THE JOURNAL OF CHEMICAL PHYSICS 132, 114510 共2010兲
Dissociation of NaCl in water from ab initio molecular dynamics
simulations
Jeff Timko, Denis Bucher,a兲 and Serdar Kuyucakb兲
School of Physics, University of Sydney, New South Wales 2006, Australia
共Received 19 December 2009; accepted 22 February 2010; published online 17 March 2010兲
We perform ab initio molecular dynamics simulations to study the dissociation of NaCl in water.
The potential of mean force 共PMF兲 between the two ions is determined using the constrained-force
method. The simulation windows corresponding to the contact and solvent-separated minima, and
the transition state in between, are further analyzed to determine the changes in the properties of
hydration waters such as coordination number, dipole moment, and orientation. The ab initio results
are compared with those obtained from classical molecular dynamics simulations of aqueous NaCl
using several common force fields. The ab initio PMF is found to have a shallower contact
minimum and a smaller transition barrier compared with the classical ones. Also the binding free
energy calculated from the ab initio PMF almost vanishes whereas it is negative for all the classical
PMFs. Water dipole moments are observed to exhibit little change during dissociation, indicating
that description of NaCl with a nonpolarizable force field may be feasible. However,
overcoordination of the ion pair at all distances remains as a serious shortcoming of the current
classical models. The ab initio results presented here provide useful guidance for alternative
parametrizations of the nonpolarizable force fields as well as the polarizable ones currently under
construction. © 2010 American Institute of Physics. 关doi:10.1063/1.3360310兴
I. INTRODUCTION
Electrolyte solutions have been traditionally described
using continuum theories as exemplified by those of Debye,
Hückel, and Onsager.1–3 In recent years, however, there has
been a steady shift toward microscopic theories urged by the
more exacting demands from biochemistry and molecular
biology. An accurate description of biomolecules requires
atomic-level modeling using classical molecular dynamics
共MD兲 simulations.4,5 Because water and ions are an integral
part of a biomolecular simulation system—and in some cases
directly involved in the function of a protein, e.g., ion
channels—their accurate representation is also an important
issue in MD simulations. The force fields currently used for
description of water and ions typically consist of Coulomb
and short-range Lennard-Jones 共LJ兲 6-12 interactions. For
computational expediency, the polarization interaction has
been left out, and its effect has been included in an average
way by increasing the partial charges on atoms. This prescription is used in all major force fields such as AMBER,6
CHARMM,7 and GROMACS.8
While the nonpolarizable force fields have been quite
successful in description of biomolecules, closer scrutiny by
comparing the MD results with ab initio calculations9–12 or
detailed experiments13–18 has revealed some discrepancies
that may require inclusion of the polarization interaction for
resolution. Ab initio MD simulations provide the most direct
tool for studying the role of polarization in biomolecular
systems. Recent developments in parallel computing have
a兲
Present address: Department of Chemistry and Biochemistry, University of
California, San Diego, La Jolla, CA 92093-0365.
b兲
Electronic mail: [email protected].
0021-9606/2010/132共11兲/114510/8/$30.00
made such investigations distinctly feasible. For example,
the ab initio methods have been used in description of enzyme reactions,19–23 computation of pKa values,24,25 and in
guiding and testing of the classical force fields.26–29
The question of the accuracy of the classical force fields
for electrolyte solutions has been raised in some recent MD
studies, where formation of NaCl and KCl clusters was observed in long MD simulations.30–34 While this behavior appears to be specific to the AMBER force field, it has nevertheless started a debate on how to construct more reliable
force fields for ions.35–37 Force field optimization typically
involves fitting the two LJ parameters for an ion to the experimental solvation free energy and the ion-water radial distribution function 共RDF兲. Not surprisingly all force fields
reproduce these quantities very well despite a considerable
variation among the parameter values. Where the force fields
differ most are the ion-ion RDFs for which there are no
experimental data.38 Thus the ab initio computation of the
potential of mean force 共PMF兲 between two ions would provide additional constraints in optimizing the force fields. An
advantage of using the ab initio results as a guide in optimizing the force fields is that it shifts the focus from global to
microscopic solvation properties. This will be especially useful in constraining the polarizable force fields currently under construction28,39 because it enables a detailed comparison
of the polarization properties of water in the solvation shell
of ions.
Here we present the results of an ab initio computation
for the dissociation of aqueous NaCl obtained from the Car–
Parrinello MD 共CPMD兲 simulations.40 We use the
constrained-MD method41 to determine the PMF between the
Na+ and Cl− ions from the contact-ion pair 共CIP兲 position to
132, 114510-1
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114510-2
J. Chem. Phys. 132, 114510 共2010兲
Timko, Bucher, and Kuyucak
the solvent-separated-ion pair 共SSIP兲 position and beyond.
