Intermolecular Energy and Virial Function (Van der Waals)
=========================================================

In this section, the virial and energy equation of Van der Waals interaction for different potential function are discussed in details.

VDW
---

This option calculates potential energy without any truncation.

``Potential Calculation``
  Interactions between atoms can be modeled with an n-6 potential, a Mie potential in which the attractive exponent is fixed. The Mie potential can be viewed as a generalized version of the 12-6 Lennard-Jones potential,

  .. math:: 

    E_{\texttt{VDW}}(r_{ij}) = C_{n_{ij}} \epsilon_{ij} \bigg[\bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^{n_{ij}} - \bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^6\bigg]

  where :math:`r_{ij}`, :math:`\epsilon_{ij}`, and :math:`\sigma_{ij}` are, respectively, the separation, minimum potential, and collision diameter for the pair of interaction sites :math:`i` and :math:`j`. The constant :math:`C_n` is a normalization factor such that the minimum of the potential remains at :math:`-\epsilon_{ij}` for all :math:`n_{ij}`. In the 12-6 potential, :math:`C_n` reduces to the familiar value of 4.

  .. math:: 
    
    C_{n_{ij}} = \bigg(\frac{n_{ij}}{n_{ij} - 6} \bigg)\bigg(\frac{n_{ij}}{6} \bigg)^{6/(n_{ij} - 6)}

``Virial Calculation``
  Virial is basically the negative derivative of energy with respect to distance, multiplied by distance.

  .. math:: 

    W_{\texttt{VDW}}(r_{ij}) = -\frac{dE_{\texttt{VDW}}(r_{ij})}{r_{ij}}\times \frac{\overrightarrow{r_{ij}}}{{r_{ij}}} = F_{\texttt{VDW}}(r_{ij}) \times \frac{\overrightarrow{r_{ij}}}{{r_{ij}}}

  Using n-6 LJ potential defined above:

  .. math::

    F_{\texttt{VDW}}(r_{ij}) = 6C_{n_{ij}} \epsilon_{ij} \bigg[\frac{n_{ij}}{6} \times \bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^{n_{ij}} - \bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^6\bigg]\times \frac{1}{{r_{ij}}}

.. note:: This option only evaluates the energy up to specified ``Rcut`` distance. Tail correction to energy and pressure can be specified to account for infinite cutoff distance.

EXP6
----

This option calculates potential energy without any truncation.

``Potential Calculation``
  Interactions between atoms can be modeled with an exp-6 (Buckingham) potential,

  .. math:: 

    E_{\texttt{VDW}}(r_{ij}) =
    \begin{cases}
      \frac{\alpha_{ij}\epsilon_{ij}}{\alpha_{ij}-6} \bigg[\frac{6}{\alpha_{ij}} \exp\bigg(\alpha_{ij} \bigg[1-\frac{r_{ij}}{R_{min,ij}} \bigg]\bigg) - {\bigg(\frac{R_{min,ij}}{r_{ij}}\bigg)}^6 \bigg] &  r_{ij} \geq R_{max,ij} \\
      \infty & r_{ij} < R_{max,ij}
    \end{cases}

  where :math:`r_{ij}`, :math:`\epsilon_{ij}`, and :math:`R_{min,ij}` are, respectively, the separation, minimum potential, and minimum potential distance for the pair of interaction sites :math:`i` and :math:`j`. 
  The constant :math:`\alpha_{ij}` is an  exponential-6 parameter. The cutoff distance :math:`R_{max,ij}` is the smallest positive value for which :math:`\frac{dE_{\texttt{VDW}}(r_{ij})}{dr_{ij}}=0`.

  .. note::
    In order to use ``Mie`` or ``Exotice`` potential file format for ``Buckingham`` potential, instead of defining :math:`R_{min}`, we define :math:`\sigma` (collision diameter or the distance, where potential is zero) 
    and GOMC will calculate the :math:`R_{min}` and :math:`R_{max}` using ``Buckingham`` potential equation. 

``Virial Calculation``
  Virial is basically the negative derivative of energy with respect to distance, multiplied by distance.

