The prerequisite is that we can analyze well-localized MOs. In other words, small ‘‘localization defects’’ are required (i.e., Hund’s localization condition should be fulﬁlled reasonably well) . Then we can analyze the hybridization of some kind of localized MOs. The choice of localized MOs and of population analysis does of course to some extent determine the numerical values of the hybridization ratios we get from such analyses, but the qualitative conclusions obtained by different methods are similar. Kutzelnigg in 1986  used Boys’ localization  and Mulliken populations . The latter have the disadvantage of exhibiting strong basis-set dependencies. A widely used, and nowadays readily available, approach is to consider the natural atomic orbital (NAO) hybridizations of natural localized Molecular Orbitals (NLMOs) within Weinhold’s natural bond orbital (NBO) scheme . We note in passing that, although the outcome of the 5 1 Chemical Bonding of Main-Group Elements p/s ratio 6 CH4 3.2 3.0 2.8 2.6 CF4 2.4 2.2 2.0 BF3 1.8 1.6 BH3 1.4 1.2 1.0 0.8 0.6 0.4 0.2 Period: 2 SiH4 GeH4 SnH4 PbH4 SiF4 AIH3 AIF3 GaH3 GeF4 GaF3 3 InH3 4 TIH3 SnF4 PbF4 InF3 5 TIF3 6 Figure 1.2 NAO/NLMO p/s hybridization ratios for hydrides and ﬂuorides of second and third period elements in their maximum oxidation state (B3LYP/def2-TZVP results). NBO scheme for d-orbital participation in main-group chemistry (see later text) might to some extent be biased by placing the outer d-orbitals into the ‘‘natural Rydberg set’’ of the NAO scheme , no such problem arises regarding the relative role of s- and p-orbitals. Figure 1.2 illustrates the NAO/NLMO p/s hybridization ratios of some simple compounds when moving down a given p-block main group. Starting with simple homoleptic hydrogen compounds in the maximum oxidation state of the central element (Figure 1.2; lone pairs are absent), we see clearly that the 2p-compounds exhibit hybridizations that correspond reasonably well to those expected from the bond angles and the usual formal considerations. This holds notably well for hydrocarbons, in line with the relative success and popularity of using bond angles to discuss hybridization (and vice versa) in organic chemistry. In contrast, the 3p- or 4p-compounds exhibit much larger ns- and lower np-character in their bonding hybrids. It is clear, that here the usual simple relations between bond angles and central-atom hybridization cease to function. Replacing hydrogen by ﬂuorine as bonding partner (Figure 1.2) leads to a drastic reduction of the p/s ratios. Obviously, hybridization defects are enhanced signiﬁcantly by the more electronegative ﬂuorine substituents [23, 24] (for as yet unclear reasons, BF3 is an exception to this rule). This goes parallel to strongly positive charges of the central atoms. As the central atom contracts, the sizes of its s- and p-orbitals become even more disparate: the contraction of the valence s-orbitals is more pronounced than that of the p-orbitals. This has further consequences for the bonding and chemistry of such compounds (see below). Note that even for a carbon compound like CF4 , hybridization defects are now already pronounced, and relations between bond angle and hybridization are not straightforward anymore. This also means that the assumptions of Bent’s rule  on the 1.2 Importance of Lack of Radial Nodes of the 2p Shell relations between bond angles and hybridization for heteroleptically substituted systems have to be modiﬁed to account for such hybridization defects. 1.2.2 The Inert-Pair Effect and its Dependence on Partial Charge of the Central Atom Hybridization defects cause a general weakening of covalent bonds. This is related to the fact that hybridization is effective in stabilizing molecules by (i) improving bonding orbital overlaps and by (ii) minimizing destabilizing antibonding interactions . Hybrids made from s- and p-orbitals with too dissimilar sizes are ineffective for both of these aspects and, thus, make poor covalent bonds (the overall beneﬁts from hybridization may also decrease further down the group ). This affects, in particular, compounds of the heavier p-block elements in their maximum oxidation state. Here the valence s-orbital has to be involved in bonding and, thus, is required to hybridize to some extent with the p-orbitals. In contrast, in lower oxidation states, the valence s-character of the heavy p-block element tends to accumulate in a non-bonding lone-pair (lp) type orbital, and the bonds tend to be made largely by essentially unhybridized p-orbitals. This is more favorable for making stable bonds. The latter relationships can be extracted easily from the p/s hybridization ratios of ammonia and its heavier homologues in Figure 1.3: whereas isovalent hybridization is still effective for ammonia itself, the segregation into a lone-pair NLMO of predominant s-character and bonds with predominant p-character is apparent for the heavier homologues. Fluorine substitution does again enhance hybridization defects, and thus isovalent hybridization is largely absent now even for NF3 . These observations provide a modern framework for rationalizing the inertpair effect, that is, the fact that the highest oxidation state becomes increasingly 16 BiF3 14 p/s ratio 12 bonds 10 SbF3 BiH3 AsF3 8 NF3 6 4 NH3 2 NH3 0 Period: PF3 PH3 PH3 NF3 2 SbH3 AsH3 PF3 3 Ione pairs AsH3 AsF3 4 SbH3 SbF3 5 BiH3 BiF 3 6 Figure 1.3 NAO/NLMO p/s hybridization ratios for bonds and lone pairs of second- and third-period hydrides and ﬂuorides in lower oxidation state (B3LYP/def2-TZVP results). 