The ab initio PMF is compared with the classical ones obtained using several common force fields. To shed further
light on the differences between the ab initio and classical
PMF results, we analyze the solvation structures at the CIP
and SSIP positions and the transition state 共TS兲 in between
them. The coordination number, dipole moment, and orientation of water molecules in the solvation shell are calculated
at those window positions and compared with the similar
quantities obtained using the CHARMM force field.
As far as we are aware, this is the first ab initio calculation of the PMF for the dissociation of the Na+ – Cl− ions in
solution 共or any other ion pair兲. In a previous CPMD simulation of the aqueous NaCl system, the ions were separated
by 8 Å and their effect on the hydrogen bond network was
investigated via the water dipole moments.42 At that distance,
the ion-ion PMF vanishes and they essentially move independent of each other.
W共r兲 = −
冕
r
F̄共r⬘兲dr⬘ + 2kBT ln r + C,
共3兲
r0
where r0 is a reference point for the PMF and C is a constant
of integration chosen such that W共r兲 vanishes at large ion-ion
separations.
The PMF is useful for detailed comparisons of the
ab initio and classical results but it does not tell whether
the ion pair can bind or not. For that purpose one needs the
binding free energy of the ion pair, which can be calculated
from the volume integral of the PMF
冋
Gb = − kBT ln
4␲
V
冕 冉
rc
exp
0
冊 册
− W共r兲 2
r dr ,
k BT
共4兲
where V = 4␲r3c / 3 is the volume occupied by the ion pair. A
negative Gb value would signal ion clustering problems in
MD simulations—the larger the Gb is 共in absolute value兲, the
faster the clustering is likely to occur.
II. COMPUTATIONAL METHODS
B. Ab initio simulations
A. PMF calculations
If simulation time is not a concern, the most straightforward way to calculate the PMF between two ions in a salt
solution is to perform a sufficiently long simulation of the
system and determine the RDF g共r兲 of the two ions from
the trajectory data. The PMF W共r兲 is then obtained from the
RDF using
W共r兲 = − kBT ln g共r兲,
共1兲
where kB is the Boltzmann constant, T is the temperature,
and r is the ion-ion separation which is the reaction coordinate. We use this method to determine the Na+ – Cl− PMF in
classical MD simulations.
For ab initio simulations, the above method is clearly not
feasible at present, and a method with biasing is required to
enable a faster sampling of the system along the reaction
coordinate. The two main methods available for this purpose
are constrained MD 共Ref. 41兲 and umbrella sampling.43 Test
calculations performed for the PMF of methane-methane association indicate that force averaging with constraint biasing has better convergence properties compared with configuration sampling with an umbrella potential.44 Indeed the
former method has been employed in a recent ab initio calculation of the PMF between a methane pair in the framework of the CPMD simulations.10 Therefore, we also choose
the constrained-MD method in the present ab initio PMF
calculations. Here the ion-ion distance is fixed at certain
separations r, and the average effective force F̄共r兲 is determined from the time average of the constraint force sampled
during the CPMD simulations. Fixing the ion-ion distance
reduces their entropy, which is equivalent to an entropic
force of magnitude −2kBT / r. This volume-entropy force
needs to be taken into account when relating the average
effective force to the PMF10
−
2kBT
dW共r兲
= F̄共r兲 −
.
dr
r
Integration of Eq. 共2兲 yields for the PMF
共2兲
The ab initio MD simulations of the NaCl solution are
performed using the CPMD code.45 Here the electronic
structure is determined from the density functional theory
共DFT兲 by solving the Kohn–Sham equations,46 and the nuclei propagate in this ab initio potential according to Newton’s equation of motion. Only the valence electrons of each
atomic species are included in the DFT calculations while
the interaction of the core with the valence electrons is described by pseudopotentials. We use norm-conserving
Goedecker-type psuedopotentials47 for Na and Cl and
Troullier–Martins48 pseudopotentials for O. For H,
a von Barth–Car analytical pseudopotential49 is employed.
The 2s and 2p semicore states of Na and 3s and 3p states of
Cl are treated explicitly as valence states. For the exchangecorrelation functional in DFT we choose BLYP,50,51 which—
despite a slight overstructuring in the oxygen-oxygen RDF—
remains as one of the best choices for ab initio simulations of
aqueous systems.52–56 The valence electrons in the Kohn–
Sham equations are expanded in a plane-wave basis set with
an energy cutoff of 80 Ry. The CPMD equations are integrated using the Verlet algorithm with a time step of 4 a.u.
