Treatment of Solvation Effects with COSMO

The Conductor-like Screening Model [89] (COSMO) is a continuum solvation model (CSM), where the solute molecule forms a cavity within the dielectric continuum of permittivity $\varepsilon$ that represents the solvent. The charge distribution of the solute polarizes the dielectric medium. The response of the medium is described by the generation of screening charges on the cavity surface.

CSMs usually require the solution of the rather complicated boundary conditions for a dielectric in order to obtain the screening charges. COSMO instead uses the much simpler boundary condition of vanishing electrostatic potential for a conductor,

$${\bf\Phi}^{tot} = 0.$$

This represents an electrostatically ideal solvent with $\varepsilon=\infty$. The vector of total electrostatic potential on the cavity surface segments is determined by the solute potential ${\bf\Phi}^{sol}$, which consist of the electronic and the nuclear part, and the vector of screening charges $\bf q$,

$${\bf\Phi}^{tot} = {\bf\Phi}^{sol}+ {\bf Aq} = 0.$$

$\bf A$ is the Coulomb matrix of the screening charge interactions. For a conductor, the boundary condition ${\bf\Phi}^{tot}=0$ defines the screening charges as

$${\bf q } = -{\bf A}^{-1}{\bf\Phi}^{sol}.$$

To take into account the finite permittivity of real solvents, the screening charges are scaled by a factor.

$$f(\varepsilon) = \frac{\varepsilon-1}{\varepsilon+\frac{1}{2}}$$

$${\bf q}^{\star} = f(\varepsilon){\bf q }$$

The deviations of this COSMO approximation from the exact solution are rather small. For strong dielectrics like water they are less than 1%, while for non-polar solvents with $\varepsilon\approx 2$ they may reach 10% of the total screening effects. However, for weak dielectrics, screening effects are small, and the absolute error therefore typically amounts to less than one kcal/mol. The dielectric energy, i.e. the free electrostatic energy gained by the solvation process, is half of the solute-solvent interaction energy.

$$E_{diel} = \frac{1}{2}f(\epsilon){\bf q}^{\dagger}{\bf\Phi}^{sol}$$

The total free energy of the solvated molecule is the sum of the energy of the isolated system calculated with the solvated wave function and the dielectric energy.

A COSMO energy calculation starts with the construction of the cavity surface grid. Within the SCF procedure, the screening charges are calculated in every cycle and the potential generated by these charges is included into the Hamiltonian. This ensures a variational optimization of both the molecular orbitals and the screening charges, which then also allows the evaluation of analytic gradients.