7 8 1 Chemical Bonding of Main-Group Elements unfavorable when moving down a given p-block main group. The ﬁrst explanations based on an energetic unavailability of the ns-orbital were quickly found to be untenable and were replaced in the 1950s by Drago’s  balance between the necessary energy for promotion to the higher valence state and the additional binding energy provided by the extra bonds in the high-valence compound. Due to a general weakening of bonds down a given group, the balance would then be shifted towards a destabilization of the higher oxidation state. The above considerations on hybridization defects tell us that, for example, an sp3 valence state will never be reached in a compound like PbX4 (cf. Figure 1.2) . Instead, the lack of effective isovalent hybridization weakens the covalent bonds in the highest oxidation state. As hybridization is largely absent in the lower oxidation states (see earlier text), essentially pure p-orbitals are then used for bonding, making relatively strong bonds and thus stabilizing the two-electron-reduced oxidation state. This also provides us with a ready explanation for why the inert-pair effect is much more pronounced for the truly inorganic compounds of a given heavy p-block element than for organoelement compounds or hydrides:  electronegative substituents like halogen, alkoxy or amido functionalities enhance hybridization defects dramatically in the highest oxidation state (cf. Figure 1.2), whereas the anyway marginal hybridization in the lower oxidation state is affected much less (cf. Figure 1.3). This explains why, for example, organolead compounds are actually much better known in their oxidation state +IV (e.g., tetraalkyl lead compounds, which for a long time, were used extensively as antiknock additives in gasoline), whereas inorganic lead(IV) compounds are either very unstable or, at least, strongly oxidizing, and lead(II) is the dominant oxidation state . Similar comparisons may be made, for example, for Tl or Bi inorganic versus organoelement compounds. And successive substitution of alkyl groups by electronegative substituents is well known to destabilize the highest oxidation state. We mention here in passing, that for the 6p-elements, the 6p/6s size differences and thus the hybridization defects are aggravated (cf. Figures 1.2, 1.3) by the relativistic contraction of the 6s-orbital (see later text), leading to much more pronounced inert-pair effects compared to the 5p-homologues. The often found designation of the inert-pair effect as a relativistic effect is, however, partly misleading: relativistic effects are comparably unimportant for the hybridization defects of the 4p- and 5p-elements. Indeed, even the term ‘‘inert-pair effect’’ is somewhat unfortunate, given that in the highest oxidation state, the valence s-orbital participates more in bonding (relative to the p-orbitals) than assumed from the usual correlations with bond angles (Figure 1.2). Related observations pertain to the stability of carbenes and their heavier homologues. It is well known that, for example, CH2 has a triplet ground state, whereas its heavier homologues feature closed-shell singlet ground states with increasing singlet-triplet gaps down the group. This can be easily rationalized in comparison with the NLMO hybridizations for NH3 and its heavier homologues (cf. Figure 1.3): isovalent hybridization is still favorable in the second period. Thus, the gain in exchange energy for a triplet state formed by the single occupation of one pure carbon 2p-orbital with 𝜋-symmetry and of one carbon-centered approximate 2s2p2 hybrid with 𝜎-symmetry is sufﬁcient to render it energetically competitive with the 1.2 Importance of Lack of Radial Nodes of the 2p Shell double occupation of the sp2 hybrid for the closed-shell singlet. In contrast, in SiH2 hybridization, defects give the 𝜎-type Si-centered orbital largely 3s-character and the Si-H bonds mostly 3p-character. It is clear that double occupation of the 𝜎-type lp is now favored in spite of the loss of exchange energy, giving a singlet ground state. Consequently, we expect that electronegative substituents (i) favor singlet over triplet ground states even for carbenes (and stabilize the singlet further for the heavier homologues) and (ii) stabilize the carbene overall compared to a corresponding tetravalent carbon compound. It is, thus, natural that in N-heterocyclic or ‘‘Arduengo-type’’ carbenes (NHCs),  the ﬁrst stable carbenes found, the carbon atom is bound to two electronegative nitrogen atoms. Although cyclic delocalization and a +M-effect of the nitrogen free electron pairs on the carbene center are often invoked to rationalize the stability of NHCs, it is clear that the electronegativity of the substituents is also important as it favors hybridization defects of the carbon 2s- and 2p-orbitals and thus stabilizes both the lower oxidation state and the singlet ground state. Mercury is the only metal that is liquid at room temperature; and it is well known that this is caused by the relativistic contraction of the 6s-shell [28, 29], which makes hybridization between 6s- and 6p- orbitals more difﬁcult and thus diminishes the binding strength. That is, relativistically enhanced hybridization defects are responsible for the low cohesion energy and low melting and boiling points of mercury (e.g., compared to cadmium). The nevertheless reasonably inert character of elemental mercury is the consequence of an inert-pair effect also in the possible reaction products: relatively weak bonds in Hg(II) compounds, speciﬁcally when bound to very electronegative substituents, are due to strong hybridization defects. NAO/NLMO hybridization ratios for linear HgX2 molecules (e.g., 6s6p0.20 5d0.14 for HgH2 , 6s6p0.06 5d0.21 for HgF2 ) suggest that 6s6p- (or 6s5d-) hybrids may be less suitable qualitative bonding descriptions in such compounds than three-center four-electron bonding involving only the 6s-orbital (cf. discussions about Natural Population Analysis, NPA ). Interestingly, relatively weak intra molecular and intermolecular interactions in Hg(II) compounds are also responsible for the widespread occurrence of Hg(I)-species with Hg–Hg bonding  and for the existence of Hg(IV)F4 under low-temperature matrix isolation conditions (see earlier text) [5, 6]. As already discussed, electronegative substituents destabilize the heavy p-block elements in their maximum oxidation state. Computationally, this may be seen clearly for calculated energies for 1,1-elimination reactions. For example, along the series PbR4 , PbR3 F, PbR2 F2 , PbRF3 , PbF4 (R = H, CH3 ), elimination of R2 , RF or F2 becomes less endothermic or more exothermic . However, at the same time, it is found, that the Pb-R and Pb-F bonds become shorter along the same series. That is, increasing hybridization defects due to electronegative substituents destabilize the Pb(IV) compounds thermochemically; yet they contract the bonds, essentially due to the smaller size of the more positively charged central atom . Indeed, as the 6s-character of the