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Continuum Solvent Non-Electrostatics


Rules for implicit solvation

3. Electrostatics are only part of the free energy of solvation


You also need to somehow account for cavitation, dispersion, solvent structural changes etc.


You can always make electrostatics better at the same time

How to account for non-electrostatic terms

  • (worst) You could ignore them completely, though this would only really be valid for systems where electrostatic effects will dominate
  • Attempt to compute them separately e.g. one property, one calculation
  • (best) Assume proportionality to the solvent-accessible surface area and parameterise microscopic surface tensions
    • Continuum solvation is inherently semiempirical, so parameterisation should not be feared.

First solvation-shell contributions

One way to approximate the Solvent Accessible Surface Area (SASA) is to “roll a ball” over the surface of the molecule and the volume it takes up defines the



  • \(\sum\limits_k^{atoms}=\) Adding up the influence of each atom \(k\)
  • \(A_k=\) Surface area exposed of the atom \(k\)
  • \(\sigma_k=\) A characteristic surface tension, based on atomic number
  • \(\sum\limits_{k'}^{atoms}\sigma_{kk'}(R)=\) A modifier for the interaction of functional groups
    • \(\sum\limits_{k'}^{atoms}=\) Adding up the influence of each other atom (\(k'\))
    • \(\sigma_{kk'}(R)=\) looks at the distance between the two atoms (\(k\) and \(k'\)) and modifies the value of \(\sigma\)
\[ G_{CDS}=\sum\limits_k^{atoms}A_k\bigg(\sigma_k+\sum\limits_{k'}^{atoms}\sigma_{kk'}(R)\bigg) \]

The value of \(\sigma_{kk'}\) varies over distance, which will result in a gradual switching behaviour. In the figure below, the short distances might represent a ketone (the switch is on) and the longer distance might represent an ether (the switch is off)


Microscopic surface tensions

This simple equation shows the non-electrostatic solvation energy \(G_{CDS}\) is equal to the sum of the exposed surface area of each atom \(A_k\) times some proportionality constant \(\sigma_k\).

\[ G_{CDS}=\sum\limits_kA_k\sigma_k \]

E.g. 1. SMx universal solvation model

These are universal, because the surface tension (\(\sigma_k\)) will change based on the solvent

  • Surface tensions are treated as functions, rather than parameters
  • The value arrives from running over a series of descriptors (\(\sum_j^{descr}\)) and taking a parameter associated with that descriptor (\(\hat{\sigma}_{Z_i}\)) and multiplying it with the descriptor (\(\xi_j\)$)
\[ \sigma_i=\sum\limits_j^{descr}\hat{\sigma}_{Z_i}\xi_j \]

The descriptors could be (incomplete list):

  • \(n=\) solvent index of refraction (is a direct measure of polarisability of the solvent)
  • \(\gamma=\) solvent macroscopic surface tension (how hard it is to cavitate the solvent)
  • \(\alpha=\) Abraham h-bonding acidity (the ability of the solvent to h-bond as a proton donor)
  • \(\beta=\) Abraham h-bonding basicity (the ability of the solvent to h-bond as a proton acceptor)

To create these parameters

Take the experimental data, subtract the electrostatics to get the \(G_{CDS}\), we know the surface area, since we can calculate it, which leaves the unknown parameters.

After doing a big multilinear regression, we can determine the universal parameters of the solvent

\[ \Delta G_{aq,\:expt}-\Delta G_{ENP}=G_{CDS}=\sum\limits_kA_k\sigma_k \]

The result

SM8 has about 72 parameters for 2500 data (H, C, N, O, F, S, P, Cl, Br based compounds) in 91 solvents.

There is a mean error of \(\sim\pm0.6\:kcal\:mol^{-1}\) for neutral species and \(\pm3-6\:kcal\:mol^{-1}\) for ions (depending on the solvent)

Examples of solvent descriptors

\(\ce{H2O}\) \(\ce{C6H6}\) \(\ce{CH2Cl2}\)
Dielectric constant (\(\varepsilon\)) 78.36 2.27 8.93
Abraham h-bonding acidity (\(\alpha\)) 0.82 0.00 0.10
Abraham h-bonding basicity (\(\beta\)) 0.38 0.141 0.05
Refractive index (\(n\)) 1.33 1.50 1.42
Surface tension (\(cal\cdot mol^{-1}\cdot\unicode{x212B}^{-2}\)) 104.71 40.62 39.15
Carbon aromaticity 0.00 1.00 0.00
Electronegative halogenicity 0.00 0.00 0.67

SM8 Performance

Mean unsigned errors (kcal/mol) for SM8 compared to other models

Solute Class Data N SM8 IEFPCM (G03/UA0) C-PCM GAMESS PB Jaguar All Equal to Mean2
Aqueous neutrals 274 0.5 4.9 1.6 0.9 2.7
Non-aqueous neutrals 666 0.6 6.0 2.8 2.3 1.5
Aqueous ions 112 3.2 12.4 8.4 4.0 8.6
Non-aqueous ions 220 4.9 8.4 8.4 8.1 8.6

  1. Benzene’s \(\pi\) cloud can actually accept a proton to some extent. 

  2. Assumes no difference in atom types.