Quantum Theory of Atoms in Molecules

Chérif F. Matta is an assistant professor of chemistry at Mount Saint Vincent University and an adjunct professor of chemistry at Dalhousie University, both in Halifax, Canada. He obtained his BSc from Alexandria University, Egypt, in 1987 and gained his PhD in theoretical chemistry from McMaster University, Hamilton, Canada in 2002. He was then a postdoctoral fellow at the University of Toronto, Canada, before being awarded an I. W. Killam Fellowship at Dalhousie University. Professor Matta has held the J. C. Polanyi Prize in Chemistry, two BioVision Next Fellowships, and a Chemistry Teaching Award, and has more than 40 papers and book chapters and two software programs to his credit. His research is in theoretical and computational chemistry with a focus on QTAIM and its applications. Russell Boyd graduated from the University of British Columbia in chemistry in 1967, receiving his PhD in theoretical chemistry from McGill University in 1971. He subsequently went to Oxford University, UK, as a postdoctoral fellow, before returning to British Columbia with a Killam Postdoctoral Fellowship at the Department of Chemistry from 1973 to 1975. He then joined Dalhousie University, Halifax, where he held the Chair of Chemistry from 1992 to 2005 and became McLeod Chair in 2001. Professor Boyd has published about 200 papers in computational and theoretical chemistry. His current interests include the effects of radiation on DNA and proteins, the mechanism by which a leading anti-tumor drug cleaves DNA, and the design of catalysts.

• Foreword viiPreface xixList of Abbreviations Appearing in this Volume xxviiList of Contributors xxxiii1 An Introduction to the Quantum Theory of Atoms in Molecules 1Chérif F. Matta and Russell J. Boyd1.1 Introduction 11.2 The Topology of the Electron Density 11.3 The Topology of the Electron Density Dictates the Form of Atoms in Molecules 51.4 The Bond and Virial Paths, and the Molecular and Virial Graphs 81.5 The Atomic Partitioning of Molecular Properties 91.6 The Nodal Surface in the Laplacian as the Reactive Surface of a Molecule 101.7 Bond Properties 101.7.1 The Electron Density at the BCP (pb) 111.7.2 The Bonded Radius of an Atom (rb), and the Bond Path Length 111.7.3 The Laplacian of the Electron Density at the BCP (∇2pb) 111.7.4 The Bond Ellipticity (є) 121.7.5 Energy Densities at the BCP 121.7.6 Electron Delocalization between Bonded Atoms: A Direct Measure of Bond Order 131.8 Atomic Properties 151.8.1 Atomic Electron Population [N(Ω)] and Charge [q(Ω)] 161.8.2 Atomic Volume [Vol.(Ω)] 161.8.3 Kinetic Energy [T(Ω)] 171.8.4 Laplacian [L(Ω)] 171.8.5 Total Atomic Energy [Ee(Ω)] 181.8.6 Atomic Dipolar Polarization [μ(Ω)] 201.8.7 Atomic Quadrupolar Polarization [Q(Ω)] 241.9 ‘‘Practical’’ Uses and Utility of QTAIM Bond and Atomic Properties 251.9.1 The Use of QTAIM Bond Critical Point Properties 251.9.2 The Use of QTAIM Atomic Properties 261.10 Steps of a Typical QTAIM Calculation 27References 30Part I Advances in Theory 352 The Lagrangian Approach to Chemistry 37Richard F. W. Bader2.1 Introduction 372.1.1 From Observation, to Physics, to QTAIM 372.2 The Lagrangian Approach 382.2.1 What is The Lagrangian Approach and What Does it Do? 382.2.2 The Lagrangian and the Action Principle – A Return to the Beginnings 392.2.3 Minimization of the Action 402.2.4 Steps in Minimizing the Action 412.3 The Action Principle in Quantum Mechanics 422.3.1 Schrödinger’s Appeal to the Action 