共⬃0.1 fs兲 and using a fictitious mass of 400 a.u. for the
electrons. Periodic boundary conditions are applied using the
simple cubic symmetry for the cell images. The temperature
is maintained at 300 K using a Nosé–Hoover thermostat
chain for the ions with a frequency of 1200 cm−1.57,58
The simulation system is constructed using a standard
cubic box of size 12.5 Å containing 64 water molecules,
which has a density of ⬃1 g / cm3. A pair of water molecules
near the center of the box is replaced with a Na+ – Cl− ion
pair, resulting in an ⬃0.9 M salt solution. Initially the ionion separation is fixed at r = 2.7 Å, corresponding to the CIP
position. This system is subjected to energy minimization
and then equilibrated for 100 ps using classical MD simulations. The resulting atomic coordinates are used as a starting
configuration for the ab initio calculations. The system is
first quenched to its Born–Oppenheimer energy surface, followed by CPMD simulations where the ion-ion separation is
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J. Chem. Phys. 132, 114510 共2010兲
Dissociation of NaCl in water
increased or decreased at a constant rate to create windows
for sampling of the constraint force. Because the force rapidly increases when the ions are pushed together, the window
positions are taken at 0.1 Å steps from r = 2.7 to 2.2 Å. The
force is better behaved when the ions are pulled apart, hence
the window positions are taken at 0.2 Å steps between 2.7
and 5.7 Å. At 4.7 Å, which corresponds to the SSIP position,
the system is again quenched to its Born–Oppenheimer energy surface before continuing with the pulling of the ions.
In this way, a total of 21 windows is generated for force
sampling. The maximum pulling distance is limited to about
half of the box size to avoid potential artifacts due to the
image ions getting too close to the sampled ions 共this is
justified below兲. For each window, the CPMD simulations
are carried out for 6 ps while keeping the reaction coordinate
r fixed. This simulation time was found to be sufficient for
the convergence of the PMF in the ab initio calculations for
a methane pair.10 The constraint force required to fix the
distance between the ions is recorded at each time step, and
the effective force acting on the ions at the window position
is determined from the average of these data. Finally the
resulting average effective force is integrated according to
Eq. 共3兲 to determine the ab initio Na+ – Cl− PMF.
C. Classical simulations
For comparison purposes we have also calculated the
Na+ – Cl− PMF for five sets of classical force field parameters, including those used in the current versions of the
CHARMM 共version 27兲, AMBER 共version 9兲, and
GROMACS 共version 3兲 force fields as well as the parameters
determined by Smith and Dang59 and Joung and Cheatham.35
We note that in the recently released version 10 of AMBER,
the parameters of Joung and Cheatham35 have been adapted.
All the AMBER results here are obtained using version 9,
which has led to the clustering problems mentioned in the
Sec. I. The TIP3P model60 is used for water molecules in all
cases except in GROMACS which uses the SPC model.61
Classical MD simulations are performed using version 2.6 of
the NAMD code.62 The simulation system consists of 3780
water molecules and 71 Na+ – Cl− ion pairs in a cubic box of
side length of 50 Å, which corresponds to a 1 M salt solution. An NpT ensemble is used with periodic boundary conditions. The temperature is maintained at 300 K through
Langevin damping with a coefficient of 5 ps−1. Similarly the
pressure is kept at 1 atm using the Langevin piston method
with a damping coefficient of 5 ps−1.
Electrostatic interactions are computed using the
particle-mesh Ewald algorithm with 50 grid points along
each direction, corresponding to a grid spacing of 1 Å. The
list of nonbonded interactions is truncated at 13.5 Å. The
short-range LJ interactions are switched off in the range of
10–12 Å using a smooth switching function. A time step of 2
fs is employed in the MD simulations, and the trajectory data
are written at 0.1 ps intervals during the production runs. For
each force field the system is first energy minimized and then
equilibrated for 500 ps. After this, production data are col-
6
4
PMF (kcal/mol)
114510-3
1
2
2
4
5
3
0
6
-2
2
3
4
5
6
r (Å)
FIG. 1. Convergence of the ab initio PMF. The dashed lines show the
individual PMFs obtained from 1 ps blocks of data 共numbered from 1 to 6
sequentially兲. The solid line shows the PMF obtained from the production
run 共2–6 ps兲.
lected for 10 ns. The Na+ – Cl− RDF is determined from the
ion trajectory data, which is then used in Eq. 共1兲 to calculate
the Na+ – Cl− PMF.
