Molecular Interactions
Description
* Contains applications to many physical systems and worked examples
* Proceeds from introductory material to advanced modern treatments
* Has relevance for new materials, biological phenomena, and energy and fuels production
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David A. Micha, PhD, is a Professor of Chemistry and Physics at the University of Florida, presently Adjunct and Emeritus, with continuing research activity. His many research interests include molecular interactions and kinetics, and quantum molecular dynamics involving energy transfer, electron transfer, light emission, reactions in gas phase collisions, and also at solid surfaces. His work has been recognized with awards from the Alfred P. Sloan Foundation and the Dreyfus Foundation, and with an Alexander von Humboldt Senior Scientist Award. Dr. Micha has been the organizer of several Pan-American Workshops and is a co-organizer of the "Sanibel Symposium on Theory and Computation for the Molecular and Materials Sciences" at the University of Florida.
Content
Preface xi
1 Fundamental Concepts 1
1.1 Molecular Interactions in Nature 2
1.2 Potential Energies for Molecular Interactions 4
1.2.1 The Concept of a Molecular Potential Energy 4
1.2.2 Theoretical Classification of Interaction Potentials 6
1.2.2.1 Small Distances 7
1.2.2.2 Intermediate Distances 8
1.2.2.3 Large Distances 8
1.2.2.4 Very Large Distances 8
1.3 Quantal Treatment and Examples of Molecular Interactions 9
1.4 Long-Range Interactions and Electrical Properties of Molecules 21
1.4.1 Electric Dipole of Molecules 21
1.4.2 Electric Polarizabilities of Molecules 22
1.4.3 Interaction Potentials from Multipoles 23
1.5 Thermodynamic Averages and Intermolecular Forces 24
1.5.1 Properties and Free Energies 24
1.5.2 Polarization in Condensed Matter 25
1.5.3 Pair Distributions and Potential of Mean-Force 26
1.6 Molecular Dynamics and Intermolecular Forces 27
1.6.1 Collisional Cross Sections 27
1.6.2 Spectroscopy of van der Waals Complexes and of Condensed Matter 28
1.7 Experimental Determination and Applications of Interaction Potential Energies 29
1.7.1 Thermodynamics Properties 30
1.7.2 Spectroscopy and Diffraction Properties 30
1.7.3 Molecular Beam and Energy Deposition Properties 30
1.7.4 Applications of Intermolecular Forces 31
References 31
2 Molecular Properties 35
2.1 Electric Multipoles of Molecules 35
2.1.1 Potential Energy of a Distribution of Charges 35
2.1.2 Cartesian Multipoles 36
2.1.3 Spherical Multipoles 37
2.1.4 Charge Distributions for an Extended System 38
2.2 Energy of a Molecule in an Electric Field 40
2.2.1 Quantal Perturbation Treatment 40
2.2.2 Static Polarizabilities 41
2.3 Dynamical Polarizabilities 43
2.3.1 General Perturbation 43
2.3.2 Periodic Perturbation Field 47
2.4 Susceptibility of an Extended Molecule 49
2.5 Changes of Reference Frame 52
2.6 Multipole Integrals from Symmetry 54
2.7 Approximations and Bounds for Polarizabilities 57
2.7.1 Physical Models 57
2.7.2 Closure Approximation and Sum Rules 58
2.7.3 Upper and Lower Bounds 59
References 60
3 Quantitative Treatment of Intermolecular Forces 63
3.1 Long Range Interaction Energies from Perturbation Theory 64
3.1.1 Interactions in the Ground Electronic States 64
3.1.2 Interactions in Excited Electronic States and in Resonance 68
3.2 Long Range Interaction Energies from Permanent and Induced Multipoles 68
3.2.1 Molecular Electrostatic Potentials 68
3.2.2 The Interaction Potential Energy at Large Distances 70
3.2.3 Electrostatic, Induction, and Dispersion Forces 73
3.2.4 Interacting Atoms and Molecules from Spherical Components of Multipoles 75
3.2.5 Interactions from Charge Densities and their Fourier Components 76
3.3 Atom-Atom, Atom-Molecule, and Molecule-Molecule Long-Range Interactions 78
3.3.1 Example of Li++Ne 78
3.3.2 Interaction of Oriented Molecular Multipoles 79
3.3.3 Example of Li++HF 80
