Non-bonded interactions
-----------------------

Non-bonded interactions in |Gromacs| are pair-additive:

.. math:: V(\mathbf{r}_1,\ldots \mathbf{r}_N) = \sum_{i<j}V_{ij}(\mathbf{r}_{ij});
          :label: eqnnbinteractions1

.. math:: \mathbf{F}_i = -\sum_j \frac{dV_{ij}(r_{ij})}{dr_{ij}} \frac{\mathbf{r}_{ij}}{r_{ij}}
          :label: eqnnbinteractions2

Since the potential only depends on the scalar distance, interactions
will be centro-symmetric, i.e. the vectorial partial force on particle
:math:`i` from the pairwise interaction :math:`V_{ij}(r_{ij})` has the
opposite direction of the partial force on particle :math:`j`. For
efficiency reasons, interactions are calculated by loops over
interactions and updating both partial forces rather than summing one
complete nonbonded force at a time. The non-bonded interactions contain
a repulsion term, a dispersion term, and a Coulomb term. The repulsion
and dispersion term are combined in either the Lennard-Jones (or 6-12
interaction), or the Buckingham (or exp-6 potential). In addition,
(partially) charged atoms act through the Coulomb term.

.. _lj:

The Lennard-Jones interaction
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The Lennard-Jones potential :math:`V_{LJ}` between two atoms equals:

.. math:: V_{LJ}({r_{ij}}) =  \frac{C_{ij}^{(12)}}{{r_{ij}}^{12}} -
                              \frac{C_{ij}^{(6)}}{{r_{ij}}^6}
          :label: eqnnblj

See also :numref:`Fig. %s <fig-lj>` The parameters :math:`C^{(12)}_{ij}` and
:math:`C^{(6)}_{ij}` depend on pairs of *atom types*; consequently they
are taken from a matrix of LJ-parameters. In the Verlet cut-off scheme,
the potential is shifted by a constant such that it is zero at the
cut-off distance.

.. _fig-lj:

.. figure:: plots/f-lj.*
   :width: 8.00000cm

   The Lennard-Jones interaction.

The force derived from this potential is:

.. math:: \mathbf{F}_i(\mathbf{r}_{ij}) = -\left( 12~\frac{C_{ij}^{(12)}}{{r_{ij}}^{13}} -
                                    6~\frac{C_{ij}^{(6)}}{{r_{ij}}^7} \right) {\frac{{\mathbf{r}_{ij}}}{{r_{ij}}}}
          :label: eqnljforce

The LJ potential may also be written in the following form:

.. math:: V_{LJ}(\mathbf{r}_{ij}) = 4\epsilon_{ij}\left(\left(\frac{\sigma_{ij}} {{r_{ij}}}\right)^{12}
          - \left(\frac{\sigma_{ij}}{{r_{ij}}}\right)^{6} \right)
          :label: eqnsigeps

In constructing the parameter matrix for the non-bonded LJ-parameters,
two types of combination rules can be used within |Gromacs|, only
geometric averages (type 1 in the input section of the force-field
file):

.. math:: \begin{array}{rcl}
          C_{ij}^{(6)}    &=& \left({C_{ii}^{(6)} \, C_{jj}^{(6)}}\right)^{1/2}    \\
          C_{ij}^{(12)}   &=& \left({C_{ii}^{(12)} \, C_{jj}^{(12)}}\right)^{1/2}
          \end{array}
          :label: eqncomb

or, alternatively the Lorentz-Berthelot rules can be used. An
arithmetic average is used to calculate :math:`\sigma_{ij}`, while a
geometric average is used to calculate :math:`\epsilon_{ij}` (type 2):

.. math:: \begin{array}{rcl}
          \sigma_{ij}   &=& \frac{1}{ 2}(\sigma_{ii} + \sigma_{jj})        \\
          \epsilon_{ij} &=& \left({\epsilon_{ii} \, \epsilon_{jj}}\right)^{1/2}
          \end{array}
          :label: eqnlorentzberthelot

finally an geometric average for both parameters can be used (type 3):

.. math:: \begin{array}{rcl}
          \sigma_{ij}   &=& \left({\sigma_{ii} \, \sigma_{jj}}\right)^{1/2}        \\
          \epsilon_{ij} &=& \left({\epsilon_{ii} \, \epsilon_{jj}}\right)^{1/2}
          \end{array}
          :label: eqnnbgeometricaverage

This last rule is used by the OPLS force field.