To study the finite size effects and justify the choice of
the maximum Na–Cl distance in the ab initio PMF, we have
also calculated the classical PMF 共for the CHARMM force
field兲 in a small box using the same system as in the CPMD
simulations. The PMF obtained from the small box overlaps
with the one obtained from the large box up to 5.5 Å and
starts deviating from it afterward. At 5.7 Å, the deviation is
still small enough. This indicates that the CPMD results presented in Figs. 1 and 3 are not influenced much by the finite
size effects.
The CIP and SSIP configurations are well populated in
free MD simulations but not the TS in between them. In
order to collect sufficient data for comparisons of the
ab initio and classical solvation structures, we have performed an additional MD simulation with the CHARMM
force field where the ion-ion separation is restrained to 3.6 Å.
For this purpose a smaller system 共54 water molecules with
one NaCl pair兲 is used and data are collected during a 5 ns
production run.
D. Solvation structure and dipole moments
An advantage of the MD simulations is that one obtains
detailed data on the microscopic solvation structure of the
ion pair at various separations. Comparison of the solvation
structures obtained from the ab initio and classical MD simulations could provide useful insights on how to improve the
classical force fields. To this end, we consider three ion-ion
configurations that are of particular importance: CIP and
SSIP positions and the TS in between them. For the ab initio
data we use the windows from the CPMD simulations that
are closest to these positions. The classical data are gathered
from the free MD simulations for the CIP and SSIP positions
using a bin of size 0.1 Å centered at those positions, and
from the constrained MD simulations for the TS as mentioned above. Using the trajectory data, we calculate the hy-
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J. Chem. Phys. 132, 114510 共2010兲
Timko, Bucher, and Kuyucak
dration numbers for each ion as well as the number of bridging waters that coordinate both ions at each ion-ion position.
The average orientations of the water molecules in the coordination shells are also computed and compared.
The dipole moments of water molecules are fixed in the
classical force fields, but in reality they could vary with the
ion-ion distance. If substantial changes were to occur in the
dipole moments of the hydration waters during dissociation,
it would be very difficult to describe this process using a
nonpolarizable force field. To check against this possibility,
we calculate the dipole moments of hydration waters from
the CPMD data for the three positions considered above. The
dipole moments are calculated using the maximally localized
Wannier function 共MLWF兲 analysis.63 The MLWF orbitals
are obtained from the Bloch orbitals available from the DFT
calculations via a unitary transformation and their spread is
minimized iteratively. Using the centers of the MLWFs and
the charges associated with them, the dipole moment can be
calculated from the classical formula
␮ = 兺 ZneRn − 兺 2eri ,
n
共5兲
0.01
0
constraint force (a.u.)
114510-4
-0.01
0.01
0
-0.01
0.01
0
-0.01
0
1
2
3
4
5
6
simulation time (ps)
FIG. 2. The constraint force between the ions 共gray point兲 and its cumulative average 共lines兲 are plotted against the simulation time for ion separations: r = 3.1 Å 共top兲, r = 3.9 Å 共middle兲, and r = 4.7 Å 共bottom兲. The
dashed line shows the convergence of the cumulative average force obtained
from the whole 6 ps of data and the solid line shows the same for only the
production data 共2–6 ps兲. Note that 1 a.u. is 1186 kcal/mol/Å.
i
where the first sum is over the O and two H atoms in water
which are at positions Rn and have ionic charges Zne, and the
second sum is over the four MLWF centers in water at positions ri with charges −2e. The MLWF analysis has been used
in numerous ab initio calculations of the dipole moments
previously.11,12,42,52,53 Also this method has been successful
in reproducing the dipole moment of water clusters
共n = 1 , . . . , 5兲 where experimental data exist.64
III. RESULTS AND DISCUSSION
A. Ab initio PMF for NaCl dissociation
The results of the ab initio PMF calculations for the
dissociation of NaCl are displayed in Fig. 1. To address the
equilibration and convergence issues, we have divided the 6
ps of CPMD data into 1 ps blocks and determined the corresponding PMF for each block. These are labeled from 1 to 6
in Fig. 1. The advantage of the block representation of the
data compared with the cumulative one is that the equilibration phase can be more clearly identified. The inspection of
the individual PMFs shows that the equilibration of the CIP
configuration is almost instantaneous but the SSIP configuration takes about 2 ps to equilibrate. Therefore, we have
excluded the first 2 ps of data as equilibration and constructed a final PMF from the remaining 4 ps of production
data 共thick curve in Fig. 1兲. The individual PMFs from three
to six are seen to fluctuate around this final PMF within a
band of ⬃1 kcal/ mol, which suggests the convergence of
the results. In the case of methane-methane PMF, the equilibration time was found to be 3 ps.10 The faster equilibration
in the NaCl PMF is reasonable because equilibration is expected to take less time for a charged pair compared with a
neutral pair.
