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Unformatted text preview: © M. S. Shell 2009 1/23 last modified 10/4/2010 Classical semiempirical force fields ChE210D Today's lecture: approximate descriptions of interatomic interactions suitable for descriptions of many organic, inorganic, and other nonreacting systems. The classical picture The classical approximation The extensive computational demands of electronic structure calculations mean that their application to even modestsized molecular systems is quite limited. Fortunately, to good approximation, we don’t need to solve the Schrodinger equation for many systems to accurate ly reproduce their properties. Instead, we can use a classical description, which ignores the motions of the electrons and describes the timeevolution of the nuclear positions alone. A classical approach uses a force field or classical potential energy function that is an approximation to the quantum ground state potential energy surface due to electronic structure and internuclear interactions, as a function of the positions of the nuclei. Classical descriptions work very well under the following conditions: • the BohrOppenheimer approximation is valid • the electronic structure is not of interest • the temperature is modest (not too low) • there is no bond breaking or forming • electrons are highly localized (metals and pibonded systems are delocalized) Basic features In the classical approximation, we describe a system by the positions and momenta of all of the atomic nuclei: g G ¡ ¢£ ¤ ,¥ ¤ ,¦ ¤ ,£ § , …, ¥ G ,¦ G ¨ © G ¡ ª« ¬,¤ ,« ,¤ ,« ®,¤ ,« ¬,§ ,…,« ,G ,« ®,G ¯ Even though technically we deal with nuclei, we can think of the fundamental particle as an atom. Unlike quantum uncertainty, each atom has a definite position g and momentum © . © M. S. Shell 2009 2/23 last modified 10/4/2010 Alternatively, we could consider the velocity g instead of the momentum, since the two simply differ by a constant mass factor. We will actually use the momentum more often, since there are several reasons why it is a more natural in statistical mechanics. A microstate is just one “configuration” of the system. In a classical system, one microstate is characterized by a list of the 3G positions ¡ ¢ and 3G momenta £ ¤ , for a total of 6G pieces of information. For a microstate ¥ we might use the notation ¦£ § ¤ ,¡ § ¤ ¨ to indicate specific values of these variables. For any microstate, we can calculate the total, potential, and kinetic energies. The potential energy function depends on the positions ©¦¡ ¤ ¨ and the kinetic energy function depends on the momenta ª¦£ ¤ ¨ . The Hamiltonian of a classical system is the function that gives the energy of a microstate: «¦£ ¤ ,¡ ¤ ¨ ¬ ª¦£ ¤ ¨ ©¦¡ ¤ ¨ The kinetic energy term simply relates to the atomic momenta: ª¦£ ¤ ¨ ¬ ® £ ¯  ° 2¥ ¯ ¯ Interactions between atoms are described by a potential energy function that depends on the positions but not the momenta of all of the atoms,...
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 Fall '09
 SHELL
 Atom, Energy, Potential Energy, M. S. Shell

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