- Water model
effects.

An alternative to the explicit water models is to use an

implicit solvation model, also known as a continuum model.**imple water models**The simplest water models treat the water molecule as rigid and rely only on non-bonded interactions. The electrostatic interaction is modeled using

Coulomb's law and the dispersion and repulsion forces using theLennard-Jones potential . The potential for models such as TIP3P and TIP4P is represented by:$E\_\{ab\}\; =\; sum\_\{i\}\; ^\{on\; a\}\; sum\_\{j\}\; ^\{on\; b\}\; frac\; \{k\_Cq\_iq\_j\}\{r\_\{ij\; +\; frac\; \{A\}\{r\_\{OO\}^\{12\; -\; frac\; \{B\}\{r\_\{OO\}^6\}$

where "k

_{C}", theelectrostatic constant , has a value of 332.1 Å·kcal/mol in the units commonly used in molecular modeling; "q_{i}" are thepartial charge s relative to the charge of the electron; "r_{ij}" is the distance between two atoms or charged sites; and "A" and "B" are the Lennard-Jones parameters. The charged sites may be on the atoms or on dummy sites (such as lone pairs). In most water models, the Lennard-Jones term applies only to the interaction between the oxygen atoms.The figure below shows the general shape of the 3- to 6-site water models. The exact geometric parameters (the OH distance and the HOH angle) vary depending on the model.

:

**3-site**The simplest models have three interaction sites, corresponding to the three atoms of the water molecule. Each atom gets assigned a point charge, and the oxygen atom also gets the Lennard-Jones parameters. The 3-site models are very popular for

molecular dynamics simulations because of their simplicity and computational efficiency. Most models use a rigid geometry matching the known geometry of the water molecule. An exception is the SPC model, which assumes an ideal tetrahedral shape (HOH angle of 109.47°) instead of the observed angle of 104.5°.The table below lists the parameters for some 3-site models.

Note, however, that the BNS and ST2 models do not use Coulomb's law directly for the electrostatic terms, but a modified version that is scaled down at short distances by multiplying it by the switching function "S(r)":

:$S(r\_\{ij\})\; =\; egin\{cases\}\; 0,\; mbox\{if\; \}r\_\{ij\}\; le\; R\_L\; \backslash \; frac\{(r\_\{12\}\; -\; R\_L)^2(3R\_U\; -\; R\_L\; -\; 2r\_\{ij\})\}\{(R\_U\; -\; R\_L)^2\},\; mbox\{if\; \}R\_L\; le\; r\_\{ij\}\; le\; R\_U\; \backslash \; 1,\; mbox\{if\; \}R\_U\; le\; r\_\{ij\}end\{cases\}$

Therefore the "R

_{L}" and "R_{U}" parameters only apply to BNS and ST2.**6-site**A 6-site model that combines all the sites of the 4- and 5-site models was developed by Nada and van der Eerden [

*H. Nada, J.P.J.M. van der Eerden, "J. Chem. Phys."*] . It was found to reproduce the structure and melting of ice better than other models.**2003**, "118", 7401. doi|10.1063/1.1562610**Other***MB model. A more abstract model resembling the

Mercedes-Benz logo that reproduces some features of water in two-dimensional systems. It is not used as such for simulations of "real" (i.e., three-dimensional) systems, but it is useful for qualitative studies and for educational purposes. [*K. A. T. Silverstein, A. D. J. Haymet, and K. A. Dill. A Simple Model of Water and the Hydrophobic Effect. "J. Am. Chem. Soc."*]**1998**, "120", 3166-3175. doi|10.1021/ja973029k

*Coarse-grained models. One- and two-site models of water have also been developed. [*S. Izvekov, G. A. Voth. Multiscale coarse graining of liquid-state systems "J. Chem. Phys.*] In coarse grain models, each site can represent several water molecules.**2005**, "123", 134105. doi|10.1063/1.2038787**Computational cost**The computational cost of a water simulation increases with the number of interaction sites in the water model. The CPU time is approximately proportional to the number of interatomic distances that need to be computed. For the 3-site model, 9 distances are required for each pair of water molecules (every atom of one molecule against every atom of the other molecule, or 3 × 3). For the 4-site model, 10 distances are required (every charged site with every charged site, plus the O-O interaction, or 3 × 3 + 1). For the 5-site model, 17 distances are required (4 × 4 + 1). Finally, for the 6-site model, 26 distances are required (5 × 5 + 1).

When using rigid water models in molecular dynamics, there is an additional cost associated with keeping the structure constrained, using constraint algorithms (although with bond lengths constrained it is often possible to increase the time step).

**ee also***

Force field (chemistry) **References**

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