It is also worthwhile to look at the constraint force at
several distances and check the convergence of its cumulative average. This is done in Fig. 2, where the constraint
force 共gray points兲 and its cumulative average 共lines兲 are
plotted against the simulation time for three ion-ion separations, r = 3.1, 3.9, and 4.7 Å. To show the effect of the equilibration, we present the cumulative average force for both the
production data 共2–6 ps, solid line in Fig. 2兲 and the whole
CPMD data 共0–6 ps, dashed line in Fig. 2兲. The constraint
force is seen to exhibit large fluctuations around a baseline,
and it would be very difficult to make a decision on the
equilibration time on the basis of these data. The cumulative
forces, in contrast, converge rather quickly and remain more
or less flat in the last 2 ps. However, they are seen to converge to different values in two cases, highlighting the importance of choosing the correct equilibration time. Because
the CPMD simulations are computationally expensive, an
optimal choice for the equilibration time is very important.
The block-data analysis of the PMF curves in Fig. 1 is visually very clear in this respect, hence it provides a valuable
guidance for separating the equilibration phase from the production data.
B. Polarization of hydration waters during dissociation
In classical nonpolarizable models, water molecules
have a fixed dipole moment, e.g., in the TIP3P model,
␮ = 2.35 D. In reality, however, water molecules have a
fairly large polarizability, and their dipole moments could
change substantially with the environment.11,12 Thus the first
question to ask in assessing the viability of using nonpolarizable models in the description of NaCl dissociation is
whether the dipole moments of hydration waters change significantly during dissociation. For this purpose, we have calculated the average dipole moment of hydration waters from
the CPMD production data using Eq. 共5兲. The data are
sampled at 50 fs intervals, and the average dipole moment is
obtained from the analysis of 80 frames. The results for the
CIP, TS, and SSIP positions—corresponding to the ion-ion
separations,
r = 2.68, 3.38, and 4.74 Å—are shown in Table I. For comparison, we show the bulk results for each ion in the last
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114510-5
J. Chem. Phys. 132, 114510 共2010兲
Dissociation of NaCl in water
TABLE I. Average dipole moments of the water molecules 共in units of
Debye, D兲 coordinating the Na+ ion only, the Cl− ion only, and both ions
simultaneously 共shared兲 at the CIP, TS, and SSIP positions. The corresponding bulk values are taken from Ref. 65 for Na+ and Ref. 66 for Cl−.
TS
SSIP
Bulk
2.89
2.93
2.79
2.87
2.86
2.77
2.77
2.90
2.81
2.81
2.87
¯
column of the table, which were computed using similar systems and the BLYP functional.65,66 Because polarizability of
water changes with the ion type, we have calculated the dipole moment of hydration waters separately for Na+ and Cl−
ions. The last row shows the result for the bridging waters
that belong to the hydration shell of both ions simultaneously.
Inspection of Table I shows that the variation in the dipole moments of hydration waters with ion-ion distance is
quite small and certainly within the accuracy of the calculations. Distributions of the dipole moments for each ion are
also found to be similar to those of the bulk calculations.65,66
Overall the results in Table I indicate that variations in water
polarization is not expected to play a significant role in dissociation of aqueous NaCl.
The reduction in the dipole moments of the shared water
appears counterintuitive—the electric field is twice as large
at the middle of the ions compared with a single ion—and
therefore requires some explanation. The average number of
shared waters is about 0.5 in the CIP and about 1 in the TS
and SSIP positions. The position and orientation of this water
indicate that it is mainly in the coordination shell of the Na+
ion in the CIP and TS positions. For example, in the CIP
position, the shared water is in the coordination shell of the
Na+ ion 100% of the time but moves in and out of the coordination shell of Cl− ion resulting in a coordination number
of 0.5. The presence of the Cl− ion actually hinders the polarization of the shared water in the CIP and TS configurations, which partly explains the smaller values of the dipole
moments for the shared waters. A more interesting situation
occurs in the SSIP position where both ions are equally well
coordinated by the shared water molecule. Yet the dipole
moment of this water still remains well below the bulk value
of 3.0 D. Inspection of the hydrogen bonds of the shared
water from the CPMD data helps to explain this counterintuitive result. The shared water makes a hydrogen bond with
PMF (kcal/mol)
Na+
Cl−
Shared
CIP
6
CPMD
AMBER
CHARMM
GROMACS
Smith and Dang
Joung and Cheatham
4
2
0
-2
2
3
4
5
FIG. 3. Comparison of the ab initio PMF 共CPMD兲 for dissociation of NaCl
with the classical ones obtained using the force fields AMBER, CHARMM,
GROMACS, Smith and Dang, and Joung and Cheatham.
