2
r  (Å)
0
1
2
3 β V 12
SR
(r)
2
4
6
8
g 12
MC
(r)
g 12
MD
(r)
4 6 8
0
1
2
3 β V 11
SR
(r)
4 6 8
0
1
2
3β V 22
SR
(r)
(a)
(b)
(d)
(c)
FIG. 1: Eﬀective McMillan-Mayer short-range pair potentia ls
extracted from explicit solvent simulations using the HNC
closure. (a) Cation anion, (b) cation cation, (c) anion anion,
(d) cation anion RDF obtained from explicit solvent MD and
implicit solvent MC simulations.
pute all ion thermodynamic properties through implicit
solvent MC simulations.
The second stage of our coarse-graining procedure con-
sists in applying LPT, in order to deduce the best ana-
lytical model of electrolyte solutions which reproduces
this molecular description. The principle of LPT is to
describe the properties of a given system in terms of
those of a well known reference system, with the diﬀer-
ence between them treated as a perturbation in the ref-
erence potential. Assuming pairwise additive potentials,
Vij = V (0)
ij + ∆V ij , a ﬁrst-order truncated expression for
the free energy density of the system βfv is obtained,
βfv ≲ βf (0)
v + 1
2β
∑
i,j
ρiρj
∫
dr g(0)
ij (r)∆V ij (r) (1)
which depends only on the free-energy density f(0)
v and
RDF g(0) of the reference ﬂuid, with β = ( kBT )− 1 and
ρi the concentration of species i. The Gibbs-Bogoliubov
inequality [15] ensures that the right-hand side of Eq. ( 1)
is actually a strict upper bound. Once a reference system
has been chosen, the expression on the right-hand side of
Eq. (
1) must be minimized with respect to the parameters
deﬁning the reference. This procedure yields the best
ﬁrst-order approximation to the free energy of the system
under consideration.
For a system of charged particles in solution, the nat-
ural reference is the PM, deﬁned in terms of the charge
and diameter (σi) of each species. In this case, the per-
turbing potentials are just the short-range eﬀective po-
tentials computed above (∆Vij = V SR
ij ). We use the
MSA [3] solution to the PM, since it provides analyti-
cal expressions for both the free energy and the RDF.
The perturbation term is evaluated using an exponential
approximation to the RDF obtained within the MSA,
g(r) = exp [ gMSA(r) − 1], which removes any unphysical
negative regions and improves the comparison with HNC
calculations.
0.9
1
1.1
1.2
1.3Φ
MC
MSA2
DHLL
Exp
0 0.5 1 1.5
c
1/2
  (mol.L
-1
)
1/2
3
4
5σ   (Å)
σ 1 (MSA-fit)
σ 2 (MSA-fit)
σ 1 (MSA2)
σ 2 (MSA2)
(a)
(b)
FIG. 2: (Color online) (a) Osmotic coeﬃcient Φ in the
McMillan-Mayer frame of reference. (diamond) MC simula-
tions, (dot dashed) MSA2, (dot) Debye H¨ uckel Limiting law
(DHLL), (cross) experiments (Ref. [18] with the McMillan-
Mayer to Lewis Randall conversion). (b) Minimization diam-
eters. (dot dashed) MSA2 and (diamond) MSA-ﬁt.
We ﬁrst used LPT for a two-component system (Na +
and Cl − free ions) within the MSA (model MSA2), for
concentrations ranging from 0.1 to 2 . 0 mol l− 1. The mini-
mization leads to almost constant diameters on the whole
range of concentration:σ1 = 3 . 67 ˚
A and σ2 = 4 . 78 ˚
A.
As shown in Fig.
2, these parameters yield osmotic co-
eﬃcients close to MC calculations only at very low con-
centration, i.e.,c ≤ 0. 1 mol l− 1 (experimental values are
given for indicative purposes only, since a perfect model
will exactly match the MC results). For molar solutions,
the LPT results diﬀer considerably from MC calculations.
This discrepancy can easily be understood by comparing
the diameters found within the MSA2 calculation with
the eﬀective potentials given in Fig.
1. The anion/cation
contact distance obtained within the MSA2 calculation
is 4. 2 ˚
A, which is in the region of the second minimum of
the eﬀective potential and corresponds to the situation
where there is a single layer of water molecules between
the ions. The ﬁrst minimum of the potential, which cor-
responds to the contact ion pair (CIP) is thus completely
ignored by the MSA2 calculation. If the MSA diameters
are directly ﬁtted to reproduce the MC osmotic pres-