A temperature of maximum density in soft sticky dipole water

Chemical Physics Letters 376 (2003) 646–652

www.elsevier.com/locate/cplett

A temperature of maximum density in soft sticky dipole water

Ming-Liang Tan a, Justin T. Fischer a, Amalendu Chandra a,Bernard R. Brooks b, Toshiko Ichiye a,b,*

a School of Molecular Biosciences, Washington State University, Pullman, WA 99164-4660, USAb Laboratory of Biophysical Chemistry, National Heart, Lung, and Blood Institute, National Institutes of Health,

Bethesda, MD 20892-8014, USA

Received 30 December 2002; in final form 23 April 2003

Published online: 10 July 2003

Abstract

A temperature of maximum density near 260 K at 1 atm has been found for the soft sticky dipole (SSD) water model

in molecular dynamics simulations. The parameters of SSD have been optimized to reproduce the density of water as

well as other structural, thermodynamic, dielectric, and dynamic properties at room temperature and 1 atm. Re-

markably, this simple model is able to reproduce the anomalous temperature dependence of the density using pa-

rameters optimized at room temperature. Furthermore, these results indicate that the tetrahedral nature of water is

important in determining this anomalous behavior.

� 2003 Elsevier B.V. All rights reserved.

1. Introduction

Due to its crucial role in chemical and biologi-

cal processes as well as its peculiar properties,

water is probably the most extensively studied li-

quid. However, its properties are still not wellunderstood at a molecular level. In the 1980s,

several rigid, pair-wise additive, non-polarizable,

three-site water potentials, such as the SPC [1],

SPC/E [2] and TIP3P [3], were proposed that were

parameterized to reproduce some water properties

near ambient conditions in computer simulations.

These models are advantageous because they are

* Corresponding author. Fax: +1-509-335-9688.

E-mail address: [email protected] (T. Ichiye).

0009-2614/$ - see front matter � 2003 Elsevier B.V. All rights reserv

doi:10.1016/S0009-2614(03)01044-3

fast in computer simulations. Remarkably, these

simple models are able to reproduce water prop-

erties far from ambient conditions [1–5]. However,

these models have done less well in reproducing

one of water�s most interesting properties, namely,a maximum in the temperature dependence of thedensity. The experimental temperature of maxi-

mum density (TMD) occurs at 277 K while the

TMD of these models, if any, is below 240 K [4,5].

One strategy for improvement is to use models

with more sites. The four-site TIP4P [6] model has

a broad maximum with a TMD of around 248 K

[5]. Moreover, the five-site ST2 [7] and TIP5P [8]

models have TMD around 300 K [7] and 273 K [8],respectively. Another strategy for improvement is

to develop polarizable water models in which

electronic polarization in response to the sur-

ed.

mail to: [email protected]

M.-L. Tan et al. / Chemical Physics Letters 376 (2003) 646–652 647

rounding molecules is explicitly accounted for. For

example, the polarizable fluctuating charge TIP4P-

FQ model exhibits TMD around 280 K [9].

Recently, progress has been made in developing

accurate polarizable water models, which was de-

scribed in detail elsewhere and references therein[9–12]. Nevertheless, additional computational

expense is needed to compute the many-body

terms explicitly and unresolved questions remain

about the effects of incomplete iteration and the

optimal method of including polarization. Thus,

simple effective pairwise models, where non-addi-

tive many-body effects are taken into account in an

average way, will continue to be widely used es-pecially in the simulations of large systems.

A single-site non-polarizable water model,

which is referred as the soft sticky dipole (SSD)

model, has been introduced recently [13]. The

SSD model is based on the hard-sphere sticky di-

pole model by Bratko et al. [14,15] and consists of

a Lennard-Jones sphere embedded with a point

dipole and a tetrahedral sticky potential, where theshort-range sticky potential describes the hydro-

gen bonding and regulates the tetrahedral coordi-

nation in the first shell of water. The sticky

potential is an effective description of the energy

terms involving the quadrupole and octopole

moments such as found in the multipole expan-

sions of Kusalik and Patey [16,17], but requires

fewer computations. The center of mass is the onlyinteraction site in the model. Evaluation of the

interaction between two SSD water molecules re-

quires computing only one distance between the

two centers of mass, four spherical angles, and the

angle between the dipole vectors. On the other

hand, three-site models like TIP3P and SPC/E re-

quire computing nine intermolecular distances and

five-site models like ST2 and TIP5P require com-puting 17 intermolecular distances. Simulations

with the SSD potential are about seven and four

times faster than with three-site models in Monte

Carlo [13] and molecular dynamics simulations

[18], respectively.

Previous Monte Carlo and molecular dynamics

simulation results show that SSD water has good

structural, thermodynamic, dielectric, and dy-namic properties at room temperature [13,18,19].

