Molecular Aggregation Structure Analysis and Molecular Simulation of Crystals and Liquids ANGELO GAVEZZOTTI University of Milano OXFORD UNIVERSITY PRESS
Contents PART I FUNDAMENTALS 1 The molecule: structure, size and shape 3 1.1 Atoms and bonds 3 1.2 Classification concepts in many particle systems 5 1.2.1 Structure 5 1.2.2 Symmetry 8 1.2.3 Order 9 1.2.4 Enthalpy and entropy 9 1.3 Must a molecule have a size? 10 1.3.1 Mass and length dimensions 10 1.3.2 Atomic radü 13 1.3.3 Molecular volume and surface 15 1.3.4 Size in terms of electron density 18 1.4 Must a molecule have a shape? 20 1.5 Historical portraits: a chemistry course in the early 1960's 24 2 Molecular vibrations and molecular force fields 30 2.1 Vibrational modes and force constants 30 2.2 Molecular mechanics 35 2.3 Evolution of molecular force fields 39 2.4 Appendix: an example of coordinate transformation 43 2.5 Historical portraits: Got a force constant? 45 3 Quantum chemistry 53 3.1 Some fundamentals of quantum mechanics 53 3.1.1 Dynamic variables, wavefunctions, operators 54 3.1.2 The Schrödinger equation and stationary states 54 3.1.3 Linear momentum 56 3.1.4 Angular momentum 57 3.1.5 Spherical harmonics 59 3.1.6 The harmonic oscillator 60 3.2 The hydrogen atom and atomic orbitals 61 3.3 Spin 63 3.4 Many-electron systems 65 3.5 Molecular orbitals: The Fock and Roothaan equations 67 3.5.1 The variational principle 69 3.5.2 Why do a molecular orbital calculation? 72
x CONTENTS 3.6 Approximate quantum chemical methods: NDO and EHT 74 3.7 Evolution of quantum chemical calculations: Beyond Hartree Fock 76 3.7.1 Configuration interaction and M011er Plesset perturbation methods 77 3.7.2 Density functional theory (DFT) methods 78 3.7.3 Other beyond-hartree Fock methods 80 3.7.4 Ethylene in perspective 81 3.8 Dimerization energies and basis set superposition error 81 3.9 Historical portraits: early experiences in quantum chemistry 82 4 The physical nature and the computer simulation of the intermolecular potential 87 4.1 Experimental facts and conceptual framework 87 4.2 The representation of the molecular charge distribution and of the electric potential 89 4.2.1 Full electron density 89 4.2.2 Central multipoles 90 4.2.3 Distributed multipoles 92 4.2.4 Point charges 92 4.3 Coulombic potential energy 94 4.4 Polarization (electrostatic induction) energy 96 4.5 Dispersion energy 99 4.6 Pauli (exchange) repulsion energy 101 4.7 Total energies versus partitioned energies 103 4.7.1 Pairwise additivity 103 4.7.2 Interpretation 104 4.8 Intermolecular hydrogen bonding 105 4.9 Simulation methods 106 4.9.1 The intermolecular atom atom model for organic crystals 106 4.9.2 Distributed multipole methods 110 4.9.3 Other density-based methods 111 4.10 Ad hoc or transferable? Force field fitting from ab initio calculations 112 5 Crystal symmetry and X-ray diffraction 120 5.1 A structural view of crystal symmetry: bottom-up crystallography 120 5.2 Space group symmetry and its mathematical representation 127 5.3 von Laue's idea, 1912 130 5.4 The structure factor 131 5.4.1 Scattering by one or two charge points 131 5.4.2 The atomic scattering factor 133
