.. module:: gpaw.solvation
.. _continuum_solvent_model:

=============================
Continuum Solvent Model (CSM)
=============================

Theoretical Background
======================

A continuum solvent model (CSM) has been added to the GPAW
implementation as described in Reference [#HW14]_. The model is based
on a smooth cavity `g(\br)` represented on a real space grid. The
electrostatic screening effect of the solvent is modeled as a
spatially varying permittivity (relative dielectric constant)
`\epsilon(\br)`, which is related to the cavity by

.. math:: \epsilon(\br) = 1 + (\epsilon_{\infty} - 1) g(\br).

The bulk static dielectric constant `\epsilon_{\infty}` is taken from
experiment for the solvent in use (e.g. `\epsilon_{\infty} \approx 80`
for water at room temperature). The electrostatic energy for the total
charge distribution `\rho(\br)` of the solute (electron density +
atomic nuclei) is calculated as

.. math:: W_{\mathrm{el}} = \frac{1}{2} \int \rho(\br) \Phi(\br) \mathrm{d}\br

where `\Phi(\br)` is the electrostatic potential in the presence of
the dielectric. It is obtained by solving the Poisson equation
including the spatially varying permittivity:

.. math:: \nabla \Big(\epsilon(\br) \nabla \Phi(\br)\Big) = -4\pi \rho(\br)

This is done numerically with a finite difference multi-grid solver as
outlined in Reference [#SSS09]_.

As not only the electrostatic interaction but also cavity formation
and short-range interactions between the solute and the solvent
contribute to the solvation Gibbs energy, additional short-range
interacitons can be included in the calculation. The ansatz chosen in
Reference [#HW14]_ is to describe the cavity formation energy and all
short-range interactions as a term proportional to the surface area of
the cavity. The surface area is not well defined for a smooth cavity
`g(\br)`. The approach of Im. *et al.* [#IBR98]_ is taken, where the
surface area `A` is calculated from the gradient of the cavity:

.. math:: A = \int \| \nabla g(\br) \| \mathrm{d}\br

The cavity formation energy and short-range interaction is then given
as `G_{\mathrm{cav+sr}} = \gamma A` with a proportionality constant
`\gamma` that has the dimensions of a surface tension.

The cavity is constructed from an effective repulsive potential
`u(\br)` via the Boltzmann equation

.. math:: g(\br) = \exp \Big(-\frac{u(\br)}{k_{\mathrm{B}} T}\Big).

The effective potential is taken as the repulsive part of the
Lennard-Jones potential and constructed from the positions
`\mathbf{R}_a` of the atomic nuclei and a set of atomic radii
`R_a^{\mathrm{vdW}}` as

.. math:: u(\br) = u_0 \sum_a \Big(\frac{R_a^{\mathrm{vdW}}}{\|\br - \mathbf{R}_a \|} \Big)^{12}.

The single free parameter `u_0` describes the strength of the repulsion at
the atomic radii. It controls the size of the cavity and can be fitted
to experimental partial molar volumes (per atom) `\bar{v}_M^{\infty}`
at infinite dilution via the compressibility equation

.. math:: \int (1 - g(\br)) \mathrm{d}\br = \bar{v}_M^{\infty} - \kappa_T k_{\mathrm{B}} T,

where `\kappa_T` is the isothermal compressibility of the bulk solvent
(usually negligible compared to `\bar{v}_M^{\infty}` except for very
small molecules). For water as a solvent, Bondi van der Waals radii
with a modified value for hydrogen (1.09 Å) are a good
choice for the atomic radii of the effective repulsive potential
[#HW14]_.

Altogether, the model has three parameters. The static dielectric
constant `\epsilon_{\infty}` of the solvent is taken directly from
experimental data. The parameter `u_0` is fitted to experimental
volumes. Afterwards, the parameter `\gamma` can be fitted to
experimental solvation Gibbs energies. Parameters for water (at room
temperature) as a solvent are given in Reference [#HW14]_. A
preliminary parametrization for dimethylsulfoxide (DMSO) is also given
there. The accuracy for aqueous solvation Gibbs energies compared to
experimental values is about 5 kJ per mole for neutral molecules and
13 kJ per mole for cations. It is not recommended to apply the model
to anions in water without further modification as short-range
interactions between anions and water are not well represented using
the parameters optimized for neutral molecules [#HW14]_.

A lot of the parts that make up the model (cavity, dielectric
function, Poisson solver, non-electrostatic interactions) can be
replaced easily by alternative implementations as they are represented
as Python classes in the source code. Some alternative models already
exist in the GPAW source code, yet they are not well tested and
therefore not recommended for production use. Nevertheless they can
serve as an example on how to add your own solvation
interactions/models to GPAW. Also refer to Section III of Reference
[#HW14]_ for a more precise description of the general framework of
continuum solvent models with a smooth cavity employed in the GPAW
implementation. In the following section, the usage of the model
described above is documented in terms of a use case.

Usage Example: Ethanol in Water
===============================

As a usage example, the calculation of the solvation Gibbs energy of
ethanol in water is demonstrated. For simplicity, the solvation Gibbs
energy is calculated as the total energy difference of a calculation
with the continuum solvent model and a gas phase calculation using a
fixed geometry of the ethanol molecule. In principle, a relaxation of
the solute can be done also with the continuum solvent model. The
following annotated script demonstrates how to perform the calculation
using the continuum solvent model:

.. literalinclude:: ethanol_in_water.py

The calculated value for the solvation Gibbs energy should be about
-4.5 kcal per mole. The experimental value is -5.0 kcal per mole
[#KCT06]_. Please refer to the :git:`solvation module source code
<gpaw/solvation>` for further reading about the usage of the
:class:`~gpaw.solvation.calculator.SolvationGPAW` calculator class or model
specific parts.

There is also a helper function to use the solvation parameters for
water as in the above example:

.. literalinclude:: kwargs_factory.py

.. autoclass:: gpaw.solvation.calculator.SolvationGPAW


References
==========

.. [#HW14] A. Held and M. Walter,
           :doi:`Simplified continuum solvent model with a smooth cavity based on volumetric data <10.1063/1.4900838>`,
           *J. Chem. Phys.* **141**, 174108 (2014).

.. [#SSS09] V. M. Sanchez, M. Sued and D. A. Scherlis,
            :doi:`First-principles molecular dynamics simulations at solid-liquid interfaces with a continuum solvent <10.1063/1.3254385>`,
            *J. Chem. Phys.* **131** 174108 (2009).

.. [#IBR98] W. Im, D. Beglov and B. Roux,
            :doi:`Continuum solvation model: Computation of electrostatic forces from numerical solutions to the Poisson-Boltzmann equation <10.1016/S0010-4655(98)00016-2>`,
            *Comput. Phys. Commun.* **111** 59 (1998).

.. [#KCT06] C. P. Kelly, C. J. Cramer and D. G. Truhlar,
            :doi:`Aqueous Solvation Free Energies of Ions and Ion-Water Clusters Based on an Accurate Value for the Absolute Aqueous Solvation Free Energy of the Proton <10.1021/jp063552y>`,
            *J. Phys. Chem. B* **110** 16066 (2006)