422.3.2 Schrödinger’s Minimization 422.3.2.1 Two Ways of Expressing the Kinetic Energy 432.3.3 Obtaining an Atom from Schrödinger’s Variation 442.3.3.1 The Role of Laplacian in the Definition of an Atom 452.3.4 Getting Chemistry from δG(Ψ, ∇Ψ; Ω) 462.4 From Schrödinger to Schwinger 482.4.1 From Dirac to Feynman and Schwinger 482.4.2 From Schwinger to an Atom in a Molecule 492.5 Molecular Structure and Structural Stability 522.5.1 Definition of Molecular Structure 522.5.2 Prediction of Structural Stability 532.6 Reflections and the Future 532.6.1 Reflections 532.6.2 The Future 55References 573 Atomic Response Properties 61Todd A. Keith3.1 Introduction 613.2 Apparent Origin-dependence of Some Atomic Response Properties 623.3 Bond Contributions to ‘‘Null’’ Molecular Properties 643.4 Bond Contributions to Atomic Charges in Neutral Molecules 703.5 Atomic Contributions to Electric Dipole Moments of Neutral Molecules 713.6 Atomic Contributions to Electric Polarizabilities 733.7 Atomic Contributions to Vibrational Infrared Absorption Intensities 783.8 Atomic Nuclear Virial Energies 823.9 Atomic Contributions to Induced Electronic Magnetic Dipole Moments 883.10 Atomic Contributions to Magnetizabilities of Closed-Shell Molecules 90References 944 QTAIM Analysis of Raman Scattering Intensities: Insights into the Relationship Between Molecular Structure and Electronic Charge Flow 95Kathleen M. Gough, Richard Dawes, Jason R. Dwyer, and Tammy L. Welshman4.1 Introduction 954.2 Background to the Problem 964.2.1 Conceptual Approach to a Solution 974.2.1.1 Experimental Measurement of Raman Scattering Intensities 974.2.1.2 Theoretical Modeling of Raman Scattering Intensities: What We Did and Why 994.3 Methodology 1004.3.1 Modeling α and ∂α/∂r 1014.3.2 Recouping α From the Wavefunction, With QTAIM 1024.3.3 Recovering ∂α/∂r From QTAIM 1034.4 Specific Examples of the Use of AIM2000 Software to Analyze Raman Intensities 1034.4.1 Modeling α in H2 1044.4.1.1 Modeling ∆α/∆r in H2 1064.4.2 Modeling α and ∆α/∆r in CH4 1064.4.3 Additional Exercises for the Interested Reader 1084.5 Patterns in α That Are Discovered Through QTAIM 1094.6 Patterns in ∂α/∂rCH That Apply Across Different Structures, Conformations, Molecular Types: What is Transferable? 1114.6.1 Patterns in ∆α/∆rCH Revealed by QTAIM 1114.6.1.1 QTAIM Analysis of ∆α/∆rCH in Small Alkanes 1114.6.1.2 What Did We Learn From QTAIM That Can be Transferred to the Other Molecules? 1134.7 What Can We Deduce From Simple Inspection of ∂α/∂rCH and ∂α/∂rCC From Gaussian? 1144.7.1 Variations in ∂α/∂rCH Among the Alkanes 1144.7.2 ∆α/∆rCH in Cycloalkanes, Bicycloalkanes, and Hedranes 1164.7.3 Patterns That Emerge in ∆α/∆rCC of Alkanes 1164.7.4 Unsaturated Hydrocarbons and the Silanes: C-H, C=C, and Si-Si Derivatives 1174.8 Conclusion 118References 1195 Topological Atom–Atom Partitioning of Molecular Exchange Energy and its Multipolar Convergence 121Michel Rafat and Paul L. A. Popelier5.1 Introduction 1215.2 Theoretical Background 1235.3 Details of Calculations 1285.4 Results and Discussion 1305.4.1 Convergence of the Exchange Energy 1305.4.2 Convergence of the Exchange Force 1365.4.3 Diagonalization of a Matrix of Exchange Moments 1365.5 Conclusion 139References 1396 The ELF Topological Analysis Contribution to Conceptual Chemistry and Phenomenological Models 141Bernard Silvi and Ronald J. Gillespie6.1 Introduction 1416.2 Why ELF and What is ELF? 1426.3 Concepts from the ELF Topology 1446.3.1 The Synaptic Order 1456.3.2 The