CHARMM, AMBER, COMPASS, DREIDING, and OPLS force fields
=======================================================

Here we only discuss formulas implemented in LAMMPS that correspond to
formulas commonly used in the CHARMM, AMBER, COMPASS, and DREIDING force
fields.  Setting coefficients is done either from special sections in an
input data file via the :doc:`read_data <read_data>` command or in the
input script with commands like :doc:`pair_coeff <pair_coeff>` or
:doc:`bond_coeff <bond_coeff>` and so on.  See the :doc:`Tools <Tools>`
doc page for additional tools that can use CHARMM, AMBER, or Materials
Studio generated files to assign force field coefficients and convert
their output into LAMMPS input. LAMMPS input scripts can also be
generated by `charmm-gui.org <https://charmm-gui.org/>`_.

CHARMM and AMBER
----------------

The `CHARMM force field
<https://mackerell.umaryland.edu/charmm_ff.shtml>`_ :ref:`(MacKerell)
<howto-MacKerell>` and `AMBER force field
<https://ambermd.org/AmberModels.php>`_ :ref:`(Cornell) <howto-Cornell>`
have potential energy function of the form

.. math::

  V & = \sum_{bonds} E_b + \sum_{angles} \!E_a + \!\overbrace{\sum_{dihedral} \!\!E_d}^{\substack{
         \text{charmm} \\
        \text{charmmfsw}
      }} +\!\!\! \sum_{impropers} \!\!\!E_i \\[.6em]
      & \quad + \!\!\!\!\!\!\!\!\!\!\underbrace{~\sum_{pairs} \left(E_{LJ}+E_{coul}\right)}_{\substack{
         \text{lj/charmm/coul/charmm} \\
        \text{lj/charmm/coul/charmm/implicit} \\
        \text{lj/charmm/coul/long} \\
        \text{lj/charmm/coul/msm} \\
         \text{lj/charmmfsw/coul/charmmfsh} \\
        \text{lj/charmmfsw/coul/long}
      }} \!\!\!\!\!\!\!\!+ \!\!\sum_{special}\! E_s + \!\!\!\!\sum_{residues} \!\!\!{\scriptstyle\mathrm{CMAP}(\phi,\psi)}


The terms are computed by bond styles (relationship between two atoms),
angle styles (between 3 atoms) , dihedral/improper styles (between 4
atoms), pair styles (non-covalently bonded pair interactions) and
special bonds. The CMAP term (see :doc:`fix cmap <fix_cmap>` command for
details) corrects for pairs of dihedral angles ("Correction MAP") to
significantly improve the structural and dynamic properties of proteins
in crystalline and solution environments :ref:`(Brooks)
<howto-Brooks>`. The AMBER force field does not include the CMAP term.

The interaction styles listed below compute force field formulas that
are consistent with common options in CHARMM or AMBER.  See each
command's documentation for the formula it computes.

* :doc:`bond_style <bond_harmonic>` harmonic
* :doc:`angle_style <angle_charmm>` charmm
* :doc:`dihedral_style <dihedral_charmm>` charmmfsh
* :doc:`dihedral_style <dihedral_charmm>` charmm
* :doc:`pair_style <pair_charmm>` lj/charmmfsw/coul/charmmfsh
* :doc:`pair_style <pair_charmm>` lj/charmmfsw/coul/long
* :doc:`pair_style <pair_charmm>` lj/charmm/coul/charmm
* :doc:`pair_style <pair_charmm>` lj/charmm/coul/charmm/implicit
* :doc:`pair_style <pair_charmm>` lj/charmm/coul/long
* :doc:`special_bonds <special_bonds>` charmm
* :doc:`special_bonds <special_bonds>` amber

The pair styles compute Lennard Jones (LJ) and Coulombic interactions
with additional switching or shifting functions that ramp the energy
and/or force smoothly to zero between an inner :math:`(a)` and outer
:math:`(b)` cutoff. The older styles with *charmm* (not *charmmfsw* or
*charmmfsh*\ ) in their name compute the LJ and Coulombic interactions
with an energy switching function (esw) :math:`S(r)` which ramps the energy
smoothly to zero between the inner and outer cutoff. This can cause
irregularities in pairwise forces (due to the discontinuous second
derivative of energy at the boundaries of the switching region), which
in some cases can result in complications in energy minimization and
detectable artifacts in MD simulations.