bonds increases on average with increasing ﬂuorine substitution, they become shorter, consistent with the smaller size of 6s- compared to 6p-orbitals. Yet the large difference in the sizes of 6s- and 9 10 1 Chemical Bonding of Main-Group Elements 6p-orbitals is at the same time responsible for the bond weakening. This leads to a breakdown of the frequently and implicitly assumed correlation between bond length and bond strength (in this case binding energy). This has been tested in more detail computationally by constructing the heavy ethane homologue Sn2 H6 (with staggered structure and direct Sn–Sn bonding) and its successively ﬂuorinesubstituted analogues Sn2 Hx F6−x (x = 0-5) . It turns out that, indeed, shorter Sn–Sn bond lengths in the ﬂuorine-substituted derivatives do not correspond to larger but, in most cases, to lower Sn–Sn binding energies. That is, contraction of the Sn–Sn bonds due to the increasingly positive central-atom charges goes parallel to weaker bonds due to hybridization defects. Correlations with Sn-Sn stretching frequencies are also in disagreement with, for example, Badger’s rule . Further computations on the analogous series with E = C, Si, Ge indicates similar behavior for Ge as found for Sn, a more complicated, different behavior for carbon because of negative hyperconjugation effects, and the most ‘‘normal’’ behavior with reasonable correlation between shorter bonds and larger dissociation energies for E = Si . We note that a similar breakdown of the correlation between bond lengths and binding energies has been found experimentally  (and was in part analyzed computationally ) for transition-metal phosphine complexes, where the more electronegative substituents give shorter but weaker TM–PX3 bonds (see also Chapter 7 in this book). Again, hybridization defects are responsible for this breakdown of the usual bond-length/bond-strength correlations. 1.2.3 Stereo-Chemically Active versus Inactive Lone Pairs A related but separate question concerns the ‘‘stereo-chemical activity’’ of the free electron pair in the oxidation state, two units below the maximum one. As we have seen earlier, simple relations between hybridization (and thus of s- or p-character of the lp) and bond angles fail for the heavier p-block compounds, because of hybridization defects. The VSEPR model assumes intrinsically that a lp will exhibit a space requirement in the coordination space of the central atom, in fact, more than a covalent single bond. Something like a ‘‘stereo-chemically inactive lp’’ is thus clearly outside the assumptions of the VSEPR model. Indeed, for coordination numbers below 6, the presence of lp is always found to exert a stereo-chemical inﬂuence, both in molecules or in extended solid-state structures. For larger coordination numbers than 6, the structural preferences are anyway not very clear-cut and dominated by ﬂuxional situations. It is thus coordination number 6 that gives rise to notable exceptions to the VSEPR model and to a very subtle balance between different inﬂuences. Among molecular systems, XeF6 is the most widely considered case [35–37]. While xenon hexaﬂuoride exhibits partly ionized bonding and a clear stereo-chemical activity of the lone pair in the solid state, gas-phase spectroscopic studies suggest that gas-phase XeF6 is a ﬂuxional system, consistent with a nonnegligible but weak stereo-chemical activity . Computational studies show that the energy difference between C3v and C2v distorted octahedral and regular Oh structures are (i) very small and (ii) very difﬁcult 1.2 Importance of Lack of Radial Nodes of the 2p Shell to calculate [36, 37]. Whereas early non-relativistic Hartree-Fock calculations give strongly distorted C3v minima and very high-lying Oh structures, both relativistic effects (because of the contraction of the Xe 5s-orbital) and electron correlation stabilize the Oh structure . The most recent and accurate calculations  used explicitly correlated CCSD(T)-F12b methods and showed the C3v minima to be about 4 kJ mol−1 below the C2v –symmetrical pseudo-rotation transition states and about 7–8 kJ mol−1 below the Oh -symmetrical stationary point on the potential energy surface (without vibrational corrections), consistent with a ﬂuxional situation in the gas phase. XeF6 is, thus, a remarkable border-line case, just barely on the side of a stereo-chemically active lp. The isoelectronic 5p-anions IF6 − and TeF6 2− appear to behave very similarly, according to computations . Notably, however, the calculations suggest that, in contrast to these 5p-species, valence-isoelectronic compounds of both lighter and heavier central atoms exhibit regular octahedral structures ! For the 3p- or 4p-species (e.g., ClF6 − , BrF6 − ), this may be attributed to the smaller size of the central atom and thus to steric crowding. In contrast, calculations for 6p-species like PoF6 2− , AtF6 − or RnF6 indicate that the large relativistic contraction of the 6s-orbital renders the lp stereo-chemically inactive and thus favors the Oh structures . Another important aspect related to the presence of stereochemically active lp are trends in inversion barriers and thus the question of the conﬁgurational stability of, for example, amines versus phosphines, carbanions versus silyl anions, and so on. The inversion barriers tend to increase sharply from the 2p- to the homologous 3p-systems. For example, ammonia is ﬂuxional with an inversion barrier of ca. 25 kJ mol−1 , in the range of zero-point vibrations . In contrast, PH3 exhibits a barrier of ca. 130 kJ mol−1 and, thus, is a much more rigid molecule . Computations suggest that the barriers further increase moderately to AsH3 and SbH3 and, then again, more sharply to BiH3 (due to the relativistic contraction of the 6s-orbital) . Analogous considerations hold for substituted amines and phosphines. Similarly, carbanions tend to have much smaller inversion barriers than the corresponding silyl anions, which is of fundamental importance for the conﬁgurational stability of enantiomers in the case of chiral substitution patterns . These differences between the 2p-species and their heavier homologues (and the relativistically enhanced barriers for the 6p-species) may be rationalized straightforwardly from the above considerations on hybridization defects: pyramidal minimum structures of amines or carbanions are expected to still exhibit to a reasonable approximation isovalent hybridization of both bonding and free electron pairs. In contrast, for phosphines or silyl anions, the lp have largely s-character and the bonds largely p-character (cf. Figure 1.3). At the trigonal transition state for inversion, the lp is required by symmetry to be a pure p-orbital, whereas the bonding orbitals have to hybridize the s- and p-orbitals. Whereas this leads to reasonably sp2 -hybridized bonding at the transition state for the 2p-elements, appreciable hybridization defects apply for the heavier congeners (with large s-character involved in the E–H bonds). As a consequence of the arising poor bonding overlap, the transition state is destabilized in the latter case, leading to substantially increased inversion barriers. 