Accurate Description of Calcium Solvation in

Concentrated Aqueous Solutions

Miriam Kohagen, Philip E. Mason, and Pavel Jungwirth∗

Institute of Organic Chemistry and Biochemistry, Academy of Sciences of the Czech

Republic, Flemingovo nam. 2, 16610 Prague 6, Czech Republic

E-mail: pavel.jungwirth@uochb.cas.cz

KEYWORDS: calcium chloride, aqueous solution, molecular dynamics, neutron scatter-

ing

Abstract

Calcium is one of the biologically most important ions, however, its accurate descrip-

tion by classical molecular dynamics simulations is complicated by strong electrostatic

and polarization interactions with surroundings due to its divalent nature. Here, we

explore the recently suggested approach for effectively accounting for polarization ef-

fects via ionic charge rescaling and develop a new and accurate parametrization of

the calcium dication. Comparison to neutron scattering and viscosity measurements

demonstrates that our model allows for an accurate description of concentrated aqueous

calcium chloride solutions. The present model should find broad use in efficient and

accurate modelling of calcium in aqueous environments, such as those encountered in

biological and technological applications.

∗To whom correspondence should be addressed

1

Introduction

Around 99 % of calcium in the human body is stored in bones and teeth in an insoluble

form.1 The remaining ”soluble calcium” has crucial tasks in signalling,2 in processes including

neurotransmitter release3 or muscle contraction,4 as well as in activation of enzymes and

hormones.5 Because of its importance both in biological and technological contexts, it has

been subject to various theoretical6–8 and experimental9–11 studies. The interactions of

calcium with the surrounding water molecules are characterized by relatively fast kinetics and

a high flexibility.12,13 In aqueous solutions a large variety of coordination numbers between

5 and 10 has been deduced experimentally,8,13–21 whereas in biological systems, such as in

the binding loops of calmodulin, a coordination number of 7 and a pentagonal bipyramidal

orientation is typically observed.22,23 The variety of observed coordination numbers is due to

the presence of several shallow local minima on the corresponding energy landscape.24 For

similar reasons, aqueous solutions of calcium chloride show an anomalous thermodynamic

behavior at high concentrations.19,25

To faithfully capture the complex features connected with calcium hydration within clas-

sical molecular dynamics (MD) simulations is rather challenging.26–32 This divalent cation

can strongly polarize surrounding water molecules (a sizable charge transfer can even oc-

cur33), which is an effect missing in typical non-polarizable MD simulations. One approach

to handle this effect for calcium and magnesium has been introduced in Ref. 26. The au-

thors distributed the charge and the van der Waals centers to multiple sites represented by

dummy atoms. Recently, it has been shown that polarization can be effectively accounted

for within non-polarizable simulations via rescaling ionic charges by the inverse square root

of the electronic part of the solvent dielectric constant (i.e., by a factor of roughly 0.75 for

water).34 In our previous work35–38 we showed that this approach, denoted as the electronic

continuum correction (ECC), leads to significant improvements in description of electrolyte

solutions, particularly containing multivalent ions.

Within standard non-polarizable simulations excessive ion clustering often occurs due

2

to overestimation of the strength of the ion-ion interactions with respect to the ion-water

interactions.38 ECC to a large extent fixes this problem yielding also a much better agree-

ment between the results from the molecular dynamics simulations and neutron scattering

data.35,36,38 Further evidence for the importance of accounting for polarization for divalent

ions comes from a study where the authors changed the charge of calcium systematically

to get the experimentally estimated free energy of hydration.39 For the calcium charges of

1.6 e and 2.0 e, they found distinctively different values for the coordination number of water

around calcium. Furthermore, the residence times of the water molecules in the first solva-

tion shell differed by one order of magnitude. Based on the above experience, we address

here a challenging problem involving a divalent cation, namely modelling the structure of a

concentrated aqueous solution of calcium chloride.

