INTERMOLECULAR FORCES

Transcription

1 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

2 CONTENTS t2. 3. Pa 8 e INTRODUCTION 1.1. Historical background 1.2. Intermolecular energy Origins of intermolecular forces Long-range energy Electrostatic energy Induction energy Dispersion energy Long-range energy: summary Short-range energy Representation of the intermolecular pair potential energy function Simple functions More flexible analytic functions 28/ Representational functions Non-spherical molecules Sources of information about intermolecular forces Gas imperfection Transport properties of dilute gases Molecular beams Spectra Solid-state properties Summary THEORETICAL CALCULATION OF INTERMOLECULAR FORCES 2.1. Introduction Long-range forces Electrostatic energy Induction energy Dispersion energy Long-range three-atom energy Combination rules Short-range forces Heitler-London methods Molecular orbital theory United atom perturbation theory Model calculations Intermediate-range energy Exchange perturbation theories Molecular orbital theory Model calculations Results 88 GAS IMPERFECTIONS *3.1. Introduction 96 *3.2. Van der Waals equation of state 99

3 CONTENTS *3.3. The virial equation of state Statistical mechanics and the virial equation Virial coefficients and intermolecular forces Quantum corrections Temperature dependence of virial coefficients Corresponding states and virial coefficients Virial coefficients for model potential energy functions Experimental measurements Low-pressure techniques High-pressure techniques Calculation of virial coefficients from experimental results Joule-Thomson coefficients Intermolecular forces from virial coefficients Use of model potential functions Inversion methods Virial coefficients of more complex molecules Calculations using model non-spherical potentials Experimental results for non-spherical molecules Mixtures Dielectric virial coefficients 156 Appendix A3.1. Gas properties in terms of virial coefficients 158 MOLECULAR COLLISIONS 4.1. Introduction Classical dynamics of molecular collisions Binary elastic collisions The equivalent one-body problem Experimental characterization of scattering: cross-sections Definition of collision cross-sections Shortcomings of classical treatment Experimental techniques Principles of molecular beam experiments Sources Velocity and state selection Detectors Centre-of-mass-laboratory coordinates Integral cross-section measurements Differential cross-section measurements 183 t4.5. Quantum mechanical treatment of elastic scattering The wave function The cross-sections Solution of the wave equation: method of partial waves Hard-sphere scattering The semi-classical phase shift Analytical evaluation of phase shifts The scattering cross-sections Scattering of identical particles 211 The determination of intermolecular forces from elastic scattering cross-sections Quality of data Interpretation of data having no oscillatory fine structure

4 r jbiblloihek FB7 fhysikol. Chemie / Chem. lethnojegie CONTENTS Techniscfte Kochstfaufe Interpretation of data for which sojme quantur structure is observed Interpretation of fully-resolved scattering data 4.7. Scattering processes in polyatomic systems Inelastic scattering cross-sections Experimental methods The scattering S-matrix: elastic scattering Scattering matrices: the general case Relation of the S-matrix to scattering cross-sections Evaluation of the S-matrix: quantum approach Close-coupling calculations The centrifugal sudden approximation The infinite-order sudden approximation The space-fixed orientation approximation The l z -conserved energy sudden (IOS) approximation A Other approximate methods 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 THE KINETIC THEORY OF NON-UNIFORM DILUTE GASES *5.1. Introduction 266 *5.2. Simple kinetic theory Viscosity Thermal conductivity Diffusion Comparison with exact kinetic theory for hard-sphere molecules Application to real gases 275 t5.3. The rigorous kinetic theory 276 t The Boltzmann equation The Maxwell-Boltzmann equilibrium velocity distribution t5.5. The Chapman-Enskog solution for non-uniform gases The zeroth-order solution The first-order solution Explicit evaluation of the thermal conductivity coefficient 292 t5.6. Transport coefficients for pure gases Formulae for special intermolecular potential models 299 t5.7. Transport coefficients for binary mixtures Formulae for special intermolecular potential models 304 t5.8. Polyatomic gases Quantum mechanical theory The semi-classical theory t The classical theory The transport coefficients of polyatomic gases The semi-classical transport coefficients The Mason-Monchick approximation Polyatomic gas mixtures 315

5 xii CONTENTS Quantum-mechanical effects 316 t5.10. The kinetic theory of dense gases The Enskog theory 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 THE TRANSPORT PROPERTIES OF GASES AND INTERMOLECULAR FORCES 6.1. Introduction The calculation of transport properties from the intermolecular potential Experimental measurements Viscosity Thermal conductivity Binary diffusion coefficient The thermal diffusion factor Extraction of collision integrals from experimental data Principle of corresponding states Traditional methods of testing intermolecular potential functions The direct determination of intermolecular forces: inversion methods The basis of the inversion procedure Polyatomic gases Simple molecular models The semi-classical and classical kinetic theories The Mason and Monchick approximation Spherically-averaged potential The principle of corresponding states Monchick and Mason's treatment of polar gases Recent developments 380 I 7. SPECTROSCOPIC MEASUREMENTS 7.1. Introduction The origins of van der Waals molecules Concentrations 390 f Lifetimes Experimental techniques 391 I Ultraviolet and infrared studies of dimers 391 f Molecular-beam electric resonance: radiofrequency and microwave spectra Other techniques, Analysis of rotation-vibration spectra to give U(r): spherical systems Introduction The Rydberg-Klein-Rees inversion method Determination of dissociation energy (well depth) 400

6 CONTENTS 7.5. Experimental results for inert gas dimers Pure components Mixtures 7.6. Analysis of spectra for non-spherical systems General approach Diatomic-monatomic dimer systems Methods of calculating bound-state energies Weakly anisotropic systems Strongly anisotropic systems 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 Introduction Static lattice properties Lattice vibrations Lattice vibrations: higher order effects Experimental measurements Results for the inert gases More complex molecules 8.4. The liquid state Liquid structure Distribution function theories of liquids Computer simulation Perturbation theories Transport properties of liquids Experimental measurements for simple liquids Liquid-state properties and intermolecular forces Results for argon More complex molecules Liquid mixtures Appendix A8.1. The virial theorem of Clausius 9. INTERMOLECULAR FORCES: THE PRESENT POSITION *9.1. Introduction * * The Lennard-Jones era The basis of recent advances Potential energy functions Accurate potential functions (Class I) * Argon Krypton Helium Neon Xenon Argon-krypton Helium-heavier inert gas interactions

7 CONTENTS 9.6. Class II potential functions Kr-Xe, Ar-Xe, Ne-Xe, Ne-Ar, Ne-Kr Alkali metal-mercury system Class III potential functions Methane f Hydrogen 509 t Other diatomic molecules Inert-gas-diatomic-molecule interactions Benzene 513 J Water *9.9. Class IV potential functions General features of potential energy functions f 9.9.i: Conformality 518 I 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.