a neighboring water molecule 74% of the time 共38% donor,
36% acceptor, and 26% none兲. This is far below the ideal
situation where the shared water would have one donor and
one acceptor. Each hydrogen bond contributes about 0.2 D to
the dipole moment of water.11 Thus due to geometrical restrictions, the average number of hydrogen bonds the shared
water makes is reduced from the possible 2 to 0.74, which
appears to be sufficient to cancel any polarization gains from
the simultaneous presence of the Na+ and Cl− ions.
C. Comparison of ab initio and classical PMFs
The ab initio PMF for the dissociation of NaCl is compared with the classical PMFs for several force fields in Fig.
3. The classical PMFs are determined to distances r = 10 Å.
Because all the classical PMFs are flat beyond 8 Å, the region 8–10 Å is used to set the zero of the potential, and thus
align the PMFs. Due to computational limitations, the ab
initio PMF could not be extended to this asymptotic region.
We have therefore aligned the ab initio PMF with the classical ones for r ⬎ 5 Å, where all PMFs bunch together and
tend to zero. This choice affects the binding energy but not
the other properties discussed in the paper. To make the comparison of PMFs easier, we also list in Table II the main
properties of each PMF, namely, the CIP, TS, and SSIP positions, the barrier height relative to the CIP minimum, and
the binding free energy obtained from Eq. 共4兲. Because the
TABLE II. Positions of the first minimum 共CIP兲, the maximum 共TS兲, and the second minimum 共SSIP兲 in the
ab initio and the classical PMFs. The column UB shows the barrier height relative to the CIP minimum, and Gb
gives the binding free energy obtained from the integration of the PMFs via Eq. 共4兲.
CPMD
AMBER
CHARMM
GROMACS
Smith and Dang
Joung and Cheatham
6
r (Å)
CIP
共Å兲
TS
共Å兲
SSIP
共Å兲
UB
共kcal/mol兲
Gb
共kcal/mol兲
2.68
2.76
2.62
2.67
2.81
2.71
3.38
3.82
3.58
3.63
3.67
3.63
4.74
5.40
5.10
5.21
5.27
5.19
1.40⫾ 0.28
3.56
3.49
3.32
2.70
3.16
−0.03⫾ 0.07
⫺0.43
⫺0.16
⫺0.13
⫺0.09
⫺0.12
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114510-6
J. Chem. Phys. 132, 114510 共2010兲
Timko, Bucher, and Kuyucak
TABLE III. Average coordination numbers determined from the CPMD and classical MD simulations with the
CHARMM force field. The row Na+ 共Cl−兲 refers to waters which coordinate only the Na+ 共Cl−兲 ion, and the row
shared refers to waters which coordinate both ions. The total is obtained by adding all three coordination
numbers. The bulk values for CPMD are taken from Ref. 65 for Na+ and Ref. 42 for Cl−.
CPMD
+
Na
Cl−
Shared
Total
CHARMM
CIP
TS
SSIP
Bulk
CIP
TS
SSIP
Bulk
2.8
4.0
0.5
7.3
3.2
5.0
1.1
9.3
4.2
4.3
1.1
9.6
5.2
6.0
¯
11.2
1.8
5.3
2.2
9.3
3.3
6.1
2.0
11.4
4.3
6.4
1.0
11.7
5.7
7.5
¯
13.2
ab initio simulations are quite short, we have included the
error bars in the ab initio calculations of free energies, which
are obtained from the block data in Fig. 1
Figure 3 shows that qualitatively the ab initio PMF is
very similar to the classical ones—a well defined CIP minimum is separated from a broader SSIP minimum by a TS
barrier. However, as is clear from Fig. 3 and quantified in
Table II, there are sizable differences between the ab initio
and classical PMFs with respect to both the positions of the
extremum points and their energies. The CIP distance is well
constrained from the ionic sizes so there is a little problem
there, but the TS and SSIP distances are markedly larger in
the classical PMFs. In this regard, AMBER shows the maximum deviation from the ab initio PMF while the others are
quite similar, with CHARMM being the closest to the
ab initio results. The shorter TS and SSIP distances in the
ab initio simulations are presumably due to the fact that water molecules are both flexible and polarizable, which is not
the case in the classical models. More important differences
occur in free energies. First, the CIP minimum is relatively
shallower and the TS barrier is much lower in the ab initio
PMF. As a result, the TS barrier height relative to the CIP
minimum is more than doubled in the classical PMFs compared with the ab initio PMF. Second the CIP and SSIP
minima are nearly degenerate in the ab initio PMF, whereas
the CIP minimum is always lower than that of SSIP in the
classical PMFs, which could be seen as a harbinger of clustering problems. Inspection of Fig. 3 and Table II shows that
AMBER exhibits the largest deviation in free energies with a
distinctly lower CIP minimum and the highest TS barrier.