CONTENTS xi 5.4.3 The molecular structure factor 134 5.4.4 The structure factor for infinite periodic systems 135 5.5 Miller indices and Bragg's law 137 5.6 The electron density in a crystal 139 5.7 The atomic prejudice 139 5.8 Structure and X-ray diffraction: Some examples 140 5.9 Historical portraits: Training of a crystallographer in the 1960s 144 6 Periodic systems: Crystal orbitals and lattice dynamics 153 6.1 The mathematical description of crystal periodicity 153 6.1.1 Equivalent positions and systematic absences in diffraction patterns 153 6.1.2 Reciprocal space, wave vector, Brillouin zone 155 6.1.3 Bloch functions 155 6.2 The electronic structure of solids 157 6.2.1 The crystal orbital approach 157 6.2.2 Band structures: Complicated but not difficult 158 6.2.3 Comparison with experiment; electronic density of states 162 6.3 Lattice dynamics and lattice vibrations 163 6.3.1 Periodic vibrations in infinite crystals 163 6.3.2 Comparison with experiment; measuring lattice-vibration frequencies 167 7 Molecular structure and macroscopic properties: Calorimetry and thermodynamics 172 7.1 Molecules and macroscopic bodies 172 7.2 Energy 174 7.2.1 The partition function: Molecules 174 7.2.2 The partition function: Macroscopic systems 176 7.2.3 Internal energy I: From statistics and quantum mechanics 177 7.2.4 Internal energy II: From thermal and mechanical experiments 179 7.3 Heat capacity 179 7.4 Entropy 180 7.4.1 Classical entropy 181 7.4.2 Statistical entropy 182 7.4.3 The calculation of entropy for chemical systems 182 7.5 Free energy and chemical equilibrium 183 7.5.1 Chemical potential 183 7.5.2 Free energy 184 7.5.3 Chemical potentials in practice 184 7.6 Thermodynamic measurements 186
xii CONTENTS 7.6.1 Heat capacity 186 7.6.2 Melting enthalpies 188 7.6.3 Sublimation enthalpies 190 7.7 Derivatives 193 8 Correlation studies in organic solids 196 8.1 The Cambridge Structural Database (CSD) of organic crystals 196 8.2 Structure correlation 198 8.3 Retrieval of molecular and crystal structures from the CSD 199 8.4 The SubHeat database 201 8.5 The geometrical categorization of intermolecular bonding 202 8.6 Space analysis of molecular packing modes 203 8.6.1 Empty space versus filled space 203 8.6.2 Close packing in crystals 204 8.7 The calculation of intermolecular energies in crystals 207 8.7.1 Lattice energies: Some basic concepts 207 8.7.2 Convergence problems in lattice sums 212 8.7.3 Sublimation entropies and vapor pressures of crystals 213 8.8 General-purpose force fields for organic crystals 214 8.9 Accuracy and reproducibility 217 8.10 Correlation between molecular and crystal properties: Fact or fiction? 220 8.10.1 Bivariate analysis 220 8.10.2 Principal component analysis 223 8.11 Acceptable crystal structures 225 8.12 Historical portraits: Lattice energies and the phase problem in the old days 225 9 The liquid state 230 9.1 Proper liquids 230 9.2 Molecular dynamics (MD) 230 9.2.1 Equations of motion 231 9.2.2 Temperature 232 9.2.3 Pressure 233 9.2.4 NPT and NVT simulations 234 9.2.5 Performance and constraints in molecular dynamics simulations 235 9.3 The Monte Carlo (MC) method 236 9.4 Structural and dynamic descriptors for liquids 237 9.4.1 Radial distribution functions 238 9.4.2 Correlation functions 241 9.5 Physicochemical properties of liquids from MD or MC simulations 243 9.5.1 Enthalpy, heat capacity and density 243
CONTENTS xiii 9.5.2 The Jorgensen school 244 9.5.3 Crystal and liquid equations of state 245 9.6 Polarizability and dielectric constants 246 9.7 Free energy simulations 247 9.8 A theme with variations 249 9.9 Water 249 10 Computers 254 10.1 Bits and pieces 254 10.2 Operating systems 256 10.3 Computer programming 258 10.4 Bugs and program checking and validation 260 10.5 Reproducibility 261 10.6 "Because it's there"? 