11 12 1 Chemical Bonding of Main-Group Elements Central-atom NPA charges Q(E) and NAO/NLMO hybridization ratios of bonds (BD) and lone pairs for minima and inversion transition states of ammonia homologuesa. Table 1.1 NH3 NH2 F NHF2 NF3 PH3 a B3LYP/def2-TZVP Q (E) BD Lone pair min TS min −1.04 −1.11 −0.38 sp3.39 p sp1.58 TS −0.51 min 0.14 TS 0.00 min TS min TS 0.58 0.44 0.02 −0.21 sp2.87 sp1.99 N-H: sp2.93 N-F: sp5.22 N-H: sp1.38 N-F: sp2.80 N-H: sp3.06 N-F: sp5.18 N-H: sp0.92 N-F: sp1.98 sp5.24 sp1.30 sp4.95 sp1.52 p sp0.93 p sp0.62 p sp0.81 p results. These considerations are supported by the NAO/NLMO hybridization ratios of NH3 versus PH3 at both minimum and transition-state structures (Table 1.1). Most notably, we may use this framework also to understand the effect of electronegative substituents on inversion barriers. For example, the barrier increases substantially from NH3 to NH2 F to NHF2 to NF3 (cf. also Ref. ), consistent with an increase in hybridization defects along this series at the planar transition-state structure (Table 1.1): differences between N–H and N–F bond hybridizations reﬂect the different H and F electronegativities, consistent with an appropriately modiﬁed version (accounting for hybridization defects, see earlier text) of Bent’s rule . Interestingly, the N–H bonds at the minimum structure are still close to idealized sp3 hybridization even in NH2 F and NHF2 , whereas the nitrogen lone pair and the N–F bond(s) feature increasing separation of the nitrogen s-character into the former. At the inversion transition state, the N–H bonds in NH3 still exhibit the nominal sp2 hybridization for a trigonal coordination. The substantial and worsening hybridization defects are particularly notable for the N–H bonds in NH2 F and NHF2 ; and then they are also signiﬁcant for the N–F bonds in NHF2 and especially in NF3 . One might therefore expect to have very large barriers for species like PF3 , AsF3 , SbF3 , or BiF3 . However, in these cases it has been found unexpectedly by computations that inversion proceeds not via a D3h -symmetrical trigonal but via a C2v -symmetrical Y-shaped transition state, leading to lower barriers and, indeed, an inverted trend with the barriers decreasing down the group . 1.2 Importance of Lack of Radial Nodes of the 2p Shell 1.2.4 The Multiple-Bond Paradigm and the Question of Bond Strengths Similar arguments may be applied to the relative instabilities and the often ‘‘trans-bent’’ structures of multiple bonds between the heavier p-block elements. Very often, these well-studied phenomena are rationalized from the singlet–triplet energy gaps of the carbene-like constituents that make up a given XX’E=E’YY’ oleﬁn analogue (or from the doublet–quartet gaps of carbyne analogues for alkyne analogues), using the framework of the Carter–Goddard–Trinquier–Malrieu (CGTM) model [44, 45]. That is, when for example the S-T gap of the carbene, silylene, and so on is large, the bonding is described as a donor–acceptor interaction of singlet closed-shell fragments, thus explaining both the relatively weak bond and the structural distortions from a planar oleﬁn-like arrangement. For small S-T gaps, one combines conceptually triplet open-shell fragments, leading to the standard description of oleﬁn-type double bonding with planar structure. However, just as we have argued above for the stabilities of NHCs and related species, it may actually be more worthwhile to put the relative sizes of s- and p-orbitals, rather than energies, into focus. We then argue via isovalent hybridization to explain the oleﬁn-like case and via increasing hybridization defects for heavier p-block central atoms and for electronegative substituents. This then also explains the inﬂuence of electronegative substituents. For example, enhanced hybridization defects for the positively charged carbon centers explain straightforwardly why the C=C double bond in C2 F4 is much weaker than that in ethylene: on one hand, the C–C sigma bonding is weakened by the incresased hybridization defects in the ﬂuorine-substituted oleﬁn; on the other hand, the singlet CF2 fragment is not affected adversely as carbon s- and p-orbitals segregate into lp and C–F bonds, respectively (see above). Similarly, the trans-bent structures of silenes, germenes, or stannenes reﬂect the lack of efﬁcient isovalent hybridization. In fact, the double bonds are sufﬁciently weak that the isolated singlet ER2 fragments become more stable down the group, which is just another manifestation of the inert-pair effect. As we have seen earlier, the latter can be conveniently discussed via hybridization defects for the high-valent compound (in the present case the oleﬁn-homologue) and via segregation of s- and p-orbitals in the low-valent situation (i.e., for the fragments). This closes the circle of arguments and shows that the consideration of the relative sizes of valence s- and p-orbitals does indeed provide a broad framework for discussing periodic trends of main-group structure, stability, and bonding. Similar considerations may be applied to the energetics and bending of formal triple bonds . Evaluation of σ- and π-bond increments reveals that for the 2p-elements of groups 15, 16, and 17, weakening of the corresponding single bonds by the so-called ‘‘lone-pair bond weakening effect’’ (LPBW)  works in favor of multiple bonding, whereas this aspect does not apply to the heavier homologues. Examples are the weak E-E single bonds of hydrazine, hydrogen peroxide, or diﬂuorine compared to diphosphine, hydrogen disulﬁde, or dichlorine, respectively. Notably, the LPBW effect is yet another result of the particularly compact, nodeless 13 14 1 Chemical Bonding of Main-Group Elements 2p-shell: the π-type lp exhibit substantial Pauli repulsion with the σ-bonding orbitals . Interestingly, from a VB point of view, such compounds frequently may be classiﬁed as charge-shift-bonding cases (See Chapter 5 in Volume 1). 