The contrast between the ab initio and classical PMFs is
succinctly summarized by the binding free energy calculations presented in the last column of Table II. Experimentally, the solubility limit of the aqueous NaCl at room temperature is 5.4 M.67 Thus no clustering should occur for the
⬃1 M NaCl solutions considered here. The binding free energy is very small 共in absolute value兲 in the ab initio case,
which more or less ensures this. But in all the classical models a substantially larger binding free energy is obtained,
which will lead to some degree of clustering depending on
the simulation time. AMBER by far yields the largest binding free energy, so it is not surprising that clustering has first
been observed for this force field in multinanosecond MD
simulations. For the other force fields, the binding free energy is relatively smaller, hence clustering is expected to
occur in a much longer time scale.
Leaving aside AMBER, which is clearly an outlier, it is
still difficult to choose among the remaining classical PMFs
the closest one to the ab initio PMF. While the Smith and
Dang parametrization yields the best results for free energies,
its CIP position—which is the best determined quantity—is
substantially off. The remaining three PMFs are very close to
each other, with the latest Joung and Cheatham parameters
offering a slight improvement over CHARMM and
GROMACS. There is clearly room for improvement of the
classical force fields. To provide more clues as to how this
may be achieved, we next provide a microscopic analysis of
the hydration structure around the Na+ and Cl− ions in
ab initio simulations and contrast them with the corresponding classical results.
D. Solvation structure during dissociation
To see how the solvation structure changes with the ionion separation in the ab initio MD simulations, we have examined the coordination shells of each ion at the CIP, TS,
and SSIP positions. For comparison, similar calculations are
performed in classical MD simulations using the CHARMM
force field. Table III gives the average coordination numbers
obtained from the ab initio 共CPMD兲 and classical
共CHARMM兲 MD simulations at each of these three positions. The cutoff values are determined from the first minimum in the ion-oxygen RDF as 3.2 Å for Na+ and 3.9 Å for
Cl−. To help with the visualization of the results, we show in
Fig. 4 typical snapshots from the ab initio and classical simulations for each configuration. Because the sampling time in
the ab initio simulations is much shorter than the classical
ones, there are considerable uncertainties in the ab initio results. Thus while they provide useful insights for classical
force fields, it is important that they are confirmed in future
ab initio studies with longer simulation times.
Before interpreting the results, we should emphasize that
the definition of shared waters is based on the ion-water oxygen distance, and not on the orientation of the water molecule. For example, coordination of a water molecule by a
Cl− ion would require pointing of the O–H bond along the
O – Cl− vector. Inspection of the orientations of the water
molecules in the production data indicates that the shared
waters are, in fact, mainly in the coordination shell of the
Na+ ion in the CIP and TS positions. True joint coordination
by both ions occurs only in the SSIP position. The snapshots
in Fig. 4 depict this situation, which is further analyzed in
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114510-7
J. Chem. Phys. 132, 114510 共2010兲
Dissociation of NaCl in water
vector and for Cl− the angle between the Cl− – H vector and
the H–O vector, where H is the water-hydrogen closest to the
Cl− ion. The same angles are calculated for the shared waters
for Na+ and Cl− coordination separately. Overall there is a
reasonable agreement between the ab initio and classical results, with the Cl− ion results faring slightly better than those
of Na+. In both ab initio and classical cases, there is quite a
good alignment between the H–O and Cl− – H vectors, and
this hardly changes with distance. The slightly higher angle
found in CHARMM is presumably due to the constraints
imposed by the higher coordination number. For the Na+ ion,
CPMD predicts larger angles compared with CHARMM, and
also there is some variation in the dipole orientations with
distance, which is not observed in CHARMM. The largest
difference between the ab initio and classical models occurs
for the shared water, especially in the SSIP position. In
CPMD, the shared water’s coordination of the Cl− ion improves with distance whereas no change is observed in
CHARMM. This may have contributed to the smaller CIP
well obtained in the ab initio PMF compared with the classical PMFs. These results suggest that the dipole orientations
could also be helpful in understanding the differences between the ab initio and classical PMFs, and as such could
play a role in constraining the force field parameters.