Buckingham potential
~~~~~~~~~~~~~~~~~~~~

The Buckingham potential has a more flexible and realistic repulsion
term than the Lennard-Jones interaction, but is also more expensive to
compute. The potential form is:

.. math:: V_{bh}({r_{ij}}) = A_{ij} \exp(-B_{ij} {r_{ij}}) -
                             \frac{C_{ij}}{{r_{ij}}^6}
          :label: eqnnbbuckingham

.. _fig-bham:

.. figure:: plots/f-bham.*
   :width: 8.00000cm

   The Buckingham interaction.

See also :numref:`Fig. %s <fig-bham>`. The force derived from this is:

.. math:: \mathbf{F}_i({r_{ij}}) = \left[ A_{ij}B_{ij}\exp(-B_{ij} {r_{ij}}) -
                                   6\frac{C_{ij}}{{r_{ij}}^7} \right] {\frac{{\mathbf{r}_{ij}}}{{r_{ij}}}}
          :label: eqnnbbuckinghamforce

.. _coul:

Coulomb interaction
~~~~~~~~~~~~~~~~~~~

The Coulomb interaction between two charge particles is given by:

.. math:: V_c({r_{ij}}) = f \frac{q_i q_j}{{\varepsilon_r}{r_{ij}}}
          :label: eqnvcoul

See also :numref:`Fig. %s <fig-coul>`, where
:math:`f = \frac{1}{4\pi \varepsilon_0} = {138.935\,458}` (see chapter :ref:`defunits`)

.. _fig-coul:

.. figure:: plots/vcrf.*
   :width: 8.00000cm

   The Coulomb interaction (for particles with equal signed charge) with
   and without reaction field. In the latter case
   :math:`{\varepsilon_r}` was 1, :math:`{\varepsilon_{rf}}` was 78, and
   :math:`r_c` was 0.9 nm. The dot-dashed line is the same as the dashed
   line, except for a constant.

The force derived from this potential is:

.. math:: \mathbf{F}_i(\mathbf{r}_{ij}) = -f \frac{q_i q_j}{{\varepsilon_r}{r_{ij}}^2}{\frac{{\mathbf{r}_{ij}}}{{r_{ij}}}}
          :label: eqnfcoul

A plain Coulomb interaction should only be used without cut-off or when
all pairs fall within the cut-off, since there is an abrupt, large
change in the force at the cut-off. In case you do want to use a
cut-off, the potential can be shifted by a constant to make the
potential the integral of the force. With the group cut-off scheme, this
shift is only applied to non-excluded pairs. With the Verlet cut-off
scheme, the shift is also applied to excluded pairs and self
interactions, which makes the potential equivalent to a reaction field
with :math:`{\varepsilon_{rf}}=1` (see below).

In |Gromacs| the relative dielectric constant :math:`{\varepsilon_r}` may
be set in the in the input for :ref:`grompp <gmx grompp>`.

.. _coulrf:

Coulomb interaction with reaction field
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The Coulomb interaction can be modified for homogeneous systems by
assuming a constant dielectric environment beyond the cut-off
:math:`r_c` with a dielectric constant of :math:`{\varepsilon_{rf}}`.
The interaction then reads:

.. math:: V_{crf} ~=~
          f \frac{q_i q_j}{{\varepsilon_r}{r_{ij}}}\left[1+\frac{{\varepsilon_{rf}}-{\varepsilon_r}}{2{\varepsilon_{rf}}+{\varepsilon_r}}
          \,\frac{{r_{ij}}^3}{r_c^3}\right]
          - f\frac{q_i q_j}{{\varepsilon_r}r_c}\,\frac{3{\varepsilon_{rf}}}{2{\varepsilon_{rf}}+{\varepsilon_r}}
          :label: eqnvcrf

in which the constant expression on the right makes the potential zero
at the cut-off :math:`r_c`. For charged cut-off spheres this corresponds
to neutralization with a homogeneous background charge. We can rewrite
:eq:`eqn. %s <eqnvcrf>` for simplicity as