Neutron scattering, especially with isotope substitution has been proven to be a reliable

tool to investigate solvation structures of ions in concentrated aqueous solutions.36,40 It also

allows for direct assessment of force field quality in terms of agreement in coordination

numbers and peak positions of radial distribution functions.38,40

In Ref. 32 a similar approach has been adopted. The authors also studied aqueous CaCl2

solution comparing neutron scattering data to simulation data. However, there are significant

differences, since in the current work the isotopic substitution for the neutron scattering data

was performed directly on the center of interest (i.e., the calcium ion in this case), whereas

in Ref. 32 the isotopic substitution was performed on the exchangeable hydrogen in the

system. This means that any data relevant to the hydration structure of the calcium ion

had to be deconvoluted from the total structural measurement. The fraction of the data

directly relevant to the calcium hydration in that data is about 3 % of the total scattering,

while the majority of the total scattering signal comes from water. The present approach

with isotopic substitution on calcium removes the bulk of these correlations that are not of

a direct interest and allows only the relevant parts of the data (directly corresponding to

calcium solvation) to be examined.

3

In this paper, we develop an accurate parametrization for the calcium ion based on

the ECC approach and use comparison to neutron scattering data19 as a measure for the

quality of our force field. The aim is to provide a calcium model that could be reliably

used in simulations of aqueous solutions of this ion in various biological and technological

applications.

Computational details

Classical molecular dynamics simulations were carried out for aqueous solutions of CaCl2.

The molar ratio used here was 4 moles of CaCl2 to 55.55 moles of water, for convenience

hereafter referred to as 4 m. Two different water models were employed, namely SPC/E41

and, to test transferability of the developed calcium force field, also TIP4P/2005.42 The

results for TIP4P/2005 water not shown in the main text, can be found in the SI (Table 8-12

and Fig. 6 and 7). The unit cell contained 5750 water molecules, 414 calcium dications,

and 818 chloride anions. Initially, the Lennard–Jones parameter from Ref. 28 were used for

chloride and the parameters from the Gromos 53a6 force field27 for calcium, see Table 1.

The mixed Lennard–Jones parameters were derived from self-parameters using the Lorentz–

Berthelot mixing rules. Additionally, the geometric averages for both sigma and epsilon

parameters, as implemented, e. g., in the Optimized-Potentials for Liquid Simulations43,44

(OPLS) force field were used. The results of the latter calculations are provided in the

Supporting Information (SI) (see Table 3-7 and Fig. 2-5). Additionally, in the same tables

and figures of the SI, results for a smaller calcium ion (A1), for a chloride ion from literature45

(A2) and for recently published parameters31 (A3) can be found. The exact values, including

the ones for the counter ions of set A1 and A2, are given in Table 2 of the SI.

All simulations were performed using the Gromacs 4.5.4 program package46 with a time

step of 1 fs employing 3D periodic boundary conditions. Simulations were carried out under

constant pressure of 1 bar, controlled by the Parrinello–Rahman barostat47 with a coupling

4

Table 1: Force field parameters of the ions employed in this study.

Caσ (Å) ε (kJ/mol) charge

full 2.8196 0.5072 +2.0ECC 2.8196 0.5072 +1.5ECCR 2.5376 0.5072 +1.5

Clσ (Å) ε (kJ/mol) charge

full 4.4499 0.4184 -1.0ECC 4.4499 0.4184 -0.75ECCR 3.7824 0.4184 -0.75

constant of 0.5 ps and constant temperature of 298 K, controlled by the velocity rescaling

thermostat48 with a coupling time of 0.5 ps. Simulations of a length of at least 30 ns were

performed for models with both scaled and unscaled charges. Data from the last 10 ns

were used for further analysis, e. g., calculations of the radial distribution functions (RDFs),

and comparison with neutron scattering data. For the description of the water molecules

constraints for bonds and angles were applied using the Lincs algorithm.49 The electrostatic

and van der Waals interactions were truncated at 1.2 nm and the long range electrostatic

interactions were accounted for by the particle mesh Ewald method.50 Since the original ionic

sizes were parametrized together with the full charge model, we refined them after charge

scaling according to the experimental number density and the neutron scattering data.19

Decreasing the van der Waals radius of calcium by 10 % and chloride by 15 % resulted in

number densities within 5 % of the experimental values,8,19 see Table 2. The results from

this size rescaling are denoted as electronic continuum correction with rescaling (ECCR)

throughout the paper. Note, that changing the size of the chloride ion influences the density

more significantly than changing the size of the calcium (see SI), since there are as twice as

many chloride ions in the simulation box than calcium ions, moreover, they have a larger

van der Waals radius.