However, the original parameters of the SSD

water model give low water density (0.957 g/cm3)

and low heat of vaporization energy at room

temperature and 1 atm, which may be improved by

optimizing the parameters used in the SSD po-

tential. Furthermore, the temperature dependence

of properties of SSD water has not been studied.For example, does SSD water have a temperature

of maximum density, one of the most well known

peculiar properties of real liquid water?

In this Letter, we further optimize the parame-

ters of the SSD water model to reproduce water

properties at room temperature and atmospheric

pressure. Then, molecular dynamics simulations

for SSD liquid water are performed at 1 atmpressure and at different temperatures ranging

from 230 to 330 K to locate the temperature of

maximum density.

2. Methods

Molecular dynamics simulations were carriedout in both the isothermal-isobaric and the mi-

crocanonical ensembles. Periodic boundary con-

ditions with the minimum image convention were

used with a cubic box of 256 water molecules. The

effect of system size was found to be minor in

previous MD simulations [18]. A spherical trun-

cation of the Lennard-Jones interaction was em-

ployed at half of the simulation box, with a longrange correction applied outside of the cutoff ra-

dius [20]. The Ewald method was used to treat the

long range dipole–dipole interactions with the di-

electric constant of the surrounding medium

�0 ¼ 1 and the convergence parameter a ¼ 6:4=Lwhere L is the box length [21,22]. The minimum

image convention was used for the real space

portion and a cut-off at k2max ¼ 54r�2 was used forthe reciprocal space portion in the Ewald sum [18].

The quaternion formulation was employed for the

rotational motion of a water molecule and the

leap-frog algorithm with a time step of 1 fs was

adapted for the integration over time. For the

starting configuration, the water molecules were

located on a face-centered cubic lattice with ran-

dom orientations.For the temperature dependence of the density,

molecular dynamics simulations were carried out

648 M.-L. Tan et al. / Chemical Physics Letters 376 (2003) 646–652

in the NPT ensemble at a pressure of 1 atm and

temperatures between 230 and 330 K at about

every 10 K. The water density was determined

from the average volume of the simulation box by

q ¼ M=ð0:6022� hV i=NÞ;where M is the molecular weight of water, N is the

number of water molecules in the cubic box, hV i isthe averaged volume and q is the density. Every

simulation at a given temperature was performed

long enough so that the average density did not

drift appreciably with simulation time. The equil-ibration and production times were 1.2 and 2.1 ns,

respectively, at T ¼ 230 and 240 K; 1.0 and 1.4 ns,

respectively, at T ¼ 250, 260, and 270 K; 0.8 and

1.2 ns, respectively, at T ¼ 280 K; and 0.6 and 0.9

ns, respectively, at T ¼ 290, 298, 310, 320 and 330

K. All properties reported here for T ¼ 298 K

except the dynamic and dielectric properties were

from the NPT simulation at T ¼ 298 K.For the dynamical and dielectric properties,

molecular dynamics simulations were carried out

in the NVE ensemble at an average temperature of

298 K. The diffusion constant was obtained by

calculating the long-time limit of the mean-square

displacement and the dielectric constant was ob-

tained by calculating the collective dipolar corre-

lation function [18]. For these simulations, thesystem was equilibrated for 800 ps and then the

diffusion and dielectric constants were calculated

over 3.3 ns.

3. Results

3.1. Optimization of parameters

Previous MD simulations in the NVE ensembleat the experimental density of 0.997 g/cm3 show

that SSD water with the original parameters [13]

has a pressure of 660 atm at room temperature,

which is somewhat higher than the experimental

pressure of 1 atm. MD simulations in the NPT

ensemble at room temperature and 1 atm yield a

density of 0.958 g/cm3, which is lower than the

experimental density of 0.997 g/cm3. Thus, theoriginal parameters were further optimized to re-

produce water density at room temperature and

atmospheric pressure, while maintaining other

properties.

A series of MD simulations were performed in

which the parameters in the Lennard-Jones, di-

pole, and sticky potentials were varied. The SSD

water density can be increased by decreasing theLennard-Jones diameter, increasing the dipole

moment, or increasing the sticky potential

strength. However, increasing the original SSD

dipole moment l ¼ 2:35 D, which is the same asthat of a TIP3P or SPC/E monomer, will increase

the already too high first peak in the O–O radial

distribution function and alter the good dielectric

properties. Thus, the Lennard-Jones and stickypotential parameters were varied at fixed l ¼ 2:35D. The original and optimized Lennard-Jones di-

ameters were r ¼ 3:051 �AA and r ¼ 3:016 �AA, re-spectively, and the original and optimized

strengths of the sticky potential parameter were

m0 ¼ 3:7284 kcal/mol and m0 ¼ 3:6613 kcal/mol,

respectively. Henceforth, the original and opti-

mized SSD parameters will be referred to as SSD0and SSD1, respectively.