3.4 Calculation of Dispersion Energies 81
3.4.1 Dispersion Energies from Molecular Polarizabilities 81
3.4.2 Combination Rules 82
3.4.3 Upper and Lower Bounds 83
3.4.4 Variational Calculation of Perturbation Terms 86
3.5 Electron Exchange and Penetration Effects at Reduced Distances 87
3.5.1 Quantitative Treatment with Electronic Density Functionals 87
3.5.2 Electronic Rearrangement and Polarization 93
3.5.3 Treatments of Electronic Exchange and Charge Transfer 98
3.6 Spin-orbit Couplings and Retardation Effects 102
3.7 Interactions in Three-Body and Many-Body Systems 103
3.7.1 Three-Body Systems 103
3.7.2 Many-Body Systems 106
References 107
4 Model Potential Functions 111
4.1 Many-Atom Structures 111
4.2 Atom-Atom Potentials 114
4.2.1 Standard Models and Their Relations 114
4.2.2 Combination Rules 116
4.2.3 Very Short-Range Potentials 117
4.2.4 Local Parametrization of Potentials 117
4.3 Atom-Molecule and Molecule-Molecule Potentials 119
4.3.1 Dependences on Orientation Angles 119
4.3.2 Potentials as Functionals of Variable Parameters 124
4.3.3 Hydrogen Bonding 124
4.3.4 Systems with Additive Anisotropic Pair-Interactions 125
4.3.5 Bond Rearrangements 125
4.4 Interactions in Extended (Many-Atom) Systems 127
4.4.1 Interaction Energies in Crystals 127
4.4.2 Interaction Energies in Liquids 131
4.5 Interaction Energies in a Liquid Solution and in Physisorption 135
4.5.1 Potential Energy of a Solute in a Liquid Solution 135
4.5.2 Potential Energies of Atoms and Molecules Adsorbed at Solid Surfaces 139
4.6 Interaction Energies in Large Molecules and in Chemisorption 143
4.6.1 Interaction Energies Among Molecular Fragments 143
4.6.2 Potential Energy Surfaces and Force Fields in Large Molecules 145
4.6.3 Potential Energy Functions of Global Variables Parametrized with Machine Learning Procedures 148
References 152
5 Intermolecular States 157
5.1 Molecular Energies for Fixed Nuclear Positions 158
5.1.1 Reference Frames 158
5.1.2 Energy Density Functionals for Fixed Nuclei 160
5.1.3 Physical Contributions to the Energy Density Functional 162
5.2 General Properties of Potentials 163
5.2.1 The Electrostatic Force Theorem 163
5.2.2 Electrostatic Forces from Approximate Wavefunctions 164
5.2.3 The Example of Hydrogenic Molecules 165
5.2.4 The Virial Theorem 166
5.2.5 Integral Form of the Virial Theorem 168
5.3 Molecular States for Moving Nuclei 169
5.3.1 Expansion in an Electronic Basis Set 169
5.3.2 Matrix Equations for Nuclear Amplitudes in Electronic States 170
5.3.3 The Flux Function and Conservation of Probability 172
5.4 Electronic Representations 172
5.4.1 The Adiabatic Representation 172
5.4.2 Hamiltonian and Momentum Couplings from Approximate Adiabatic Wavefunctions 173
5.4.3 Nonadiabatic Representations 174
5.4.4 The Two-state Case 175
5.4.5 The Fixed-nuclei, Adiabatic, and Condon Approximations 176
5.5 Electronic Rearrangement for Changing Conformations 180
5.5.1 Construction of Molecular Electronic States from Atomic States: Multistate Cases 180
5.5.2 The Noncrossing Rule 181
5.5.3 Crossings in Several Dimensions: Conical Intersections and Seams 184
5.5.4 The Geometrical Phase and Generalizations 189
References 192
6 Many-Electron Treatments 195
6.1 Many-Electron States 195
6.1.1 Electronic Exchange and Charge Transfer 195
6.1.2 Many-Electron Descriptions and Limitations 198
6.1.3 Properties and Electronic Density Matrices 203
6.1.4 Orbital Basis Sets 205
6.2 Supermolecule Methods 209
6.2.1 The Configuration Interaction Procedure for Molecular Potential Energies 209
6.2.2 Perturbation Expansions 215
6.2.3 Coupled-Cluster Expansions 218
6.3 Many-Atom Methods 222
6.3.1 The Generalized Valence-Bond Method 222
6.3.2 Symmetry-Adapted Perturbation Theory 225
6.4 The Density Functional Approach to Intermolecular Forces 228
6.4.1 Functionals for Interacting Closed- and Open-Shell Molecules 228
6.4.2 Electronic Exchange and Correlation from the Adiabatic-Connection Relation 232