1.2.5 Inﬂuence of Hybridization Defects on Magnetic-Resonance Parameters For the heavier p-block elements, hybridization defects lead to enhanced nscharacter in bonds, more than one might expect from structural considerations (see earlier). This holds for the highest oxidation state of the central atom, whereas the s-character concentrates into lps for the lower oxidation states, leaving predominantly p-character for the bonding orbitals. One quantity that is known to be particularly strongly connected to the s-character in bonds is indirect NMR spin–spin coupling constants . The Fermi-contact part of such coupling constants depends crucially on the s-character of the valence orbitals involved in bonding as the communication between the nuclear spin moments happens exclusively via the spherically symmetrical part of the spin density around a given nucleus. Larger s-character enhances the communication, leading to larger coupling constants in absolute terms (the sign of the coupling depends also on the relative signs of the two nuclear g-values) for compounds, where the two atoms involved are in their maximum oxidation state. Matters become more complicated when lone pairs are present; then, the bonds have largely p-character. The coupling constants tend to be smaller and are also potentially inﬂuenced by involvement of the lp and by other coupling terms. It has furthermore been demonstrated , that a related Fermi-contact mechanism dominates spin-orbit effects on the NMR chemical shifts of nuclei connected to heavy-atom centers (the so-called ‘‘heavy-atom effect on the light-atom shielding,’’ HALA , even though the NMR nucleus may also, in fact, be a heavy atom). Here s-character in bonding has also been found to be extremely important in deﬁning the magnitude of such ‘‘spin-orbit shifts’’ (SO shifts). Therefore, the SO shifts are large for p-block elements in their maximum oxidation state (even larger for hydrogen [52, 53]), whereas the SO effects are typically small in lower oxidation states because of the predominant p-character in the bonds [54, 55]. As we have learned earlier, electronegative substituents enhance hybridization defects and, thus, increase the s-character in the bonds. This explains why, for example, substitution of the hydrogen atoms in CH3 I by ﬂuorine atoms in CF3 I does essentially double the spin-orbit effects on the 13 C shifts because of the heavy iodine substituent, from about −30 ppm to about −60 ppm, reducing the high-frequency shift in the 13 C spectrum from about 145 ppm at the nonrelativistic level to about 115 ppm upon inclusion of spin-orbit coupling . The NPA charge on carbon increases from −0.70 in CH3 I to +0.85 in CF3 I, and consequently, the p/s hybridization ratio of the C-I bond (NAO/NLMO value) decreases from sp4.14 to sp1.96 (B3LYP/def2-TZVP results). This explains the more efﬁcient Fermi-contact mechanism (direct contributions from ﬂuorine SO coupling are small). Along the same line of argument, the more-than-linear increase of 13 C spin-orbit shifts with 1.3 The Role of the Outer d-Orbitals in Bonding n along the series CH4−n In , compared to the only linear increase along the series CBr4−n In , has been rationalized; whereas the positive charge on carbon and, thus, the carbon s-character of the C–I bonds increases with n in the former series, it stays relatively constant in the latter series (because of the similar electronegativities of Br and I) . Other properties in magnetic resonance may be mentioned here, for example, hyperﬁne couplings. As the isotropic hyperﬁne coupling also depends crucially on the spherical spin-density distribution around the nucleus in question, s-character in bonding and, thus, hybridization defects will be important. Obviously, for openshell radicals the s-character of the singly occupied MO(s) is the most crucial aspect, but spin polarization of doubly occupied MOs with core or valence scharacter may also be relevant (e.g., when the singly occupied molecular orbital is of pure p-character at the given atom). 1.3 The Role of the Outer d-Orbitals in Bonding Whether or not the outer d-orbitals are true valence orbitals for the heavier p-block elements and, thus, allow the octet rule to be violated in ‘‘hypervalent compounds’’ has been one of the most controversial questions in main-group bonding for decades. Whereas, during the past 15 years, the balance has dipped clearly to the side of the d-orbitals acting only as polarization functions rather than having a true valence-orbital character, it is worthwhile to consider here brieﬂy why the controversy has lasted so long. We will then put the hypervalency issue particularly into the context of radial nodes. Other aspects of the topic are discussed in Chapter 3 in this book. In the early days of discussions of main-group bonding, it was probably Pauling’s electroneutrality principle  that favored a picture of the outer d-orbitals as inﬂuential valence orbitals: resonance structures with appreciably positive formal charges on the central p-block main-group atom were considered to be unlikely and disfavored. Therefore, for example, a semipolar resonance structure for the sulfate ion with only single S–O bonds, a dipositive formal charge on sulfur, and a negative formal charge on each of the oxygen atoms, consistent with the original suggestion by Lewis , was disregarded in the 1960s or 1970s. Hypervalent resonance structures with four S=O double bonds have dominated the inorganic-chemistry textbooks (in fact, they still do so today!). On the other hand, the discovery of noble-gas compounds in the early 1960s gave a boost to three-center four-electron bonding models  and MO-based scenarios that avoided ‘‘true’’ hypervalency. In the 1970s, when the ﬁrst realistic ab initio wave functions could be obtained, bonding analyses were inevitably done using Mulliken’s