FIG. 4. The snapshots of the hydration structures at the CIP 共top panel兲, TS
共middle panel兲, and SSIP 共bottom panel兲 positions obtained from the
ab initio 共left兲 and classical 共right兲 MD simulations. In all pictures, the Na+
ion is on the left of the Cl− ion. Shared water molecules are indicated by
dashed lines.
Table IV below. Thus it is more realistic to include the
shared coordination number in that of Na+ when comparing
the CIP and TS results. With this provision, we can summarize the results in Table III as follows. 共i兲 At the CIP and TS
positions, both ions are overcoordinated by about one water
molecule in classical MD relative to ab initio MD. 共ii兲 At the
SSIP position, the Na+ and shared coordination numbers
match but the Cl− ion is overcoordinated by two water molecules in classical MD. 共iii兲 Remarkably, while the total coordination number for the NaCl system increases with ionion separation, the classical system remains overcoordinated
by two water molecules compared with the ab initio results
at all distances. This suggests that more attention should be
paid in getting the coordination numbers right in the parametrization of the classical force fields.
Orientation of the dipole moments of the hydration waters is clearly an important consideration in deciphering the
changes in the microscopic solvation structure during NaCl
dissociation. In Table IV we compare the orientations of water molecules obtained from ab initio and classical MD simulations in the same way as the coordination numbers in Table
III. The angles plotted in Table IV are as follows. For Na+
the angle between the Na+ – O vector and the water-dipole
IV. CONCLUSIONS
In this paper, we have presented the first ab initio calculation of the PMF for the dissociation of aqueous NaCl. Because the PMF 共or the corresponding RDF兲 is not a measurable quantity, it has not been used in parametrization of the
classical force fields in the past. Detailed comparisons of the
ab initio PMF with the classical PMFs obtained using several
force fields show that they have problems with regard to both
the free energy profiles 共roughly doubled兲 and positions of
the extremum points 共further away兲. This results in appreciable binding free energies in all classical force fields,
which could cause clustering of NaCl in long MD simulations in contradiction with experimental solubility results. To
get a better understanding of the causes of these discrepancies, we have investigated the microscopic solvation structure in the ab initio MD simulations—i.e., coordination numbers, dipole moments, and orientations—and compared them
with similar quantities obtained from the classical MD simulations. No significant variations have been observed in dipole moments of hydration waters with ion-ion distance,
TABLE IV. Average angular orientations of the dipole moments 共in degrees兲 of the coordinating waters obtained from the ab initio and classical MD simulations. The last two rows give the same orientations for the
shared waters which coordinate both ions simultaneously. Standard deviations are given in brackets. See the text
for the definition of the angles.
CPMD
Na+
Cl−
Shared Na+
Shared Cl−
CIP
TS
41 共21兲
6 共4兲
48 共22兲
14 共1兲
51 共20兲
8 共4兲
58 共22兲
12 共4兲
CHARMM
SSIP
45
6
53
5
共20兲
共4兲
共17兲
共3兲
CIP
TS
SSIP
35 共19兲
10 共5兲
37 共22兲
11 共4兲
34 共18兲
11 共5兲
42 共20兲
11 共4兲
34 共18兲
11 共5兲
29 共17兲
11 共5兲
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114510-8
which suggests that NaCl dissociation may be described using nonpolarizable force fields.
As pointed out in previous comparisons of ab initio and
classical MD results,68 overcoordination of ions is a significant problem for the current classical force fields. Here we
have shown that this problem persists in the aqueous NaCl
system at various ion-ion separations. It is possible that the
problems found in the classical PMFs are also caused by
overcoordination. For example, a larger number of water
molecules in the coordination shell could push the extremum
points 共TS and SSIP兲 further away and also result in a more
stable CIP configuration. Therefore, it is important to consider this possibility in future developments of force fields
and devise ways to incorporate the coordination numbers in
the parametrization process. If such attempts fail to produce
a nonpolarizable force field consistent with the experiments
and the ab initio results, then inclusion of polarization will
be essential for description of NaCl dissociation. The ab initio results presented here will be especially useful in guiding
the efforts in construction of such polarizable force fields.
ACKNOWLEDGMENTS
This work was supported by grants from the Australian
Research Council. Calculations were performed using the
SGI Altix clusters at the National Computational Infrastructure 共Canberra兲 and the Australian Center for Advanced
Computing and Communications 共Sydney兲.
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