Subsequent simulations at constant temperature and volume with a length of 50 ns were

performed in order to obtain equilibrated input structures for viscosity calculations and

5

calculations of the residence times of water molecules in the calcium solvent shell. The last

snapshot of each of these equilibrium runs was taken as input for non-equilibrium simulations

to calculate viscosities.51,52 For systems employing the SPC/E water model, three acceler-

ation rates of 0.001, 0.0025, and 0.005 nm/ps2 were tested. The corresponding trajectories

have a length of at least 20 ns and the last 10 ns were taken for the viscosity calculations.

Based on the results, an acceleration rate of 0.0025 nm/ps2 was determined as an appropriate

choice and used also for the simulations with the TIP4P/2005 water model (see Table 10 in

the SI).

Results and discussion

Figure 1: Representative snapshots of the system with full charges (left) and the ECCRsystem (right) after a simulation time of 30 ns. The calcium ions are colored in red (oryellow, when paired with chloride) and the chloride ions in green (or violet, when pairedwith calcium).

We find it useful to start this section by qualitative considerations before moving to actual

numbers. In Figure 1, representative snapshots from simulations with full charges (left hand

side) and within the ECCR approach (right hand side) are shown. The most visible difference

6

between both systems is the number of ion pairs, shown in yellow and violet color fashion.

On the right hand side various contact ion pairs can be seen, while on the left hand side

the ions are more evenly distributed and only a few contact ion pairs can be found. This

observation is quantified via the calculated coordination numbers (see Table 3), discussed

below.

The increased ion pairing within the ECCR approach contrasts observations from earlier

studies of systems with divalent anions (namely lithium sulfate and potassium carbonate),

where excessive ion pairing or clustering was detected for models with the full charges.35,38

The moderate amount of ion pairing and lack of clustering in the present system with divalent

cations is likely due to a much larger mismatch between charge densities of cations vs.

anions53 compared to the previously studied solutions containing divalent anions.35,38 In the

following discussion we show, that the more correct description of the interactions within the

ECCR approach leads to an improvement in modeling of observable properties, in comparison

with the neutron scattering data and experimental viscosities.

Table 2: Average particle densities. Standard deviations < 0.0007 atoms/Å3.

system particle densityfull 0.091ECC 0.080ECCR 0.091Ref. 19 (exp.) 0.096Ref. 8 (exp.) 0.094

Table 3: First-shell coordination numbers (CNH2O and CNCl), first- and second-shell RDFpeak positions for calcium and chloride (r1,2(Ca–Cl)) given in Å.

system CNH2O CNCl r1(Ca–Cl) r2(Ca–Cl)full 7.76 ± 0.02 0.08 ± 0.02 2.85 ± 0.01 4.4–6.0ECC 6.36 ± 0.02 0.22 ± 0.02 3.02 ± 0.01 4.4–6.2ECCR 5.22 ± 0.02 0.73 ± 0.02 2.56 ± 0.01 3.9–5.5

Neutron scattering experiments were performed in a previous study in heavy water with

isotopically labeled calcium (natCa/44Ca).19 This approach makes it possible to evaluate the

first order differences (for exact definition see Ref. 19 and SI), which allows to obtain with

7

good accuracy the peak positions of the calcium–oxygen and calcium–hydrogen peaks in the

first solvation shell. The results of the first order difference functions both in the r-space

(∆G(r)) and in the Q-space (∆S(Q)), are shown in Figure 2. Here, Q is the radial scalar

part of the scattering vector.