population analysis . As we now know, the Mulliken populations are not only very basis-set dependent (earlier text), but they also tend to provide a much too covalent picture when the electronegativities of the bonding partners differ signiﬁcantly. This ‘‘cemented’’ the electroneutrality-principle point of view for many more years. More up-to-date 15 16 1 Chemical Bonding of Main-Group Elements methods, either of population-analysis type (e.g., natural population analysis ) or of real-space type (e.g., Bader’s quantum theory of atoms in molecules, QTAIMs ), give a more ionic picture in general and for hypervalent p-block compounds, in particular. This supports multicenter bonding and negative hyperconjugation in MO language or, equivalently, partially ionic resonance structures and no-bond double-bond resonance in VB language. d-Orbital participation, thus, is inevitably computed to be much less pronounced than suggested by Mulliken populations. As partial atomic charges are no true observables (no, not even the QTAIM ones!), none of these schemes alone can provide a deﬁnite answer to the true covalency and, thus, to the quantitative involvement of the d-orbitals in bonding. In fact, one can ﬁnd weaknesses in essentially any of the analysis methods, for example, a bias against d-orbital participation in the NBO scheme (because of the classiﬁcation of the d-functions with the ‘‘natural Rydberg set’’ ) or a possibly too ionic description by QTAIM charges . Yet the accumulated information provided by a wide range of the most reﬁned models available does, indeed, provide a much less covalent, less electro-neutral picture and, thus, a smaller involvement of the outer d-orbitals in bonding than assumed before. We may place the issue of hypervalent bonding into the more general context of the discussion throughout this chapter: hypervalent p-block compounds are stabilized by semipolar or partially ionic resonance structures in VB language or by multicenter bonding in MO language. Either way, a partial positive charge on the central atom is desirable. This is favored by (i) a high electronegativity of the substituents and by (ii) a low electronegativity of the central atom. This is why ﬂuorine substitution is most favorable in stabilizing hypervalent compounds, followed by oxygen substitution. At the same time, it is clear that the high electronegativity and small size of the 2p-elements makes them rather unfavorable as central atoms, consistent with observation. The nodelessness of the 2p-shell renders the 2p-elements electronegative and small and, thereby, helps to fulﬁll the octet rule in the 2p-series. In contrast, the heavier p-block elements are larger and less electronegative and, thus, favored as central atoms. In other words, the nodal structure of the valence np-orbitals, rather than the availability of the outer nd-orbitals, is decisive for the apparent violation of the octet rule for the heavier elements. Of course, the octet is usually not actually violated. Multicenter bonding models require some MOs that are essentially nonbonding and concentrated only on the substituents, and thus, the number of electrons in the valence shell of the central atom rarely exceeds the octet. However, here we should distinguish, between what Musher  more than 40 years ago termed hypervalent compounds of ﬁrst and second kind, respectively. In the ﬁrst class, the central atom is not in its maximum oxidation state, and thus, the central-atom ns-character concentrates in a lp. Then, as we have discussed in detail above, the bonds are made mainly from np-orbitals of the central atom, and thus, the assumptions of the usual three-center-four-electron bonding models are nicely fulﬁlled. In contrast, hypervalent compounds of the second kind exhibit the maximum oxidation state and, thus, necessarily involve the ns-orbitals fully in bonding. One thus sees (i) extensive hybridization defects 1.4 Secondary Periodicities: Incomplete-Screening and Relativistic Effects (see earlier text), (ii) a more complicated bonding pattern in which the ns-orbitals are also involved in multicenter delocalization , and c) often a subtle ‘‘true’’ hypervalency. The latter point has been demonstrated most convincingly by Häser  in his unique analysis scheme: Häser’s method is based on projecting the one-particle density matrix onto spheres around the atom. Expansion of the radius of the sphere around the central atom in, for example, PF5 or SF6 followed by integration of the AO contributions to the sphere populations up to a certain radius around the central atom indicated that the octet was violated, albeit very slightly, in contrast to hypervalent compounds of the ﬁrst kind (e.g. XeF2 ), where no octet violation could be registered by this approach. Closer analysis indicated populations with d-symmetry around the central atom in hypervalent compounds of the second kind . However, these populations did not possess true valence-orbital character but essentially described the outermost rim of the attractive potential trough of the atom, already close to the substituents. Interesting further insights were obtained regarding the multicenter delocalization in, for example, PF5 . Indeed, analyses of such species by natural resonance theory (NRT , based on a superposition of Lewis structures from strictly localized NBOs) also indicate slight violations of the octet in contrast to compounds of the ﬁrst kind. An aspect that deserves further scrutiny is to what extent the octet may also be violated when multicenter delocalization involves predominantly π-type orbitals on the substituents (in the sulfate ion or iso-electronic ions little octet violation mentioned earlier seems to be apparent from NRT, but more systems need to be analyzed). Unfortunately Häser’s approach has not been pursued further, because of his early demise. The method appears to be very powerful in the context of hypervalency discussions or for monitoring hybridization. 