Water model

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A water model is defined by its geometry, together with other parameters such as the atomic charges and Lennard-Jones parameters.
A water model is defined by its geometry, together with other parameters such as the atomic charges and Lennard-Jones parameters.

In computational chemistry, classical water models are used for the simulation of liquid water and aqueous solutions (with so-called explicit solvent, as opposed to the implicit solvation models). These models generally use the approximations of molecular mechanics. Many different models have been proposed; they can be classified by the number of points used to define the model (atoms plus dummy sites), whether the structure is rigid or flexible, and whether the model includes polarization effects.

Contents

[edit] Simple 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 the Lennard-Jones potential.

E_{ab} = \sum_{i} ^{on\ a} \sum_{j} ^{on\ b} \frac {q_iq_j e^2}{r_{ij}} + \frac {A}{r_{OO}^{12}} - \frac {B}{r_{OO}^6}

The charged sites may be on the atoms or on dummy sites (such as lone pairs). The Lennard-Jones term typically applies to the oxygen atoms only.

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.

[edit] 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.

TIPS[1] SPC[2] TIP3P[3] SPC/E[4]
r(OH), Å 0.9572 1.0 0.9572 1.0
HOH, deg 104.52 109.47 104.52 109.47
A × 10−3, kcal Å12/mol 580.0 629.4 582.0 629.4
B, kcal Å6/mol 525.0 625.5 595.0 625.5
q(O) −0.80 −0.82 −0.834 −0.8476
q(H) +0.40 +0.41 +0.417 +0.4238

The SPC/E model adds an average polarization correction to the potential energy function:

E_{pol} = \frac 1 2 \sum_{i} \frac {(\mu - \mu^0)^2}{\alpha_i}

where μ is the dipole of the effectively polarized water molecule (2.35 D for the SPC/E model), μ0 is the dipole moment of an isolated water molecule (1.85 D from experiment), and αi is an isotropic polarizability constant, with a value of 1.608 × 10−40 F m. Since the charges in the model are constant, this correction just results in adding 1.25 kcal/mol (5.22 kJ/mol) to the total energy. The SPC/E model results in a better density and diffusion constant than the SPC model.

Other models:

  • Fergunson (flex. SPC)
  • CVFF (flex.)

[edit] 4-site

The 4-site models place the negative charge on a dummy atom (labeled M in the figure) placed near the oxygen along the bisector of the HOH angle. This improves the electrostatic distribution around the water molecule. The fist model to use this approach was the Bernal-Fowler model published in 1933, which may also be the earliest water model. However, the BF model doesn't reproduce well the bulk properties of water, such as density and heat of vaporization, and is therefore only of historical interest. This is a consequence of the parameterization method; newer models, developed after modern computers became available, were parameterized by running Metropolis Monte Carlo or molecular dynamics simulations and adjusting the parameters until the bulk properties are reproduced well enough.

  • BF[5]
  • TIPS2
  • TIP4P[3]
  • TIP4PF (flexible)
  • TIP4P-Ew
  • TIP4P-Ice

[edit] 5-site

The 5-site models place the negative charge on dummy atoms (labeled L) representing the lone pairs of the oxygen atom. In the case of the TIP5P model, this results in a better geometry for the water dimer, a more "tetrahedral" water structure that better reproduces the experimental radial distribution functions from neutron diffraction, and the temperature of maximum density of water.

[edit] 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[8]. It was found to reproduce the structure and melting of ice better than other models.

[edit] Other

  • Mercedes-Benz. 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.

[edit] 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).

[edit] See also

[edit] References

  1. ^ Jorgensen, W. L. Quantum and statistical mechanical studies of liquids. 10. Transferable intermolecular potential functions for water, alcohols, and ethers. Application to liquid water. J. Am. Chem. Soc. 1981, 103, 335-340.
  2. ^ H.J.C. Berendsen, J.P.M. Postma, W.F. van Gunsteren, and J. Hermans, In Intermolecular Forces, edited by B. Pullman (Reidel, Dordrecht, 1981), p. 331.
  3. ^ a b Jorgensen, W. L.; Chandrasekhar, J.; Madura, J.; Impey, R. W.; Klein, M. L. Comparison of simple potential functions for simulating liquid water. J. Chem. Phys 1983, 79, 926-935. DOI:10.1063/1.445869
  4. ^ H. J. C. Berendsen, J. R. Grigera, and T. P. Straatsma. The Missing Term in Effective Pair Potentials. J. Phys. Chem 1987, 91, 6269-6271. DOI:10.1021/j100308a038
  5. ^ Bernal, J. D.; Fowler, R.H. J. Chem. Phys. 1933, 1, 515.
  6. ^ F.H. Stillinger, A. Rahman, J. Chem. Phys. 60, 1545 (1974)
  7. ^ Mahoney, M. W.; Jorgensen, W. L. A five-site model liquid water and the reproduction of the density anomaly by rigid, non-polarizable models. J. Chem. Phys. 2000, 112, 8910-8922.
  8. ^ H. Nada, J.P.J.M. van der Eerden, J. Chem. Phys. 2003, 118, 7401. DOI:10.1063/1.1562610
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