8

Figure 2: First order difference function in the Q-space (left hand side) and first orderdifference function in the r-space (right hand side) for the simulations with the full charges,scaled charges (ECC) and with the ECC approach and smaller ions (ECCR) in red and theexperimentally measured curve in black. The insets show the results with the TIP4P/2005water model. The error on the experimental values is ±0.0006 barns/str. For the sake ofclarity error bars are omitted in the insets and only shown for every 10th data point in thefigures.

In principle, both functions contain the same information, since the r-space representation

is the Fourier transform of the Q-space representation. Nevertheless, both curves are shown

9

here, since the experimental data were determined for a limited range within the Q-space.

Furthermore, the more intuitive representation is in the r-space and the two representations

emphasize different structural aspects. On the one hand, comparison between calculated

and measured Q-space data gives a better feeling whether the structural peak positions are

accurately reproduced, since these correspond to the correct phasing in Q-space. On the

other hand, the correct peak heights can be more easily determined in r-space.

The basic features of the experimental spectrum in the Q-space are the primary peak at

Q ≈ 3 Å−1 preceded by a sharp dip at Q ≈ 2 Å−1 and followed by several smaller oscillations

at higher Q-values. Transforming this function into the r-space results in two distinct peaks

at 2.38 Å and 3.0 Å and a broader one between 4.3 and 5.4 Å. The first feature is mostly

due to the oxygen atoms and the second one due to the hydrogen atoms of water molecules

in the first solvation shell of calcium. Note, that the first peak also contains the contribution

from contact ion pairs with chloride (if present). This general picture obtained from neutron

scattering is also found in all the simulations presented here, being consistent with previous

experimental and theoretical studies (see Table 4 in the main paper and Table 1 in the SI),

although some of the force fields seem to have problems to reproduce the second solvent

peak at all, as stated in Ref. 26.

Table 4: Sum of first-shell coordination numbers (CNH2O+Cl) and first-shell RDF peak posi-tions (in Å) for calcium and oxygen (r1(Ca–O)) including experimental results reported inliterature based on X-ray diffraction (XRD) and neutron scattering (ND).

system CNH2O+Cl r1(Ca–O)full 7.84 ± 0.04 2.46 ± 0.01ECC 6.58 ± 0.04 2.49 ± 0.01ECCR 5.95 ± 0.04 2.35 ± 0.01Ref. 8 (XRD) 5.9 ± 0.3 2.45Ref. 8 (ND) 6.5 ± 0.2 2.46 ± 0.02Ref. 17 (XRD) 6.8 2.43Ref. 19 (ND) 7.3 2.40 ± 0.01

The values in the first row of Table 4 correspond to the parameter set with full charges.

All three peaks can be qualitatively reproduced, but they are shifted significantly to larger

10

distances, which is also indicated by a phase shift in Q-space (Figure 2). Moreover, the

heights for the Ca–O and Ca–D peak are overestimated. Scaling the ionic charges results in

a decrease of these peak heights but the peak positions stay almost unchanged, see middle

panels in Figure 2 and also Table 4. In order to get the peak positions right as well, the size of

the calcium ion must also be reduced on top of the charge scaling. This approach, denoted

here as ECCR, leads to a very good agreement between the experimental and calculated

∆G(r), results shown in the last row of Figure 2.

In order to test the transferability of our parameters to other water models, we performed

additional simulations employing the TIP4P/2005 water model.42 Although all water force

fields of course aim at describing the same system (i.e., the "tap" water), transferability

is not granted automatically, since they have been fitted to different experimental proper-

ties to suit distinct purposes, e. g., description of different water phases.42 We choose the

TIP4P/2005 water model for comparison, since it is more general than SPC/E, having been

fitted to measurable properties over wide temperature and pressure ranges. From the insets

of Figure 2, it can be seen that the same trends in the comparison with the neutron scatter-

ing data can be observed with TIP4P/2005 as with the SPC/E water model. At first sight,

for the ECCR approach the comparison seems to be even better with the TIP4P/2005 water

model, but the peak ratio of the first and second peak shows a larger mismatch than for

SPC/E and, more importantly, the second peak is too small. For each oxygen atom in the

first solvation shell, two hydrogen atoms contribute to the second peak, which results in a

well-defined ratio between the peak heights of the first and second peak. The mismatch in

the ratio is a clear indication that the amount of water molecules in the first solvation shell

is too small, whereas the number of counter ions in the first solvation shell is too large in

TIP4P/2005 water (see Table 3 in the main text and Table 8 in the SI). Further one may

get the impression, that scaling only the van der Waals radii could be enough to improve

the agreement between the experiment. Such simulations (data not shown) result, however,

in extensive ion pairing and clustering. This manifests itself in a coordination number of

11

chloride around calcium of about two and a peak ratio of the first and second peak in r-space

representation which differs significantly from experiment.

