INTERMOLECULAR FORCES Their Origin and Determination By GEOFFREY C. MAITLAND Senior Research Scientist Schlumberger Cambridge Research, Cambridge MAURICE RIGBY Lecturer in the Department of Chemistry King's College, University of London E. BRIAN SMITH Fellow of St. Catherine's College, Oxford and Lecturer in Physical Chemistry, University of Oxford and WILLIAM A. WAKEHAM Professor of Chemical Physics Imperial College of Science and Technology, University of London ek FB7 Physlkol. Chemie / Chsm. Tedwologia Hochschule L 3 CLARENDON PRESS OXFORD Inventor
CONTENTS t2. 3. Pa 8 e INTRODUCTION 1.1. Historical background 1.2. Intermolecular energy 1 4 1.3. Origins of intermolecular forces 7 1.4. Long-range energy 10 1.4.1. Electrostatic energy 10 1.4.2. Induction energy 14 1.4.3. Dispersion energy 16 1.4.4. Long-range energy: summary 20 1.5. Short-range energy 23 1.6. Representation of the intermolecular pair potential energy function. 28 1.6.1. Simple functions 1.6.2. More flexible analytic functions 28/ 30 1.6.3. Representational functions 1.6.4. Non-spherical molecules 31 31 1.7. Sources of information about intermolecular forces 1.7.1. Gas imperfection 32 32 1.7.2. Transport properties of dilute gases 34 1.7.3. Molecular beams 36 1.7.4. Spectra 39 1.8. 1.7.5. Solid-state properties Summary 41 43 THEORETICAL CALCULATION OF INTERMOLECULAR FORCES 2.1. Introduction 45 2.2. Long-range forces 47 2.2.1. Electrostatic energy 48 1 2.2.2. Induction energy 53 2.2.3. Dispersion energy 57 2.2.4. Long-range three-atom energy 76 2.2.5. Combination rules 77 2.3. Short-range forces 77 2.3.1. Heitler-London methods 78 2.3.2. Molecular orbital theory 80 2.3.3. United atom perturbation theory 82 2.3.4. Model calculations 83 2.4. Intermediate-range energy 86 2.4.1. Exchange perturbation theories 86 2.4.2. Molecular orbital theory 87 2.4.3. Model calculations 87 2.5. Results 88 GAS IMPERFECTIONS *3.1. Introduction 96 *3.2. Van der Waals equation of state 99
CONTENTS *3.3. The virial equation of state 101 3.4. Statistical mechanics and the virial equation 102 3.5. Virial coefficients and intermolecular forces 108 3.6. Quantum corrections 114 3.7. Temperature dependence of virial coefficients 116 3.8. Corresponding states and virial coefficients 118 3.9. Virial coefficients for model potential energy functions 121 3.10. Experimental measurements 124 3.10.1. Low-pressure techniques 125 3.10.2. High-pressure techniques 128 3.10.3. Calculation of virial coefficients from experimental results 29 3.10.4. Joule-Thomson coefficients 130 3.11. Intermolecular forces from virial coefficients 132 3.11.1. Use of model potential functions 132 3.11.2. Inversion methods 136 3.12. Virial coefficients of more complex molecules 143 3.12.1. Calculations using model non-spherical potentials 145 3.12.2. Experimental results for non-spherical molecules 152 3.13. Mixtures 154 3.14. Dielectric virial coefficients 156 Appendix A3.1. Gas properties in terms of virial coefficients 158 MOLECULAR COLLISIONS 4.1. Introduction 162 4.2. Classical dynamics of molecular collisions 163 4.2.1. Binary elastic collisions 163 4.2.2. The equivalent one-body problem. 165 4.3. Experimental characterization of scattering: cross-sections 170 4.3.1. Definition of collision cross-sections 170 4.3.2. Shortcomings of classical treatment 173 4.4. Experimental techniques 4.4.1. Principles of molecular beam experiments 177 177 4.4.2. Sources 178 4.4.3. Velocity and state selection 180 4.4.4. Detectors 180 4.4.5. Centre-of-mass-laboratory coordinates 181 4.4.6. Integral cross-section measurements 181 4.4.7. Differential cross-section measurements 183 t4.5. Quantum mechanical treatment of elastic scattering 184 4.5.1. The wave function 185 4.5.2. The cross-sections 186 4.5.3. Solution of the wave equation: method of partial waves 187 4.5.4. Hard-sphere scattering 190 4.5.5. The semi-classical phase shift 192 4.5.6. Analytical evaluation of phase shifts 194 4.6. 4.5.7. The scattering cross-sections 198 4.5.8. Scattering of identical particles 211 The determination of intermolecular forces from elastic scattering cross-sections 213 4.6.1. Quality of data 4.6.2. Interpretation of data having no oscillatory fine structure 213 213