.. grid:: 1 1 2 2

   .. grid-item::

      .. math::

         LJ(r) &= 4 \epsilon \left[ \left(\frac{\sigma}{r}\right)^{12} -
                  \left(\frac{\sigma}{r}\right)^6 \right]\\[.6em]
         C(r) &= \frac{C q_i q_j}{ \epsilon r}\\[.6em]
         S(r) &=  \frac{ \left(b^2 - r^2\right)^2 \left(b^2 + 2r^2 - 3{a^2}\right)}
                 { \left(b^2 - a^2\right)^3 }\\[.6em]
         E_{LJ}(r) &=  \begin{cases}
           LJ(r), & r \leq a \\
           LJ(r) S(r), & a < r \leq b \\
           0, &r > b
         \end{cases} \\[.6em]
         E_{coul}(r) &=  \begin{cases}
           C(r), & r \leq a \\
           C(r) S(r), & a < r \leq b \\
           0, & r > b
         \end{cases}

   .. grid-item::

      .. image:: img/howto_charmm_ELJ.png
         :align: center

The newer styles with *charmmfsw* or *charmmfsh* in their name replace
energy switching with force switching (fsw) for LJ interactions and
force shifting (fsh) functions for Coulombic interactions
:ref:`(Steinbach) <howto-Steinbach>`

.. grid:: 1 1 2 2

   .. grid-item::

      .. math::

           E_{LJ}(r) = & \begin{cases}
       4  \epsilon \sigma^6  \left(\frac{\displaystyle\sigma
         ^6-r^6}{\displaystyle r^{12}}-\frac{\displaystyle\sigma ^6}{\displaystyle a^6
         b^6}+\frac{\displaystyle 1}{\displaystyle a^3 b^3}\right) & r\leq a \\
       \frac{\displaystyle 4 \epsilon \sigma^6   \left(\sigma ^6
         \left(b^6-r^6\right)^2-b^3 r^6 \left(a^3+b^3\right)
         \left(b^3-r^3\right)^2\right)}{\displaystyle b^6 r^{12}
         \left(b^6-a^6\right)} & a<r \leq b\\
         0, & r>b
        \end{cases}\\[.6em]
         E_{coul}(r) & =  \begin{cases}
              C(r) \frac{\displaystyle (b-r)^2}{\displaystyle r b^2}, &  r \leq b \\
              0, & r > b
            \end{cases}

   .. grid-item::
      .. image:: img/howto_charmmfsw_ELJ.png
         :align: center

These styles are used by LAMMPS input scripts generated by
https://charmm-gui.org/ :ref:`(Brooks) <howto-Brooks>`.

.. note::

   For CHARMM, newer *charmmfsw* or *charmmfsh* styles were released in
   March 2017.  We recommend they be used instead of the older *charmm*
   styles.  See discussion of the differences on the :doc:`pair charmm
   <pair_charmm>` and :doc:`dihedral charmm <dihedral_charmm>` doc
   pages.

.. note::

  The TIP3P water model is strongly recommended for use with the CHARMM
  force field. In fact, `"using the SPC model with CHARMM parameters is
  a bad idea"
  <https://matsci.org/t/using-spc-water-with-charmm-ff/24715>`_ and `"to
  enable TIP4P style water in CHARMM, you would have to write a new pair
  style"
  <https://matsci.org/t/hybrid-pair-styles-for-charmm-and-tip4p-ew/32609>`_
  . LAMMPS input scripts generated by Solution Builder on https://charmm-gui.org
  use TIP3P molecules for solvation.  Any other water model can and
  probably will lead to false conclusions.

COMPASS
-------

COMPASS is a general force field for atomistic simulation of common
organic molecules, inorganic small molecules, and polymers which was
developed using ab initio and empirical parameterization techniques
:ref:`(Sun) <howto-Sun>`.  See the :doc:`Tools <Tools>` page for the
msi2lmp tool for creating LAMMPS template input and data files from
BIOVIA's Materials Studio files.  Please note that the msi2lmp tool is
very old and largely unmaintained, so it does not support all features
of Materials Studio provided force field files, especially additions
during the last decade.  You should watch the output carefully and
compare results, where possible.  See :ref:`(Sun) <howto-Sun>` for a
description of the COMPASS force field.

These interaction styles listed below compute force field formulas that
are consistent with the COMPASS force field.  See each command's
documentation for the formula it computes.

* :doc:`bond_style <bond_class2>` class2
* :doc:`angle_style <angle_class2>` class2
* :doc:`dihedral_style <dihedral_class2>` class2
* :doc:`improper_style <improper_class2>` class2

* :doc:`pair_style <pair_class2>` lj/class2
* :doc:`pair_style <pair_class2>` lj/class2/coul/cut
* :doc:`pair_style <pair_class2>` lj/class2/coul/long

* :doc:`special_bonds <special_bonds>` lj/coul 0 0 1

DREIDING
--------

DREIDING is a generic force field developed by the `Goddard group
<http://www.wag.caltech.edu>`_ at Caltech and is useful for predicting
structures and dynamics of organic, biological and main-group inorganic
molecules.  The philosophy in DREIDING is to use general force constants
and geometry parameters based on simple hybridization considerations,
rather than individual force constants and geometric parameters that
depend on the particular combinations of atoms involved in the bond,
angle, or torsion terms.  DREIDING has an :doc:`explicit hydrogen bond
term <pair_hbond_dreiding>` to describe interactions involving a
hydrogen atom on very electronegative atoms (N, O, F).  Unlike CHARMM or
AMBER, the DREIDING force field has not been parameterized for
considering solvents (like water) and has no rules for assigning
(partial) charges.  That will seriously limit its accuracy when used for
simulating systems where those matter.