1.4 Secondary Periodicities: Incomplete-Screening and Relativistic Effects So far the nodal structure of the valence s- and p-orbitals themselves has been in our focus, allowing us to explain the special role of the 2p-elements compared to their heavier homologues. The further modulations of chemical and physical properties as we descend to a given group from period 3 on are often summarized under the term ‘‘secondary periodicity’’ [65, 66]. The main inﬂuences here are incomplete screening of nuclear charge by ﬁlled core or semi-core shells and the effects of special relativity. The former reﬂect shell structure of the atom as a whole and are already important for differences and similarities of the homologous third and fourth period elements, whereas the latter become crucial mainly for the chemistry of the sixth period elements. These aspects have been discussed in detail in various review articles (see, e.g., Refs [16, 28, 67]), and we, thus, touch them only brieﬂy. The most well-known incomplete-screening effect in the periodic table, the lanthanide contraction, is due to the successive ﬁlling of the 4f-shell by 14 electrons, leading to a contraction of the size of the later lanthanides and of the elements 17 18 1 Chemical Bonding of Main-Group Elements following them in the sixth period. This lanthanide contraction is most notable for the early 5d transition metals (e.g., regarding the similarity of the properties of Zr and Hf). It is still effective for the late 5d- and early 6p-elements, but in this case it is overshadowed by the relativistic contraction of the 6s-shell (see below) . The combined action of lanthanide contraction and relativity is responsible for a somewhat higher electronegativity of, for example, Pb versus Sn. As discussed above, the relativistic contraction of the 6s-shell is responsible for an enhancement of the inert-pair effect for the 6p-elements. Somewhat less well known but nevertheless important is the incomplete screening of nuclear charge by ﬁlling of the 3d-shell (sometimes called the ‘‘scandide contraction’’ [28, 65]). It is responsible for a contraction of the valence orbitals of the 4p-elements, in particular of those following directly on the 3d-series. This leads to a pronounced similarity of the electronegativities (in fact a slightly larger one for the 4p-element in the modern scales) and covalent radii of Al versus Ga, Si versus Ge, P versus As, and so on. For example, in covalent bonds to carbon or hydrogen, Ga is slightly smaller than Al (even more so for element-element bonds). In contrast, the slightly higher electronegativity of Ga reduces electrostatic interactions in more ionic compounds, and thus, a more ‘‘normal’’ behavior with larger bond lengths for Ga is found, for example, for halides. Whereas this has not been studied in detail so far, the scandide contraction probably enhances hybridization defects for the early 4p-elements as it likely affects the 4s-orbitals more than the 4p-orbitals and, thus, may overall increase the relevant differences in their radial extent. This may explain certain irregularities in the chemical properties for 3p- versus 4p- homologues, even further to the right in the p-block, that is, for S versus Se or Cl versus Br. It is interesting that incomplete screening of nuclear charge appears to be most pronouncedly caused by those shells that have no radial node such as the 3d- and 4f-shells. It is so far unclear why, for example, the 4d-contraction should be less effective than the 3d-contraction (the actinide contraction would be smaller than the lanthanide contraction without relativistic effects but is signiﬁcantly enhanced by the latter, thus becoming overall larger ). We may even consider a ‘‘2p-contraction’’ in the comparison of ionization energies and electronegativities for Li versus Na or Be versus Mg. It has been shown computationally [69, 70], that the question whether Li or Na is more electronegative depends on the bonding situation: while 𝜎-bonded ligands with some covalency (e.g., hydride or alkyl) give rise to a somewhat higher electronegativity of Na compared to Li, more electronegative ligands with some π-donor character (e.g., NH2 , OH, halogen, Cp) give more positive Na than Li charges. Closer analysis indicates that the Li 2s-orbital is very well shielded by the 1s core shell, whereas the 3s-orbital of Na experiences more incomplete shielding by the 2p-shell . Together with the opposing effect of the larger radius of Na compared to Li, this leads to these somewhat unusual and subtle trends. Pyykkö has analyzed them further using ‘‘pseudo atoms’’ in which core shells have been deleted and the nuclear charge has been reduced, in analogy to earlier related studies for the lanthanide and scandide contractions 1.5 ‘‘Honorary d-Elements’’ [65, 69, 71]. Notably, the observation of ‘‘inverted lithium-sodium electronegativities’’ by computations had been motivated by peculiar anomalies in the gas-phase thermochemistry of the group 1 elements . 1.5 ‘‘Honorary d-Elements’’: the Peculiarities of Structure and Bonding of the Heavy Group 2 Elements We have already concluded that the outer d-orbitals of the p-block elements are polarization functions but no true valence orbitals. This is well-known to be different for the inner d-orbitals of the transition metals, which dominate covalent bonding in the transition series. The heavier group 2 elements (Ca, Sr, Ba, Ra) also exhibit such inner (n-1)d-orbitals. Whereas these play essentially no role in bonding for the bulk metals or for neutral metal clusters, which are clearly dominated by the valence s-orbitals of the elements, matters become different with increasing positive charge on a given heavy group 2 element. Indeed, as we proceed to the most electronegative bonding partners (e.g., ﬂuorine or oxygen), the absolute role of covalent bonding decreases, but the relative role of the (n-1)d-orbitals in the diminishing covalent bonding contributions increases to the extent that, for a molecule like BaF2 , covalent bonding is almost entirely due to the Ba 5d-orbitals. This places these elements at the border line with the early transition metals and the lanthanides, and Pyykkö has coined the term ‘‘honorary d-elements’’ to indicate this . The importance of d-orbital involvement in bonding for the heavy alkaline earth elements is not a purely theoretical matter but manifests itself in structural peculiarities. The oldest example are observations in the 1960s that the molecular dihalides of the heavier group 2 elements in the gas phase may actually exhibit bent rather than linear structures whereas the Be or Mg dihalides are all clearly linear (see Ref.  for