The calculated liquid water properties with

SSD1 parameters at 298 K and 1 atm are an im-

provement over those with SSD0 parameters,

compared with experimental data (Table 1). The

water density q has been improved from 0.958 g/

cm3 for SSD0 to 0.990 g/cm3 for SSD1, which is

close to the experimental density of 0.997 g/cm3.The average intermolecular energy, E, has alsobeen improved from )9.23 to )9.61 kcal/mol,

compared to the experimental value of )9.86 kcal/mol [23] obtained by subtracting out the PV work

for expansion to the gas volume from the enthalpy

of vaporization [24].

The overall structure of SSD1 water, as seen in

the radial distribution functions gOOðrÞ, gOHðrÞ,and gHHðrÞ (Fig. 1), is in reasonably good agree-ment with recent neutron diffraction [25] and

X-ray scattering [26,27] measurements. The O–O

radial distribution (Fig. 1a) has two well-defined

peaks: the first is composed of the nearest neigh-

bors and the second is due to long range tetrahe-

dral order. For the first peak, the location rmax 1and the coordination number (found by integrat-ing the O–O radial distribution to 3.36 �AA, thefirst experimental minimum [25]) are close to

Table 1

Computed and experimental properties of water at 298 K

Model P(atm)

q(g/cm3)

�E(kcal/mol)

rmax 1(�AA)

Coordination number � D(10�5 cm2 s�1)

SSD0 660 0.997 9.60 2.78 4.7 80 2.24

SSD0 1 0.958 9.23 2.78 4.6 66 2.37

SSD1 1 0.990 9.61 2.74 4.6 72 2.13

SPC/E 1 1.007a 9.89b 2.75 4.7 68c 2.49d

TIP3P 1 1.002e 9.82e 2.77e 4.9 97d 5.06d

TIP4P 1 1.001e 10.06e 2.76e – 52f 3.29d

TIP5P 1 0.999e 9.87e 2.73e – 82e 2.62d

Expt 1 0.997g 9.86g 2.73h ; i 4.5h 78.3 j 2.30k

a Interpolated from [4].b From [2], T ¼ 306 K.c From [33].d From [34].e From [8].f From [35].g From [23,31].h From [25].i From [27].j From [30].k From [28,29].


experiment (Table 1), even though the peak issomewhat higher than the experimental peak.

More importantly, the presence of the second peak

for SSD1, which is not found in TIP3P, shows that

SSD1 has better tetrahedral structure than TIP3P

water. The second peak still exhibits a small flat-

tened area near 4.0–4.2 �AA and the peak is shifted to

larger r compared to SPC/E water and experi-

mental data, as in previous MC simulations ofSSD0 [13]. This may be partially improved by in-

creasing the strength of dipole and sticky poten-

tial, but stronger dipolar and sticky interactions

would sacrifice good dielectric properties and led

to an even higher first peak of O–O radial distri-

bution function. Nevertheless, the integration of

O–O radial distribution to 5.58 �AA yields coordi-

nation number 23.9, which is very close to thevalue of 23.5 obtained by integrating experimental

O–O radial distribution to second minimum of

5.58 �AA [25]. The overall structure of SSD1 water is

comparable to experiment, and more structured

than TIP3P, although it is not as good as SPC/E.

Dynamical and dielectric properties were also

investigated (Table 1). The translational self-diffu-

sion coefficient D of SSD1 is 2:13� 10�5 cm2/s,which is in good agreement with the experimental

value 2:30� 10�5 cm2/s [28,29]. The calculated di-electric constant � of SSD1 is 72, which is also ingood agreement with the experimental value 78 [30].

The SSD1 parameters improve SSD liquid water

properties at 298 K and 1 atm by reducing the di-

ameter of the Lennard-Jones sphere and decreasing

the strength of the sticky potential. This exemplifies

how in this model, the strength of the sticky po-

tential, which controls the hydrogen bond strength,can be varied independently of the dipole moment,

which controls the dielectric properties. In partic-

ular, the decrease in the diameter of Lennard-Jones

sphere, which is necessary to increase the density,

can be balanced by the increase in the strength of

the sticky potential so that the strength of the hy-

drogen bond at the minimum distance remains

reasonable. This is an advantage over multi-sitemodels, in which the dipole moment and the hy-

drogen bond energy are coupled because they are

both determined by the partial charges.

3.2. Temperature of maximum density

The temperature dependence of SSD1 water

density was studied in the temperature range be-tween 230 and 330 K and was compared with ex-

Fig. 2. Temperature dependence of water density at 1 atm.

