# Computational Chemistry/Continuum solvation models

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### Introduction to Continuum Solvation Models

Most reactions we are concerned with actually go on in some kind of fluid medium rather than in the gas phase at zero degrees Kelvin. (There is the interesting but impractical philosophical point that the commonest reaction in the universe is H2 + proton >> H3+ at 3 degrees Kelvin, (the remnant of the big bang temperature), and very low pressure. Most of the universe is made of Hydrogen and most of its matter is in the vast spaces between the stars in galaxies.)
There are ways of dealing with the solvent medium problem, none of them fully satisfactory but some of them very good for dilute solutions. Read the electrochemistry and ionic atmosphere sections of Atkins' Physical Chemistry for a detailed discussion of some of the issues.

### Continuum plus Quantum Mechanics

Our method of choice will abandon a description of the solvent as discrete molecules but will replace it by a continuous dielectric. This is like a jelly which can be electrostatically strained by non zero potentials which both screen and interact with the quantum part of the system.

#### The Dielectric Constant

The dielectric constant, a dimensionless number which is a ratio of electrical permittivities against the permittivity of free space determines the chemical characteristics of the solvent. Large numbers are polar solvents. Small numbers apolar. Large numbers mean that the Coulomb terms are attenuated and though the energy still goes off as 1/r the values are smaller than in free space.
The following table is a conflation of data from the following primary and secondary sources which need to be consulted over the precise values to use in research.

#### Citations for Dielectric Constant Sources

M. Witanowski, W. Sicinska, Z. Biedrzycka, Z. Grabowski, and G. A. Webb, J. Chem. Soc., Perkin Trans. 2, 619 (1996); A. d' Aprano, A. Capalbi, M. Iammarino, V. Mauro, A. Princi, B. Sesta, J. Solution Chem., Vol. 24, 227, (1995); P. W. Atkins, Physical Chemistry, Oxford University Press (4th Edition); and CRC Handbook of Chemistry and Physics, ed. D. R. Lide, 77th edition, (CRC Press, Boca Raton,1996).
The numbers do not always agree, even though they nominally have up three decimal points of accuracy. A single list of recommended numbers has somewhat arbitarially been chosen from the above sources.
Table of Relative permittivities, (dielectric constant), at 25 deg. C.
| Free space | 1.00 | Ammonia (liquid) | 16.9~ |
| Methane | 1.70 | Acetone | 19.75 |
| Cyclohexane | 1.87 | Ethanol | 24.20 |
| Dioxane | 2.19 | Methanol | 30.71 |
| Carbon tetrachloride | 2.21 | Nitrobenzene | 34.82 |
| Benzene | 2.25 | CH$_3$CN | 36.05 |
| Diethyl-ether | 3.89 | CH$_3$NO$_2$ | 36.48 |
| CHCl$_3$ | 4.806 | DMSO | 45.80 |
| CH$_2$Cl$_2$ | 8.54 | Water | 78.54 |
| Hydrogen sulphide (liquid) | 9.26 | |
The molecule lives in a shape where the dielectric is 1, i.e. free space, and outside the quantum zone a continuous dielectric with a permittivity chosen according to the system being modelled. The energies of interaction are most negative when the dielectric constant is high. This can be rationalized by image charges in the dielectric being large near the junction and are screened away rapidly but are near to their opposite values in polar groups in the molecule. Clearly energy per atom, when decomposed will follow the atoms with highest Mulliken charge population.
Such a calculation would allow the prediction of the $\Delta H$ of the acetic acids discussed earlier but of course would still give the wrong answer because of the entropic terms being entirely absent from the model.
The solvation model for Macromodel uses functional group derived charges but of course explicitly uses the molecular mechanics shape of the molecule.

### Dielectric Screening

For some molecular mechanics application, particularly the calculation of long-range Coulomb forces inside macromolecules, a dielectric constant other than 1 may be used. (A constant of 4 is often used in protein modelling). A true quantum treatment would allow for the internal screening by the electrons but molecular mechanics can only deal with this by some kind of fix. Typically a dielectric constant equivalent to an alkane is used.
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