Classical_force_fields

Classical_force_fields - © M S Shell 2009 1/23 last...

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Unformatted text preview: © M. S. Shell 2009 1/23 last modified 10/4/2010 Classical semi-empirical force fields ChE210D Today's lecture: approximate descriptions of interatomic interactions suitable for descriptions of many organic, inorganic, and other non-reacting systems. The classical picture The classical approximation The extensive computational demands of electronic structure calculations mean that their application to even modest-sized 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 time-evolution 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 Bohr-Oppenheimer 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 pi-bonded 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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Classical_force_fields - © M S Shell 2009 1/23 last...

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