From neutron scattering experiments, it was found that the first and second peak, rep-

resenting the first and the second solvation shell, are well separated by a deep minimum.19

Qualitatively, this can also be seen for the calculated radial distribution functions in Fig-

ure 3, nevertheless from the zoom-ins of the first minimum, shown in the inset, it becomes

clear, that there could be a non-zero exchange of water molecules between the first and the

second solvation shell. The same can be observed for the calcium-chloride radial distribution

function (Figure 4, lower right panel). To identify and quantify this exchange, we evaluated

water residence times and exchange rates (defined as the reciprocal of the residence time).54

Figure 3: Calcium–oxygen RDFs (solid lines) and running coordination numbers (dashedlines) for the full charged system (in black), the ECC system (in red) and the ECCR system(in blue).

12

Figure 4: Calcium–chloride RDFs (solid lines) and running coordination numbers (dashedlines) for the full charged system (in black), the ECC system (in red) and the ECCR system(in blue).

The resulting correlation functions were fitted by exponential functions, with the esti-

mated exchange rates and correlation times presented in Table 5.

Table 5: Estimates of exchange rates (in 1/s) and residence times (in ps) for water from thecorresponding autocorrelation functions (kH2O and τH2O), including experimental12,55 andtheoretical26,56,57 results reported in literature.

system kH2O τH2O

full ∼ 2.5 · 109 ∼ 400ECC ∼ 3.7 · 1010 ∼ 72ECCR ∼ 1.2 · 109 ∼ 820Ref. 55 (exp.) ∼ 5 · 108 ∼ 2000Ref. 12 (exp.) > 1010 < 100Ref. 56 (theo.) 2.3− 4.6 · 1010 21.3 – 42.6Ref. 26 (theo.) 5.0 · 1010 20Ref. 57 (theo.) 1.1 · 1010 89

Scaling the charges increases the water exchange between the first and the second solva-

tion shell more than tenfold and decreases the residence time by half an order of magnitude,

whereas the subsequent reduction of the ionic van der Waals radii has an opposite effect.

The calculated values of the residence times of 70-800 ps are all within the estimated ex-

13

perimental range.12,55 In comparison, the residence times available from QM/MM molecular

dynamics simulations56,57 are typically shorter, but it should to be kept in mind, that these

simulations lack sufficient statistics to give reliable results on this time scale. The exchange

of chloride in the first solvation shell of calcium is much slower compared to water exchange,

therefore no reliable residence times for the system with the full charges could be estimated

(the corresponding correlation functions are displayed in the SI). Nevertheless, a qualitative

comparison can be made. Scaling the charges within the ECC approach increases the dynam-

ics largely, whereas the additional reduction of the van der Waals radii (ECCR approach)

slows it down, but it still remains faster than that with full charges.

The running coordination numbers for water and chloride around calcium are displayed

as dashed lines in Figure 3 and 4. For direct comparison with experiment, the sum of both

coordination numbers (CNH2O+Cl, second column in Table 4) was evaluated. The reason

is, that even with first order isotope substitution experiments, it was not possible to fully

quantify ion pairing from the structure of the first hydration shell of calcium, since the

oxygen and the chloride peaks sit on top of each other.8 Comparison with X-ray diffraction

is further complicated by the fact that the coordination number may depend on the model

assumed for refinement of experimental data. Therefore, also for data from Ref. 17 the

sum of CNH2O+Cl is provided in Table 4. For clarity, the coordination numbers and peak

positions, discussed in the following, are summarized in Tables 3 and 4.