r jbiblloihek FB7 fhysikol. Chemie / Chem. lethnojegie CONTENTS Techniscfte Kochstfaufe 4.6.3. Interpretation of data for which sojme quantur structure is observed 4.6.4. Interpretation of fully-resolved scattering data 4.7. Scattering processes in polyatomic systems 4.7.1. Inelastic scattering cross-sections 4.7.2. Experimental methods 4.7.3. The scattering S-matrix: elastic scattering 4.7.4. Scattering matrices: the general case 4.7.5. Relation of the S-matrix to scattering cross-sections 4.7.6. Evaluation of the S-matrix: quantum approach 4.7.7. Close-coupling calculations 4.7.8. The centrifugal sudden approximation 4.7.9. The infinite-order sudden approximation 4.7.10. The space-fixed orientation approximation 4.7.11. The l z -conserved energy sudden (IOS) approximation A.1.12. Other approximate methods 4.7.13. Criteria for choosing approximate methods 4.8. The determination of intermolecular forces from inelastic crosssection measurements Appendix A4.1. The JWKB approximation Appendix A4.2. Phase shifts for potentials of the form U(r) = C n r~ n 217 225 226 228 232 234 235 237 239 241 242 247 248 248 249 253 258 261 5. THE KINETIC THEORY OF NON-UNIFORM DILUTE GASES *5.1. Introduction 266 *5.2. Simple kinetic theory 267 5.2.1. Viscosity 269 5.2.2. Thermal conductivity 272 5.2.3. Diffusion 273 5.2.4. Comparison with exact kinetic theory for hard-sphere molecules 274 5.2.5. Application to real gases 275 t5.3. The rigorous kinetic theory 276 t5.4. 5.3.1. The Boltzmann equation The Maxwell-Boltzmann equilibrium velocity distribution 276 280 t5.5. The Chapman-Enskog solution for non-uniform gases 5.5.1. The zeroth-order solution 282 287 5.5.2. The first-order solution 288 5.5.3. Explicit evaluation of the thermal conductivity coefficient 292 t5.6. Transport coefficients for pure gases 297 5.6.1. Formulae for special intermolecular potential models 299 t5.7. Transport coefficients for binary mixtures 301 5.7.1. Formulae for special intermolecular potential models 304 t5.8. Polyatomic gases 305 5.8.1. Quantum mechanical theory 5.8.2. The semi-classical theory 306 307 t5.9. 5.8.3. The classical theory The transport coefficients of polyatomic gases 308 308 5.9.1. The semi-classical transport coefficients 5.9.2. The Mason-Monchick approximation 308 314 5.9.3. Polyatomic gas mixtures 315
xii CONTENTS 5.9.4. Quantum-mechanical effects 316 t5.10. The kinetic theory of dense gases 320 5.10.1. The Enskog theory 321 5.10.2. The general kinetic theory of dense gases 322 tappendix A5.1. Integrals occurring in the kinetic theory 323 tappendix A5.2. Kinetic theory formulae for the transport properties of monatomic gases and gas mixtures 324 tappendix A5.3. Semi-classical formulae for the transport coefficients of dilute polyatomic gases and gas mixtures 336 6. THE TRANSPORT PROPERTIES OF GASES AND INTERMOLECULAR FORCES 6.1. Introduction 341 6.2. The calculation of transport properties from the intermolecular potential 343 6.3. Experimental measurements 346 6.3.1. Viscosity 347 6.3.2. Thermal conductivity 350 6.3.3. Binary diffusion coefficient 351 6.3.4. The thermal diffusion factor 353 6.4. Extraction of collision integrals from experimental data 355 6.5. Principle of corresponding states 357 6.6. Traditional methods of testing intermolecular potential functions 359 6.7. The direct determination of intermolecular forces: inversion methods 361 6.8. The basis of the inversion procedure 368 6.9. Polyatomic gases 371 6.9.1. Simple molecular models 373 6.9.2. The semi-classical and classical kinetic theories 374 6.9.3. The Mason and Monchick approximation 375 6.9.4. Spherically-averaged potential 376 6.9.5. The principle of corresponding states 376 6.9.6. Monchick and Mason's treatment of polar gases 377 6.9.7. Recent developments 380 I 7. SPECTROSCOPIC MEASUREMENTS 7.1. Introduction 387 7.2. The origins of van der Waals molecules 388 7.2.1. Concentrations 390 f 7.2.2. Lifetimes 390 7.3. Experimental techniques 391 I 7.3.1. Ultraviolet and infrared studies of dimers 391 f 7.3.2. Molecular-beam electric resonance: radiofrequency and microwave spectra 392 7.3.3. Other techniques, 394 7.4. Analysis of rotation-vibration spectra to give U(r): spherical systems 395 7.4.1. Introduction 395 7.4.2. The Rydberg-Klein-Rees inversion method 396 7.4.3. Determination of dissociation energy (well depth) 400