Localization Domains 1456.3.3 ELF Population Analysis 1476.4 VSEPR Electron Domains and the Volume of ELF Basins 1496.5 Examples of the Correspondence Between ELF Basins and the Domains of the VSEPR Model 1536.5.1 Octet Molecules 1536.5.1.1 Hydrides (CH4, NH3, H2O) 1536.5.1.2 AX4 (CH4, CF4, SiCl4) 1546.5.1.3 AX3E and AX2E2 (NCl3, OCl2) 1546.5.2 Hypervalent Molecules 1556.5.2.1 PCl5 and SF6 1556.5.2.2 SF4 and ClF3 1556.5.2.3 AX7 and AX6E Molecules 1556.5.3 Multiple Bonds 1566.5.3.1 C2H4 and C2H2 1566.5.3.2 Si2Me4 and Si2Me2 1576.6 Conclusions 158References 159Part II Solid State and Surfaces 1637 Solid State Applications of QTAIM and the Source Function – Molecular Crystals, Surfaces, Host–Guest Systems and Molecular Complexes 165Carlo Gatti7.1 Introduction 1657.2 QTAIM Applied to Solids – the TOPOND Package 1667.2.1 QTAIM Applied to Experimental Densities: TOPXD and XD Packages 1687.3 QTAIM Applied to Molecular Crystals 1707.3.1 Urea 1717.3.1.1 Urea: Packing Effects 1727.4 QTAIM Applied to Surfaces 1797.4.1 Si(111)(1*1) Clean and Hydrogen-covered Surfaces 1807.4.2 Si(111)(2*1) Reconstructed Surface 1847.5 QTAIM Applied to Host–Guest Systems 1867.5.1 Type I Inorganic Clathrates A8Ga16Ge30 (A=Sr, Ba) 1867.5.2 Sodium Electrosodalite 1907.6 The Source Function: Theory 1927.6.1 The Source Function and Chemical Transferability 1947.6.2 Chemical Information from the Source Function: Long and Short-range Bonding Effects in Molecular Complexes 1967.6.3 The Source Function: Latest Developments 201References 2028 Topology and Properties of the Electron Density in Solids 207Víctor Luaña, Miguel A. Blanco, Aurora Costales, Paula Mori-Sánchez, and Angel Martín Penda´s8.1 Introduction 2078.2 The Electron Density Topology and the Atomic Basin Shape 2098.3 Crystalline Isostructural Families and Topological Polymorphism 2138.4 Topological Classification of Crystals 2158.5 Bond Properties – Continuity from the Molecular to the Crystalline Regime 2178.6 Basin Partition of the Thermodynamic Properties 2198.7 Obtaining the Electron Density of Crystals 222References 2279 Atoms in Molecules Theory for Exploring the Nature of the Active Sites on Surfaces 231Yosslen Aray, Jesus Rodríguez, and David Vega9.1 Introduction 2319.2 Implementing the Determination of the Topological Properties of p(r) from a Three-dimensional Grid 2319.3 An Application to Nanocatalyts – Exploring the Structure of the Hydrodesulfurization MoS2 Catalysts 2369.3.1 Catalyst Models 2379.3.2 The Full p(r) Topology of the MoS2 Bulk 2419.3.3 The p(r) Topology of the MoS2 Edges 245References 254Part III Experimental Electron Densities and Biological Molecules 25710 Interpretation of Experimental Electron Densities by Combination of the QTAMC and DFT 259Vladimir G. Tsirelson10.1 Introduction 25910.2 Specificity of the Experimental Electron Density 26110.3 Approximate Electronic Energy Densities 26210.3.1 Kinetic and Potential Energy Densities 26210.3.2 Exchange and Correlation Energy Densities 27110.4 The Integrated Energy Quantities 27510.5 Concluding Remarks 276References 27811 Topological Analysis of Proteins as Derived from Medium and Highresolution Electron Density: Applications to Electrostatic Properties 285Laurence Leherte, Benoȋt Guillot, Daniel P. Vercauteren, Virginie Pichon-Pesme, Christian Jelsch, Angélique Lagoutte, and Claude Lecomte11.1 Introduction 28511.2 Methodology and Technical Details 28711.2.1 Ultra-high X-ray Resolution Approach 28711.2.2 Medium-resolution Approach 