While we use exactly the same data as the previous study,19 our r-space interpretations

are slightly different both in terms of peak positions and coordination numbers. This is

because in the original study a Lorch window function was used to help reduce termination

errors in the Fourier transform process. While this procedure indeed lowers termination

errors, it also reduces the Q-range, and hence the r-space resolution of the data. This has

the effect of peak broadening and, if sharp peaks are close to each other, it can result in peak

shifts. Here we do not use such a window function and perform the comparison between

the experimental and simulation data both in Q- and r-spaces. As the peak positions of the

14

first two r-space peaks of the calcium hydration are mostly determined by the Q-space data

above 5 Å−1 and the fit between the experiment and simulation is the best for the ECCR

simulation (see Figure 2), we can firmly conclude that the ECCR simulation provides the

best representation of calcium hydration. Furthermore, the ECCR coordination number is

in excellent agreement with the data from Ref. 8, being only slightly lower than the value

from Ref. 19. Further support for a coordination number of calcium around six comes from

Car-Parrinello-based constrained molecular dynamics simulations described in Ref. 24. The

authors calculated the free-energy profile employing the calcium-water oxygen coordination

number as a reaction coordinate. They found several shallow local minima, which indicates

that the hydration shell around calcium is highly variable. Most importantly, the deepest

minimum was found for a coordination number of 6.1.

In order to get a more detailed insight into how changes in treatment of interactions

between different components influence the ratio of water molecules to chloride ions in the

first solvation shell of calcium, i. e., the ion-pairing, both coordination numbers CNH2O and

CNCl, were evaluated. Reducing the charge of the ions (ECC) decreases the coordination

number of water (CNH2O) in the Ca2+ shell by about one, whereas the coordination number

of chloride CNCl slightly increases. Reducing the van der Waals radii (ECCR) leads to a

further decrease of around 0.6 in CNH2O and a further increase in CNCl. Comparing to the

neutron scattering experiments this increase in CNCl may be slightly too high, which can also

be seen in the corresponding r-space representations (Figure 2). The ratio of the first peak

to the second peak changes by rescaling the van der Waals radii, i. e., the first peak increases

more than the second one. Support for the fact, that there is a certain amount of chloride in

the first solvation shell of calcium comes also from X-ray diffraction experiments,8,17 which

with appropriate refinement yield a value around one for CNCl. Additionally, from the same

experiments information about Ca–Cl distances can be gained. The two experiments8,17

yield positions of the first peaks of 2.75 Å or 2.74 Å, which are only slightly larger values

than we found. These experiments8,17 also show a peak at 5 ± 0.4 Å or around 4.86 Å,

15

which can be attributed to solvent separated ion pairs, in reasonable agreement with our

simulation results.

For further comparison with experiment we calculated viscosities of the investigated so-

lutions using a non-equilibrium approach, namely the periodic perturbation method.51,52

Non-equilibrium molecular dynamics simulations were used previously for aqueous ionic liq-

uid solutions with viscosities up to 40 mPa s58 and for neat ionic liquids up to viscosities

of 250 mPa s,59 as well as for simpler systems like pure water.51 Our results for CaCl2(aq),

together with experimental comparison,60–62 are given in Table 6.

Table 6: Viscosities values in mPa s for the simulations calculated by the periodic pertur-bation method (PPM)51 with an acceleration rate of 0.0025 nm/ps2, including experimentalvalues at different concentrations given in parentheses measured at room temperature.

systemfull 11.0 ± 1.0ECC 1.42 ± 0.05ECCR 2.9 ± 0.1Ref. 62 (exp./3.86 m) 3.169Ref. 61 (exp./4 m) 3.97Ref. 60 (exp./4.054 m) 4.22Ref. 62 (exp./4.277 m) 3.745

It can be seen that the calculation with full charges grossly overestimates the experi-

mental viscosity. The ECC approach reduces this mismatch, but now underestimates the

experimental value, while the ECCR viscosity is practically right on target. This trend is

partly consistent with the calculated exchange times of water (going from full charges to