CONTENTS 7.5. Experimental results for inert gas dimers 7.5.1. Pure components 7.5.2. Mixtures 7.6. Analysis of spectra for non-spherical systems 7.6.1. General approach 7.6.2. Diatomic-monatomic dimer systems 7.6.3. Methods of calculating bound-state energies 7.6.4. Weakly anisotropic systems 7.6.5. Strongly anisotropic systems 7.6.6. Dimers of diatomic and polyatomic molecules 7.7. Sensitivity of alternative spectroscopic and scattering processes to the intermolecular potential 8. CONDENSED PHASES 8.1. Introduction 8.2. Many body forces 8.3. The solid state 8.3.1. Introduction 8.3.2. Static lattice properties 8.3.3. Lattice vibrations 8.3.4. Lattice vibrations: higher order effects 8.3.5. Experimental measurements 8.3.6. Results for the inert gases 8.3.7. More complex molecules 8.4. The liquid state 8.4.1. Liquid structure 8.4.2. Distribution function theories of liquids 8.4.3. Computer simulation 8.4.4. Perturbation theories 8.4.5. Transport properties of liquids 8.4.6. Experimental measurements for simple liquids 8.4.7. Liquid-state properties and intermolecular forces 8.4.8. Results for argon 8.4.9. More complex molecules 8.4.10. Liquid mixtures Appendix A8.1. The virial theorem of Clausius 9. INTERMOLECULAR FORCES: THE PRESENT POSITION *9.1. Introduction *9.2. 9.3. *9.4. 9.5. The Lennard-Jones era The basis of recent advances Potential energy functions Accurate potential functions (Class I) *9.5.1. Argon 9.5.2. Krypton 9.5.3. Helium 9.5.4. Neon 9.5.5. Xenon 9.5.6. Argon-krypton 9.5.7. Helium-heavier inert gas interactions 403 403 405 407 407 409 414 415 426 429 429 437 437 440 440 441 445 450 452 454 455 456 456 459 462 465 469 471 473 475 475 477 480 485 486 489 492 493 493 500 501 502 503 503 504
CONTENTS 9.6. Class II potential functions 506 9.6.1. Kr-Xe, Ar-Xe, Ne-Xe, Ne-Ar, Ne-Kr 9.6.2. Alkali metal-mercury system 506 506 9.7. Class III potential functions 9.7.1. Methane 508 508 f 9.7.2. Hydrogen 509 t 9.7.3. Other diatomic molecules 510 9.7.4. Inert-gas-diatomic-molecule interactions 512 9.7.5. Benzene 513 J 9.7.6. Water 513 9.8. *9.9. Class IV potential functions General features of potential energy functions 516 518 f 9.9.i: Conformality 518 I 9.9.2. Combining rules 519 S *9.10. Concluding remarks 523 Appendix 1. Intermolecular pair potential models Appendix 2. Collision integrals and second virial coefficients for 531 n 6 and n(r) 6 potentials Appendix 3. The extended law of corresponding states correlation 538 I of thermophysical properties Appendix 4. Recommended values for thermophysical properties of 564.some representative gases 568 Appendix 5. Characteristic parameters of some simple substances 573 Appendix 6. Pitzer's acentric factor: corresponding states for second virial coefficients 574 Appendix 7. Lattice sums, L*, for some cubic lattices 575 Appendix 8. v Collision integrals and second virial coefficients for non-spherical intermolecular potential models 577 Appendix 9. The intermolecular pair potential for argon 581 Appendix 10. Parameters of the n(r) 6 potential model for some gases 582 Appendix 11. FORTRAN-IV computer program for the evaluation of second virial coefficients 583 Appendix 12. FORTRAN-IV computer programme for the evaluation of collision integrals 590 Appendix 13. Inversion functions for gas phase properties 601 Substance index 605 Index 606 * These sections provide a suitable introduction for readers approaching the subject for the first time. t These sections may not be necessary for all users on a first reading.