Fluctuation Solution Theory Properties from Molecular Simulation - Fluctuation Theory of Solutions - page 140

6.3.3 d isTance s hiFTing
Distance shifting is another method employed with data on systems resembling
real molecules. Both Perera and Sokolić (2004) and Hess and Van der Vegt (2009)
attempt to correct the RDFs obtained from simulation by rescaling according to

() 0

gr () =α () rg () r (6.14)

ij ij ij

Here α ij is chosen in order to enforce that g ij ( r ) approaches unity at long distances.
Perera and Sokolić (2004) presented NpT simulations of the water + acetone binary
mixture. Although a correction of the order 1/ N is required, the result g ij ( r ) → 1 (for
r → ∞) is often valid for simple fluids after a few molecular diameters. This can be
realistic in simulation boxes with a few hundred particles. However, for systems
characterized by microscopic aggregation, the RDFs decay in irregular fashion with
the range of correlations in the RDF differing from that of pair interactions, even at
conditions remote from a critical point. The apparent problem in evaluating these
quantities is the upper bound of the integral relative to the range of the correlations
described by the simulations. If the system is large enough, one may consider that
the correct asymptotic behavior is attained at some cutoff R lim smaller than the half-
box length L box /2. Accordingly, the KBI can be computed by replacing the infinite
upper bound by R lim , as in truncation methods. Though this may be satisfactory for
simple fluids, it is probably incorrect for fluids with long-range correlations. If the
upper bound is not large enough to capture the correct asymptotic behavior, it will
lead to incorrect estimation of the KBIs. For water/solute systems, these correlations
seem to extend for more than five to six water diameters, which is too large even for
a system with N = 1024.
Since the L box /2 values of all partial g ij ( L box /2) are always close to unity, Perera
and Sokolić (2004) restored the correct asymptotic value at the natural half-box cut-
off by shifting the value to unity. The expression used is

1

() 0

gr () = α () rg (), r α ( ) r = (6.15)

( )

ij ij ij ij

() ( 0

1 + rL 2 g L 21 ) −

box i j box

where g ij (0) ( r ) is the uncorrected RDF. This procedure leaves values of the RDF at
contact nearly unchanged if g ij (0) ( L box /2) is close to unity. Perera and Sokolić (2004)
found that N = 864 is just enough to satisfy this condition, though N = 2048 is much
better. The merit of this equation is the use of all r < L box /2 values in the calculation
of the KBI, to avoid artifacts in evaluations of the canonical ensemble KBIs.
Hess and Van der Vegt (2009) studied cation-binding affinity with carboxylate
ions. They computed the excess coordination numbers, N ij , defined in Section 1.1.5
in Chapter 1, for water ( w ) or cations ( c ) about cations,

∞

∫

2

N = ρπ 4 (()) g r − 1 r dr (6.16)

jc c jc

0