Electrostatic Interactions
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For simplicity, electrostatic interactions are usually treated in
molecular mechanics by using a partial atomic charge model.
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A charge is associated with each atom, and will typically be a
fraction of an electronic charge (positive or negative).
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It is important to remember that this simple model may not
reproduce the electrostatic properties of a molecule correctly.
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A complete representation would require charges to be located at
other positions in addition to those on the atoms.
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For example, to reproduce the quadrupole moment of N2
(which is important in determining its interactions), at least 3 charges are
required (a charge of –q at each
nucleus, and +2q at the centre of
mass. Five charges give an even better description.
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However, for modelling large molecules, simple schemes involving
charges on the nuclei alone are typically used.
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It is important to realise that the charges giving the best
description of a molecule are likely to be different for different
conformations of a molecule.
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However, the atomic partial charges in a typical molecular
mechanics potential function are fixed and do not change. They also do not
change in response to changes in the environment, which is another weakness of
the partial atomic charge model.
Partial atomic charges can be calculated
in a number of ways.
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A simple, rapid method for small molecules is Mulliken population
analysis – this is a simple method for partitioning electron density between
atoms, based on calculated molecular orbitals. Mulliken charges are produced
straightforwardly from an ab initio calculation. However, they give a poor
description of a molecule’s electrostatic potential, and are highly dependent
on the basis set used in the calculation. In some instances, they can give a
highly unrealistic picture of the electronic distribution in molecules (e.g.
Leach pgs. 79-80 & 189; Grant & Richards pgs. 27-28, 41-43).
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Better atomic charges can be obtained for small molecules by
fitting them to the electrostatic potential (calculated by ab initio MO
methods, for example – the Hartree-Fock method with the 6-31G(d) basis set
(denoted HF/6-31G(d)) gives good results). The electrostatic potential is
calculated at a series of points around the molecule (e.g. on the surface
defined by the atomic van der Waals radii). Can also require the dipole moment to
be reproduced correctly by the fitted charges. This method can work well,
particularly for small polar molecules. For large molecules, the charges of
atoms in the interior can be poorly determined, leading to artificially high
charges for e.g. buried carbon atoms. Restraints can be applied to keep charges
on some atoms within a reasonable range (e.g. the RESP procedure used in
parameterization of the AMBER force field).
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Alternatively, charges can be found by fitting to interaction
energies calculated at ab initio levels, e.g. the energy of interaction of a
water molecule and another molecule (e.g. formamide) would be calculated at the
HF/6-31G(d) level, and MM charges for the atoms fitted to reproduce the
interaction energy. This is the approach used in the CHARMM force field).
Typically, the charges would be scaled (increased) somewhat to reflect the fact
that molecules in solution will be more polarized than in the gas phase.
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Non-bonded parameters for small molecules can be derived by fitted
to them to experimental results (e.g. for the properties of liquids),
calculating liquid properties by e.g. Monte Carlo or molecular dynamics
simulations.
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In all the above cases, the charges found for a functional group in
a small molecule will usually be treated as being transferable to larger
molecules bearing the same group.
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It is often desirable to calculate atomic charges very quickly,
e.g. to estimate the properties or interactions of very large numbers of novel
small molecules, e.g. in docking them to a protein target. In this situation,
it is not possible to carry out demanding ab initio calculations. Instead,
quick approximate methods are used, e.g. based only on the electronegativity of
atoms in the molecule, and their connectivity.