CHARMM Force Field
------------------

In ABCrystal, the function to assess the crystal stability is enthalpy:

.. math::
    H = U + PV

Here, :math:`P` is the external pressure in bar, :math:`V` is the cell volume, and :math:`U` is the internal potential energy described by CHARMM force field:

.. math::
    U = \sum_{I}\sum_{J<I}u_{IJ\mathbf{0}}+\frac{1}{2}\sum_{\mathbf{n}\ne\mathbf{0}}\sum_{I}\sum_{J}u_{IJ\mathbf{n}}

.. math::
    \begin{split}
    u_{IJ\mathbf{n}} = & \sum_{i \in I}\sum_{j \in J}\left(
    \frac{e^{2}}{4\pi\epsilon_{0}}\frac{q_{i}q_{j}}{r_{ij\mathbf{n}}}
    +4\epsilon_{ij}\left(\left(\frac{\sigma_{ij}}{r_{ij\mathbf{n}}}\right)^{12}-\left(\frac{\sigma_{{ij}}}{r_{ij\mathbf{n}}}\right)^{6}\right)
    \right) \\
    \end{split}

.. math::
    r_{ij\mathbf{n}} = \left|\mathbf{r}_i-\mathbf{r}_j+n_1\mathbf{T}_1+n_2\mathbf{T}_2+n_3\mathbf{T}_3 \right|

In this formula, :math:`I` and :math:`J` are indices for molecules; :math:`i` and :math:`j` are indices for atoms in molecule :math:`I` and :math:`J`, respectively; :math:`N` is the total number of molecules; :math:`\mathbf{T}_1,\mathbf{T}_2,\mathbf{T}_3` are the 3 vectors describing the cell; :math:`\mathbf{n}` is 3 integers :math:`n_1,n_2,n_3`. The first term describes the intermolecular Coulomb and Lennard-Jones interactions; the second term describes the interactions between molecules and external static electric field. For a molecule, you have to provide CHARMM parameters: charge :math:`q`, Lennard-Jones well depth :math:`\epsilon` and width :math:`\sigma`. Their units are:

+-------------------+------------------+
| Parameter         | Unit             |
+===================+==================+
| :math:`q`         | e                |
+-------------------+------------------+
| :math:`\epsilon`  | kJ mol :sup:`-1` |
+-------------------+------------------+
| :math:`\sigma`    | Å                |
+-------------------+------------------+
| :math:`P`         | bar              |
+-------------------+------------------+

For many molecules, the CHARMM parameters :math:`q`, :math:`\epsilon`, and :math:`\sigma` have been provided in ABCrystal distributions ``misc/charmm36``. You can directly use these parameter files.

.. tip::

    These CHARMM parameter files are provided in: ``misc/charmm36``. For example, the parameter file for water :math:`\mathrm{H}_2\mathrm{O}` and potassium cation :math:`\mathrm{K}^{+}` is ``tip4p.xyz`` and ``k.xyz``, respectively.