.. math:: V_{crf} ~=~     f \frac{q_i q_j}{{\varepsilon_r}}\left[\frac{1}{{r_{ij}}} + k_{rf}~ {r_{ij}}^2 -c_{rf}\right]
          :label: eqnvcrfrewrite

with

.. math:: \begin{aligned}
          k_{rf}  &=&     \frac{1}{r_c^3}\,\frac{{\varepsilon_{rf}}-{\varepsilon_r}}{(2{\varepsilon_{rf}}+{\varepsilon_r})}
          \end{aligned}
          :label: eqnkrf

.. math:: \begin{aligned}
          c_{rf}  &=&     \frac{1}{r_c}+k_{rf}\,r_c^2 ~=~ \frac{1}{r_c}\,\frac{3{\varepsilon_{rf}}}{(2{\varepsilon_{rf}}+{\varepsilon_r})}
          \end{aligned}
          :label: eqncrf

For large :math:`{\varepsilon_{rf}}` the :math:`k_{rf}` goes to
:math:`r_c^{-3}/2`, while for :math:`{\varepsilon_{rf}}` =
:math:`{\varepsilon_r}` the correction vanishes. In :numref:`Fig. %s <fig-coul>` the
modified interaction is plotted, and it is clear that the derivative
with respect to :math:`{r_{ij}}` (= -force) goes to zero at the cut-off
distance. The force derived from this potential reads:

.. math:: \mathbf{F}_i(\mathbf{r}_{ij}) = -f \frac{q_i q_j}{{\varepsilon_r}}\left[\frac{1}{{r_{ij}}^2} - 2 k_{rf}{r_{ij}}\right]{\frac{{\mathbf{r}_{ij}}}{{r_{ij}}}}
          :label: eqnfcrf

The reaction-field correction should also be applied to all excluded
atoms pairs, including self pairs, in which case the normal Coulomb term
in :eq:`eqns. %s <eqnvcrf>` and :eq:`%s <eqnfcrf>` is absent.

.. _modnbint:

Modified non-bonded interactions
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In |Gromacs|, the non-bonded potentials can be modified by a shift
function, also called a force-switch function, since it switches the
force to zero at the cut-off. The purpose of this is to replace the
truncated forces by forces that are continuous and have continuous
derivatives at the cut-off radius. With such forces the time integration
produces smaller errors. But note that for Lennard-Jones interactions
these errors are usually smaller than other errors, such as integration
errors at the repulsive part of the potential. For Coulomb interactions
we advise against using a shifted potential and for use of a reaction
field or a proper long-range method such as PME.

There is *no* fundamental difference between a switch function (which
multiplies the potential with a function) and a shift function (which
adds a function to the force or potential) \ :ref:`72 <refSpoel2006a>`. The
switch function is a special case of the shift function, which we apply
to the *force function* :math:`F(r)`, related to the electrostatic or
van der Waals force acting on particle :math:`i` by particle :math:`j`
as:

.. math:: \mathbf{F}_i = c \, F(r_{ij}) \frac{\mathbf{r}_{ij}}{r_{ij}}
          :label: eqnswitch

For pure Coulomb or Lennard-Jones interactions
:math:`F(r) = F_\alpha(r) = \alpha \, r^{-(\alpha+1)}`. The switched
force :math:`F_s(r)` can generally be written as:

.. math::  \begin{array}{rcl}
           F_s(r)~=&~F_\alpha(r)   & r < r_1               \\
           F_s(r)~=&~F_\alpha(r)+S(r)      & r_1 \le r < r_c       \\
           F_s(r)~=&~0             & r_c \le r     
           \end{array}
           :label: eqnswitchforce

When :math:`r_1=0` this is a traditional shift function, otherwise it
acts as a switch function. The corresponding shifted potential function
then reads:

.. math:: V_s(r) =  \int^{\infty}_r~F_s(x)\, dx
          :label: eqnswitchpotential

The |Gromacs| **force switch** function :math:`S_F(r)` should be smooth at
the boundaries, therefore the following boundary conditions are imposed
on the switch function:

.. math:: \begin{array}{rcl}
          S_F(r_1)          &=&0            \\
          S_F'(r_1)         &=&0            \\
          S_F(r_c)          &=&-F_\alpha(r_c)       \\
          S_F'(r_c)         &=&-F_\alpha'(r_c)
          \end{array}
          :label: eqnswitchforcefunction

A 3\ :math:`^{rd}` degree polynomial of the form

.. math:: S_F(r) = A(r-r_1)^2 + B(r-r_1)^3
          :label: eqnswitchforcepoly

fulfills these requirements. The constants A and B are given by the
boundary condition at :math:`r_c`:

.. math:: \begin{array}{rcl}
          A &~=~& -\alpha \, \displaystyle
                  \frac{(\alpha+4)r_c~-~(\alpha+1)r_1} {r_c^{\alpha+2}~(r_c-r_1)^2} \\
          B &~=~& \alpha \, \displaystyle
                  \frac{(\alpha+3)r_c~-~(\alpha+1)r_1}{r_c^{\alpha+2}~(r_c-r_1)^3}
          \end{array}
          :label: eqnforceswitchboundary

Thus the total force function is:

.. math:: F_s(r) = \frac{\alpha}{r^{\alpha+1}} + A(r-r_1)^2 + B(r-r_1)^3
          :label: eqnswitchfinalforce

and the potential function reads:

.. math:: V_s(r) = \frac{1}{r^\alpha} - \frac{A}{3} (r-r_1)^3 - \frac{B}{4} (r-r_1)^4 - C
          :label: eqnswitchfinalpotential

where

.. math:: C =  \frac{1}{r_c^\alpha} - \frac{A}{3} (r_c-r_1)^3 - \frac{B}{4} (r_c-r_1)^4
          :label: eqnswitchpotentialexp

The |Gromacs| **potential-switch** function :math:`S_V(r)` scales the
potential between :math:`r_1` and :math:`r_c`, and has similar boundary
conditions, intended to produce smoothly-varying potential and forces:

.. math:: \begin{array}{rcl}
          S_V(r_1)          &=&1 \\
          S_V'(r_1)         &=&0 \\
          S_V''(r_1)        &=&0 \\
          S_V(r_c)          &=&0 \\
          S_V'(r_c)         &=&0 \\
          S_V''(r_c)        &=&0
          \end{array}
          :label: eqnpotentialswitch

The fifth-degree polynomial that has these properties is

.. math:: S_V(r; r_1, r_c) = \frac{1 - 10(r-r_1)^3(r_c-r_1)^2 + 15(r-r_1)^4(r_c-r_1) - 6(r-r_1)}{(r_c-r_1)^5}
          :label: eqn5polynomal

This implementation is found in several other simulation
packages,\ :ref:`73 <refOhmine1988>`\ :ref:`75 <refGuenot1993>` but
differs from that in CHARMM.\ :ref:`76 <refSteinbach1994>` Switching the
potential leads to artificially large forces in the switching region,
therefore it is not recommended to switch Coulomb interactions using
this function,\ :ref:`72 <refSpoel2006a>` but switching Lennard-Jones
interactions using this function produces acceptable results.

Modified short-range interactions with Ewald summation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

When Ewald summation or particle-mesh Ewald is used to calculate the
long-range interactions, the short-range Coulomb potential must also be
modified. Here the potential is switched to (nearly) zero at the
cut-off, instead of the force. In this case the short range potential is
given by:

.. math:: V(r) = f \frac{\mbox{erfc}(\beta r_{ij})}{r_{ij}} q_i q_j,
          :label: eqnewaldsrmod

where :math:`\beta` is a parameter that determines the relative weight
between the direct space sum and the reciprocal space sum and
erfc\ :math:`(x)` is the complementary error function. For further
details on long-range electrostatics, see sec. :ref:`lrelstat`.