28911.2.2.1 Promolecular Electron Density Distribution Calculated from Structure Factors 28911.2.2.2 Promolecular Electron Density Distribution Calculated from Atoms 29011.2.3 A Test System – Human Aldose Reductase 29111.3 Topological Properties of Multipolar Electron Density Database 29411.4 Analysis of Local Maxima in Experimental and Promolecular Mediumresolution Electron Density Distributions 29811.4.1 Experimental and Promolecular Electron Density Distributions Calculated from Structure Factors 29911.4.2 Promolecular Electron Density Distributions Calculated from Atoms (PASA Model) 30111.5 Calculation of Electrostatic Properties from Atomic and Fragment Representations of Human Aldose Reductase 30511.5.1 Medium- and High-resolution Approaches of Electrostatic Potential Computations 30711.5.2 Electrostatic Potential Comparisons 30911.5.3 Electrostatic Interaction Energies 31211.6 Conclusions and Perspectives 312References 31412 Fragment Transferability Studied Theoretically and Experimentally with QTAIM – Implications for Electron Density and Invariom Modeling 317Peter Luger and Birger Dittrich12.1 Introduction 31712.2 Experimental Electron-density Studies 31812.2.1 Experimental Requirements 31812.2.2 Recent Experimental Advances 31912.2.2.1 Synchrotron Radiation Compared with Laboratory Sources 31912.2.2.2 Data Collection at Ultra-low Temperatures (10–20 K) 32112.3 Studying Transferability with QTAIM – Atomic and Bond Topological Properties of Amino Acids and Oligopeptides 32312.4 Invariom Modeling 32812.4.1 Invariom Notation, Choice of Model Compounds, and Practical Considerations 33012.4.2 Support for Pseudoatom Fragments from QTAIM 33112.5 Applications of Aspherical Invariom Scattering Factors 33412.5.1 Molecular Geometry and Anisotropic Displacement Properties 33412.5.2 Using the Enhanced Multipole Model Anomalous Dispersion Signal 33512.5.3 Modeling the Electron Density of Oligopeptide and Protein Molecules 33612.6 Conclusion 338References 339Part IV Chemical Bonding and Reactivity 34313 Interactions Involving Metals – From ‘‘Chemical Categories’’ to QTAIM, and Backwards 345Piero Macchi and Angelo Sironi13.1 Introduction 34513.2 The Electron Density in Isolated Metal Atoms – Hints of Anomalies 34513.3 Two-center Bonding 34913.3.1 The Dative Bond 35013.3.1.1 Metal Carbonyls 35113.3.1.2 Donor–Acceptor Interactions of Heavy Elements 35213.3.2 Direct Metal–Metal Bonding 35213.4 Three-center Bonding 35613.4.1 π-Complexes 35713.4.2 σ-Complexes 36313.4.2.1 Dihydrogen and Dihydride Coordination 36413.4.2.2 Agostic Interactions 36413.4.2.3 Hydride Bridges 36713.4.3 Carbonyl-supported Metal–Metal Interactions 37013.5 Concluding Remarks 371References 37214 Applications of the Quantum Theory of Atoms in Molecules in Organic Chemistry – Charge Distribution, Conformational Analysis and Molecular Interactions 375Jesús Hernández-Trujillo, Fernando Cortés-Guzmn, and Gabriel Cuevas14.1 Introduction 37514.2 Electron Delocalization 37514.2.1 The Pair-density 37514.2.2 3JHH Coupling Constants and Electron Delocalization 37814.3 Conformational Equilibria 38014.3.1 Rotational barriers 38014.3.1.1 Rotational Barrier of Ethane 38014.3.1.2 Rotational Barrier of 1,2-Disubstituted Ethanes 38214.3.2 Anomeric Effect on Heterocyclohexanes 38614.4 Aromatic Molecules 39114.4.1 Electronic Structure of Polybenzenoid Hydrocarbons 39114.5 Conclusions 395References 39615 Aromaticity Analysis by Means of the Quantum Theory of Atoms in Molecules 399Eduard Matito, Jordi