Simple Water Models

Theoretical studies of water frequently employ simple empirical models. Computer simulations can then predict properties of water which are to be compared with measurements.

In the commonly used interaction-site models (SPC, TIP3P, and TIP4P), a rigid geometry is adopted, and the interaction between them is calculated via pairwise summation. These models typically use three to five interaction sites. Internal changes in flexible conformation of water are often regarded too cumbersome for computer simulation studies.

The TIP3P [12] and SPC [1] models use three sites for the electrostatic interactions. The partial positive charges on the hydrogen atoms are balanced by an appropriately negative charge located on the oxygen atom. The intermolecular interaction between two water molecules is computed using a Lennard-Jones type potential with just a single interaction point per molecule centered on the oxygen atom. No van der Waals interactions involving the hydrogen atoms are calculated. The TIP3P and SPC models differ slightly in parameters. Table 1 summarizes several water models, which also includes SPC/E [2], an updated version of the SPC model. The four-site models such as that of Bernal and Fowler [3] and TIP4P [12] model shift the negative charge from the oxygen atom to a point along the bisector of the HOH angle towards the hydrogen atoms (Fig. 5). The parameters for these two models are also given in Table 1.

Table 1: A comparison of various water models. For the ST2 potential, q(M) is the charge on the lone pairs, which are at a distance 0.8 Å from the oxygen atom (see Fig. 6) [12].

	SPC	SPC/E	TIP3P	BF	TIP4P	ST2
r(OH), $\AA$	1.0	1.0	0.9572	0.96	0.9572	1.0
HOH, $\deg$	109.47	109.47	104.52	105.7	104.52	109.47
A$\times10^{-3}$,kcal $\AA^{12}$/mol	629.4	629.4	582.0	560.4	600	238.7
C, kcal $\AA^{6}$/mol	625.5	625.5	595.0	837.0	610.0	268.9
q(O)	-0.82	-0.8472	-0.834	0.0	0.0	0.0
q(H)	0.41	0.4238	0.417	0.49	0.52	0.2375
q(M)	0.0	0.0	0.0	-0.98	-1.04	0
q(LP)	0	0	0	0	0	-0.2375
r(OM), $\AA$	0.0	0.0	0.0	0.15	0.15	0.8

Figure 5: Some simple water models.

The most commonly used five-site model is the ST2 potential of Stillinger and Rahman [25]. Here, charges are placed on the hydrogen atoms and on the two lone-pair sites on the oxygen. The electrostatic contribution is modulated so that for oxygen-oxygen distances below $2.016$ Å it is zero, and for distances greater than $3.1287$ Å it takes its full value. Between these two distances the electrostatic contribution is modulated using a function that smoothly varies from 0.0 at the shorter distance to 1.0 at the longer distance.

The experimental determined dipole moment of a water molecule in the gas phase is $1.85$ D. The dipole moment of an individual water molecule calculated with any of these simple models is significantly higher. For example, the dipole moment in SPC is $2.27$ D and in TIP4P it is $2.18$ D. These values are closer to the effective dipole moment of liquid water, which is approximately $2.6$ D. These models are thus all effective mean-field, pairwise, models. The simple water models are usually modified until the desired level of agreement between the selected experiments and theory is achieved.

Thermodynamic and structural properties, such as density, radial distribution function, enthalpy of vaporisation, heat capacity, diffusion coefficient and dielectric constant, are usually used in fitting these water model parameters. It is found that some properties such as the density and the enthalpy of vaporisation are predicted rather well by all models, but values for other properties such as the dielectric constant fared less well [12]. When comparing different models, one should also take into account the computational cost.

The rigid model of water is an approximation that some properties can not be calculated. Only when internal flexibility is included the vibrational spectrum can be calculated and compared with experiment. Flexibility is most easily incorporated by adding bond-stretching and angle-bending terms to the potential function in a rigid model. For example, Ferguston has developed a flexible model for water that is based upon the SPC model [6]. The partial charges and van der Waals parameters in this model are slightly different from those in the rigid model, and flexibility is achieved using cubic and harmonic bond-stretching terms and a harmonic angle-bending term. The calculated values are compared favorably with experimental results for a wide range of thermodynamic and structural properties, including the dielectric constant and self-diffusion coefficient.