further literature). Different spectroscopic or electron diffraction techniques gave partly conﬂicting results, and the bent nature of molecules like BaF2 was later established beyond doubt by up-to-date computational methods . Such calculations provided evidence that even smaller bending angles and larger linearization energies could be obtained when replacing halogen or other π-donor ligands by pure σ-donors like hydride  or alkyl  ligands. Indeed, experimental evidence for bent group 2 or lanthanide(II) dialkyl complexes is now available . Bending of the group 2 metallocenes had been discussed even earlier. However, from today’s perspective, these metallocenes are at best quasi-linear with very shallow bending potential curves, as π-bonding actually favors linear arrangements . Detailed bonding analyses gave a rather interesting picture of those factors that favor the bent or linear structures:  linearity is obviously enhanced by Pauli or electrostatic repulsion between the M–X bonds or between X− anions, respectively, in agreement with the assumptions of the VSEPR model. Two apparently very different factors favoring bending had been discussed controversially for decades, (i) involvement of (n-1)d-orbitals in 𝜎-bonds to the ligands (see earlier text) and 19 20 1 Chemical Bonding of Main-Group Elements (ii) ‘‘inverse’’ polarization of the M2+ cations, leading to an electrostatic stabilization of the bent structure. Computations [67, 68] indicated that (i) both factors contribute to the bending and (ii) they are not strictly separable. The latter point may be appreciated from the fact that the polarizability of the cation is dominated by its outermost (n−1)p-shell. The (n−1)d-orbitals, thus, act as the primary polarization functions of the semi-core orbitals. At the same time, bending due to involvement of the valence (n−1)d-orbitals in covalent bonding will clearly lead to Pauli repulsion with the core and thus deformation of the penultimate p-shell. Therefore the covalent and core-polarization aspects of bending are closely interrelated as could be shown by a number of analysis procedures [67, 68]. The complications are due to the fact that outermost core (n−1)p- and valence (n−1)d-orbitals share the same principal quantum number. The role of π-bonding is also subtle: for the given group 2 examples with ‘‘weak π-donor’’ halide or related ligands, π-bonding clearly favors the linear structures . For more covalent transition-metal examples with strong donor ligands like ZrO2 , overlap of the π-type oxygen lp with an in-plane Zr d-orbital actually favors even more pronounced bending . These aspects show up also for more complex heteroleptic complexes. Indeed, the ‘‘non-VSEPR’’ structures are not restricted to exotic dicoordinate species but extend also to higher coordination numbers. For example, both computations and experiments indicate that rather peculiar structures may be favored for dimeric M2 X4 systems [78, 79] for exactly the same reasons that also account for the bending of the monomers. This author has argued that many peculiar bulk solid-state structures of heavy alkaline-earth dihalides or dihydrides (and of many early transition-metal compounds) also reﬂect the involvement of the (n−1)d-orbitals in σ-bonding . To complete the link to the transition metals, we note in passing that the peculiar distorted trigonal prismatic structures of species like WH6 or W(CH3 )6 or the preferences for square pyramidal rather than trigonal bipyramidal structures of TaH5 or Ta(CH3 )5 may be rationalized along similar lines (again 𝜋-bonding for analogous halide complexes gives the ‘‘classical’’ VSEPR structures in those cases). Importantly, however, covalency is much more important for such d0 species further to the right in the periodic table, and at the same time, core polarizability is expected to be much less pronounced than for group 2 species. Therefore, a covalent interpretation of such structures by d-orbital participation in σ- (and partly π-) bonding is much more reasonable than arguing via core polarization as has been done within the framework of a proposed ‘‘extended VSEPR model’’  based on the Laplacian of the charge density. More detailed arguments may be found in Ref. . We note that a decomposition of the Laplacian into strictly localized NBOs for complexes like Me3 NbCl2 and Me2 NbCl3 conﬁrms the ‘‘covalent’’ rationalization;  it turns out that maxima in the negative Laplacian, previously tagged ‘‘core shell charge concentrations’’ within extended VSEPR studies, in fact, reﬂect the backside lobes of the (n−1)d-orbitals involved in covalent σ-bonding. This is consistent with the fact that the ‘‘charge concentrations’’ are even present when a frozen core is used . References 1.6 Concluding Remarks In this chapter, we have tried to emphasize general aspects of main-group chemical bonding, with particular emphasis on periodic trends. The periodic table remains the most important classiﬁcation tool in chemistry, and it is crucial to understand even subtle secondary periodicities if one is to make efﬁcient use of the various elements for different chemical applications. The radial nodal structure of the valence orbitals has been pointed out to account for more of the known trends than most practitioners of chemistry are aware of. For example, the inversion barriers of phosphines or silyl anions, the dependence of the inert-pair effect on the electronegativity of the substituents, the stability of carbene- or carbyne-type species or of multiple bonds between heavy main-group elements are all intricately linked to hybridization defects of s- and p-valence orbitals of disparate sizes. Even the question of hypervalency is closely connected to the effects of ‘‘primogenic repulsion’’. Further important inﬂuences on the periodic trends arise from partial screening of nuclear charge (including lanthanide/actinide contraction, scandide contraction, and even a 2p-contraction) and from the effects of special relativity. Various aspects of main-group bonding are covered in more detail elsewhere in this book. References 1. See, e.g.: (a) Mahan, A.K. (1984) Phys. 2. 3. 4. 5. 6. 7. Rev. B, 29, 5982–5985; (b) Ahuja, R., Eriksson, O., and Johansson, B. (2000) Phys. Rev. B, 63, 014102/1–014102/3. 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