6.4.3 Issues with DFT, and the Alternative Optimized Effective Potential Approach 238
6.5 Spin-Orbit Couplings and Relativistic Effects in Molecular Interactions 243
6.5.1 Spin-Orbit Couplings 243
6.5.2 Spin-Orbit Effects on Interaction Energies 245
References 247
7 Interactions Between Two Many-Atom Systems 255
7.1 Long-range Interactions of Large Molecules 255
7.1.1 Interactions from Charge Density Operators 255
7.1.2 Electrostatic, Induction, and Dispersion Interactions 258
7.1.3 Population Analyses of Charge and Polarization Densities 260
7.1.4 Long-range Interactions from Dynamical Susceptibilities 262
7.2 Energetics of a Large Molecule in a Medium 265
7.2.1 Solute-Solvent Interactions 265
7.2.2 Solvation Energetics for Short Solute-Solvent Distances 268
7.2.3 Embedding of a Molecular Fragment and the QM/MM Treatment 270
7.3 Energies from Partitioned Charge Densities 272
7.3.1 Partitioning of Electronic Densities 272
7.3.2 Expansions of Electronic Density Operators 274
7.3.3 Expansion in a Basis Set of Localized Functions 277
7.3.4 Expansion in a Basis Set of Plane Waves 279
7.4 Models of Hydrocarbon Chains and of Excited Dielectrics 281
7.4.1 Two Interacting Saturated Hydrocarbon Compounds: Chains and Cyclic Structures 281
7.4.2 Two Interacting Conjugated Hydrocarbon Chains 284
7.4.3 Electronic Excitations in Condensed Matter 289
7.5 Density Functional Treatments for All Ranges 291
7.5.1 Dispersion-Corrected Density Functional Treatments 291
7.5.2 Long-range Interactions from Nonlocal Functionals 294
7.5.3 Embedding of Atomic Groups with DFT 297
7.6 Artificial Intelligence Learning Methods for Many-Atom Interaction Energies 300
References 303
8 Interaction of Molecules with Surfaces 309
8.1 Interaction of a Molecule with a Solid Surface 309
8.1.1 Interaction Potential Energies at Surfaces 309
8.1.2 Electronic States at Surfaces 314
8.1.3 Electronic Susceptibilities at Surfaces 319
8.1.4 Electronic Susceptibilities for Metals and Semiconductors 321
8.2 Interactions with a Dielectric Surface 324
8.2.1 Long-range Interactions 324
8.2.2 Short and Intermediate Ranges 329
8.3 Continuum Models 332
8.3.1 Summations Over Lattice Cell Units 332
8.3.2 Surface Electric Dipole Layers 333
8.3.3 Adsorbate Monolayers 335
8.4 Nonbonding Interactions at a Metal Surface 337
8.4.1 Electronic Energies for Varying Molecule-Surface Distances 337
8.4.2 Potential Energy Functions and Physisorption Energies 341
8.4.3 Embedding Models for Physisorption 347
8.5 Chemisorption 349
8.5.1 Models of Chemisorption 349
8.5.2 Charge Transfer at a Metal Surface 354
8.5.3 Dissociation and Reactions at a Metal Surface from Density Functionals 359
8.6 Interactions with Biomolecular Surfaces 363
References 367
Index 373
1
Fundamental Concepts
CONTENTS
- 1.1 Molecular Interactions in Nature
- 1.2 Potential Energies for Molecular Interactions
- 1.3 Quantal Treatment and Examples of Molecular Interactions
- 1.4 Long-Range Interactions and Electrical Properties of Molecules
- 1.5 Thermodynamic Averages and Intermolecular Forces
- 1.6 Molecular Dynamics and Intermolecular Forces
- 1.7 Experimental Determination and Applications of Interaction Potential Energies
- References
1.1 Molecular Interactions in Nature
Many materials and the components of living organisms in nature are made up of aggregates of atomic and molecular units. Fluids, molecular and atomic solids, polymers, and proteins are examples. The properties of these objects can be described from first principles of quantum mechanics and statistical mechanics, after their composition in terms of electrons and nuclei have been specified. This requires a theoretical framework to describe the structure of atoms and molecules and the way the atoms and molecules interact. In some cases, the interactions affect but do not change the conformations of molecules, which can then be taken as the basic building units of the objects being studied.