Results are shown for SSD1, SPC/E [4], TIP3P [5], TIP4P [5],

TIP5P [8], and experiment [31].

Fig. 1. Radial distribution functions for liquid water at

T ¼ 298 K and P ¼ 1 atm for (a) O–O, (b) O–H, and (c) H–H.

Results are shown for SSD1, SPC/E, TIP3P, and experiment.


perimental data [31] and simulations of TIPnP

[5,8] and SPC/E [4] models (Fig. 2). The SSD1water model reproduces the experimental water

density near room temperature and at lower tem-

peratures, although the density decreases toorapidly as temperature increases above the TMD.

This is a common feature of non-polarizable water

models [4,5,8], which may be improved by allow-

ing electronic polarization of the models or

perhaps by making a softer repulsive wall. Nev-

ertheless, the temperature of maximum density for

SSD water was found to be around 260 K, which is

shifted somewhat lower than the experimentalTMD at 277 K. This is better than the three-site

TIP3P model, in which the densities monotonically

increase with decreasing temperature over the

temperature range discussed here [5], the three-site

SPC/E, in which the TMD is 235 K [4], and the

five-site ST2, in which the TMD is around 300 K

[7]. The SSD1 TMD is better than that of the four-

site TIP4P, which is around 248 K (Fig. 2) and theSSD1 density is generally closer to experiment

than TIP4P below about 300 K. Only the TMD of

the computationally intensive five-site TIP5P

model at 273 K [8] is significantly better.

The existence and value of the TMD is most

likely determined by the degree of tetrahedrality

[8]. Amongst the multi-site models, there is a

general trend in which increased tetrahedral or-dering as seen in the radial distribution functions

correlates with higher values of the TMD. In

particular, the TIP3P model does not have the

characteristic tetrahedral second peak in the O–O

distribution and also does not appear to have a


TMD while at the other extreme, the ST2 model is

generally thought to be too structured and has a

TMD above the experimental value. Since the

SSD1 potential has an explicitly tetrahedral po-

tential energy term, the results here further sup-

port the idea that tetrahedrality is important forthe TMD.

In terms of the interaction potential, the exis-

tence of TMD and other good properties of SSD1

water are most likely due to a good description of

the hydrogen bond interaction, which leads to the

overall tetrahedral structure of water. The angular

dependence of the energy of the SSD1 hydrogen

bonded dimer at the optimized oxygen–oxygendistance rOO was compared with TIPnP and SPC/Emodels (Fig. 3). The minimum in the hydrogen

bond energy for the SSD model is at a distance

rOO ¼ 2:74 �AA and a tilt angle h ¼ 47�, which is

more bent than the three-site models TIP3P with

h ¼ 27� and SPC/E with h ¼ 26�, and comparableto four-site model TIP4P with h ¼ 46� and five-sitemodel TIP5P with h ¼ 51�. The experimental va-lue for the gas phase D2O dimer [32] is h ¼ 57�.This good SSD dimer structure, governed mostly

by the angular sticky potential, gives a more tet-

rahedral network and is closer to real water than

the three-site models. This is further manifested in

the appearance of the second peak of the O–O

radial distribution function (Fig. 1a).

Fig. 3. Energy of linear water dimer as a function of dimer

angle h. Results are shown at the optimized O–O distances for

SSD1 at rOO ¼ 2:74 �AA, SPC/E at rOO ¼ 2:74 �AA, TIP3P at

rOO ¼ 2:74 �AA, TIP4P at rOO ¼ 2:76 �AA, and TIP5P at rOO ¼ 2:68�AA.

Overall, the sticky potential of the SSD1 model

describes the short range hydrogen bonding in-

teraction quite well by mimicking higher order

terms in a multipole expansion of electrostatic in-

teractions. A true multipole expansion requires

terms up to at least octupoles [16,17] and thus iscomputationally much more expensive than the

SSD model.

4. Conclusions

Classical molecular dynamics simulations of

SSD water with optimized parameters give goodstructural, thermodynamic, dielectric and dynamic

properties at 298 K and 1 atm. The temperature

dependence of the density of SSD water has also

been investigated and a temperature of maximum

density has been found at around 260 K at 1 atm.

The results are in good agreement with the ex-

perimental results. The success of this model may

be attributed to the short range sticky potentialintroduced in SSD model, which gives a good de-

scription of the tetrahedral hydrogen bonding of

liquid water.

As a one-site effective pairwise additive model,

SSD-type potentials significantly increase the

computational speed in computer simulations over

multi-site models while modeling many properties

quite well. Clearly, these features of SSD poten-tials will benefit simulations of biological macro-

molecules in solution.

Acknowledgements

We are grateful to the National Science Foun-

dation for the support of this work through Grantnumber MCB-0131780.

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A temperature of maximum density in soft sticky dipole water

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