Continuum Solvent Non-Electrostatics¶

Abstract

Rules for implicit solvation¶

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

Consequence:¶

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

Tools:¶

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

Where:

$\sum\limits_k^{atoms}=$ 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

Benzene’s $\pi$ cloud can actually accept a proton to some extent. ↩
Assumes no difference in atom types. ↩

	\(\ce{H2O}\)	\(\ce{C6H6}\)	\(\ce{CH2Cl2}\)
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