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Fundamental Concepts
CONTENTS
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...