Poater, and Miquel Solà15.1 Introduction 39915.2 The Fermi Hole and the Delocalization Index 40115.3 Electron Delocalization in Aromatic Systems 40315.4 Aromaticity Electronic Criteria Based on QTAIM 40415.4.1 The para-Delocalization Index (PDI) 40415.4.2 The Aromatic Fluctuation Index (FLU) 40615.4.3 The π-Fluctuation Aromatic Index (FLUπ) 40715.5 Applications of QTAIM to Aromaticity Analysis 40915.5.1 Aromaticity of Buckybowls and Fullerenes 40915.5.2 Effect of Substituents on Aromaticity 41215.5.3 Assessment of Clar’s Aromatic π-Sextet Rule 41615.5.4 Aromaticity Along the Diels–Alder Reaction. The Failure of Some Aromaticity Indexes 41815.6 Conclusions 419References 42116 Topological Properties of the Electron Distribution in Hydrogen-bonded Systems 425Ignasi Mata, Ibon Alkorta, Enrique Espinosa, Elies Molins, and José Elguero16.1 Introduction 42516.2 Topological Properties of the Hydrogen Bond 42616.2.1 Topological Properties at the Bond Critical Point (BCP) 42616.2.2 Integrated Properties 42916.3 Energy Properties at the Bond Critical Point (BCP) 43116.4 Topological Properties and Interaction Energy 43516.5 Electron Localization Function, n(r) 43816.6 Complete Interaction Range 44016.6.1 Dependence of Topological and Energy Properties on the Interaction Distance 44016.6.2 Perturbed Systems 44816.7 Concluding Remarks 450References 45017 Relationships between QTAIM and the Decomposition of the Interaction Energy – Comparison of Different Kinds of Hydrogen Bond 453Sławomir J. Grabowski17.1 Introduction 45317.2 Diversity of Hydrogen-bonding Interactions 45617.3 The Decomposition of the Interaction Energy 45917.4 Relationships between the Topological and Energy Properties of Hydrogen Bonds 46017.5 Various Other Interactions Related to Hydrogen Bonds 46417.5.1 H+…π Interactions 46417.5.2 Hydride Bonds 46617.6 Summary 467References 468Part V Application to Biological Sciences and Drug Design 47118 QTAIM in Drug Discovery and Protein Modeling 473Nagamani Sukumar and Curt M. Breneman18.1 QSAR and Drug Discovery 47318.2 Electron Density as the Basic Variable 47418.3 Atom Typing Scheme and Generation of the Transferable Atom Equivalent (TAE) Library 47618.4 TAE Reconstruction and Descriptor Generation 47818.5 QTAIM-based Descriptors 48018.5.1 TAE Descriptors 48218.5.2 RECON Autocorrelation Descriptors 48518.5.3 PEST Shape–Property Hybrid Descriptors 48518.5.4 Electron Density-based Molecular Similarity Analysis 48718.6 Sample Applications 48918.6.1 QSAR/QSPR with TAE Descriptors 48918.6.2 Protein Modeling with TAE Descriptors 49118.7 Conclusions 492References 49419 Fleshing-out Pharmacophores with Volume Rendering of the Laplacian of the Charge Density and Hyperwall Visualization Technology 499Preston J. MacDougall and Christopher E. Henze19.1 Introduction 49919.2 Computational and Visualization Methods 50119.2.1 Computational Details 50119.2.2 Volume Rendering of the Laplacian of the Charge Density 50119.2.3 The Hyperwall 50519.2.4 Hyper-interactive Molecular Visualization 50519.3 Subatomic Pharmacophore Insights 50719.3.1 Hydrogen-bonding Donor Sites 50719.3.2 Inner-valence Shell Charge Concentration (i-VSCC) Features in Transition-metal Atoms 50919.3.3 Misdirected Valence in the Ligand Sphere of Transition-metal Complexes 51119.4 Conclusion 513References 514Index 515

"…a handsome text that serves to create a one-stop reference point exploring the quantum theory of atoms and molecules in complete detail." (Electric Review, May 2007)