262 PART II THE FRONTIER 11 Structure-property and structure activity relationships 269 11.1 Fundamental research and applied technology 269 11.2 The structure activity dogma 270 11.3 Crystal dissolution 273 11.4 Thermal properties 275 11.4.1 Thermal expansion coefficients 275 11.4.2 Heat capacity and heat transport 277 11.5 Strain and stress, elastic and viscous properties 278 11.6 Optical, electric and magnetic properties 284 11.6.1 Color 284 11.6.2 Optical properties and the polar axis 289 11.6.3 Electric and magnetic properties 292 12 Intermolecular bonding 296 12.1 The decline of the intermolecular atom atom bond 296 12.1.1 The Feynman Ehrenfest chemical bond 296 12.1.2 More familiar models: distance energy analyses 298 12.2 Full density models: the SCDS Pixel method 304 12.2.1 Theory 304 12.2.2 Coulombic energy 305 12.2.3 Polarization energy 306 12.2.4 Dispersion energy 307 12.2.5 Repulsion energy 308 12.2.6 Total energies and parameters 308 12.2.7 The generation of crystal coordinates 309 12.2.8 Pixel calculations: General features 310
xiv CONTENTS 12.2.9 Pixel theory: Pros and cons 314 12.3 Systematic application of the Pixel theory to intermolecular bonding 315 12.3.1 A glossary of intermolecular recognition modes 315 12.3.1.1 Interactions not involving hydrogen 316 12.3.1.2 C H X interactions 318 12.3.1.3 The 0 H X and N H- X hydrogen Bond 319 12.3.1.4 n-interactions 323 12.3.2 Crystal energies 325 12.4 Directed bonds versus diffuse bonding 326 13 Phase equilibria, phase changes, and mesophases: Analysis and simulation 330 13.1 Things and molecules 330 13.2 Basic thermodynamic functions 331 13.3 Melting 332 13.4 Solid liquid equilibrium and nucleation from the melt 338 13.5 Vapor liquid and vapor solid equilibrium 341 13.6 Glasses 342 13.7 Liquid crystals 345 13.8 Nucleation and growth from solution: Experiments 347 13.8.1 Overview 347 13.8.2 Light scattering, calorimetry 348 13.8.3 Chemical spectroscopy 349 13.8.4 X-ray scattering and diffraction 350 13.9 Crystal growth and morphology 351 13.9.1 Crystal faces, attachments energies, and morphology prediction 351 13.9.2 Electron micrography and atomic force microscopy (AFM) 355 13.10 Evolutionary molecular simulation 356 14 Crystal polymorphism and crystal structure prediction 367 14.1 A fundamental fact 367 14.2 What are crystal polymorphs? 368 14.2.1 The taxonomy of organic crystals 368 14.2.2 Phenomenology of crystal polymorphism 369 14.2.3 Crystal structure fingerprints: Detecting real polymorphism 375 14.2.4 Analysis of crystal polymorphism by Pixel and quantum chemical calculations 379 14.3 The construction of crystal structures by computer 383 14.3.1 "Brate force" approach 384
CONTENTS xv 14.3.2 The "Prom" sequential approach 385 14.3.3 Molecular clusters with one symmetry Operator 385 14.3.4 Combination of two or three symmetry operators 387 14.3.5 The translation search: Sorting and ranking 389 14.3.6 The Prom algorithm: Pros and cons 389 14.3.7 Some examples of crystal structure generation 390 14.4 Crystal structure prediction by computer 395 14.4.1 The aims 397 14.4.2 The tools 398 14.4.3 Are crystal structures predictable? 398 15 Epilogue: A theory of crystallization? 405 15.1 Laws and theories 405 15.2 Aggregation stages 406 15.2.1 Oligomers 407 15.2.2 Nanoparticles 407 15.2.3 Mesoparticles 410 15.3 Macroscopic crystals 410 15.4 Thermodynarnics, kinetics, and symmetry 411 15.5 The language of the theory 416 Index 419 Supplementary material OPiX, an open-code computer program package for crystal packing analysis, polymorph generation and prediction, and Pixel-SCDS calculation; manuals and source Codes Other open-source program Codes for the calculations described in the book The SubHeat database (see in Section 8.4) The polymorph database (see in Section 14.2) Details of calculations (lists of atomic coordinates, etc.) This material is available at the author's website: http://users.unimi.it/gavezzot