Once the interaction forces in a molecular system are known, the equations of motion of classical or quantum mechanics, implemented with physical boundary conditions, can be used to derive thermodynamic equilibrium and nonequilibrium properties from first principles. The response of the system to external factors, such as light and interacting species, can be described by their electromagnetic and chemical reactivity properties. The reverse is also true. Information about molecular interaction energies can be derived from measurements of thermodynamic, kinetic, and electromagnetic properties of matter, and from experiments specially devised to extract interaction energies, such as crossed molecular beam and photodissociation experiments.
Early treatments that proceed from atomic and molecular structure to the calculation of intermolecular forces and properties of molecular systems have been covered in several books going back over 50 years [1-4] and have been expanded to incorporate results of more extensive calculations [5-13] made possible by continuous improvements in computational power. The present work introduces and updates theoretical concepts and methods needed to model structures and properties and provides links to the more recent computational developments.
A chronological Table 1.1 follows with some important early discoveries on molecular interactions. They start with the acceptance of the existence of molecules and understanding of how they interact, as early as the nineteenth century. The advent of quantum theory and its applications during the first half of the twentieth century has then lead to many of the quantitative concepts about molecular structure and properties and how to calculate them, as well as ways to describe their interaction energies. Statistical mechanics has also provided thermodynamical and kinetics values of properties to be compared with experimental measurements.
Table 1.1 Chronology of some early discoveries on molecular interactions.
Year Authors Subject 1857 R. Clausius Distance dependence of interaction potentials 1868 J.C. Maxwell Simple molecular transport theory 1872 L. Boltzmann Transport theory for fluids 1873 J.D. van der Waals Equation of state for real gases 1905 P. Langevin Ion-molecule interaction potential 1912 P. Debye Dielectric properties of fluids 1924 J.E. Lennard-Jones Analytic interaction potentials 1925 J. Franck and E.U. Condon Molecular photoexcitations 1927 W. Heitler and F. London Chemical bonding in H2 1927 M. Born and J.D. Oppenheimer Quantum theory of molecules 1930 F. London Quantal calculation of dispersion forces 1931 J.C. Slater and J.G. Kirkwood Variational calculation of dispersion forces 1932 J.H. Van Vleck Electric and magnetic susceptibilities of molecules 1933 P.K.L. Drude Optical properties of fluids 1939 R. Feynman Hellmann-Feynman or force theorem 1943 B.M. Axilrod and E. Teller Three-atom interaction potentials ? ? ?The link between electrons and nuclei and potential energy functions related to molecular structure is provided by quantum chemistry, and thermodynamical properties follow by using statistical mechanics. Steady-state (or transport) properties as well as nonequilibrium properties, including reactivity, can be obtained from molecular dynamics, as shown in the block diagram illustrated in Figure 1.1. Spectroscopic properties follow from the electrodynamics of molecular systems. Transport in gases follow from molecular collision cross sections.
Figure 1.1 Properties derived from interacting electrons, nuclei, and photons.
1.2 Potential Energies for Molecular Interactions
1.2.1 The Concept of a Molecular Potential Energy
The large difference between electron and nuclear masses leads to a qualitative difference between electronic and nuclear motions. Nuclei move slowly compared to electronic motions and this allows the formation of molecules, with positively charged nuclei held together by the negative electron distributions. This leads to stable molecular systems when total energies are not too high compared to electronic binding energies. We consider electrons bound by Coulomb forces inside a finite region of space, and separately deal with (i) bound nuclei, where all particles are restricted by their own Coulomb forces to a finite region of space, and (ii) unbound nuclei with attached electrons, where some of the particles (atoms, molecules, etc.) are involved in a collision event.
For molecules occupying a finite region of space, electrons and nuclei move under their own Coulomb forces. Since the magnitude of electron and nuclear charges are comparable, their Coulomb forces are similar so that Fe ~ Fn and in terms of masses and accelerations, me?ve/?t ~ mn?vn/?t. In the time interval ?t one therefore finds comparable momentum changes, me?ve ~ mn?vn. Provided electronic and nuclear momenta are comparable to begin with and a system is observed over short times, one finds that over time pe = meve ~ pn = mnvn. Hence, since mn > 2000 me, one finds that velocities satisfy
and to a first approximation nuclei can be assumed to be at rest while electrons move around them. Fixing the nuclear positions, the molecular energies become functions of the nuclear coordinates and provide the potential energies for the nuclear motions; hence, they can be referred to as molecular potential energies. This is the Born-Oppenheimer picture of molecular structure [14, 15]. These authors showed, using quantum mechanical perturbation theory, that for bound molecular states, the potential energy correction due to nuclear motion goes as (me/mn)1/2, while the correction to molecular...