ECC) (Table 5) and the correlation functions for the chloride ions (SI). The viscosity as well

as the correlation functions (and therefore the exchange times) are indicators of the interac-

tion strength between the particles in the solution. The trends can be justified intuitively,

since the strength of interactions involving ions is the strongest in the full charged system,

weaker within the ECCR approach, and weakest for ECC. The much too high viscosity (and

the much longer correlation times, especially for the chloride ions) using the full charges,

are a strong indication, that the intramolecular interactions are overestimated although no

16

extended ion pairing, as in the case of lithium sulfate38 and potassium carbonate,35 can be

detected. The agreement of the calculated viscosity with experiment for ECCR is remarkable

(Table 6), especially considering that the calculated viscosity of pure SPC/E water is around

0.642 mPa s,51 which is about 20 % lower than the experimental value.

Additionally, we evaluated the self-diffusion coefficients for the ions and water molecules

(Table 7).

Table 7: Self-diffusion coefficients of calcium (DCa) and chloride ion (DCl) as well as forwater (DH2O) calculated from the corresponding center-of-mass mean square displacement,including experimental values63 (concentration given in parentheses) measured at room tem-perature. Values given in 10−5cm2/s.

system DCa DCl DH2O

full 0.067 ± 0.001 0.15 ± 0.01 0.285 ± 0.002ECC 0.55 ± 0.02 1.15 ± 0.01 1.57 ± 0.04ECCR 0.259 ± 0.003 0.56 ± 0.03 0.84 ± 0.01Ref. 63 (exp./4.02 m) 0.225 ± 0.002 0.447 ± 0.015 −

The self-diffusion coefficients were calculated from the corresponding center-of- mass

mean square displacements. The same trends as for the viscosities can be seen. In the

system with the full charges the self-diffusion coefficients for chloride and calcium are much

too small compared to experiment. Scaling the charges increases the self-diffusion coefficient,

leading to values around twice as large as the experimental value. Again, the adjustment of

the radii (ECCR approach) results in values in good agreement with experiment.

Conclusions

In this study, we present a new accurate force field model for aqueous calcium, which is

among the most important divalent ions in biological systems. Our approach is based on

a physically well-justified assumption, that the usually neglected polarization effects are

important for aqueous systems involving multivalent ions. Polarization can be effectively

accounted for within technically non-polarizable simulations by scaling the ionic charges by

1/√εel, i. e., by the inverse square root of the electronic part of the dielectric constant of the

17

solvent.34 Comparison with high quality neutron scattering data shows that scaling of the

ionic charges leads to a better agreement with the scattering data, but the density is too low

(by around 16 %). Therefore, further adjustment is needed. For this reason, we additionally

reduce the van der Waals radii of both ions (by 10 % for calcium and 15 % for chloride),

which results in a quantitative agreement with the neutron scattering data.19

Further support for the new model comes from comparison to experimental coordination

numbers and viscosities. In addition, robustness of the present approach is checked by em-

ploying another water model (TIP4P/2005 instead of SPC/E) and by testing different mixing

rules for Lennard–Jones interactions (see SI). The present refined charge scaled model of cal-

cium thus provides a significantly improved description of its concentrated aqueous solutions

in comparison with existing non-polarizable force fields, including a sophisticated, most re-

cently developed one31 (for a detailed comparison see SI). The present parametrization for

calcium should lead to better description of not only simple aqueous solutions but also more

complex biologically or technologically relevant system involving this important dication.

Acknowledgement

We are grateful to Adrian Barnes for valuable discussions and for kindly providing the original

neutron scattering data. This work was supported by the Czech Science Foundation (grant

number P208/12/G016) and the Czech Ministry of Education (grant number LH12001).

MK thanks for funding within the framework of the IOCB Postdoctoral Project program.

Allocation of computer time from the National supercomputing center IT4Innovations in

Ostrava is highly appreciated.

Supporting Information Available

Supporting Information Available: In the SI further results from simulations of additional

parameter sets are presented as well as from simulations with an other water model. This

material is available free of charge via the Internet at http://pubs.acs.org/.

18

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