See :ref:`(Mayo) <howto-Mayo>` for a description of the DREIDING force field

The interaction styles listed below compute force field formulas that
are consistent with the DREIDING force field.  See each command's
documentation for the formula it computes.

* :doc:`bond_style <bond_harmonic>` harmonic
* :doc:`bond_style <bond_morse>` morse

* :doc:`angle_style <angle_cosine_squared>` cosine/squared
* :doc:`angle_style <angle_harmonic>` harmonic
* :doc:`angle_style <angle_cosine>` cosine
* :doc:`angle_style <angle_cosine_periodic>` cosine/periodic

* :doc:`dihedral_style <dihedral_charmm>` charmm
* :doc:`improper_style <improper_umbrella>` umbrella

* :doc:`pair_style <pair_buck>` buck
* :doc:`pair_style <pair_buck>` buck/coul/cut
* :doc:`pair_style <pair_buck>` buck/coul/long
* :doc:`pair_style <pair_lj>` lj/cut
* :doc:`pair_style <pair_lj_cut_coul>` lj/cut/coul/cut
* :doc:`pair_style <pair_lj_cut_coul>` lj/cut/coul/long

* :doc:`pair_style <pair_hbond_dreiding>` hbond/dreiding/lj
* :doc:`pair_style <pair_hbond_dreiding>` hbond/dreiding/morse

* :doc:`special_bonds <special_bonds>` dreiding

OPLS
----

OPLS (Optimized Potentials for Liquid Simulations) is a general force
field for atomistic simulation of organic molecules in solvent.  It was
developed by the `Jorgensen group
<https://traken.chem.yale.edu/oplsaam.html>`_ at Purdue University and
later at Yale University.  Multiple versions of the OPLS parameters
exist for united atom representations (OPLS-UA) and for all-atom
representations (OPLS-AA).

This force field is based on atom types mapped to specific functional
groups in organic and biological molecules.  Each atom includes a
static, partial atomic charge reflecting the oxidation state of the
element derived from its bonded neighbors :ref:`(Jorgensen)
<howto-jorgensen>` and computed based on increments determined by the
atom type of the atoms bond to it.

The interaction styles listed below compute force field formulas that
are fully or in part consistent with the OPLS style force fields.  See
each command's documentation for the formula it computes.  Some are only
compatible with a subset of OPLS interactions.

* :doc:`bond_style <bond_harmonic>` harmonic
* :doc:`angle_style <angle_harmonic>` harmonic
* :doc:`dihedral_style <dihedral_opls>` opls
* :doc:`improper_style <improper_cvff>` cvff
* :doc:`improper_style <improper_fourier>` fourier
* :doc:`improper_style <improper_harmonic>` harmonic
* :doc:`pair_style <pair_lj_cut_coul>` lj/cut/coul/cut
* :doc:`pair_style <pair_lj_cut_coul>` lj/cut/coul/long
* :doc:`pair_modify <pair_modify>` geometric
* :doc:`special_bonds <special_bonds>` lj/coul 0.0 0.0 0.5

----------

.. _howto-MacKerell:

**(MacKerell)** MacKerell, Bashford, Bellott, Dunbrack, Evanseck, Field, Fischer, Gao, Guo, Ha, et al (1998).  J Phys Chem, 102, 3586 . https://doi.org/10.1021/jp973084f

.. _howto-Cornell:

**(Cornell)** Cornell, Cieplak, Bayly, Gould, Merz, Ferguson, Spellmeyer, Fox, Caldwell, Kollman (1995).  JACS 117, 5179-5197. https://doi.org/10.1021/ja00124a002

.. _howto-Steinbach:

**(Steinbach)** Steinbach, Brooks (1994). J Comput Chem, 15, 667. https://doi.org/10.1002/jcc.540150702

.. _howto-Brooks:

**(Brooks)** Brooks, et al (2009). J Comput Chem, 30, 1545. https://onlinelibrary.wiley.com/doi/10.1002/jcc.21287

.. _howto-Sun:

**(Sun)** Sun (1998). J. Phys. Chem. B, 102, 7338-7364. https://doi.org/10.1021/jp980939v

.. _howto-Mayo:

**(Mayo)** Mayo, Olfason, Goddard III (1990). J Phys Chem, 94, 8897-8909. https://doi.org/10.1021/j100389a010

.. _howto-Jorgensen:

**(Jorgensen)** Jorgensen, Tirado-Rives (1988). J Am Chem Soc, 110, 1657-1666. https://doi.org/10.1021/ja00214a001
