## Quantum drude oscillators for accurate many-body intermolecular forces

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##### Date

2010##### Author

Jones, Andrew

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One of the important early applications of Quantum Mechanics was to explain the
Van-der-Waal’s 1/R6 potential that is observed experimentally between two neutral
species, such as noble gas atoms, in terms of correlated uncertainty between interacting
dipoles, an effect that does not occur in the classical limit [London-Eisenschitz,1930].
When many-body correlations and higher-multipole interactions are taken into account
they yield additional many-body and higher-multipole dispersion terms.
Dispersion energies are closely related to electrostatic interactions and polarisation
[Hirschfelder-Curtiss-Bird,1954]. Hydrogen bonding, the dominant force in water, is an
example of an electrostatic effect, which is also strongly modified by polarisation effects.
The behaviour of ions is also strongly influenced by polarisation. Where hydrogen
bonding is disrupted, dispersion tends to act as a more constant cohesive force. It
is the only attractive force that exists between hydrophobes, for example. Thus all
three are important for understanding the detailed behaviour of water, and effects
that happen in water, such as the solvation of ions, hydrophobic de-wetting, and thus
biological nano-structures.
Current molecular simulation methods rarely go beyond pair-wise potentials, and
so lose the rich detail of many-body polarisation and dispersion that would permit
a force field to be transferable between different environments. Empirical force-fields
fitted in the gas phase, which is dominated by two-body interactions, generally do not
perform well in the condensed (many-body) phases. The leading omitted dispersion
term is the Axilrod-Teller-Muto 3-body potential, which does not feature in standard
biophysical force-fields. Polarization is also usually ommitted, but it is sometimes
included in next-generation force-fields following seminal work by Cochran [1971]. In
practice, many-body forces are approximated using two-body potentials fitted to reflect
bulk behaviour, but these are not transferable because they do not reproduce detailed
behaviour well, resulting in spurious results near inhomogeneities, such as solvated
hydrophobes and ions, surfaces and interfaces.
The Quantum Drude Oscillator model (QDO) unifies many-body, multipole
polarisation and dispersion, intrinsically treating them on an equal footing, potentially leading to simpler, more accurate, and more transferable force fields when it is applied
in molecular simulations. The Drude Oscillator is simply a model atom wherein a
single pseudoelectron is bound harmonically to a single pseudonucleus, that interacts
via damped coulomb interactions [Drude,1900].
Path Integral [Feynman-Hibbs,1965] Molecular Dynamics (PIMD) can, in principle,
provide an exact treatment for moving molecules at finite temperature on the Born-
Oppenheimer surface due to their pseudo-electrons. PIMD can be applied to large
systems, as it scales like N log(N), with multiplicative prefactor P that can be
effectively parallelized away on modern supercomputers. There are other ways to
treat dispersion, but all are computationally intensive and cannot be applied to large
systems. These include, for example, Density Functional Theory provides an existence
proof that a functional exists to include dispersion, but we dont know the functional.
We outline the existing methods, and then present new density matrices to improve
the discretisation of the path integral.
Diffusion Monte Carlo (DMC), first proposed by Fermi, allows the fast computation
of high-accuracy energies for static nuclear configurations, making it a useful method for
model development, such as fitting repulsion potentials, but there is no straightforward
way to generate forces. We derived new methods and trial wavefunctions for DMC,
allowing the computation of energies for much larger systems to high accuracy.
A Quantum Drude model of Xenon, fit in the gas-phase, was simulated in the
condensed-phase using both DMC and PIMD. The new DMC methods allowed for
calculation of the bulk modulus and lattice constant of FCC-solid Xenon. Both were
in excellent agreement with experiment even though this model was fitted in the gasphase,
demonstrating the power of Quantum Drudes to build transferable models by
capturing many-body effects. We also used the Xenon model to test the new PIMD
methods.
Finally, we present the outline of a new QDO model of water, including QDO
parameters fitted to the polarisabilities and dispersion coefficients of water.