  .. math:: 

    W_{\texttt{VDW}}(r_{ij}) = -\frac{dE_{\texttt{VDW}}(r_{ij})}{r_{ij}}\times \frac{\overrightarrow{r_{ij}}}{{r_{ij}}} = F_{\texttt{VDW}}(r_{ij}) \times \frac{\overrightarrow{r_{ij}}}{{r_{ij}}}

  Using exp-6 potential defined above:

  .. math::
    F_{\texttt{VDW}}(r_{ij}) =
    \begin{cases}
      \frac{6 \alpha_{ij}\epsilon_{ij}}{r_{ij}\big(\alpha_{ij}-6\big)} \bigg[\frac{r_{ij}}{R{min,ij}} \exp\bigg(\alpha_{ij} \bigg[1-\frac{r_{ij}}{R_{min,ij}} \bigg]\bigg) - {\bigg(\frac{R_{min,ij}}{r_{ij}}\bigg)}^6 \bigg] &  r_{ij} \geq R_{max,ij} \\
      \infty & r_{ij} < R_{max,ij}
    \end{cases}

.. note:: This option only evaluates the energy up to specified ``Rcut`` distance. Tail correction to energy and pressure can be specified to account for infinite cutoff distance.

.. figure:: static/VDW_Exp6.png
  :figwidth: 100%
  :width: 100%
  :align: center

  Graph of Van der Waals interaction for comparison of ``VDW`` and ``EXP6`` potentials.


SHIFT
-----
This option forces the potential energy to be zero at ``Rcut`` distance.

``Potential Calculation``
  Interactions between atoms can be modeled with an n-6 potential,
  
  .. math:: 

    E_{\texttt{VDW}}(r_{ij}) = C_{n_{ij}} \epsilon_{ij} \bigg[\bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^{n_{ij}} - \bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^6\bigg] - C_{n_{ij}} \epsilon_{ij} \bigg[\bigg(\frac{\sigma_{ij}}{r_{cut}}\bigg)^{n_{ij}} - \bigg(\frac{\sigma_{ij}}{r_{cut}}\bigg)^6\bigg]

  where :math:`r_{ij}`, :math:`\epsilon_{ij}`, and :math:`\sigma_{ij}` are, respectively, the separation, minimum potential, and collision diameter for the pair of interaction sites :math:`i` and :math:`j`. The constant :math:`C_n` is a normalization factor according to Eq. 3, such that the minimum of the potential remains at :math:`-\epsilon_{ij}` for all :math:`n_{ij}`. In the 12-6 potential, :math:`C_n` reduces to the familiar value of 4.

``Virial Calculation``
  Virial is basically the negative derivative of energy with respect to distance, multiplied by distance.

  .. math:: 

    W_{\texttt{VDW}}(r_{ij}) = -\frac{dE_{\texttt{VDW}}(r_{ij})}{r_{ij}}\times \frac{\overrightarrow{r_{ij}}}{{r_{ij}}} = F_{\texttt{VDW}}(r_{ij}) \times \frac{\overrightarrow{r_{ij}}}{{r_{ij}}}

  Using ``SHIFT`` potential function defined above:

  .. math::

    F_{\texttt{VDW}}(r_{ij}) = 6C_{n_{ij}} \epsilon_{ij} \bigg[\frac{n_{ij}}{6} \times \bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^{n_{ij}} - \bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^6\bigg]\times \frac{1}{{r_{ij}}}

  .. figure:: static/VDW_SHIFT.png

    Graph of Van der Waals potential with and without the application of the ``SHIFT`` function. With the ``SHIFT`` function active, the potential by force was reduced to 0.0 at the ``Rcut`` distance. With the ``SHIFT`` function, there is a discontinuity where the potential is truncated.

SWITCH
------
This option in ``CHARMM`` or ``EXOTIC`` force field smoothly forces the potential energy to be zero at Rcut distance and starts modifying the potential at Rswitch distance.

``Potential Calculation``
  Interactions between atoms can be modeled with an n-6 potential,

  .. math::
  
    E_{\texttt{VDW}}(r_{ij}) = C_{n_{ij}} \epsilon_{ij} \bigg[\bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^{n_{ij}} - \bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^6\bigg]\times \varphi_E(r_{ij})

  where :math:`r_{ij}`, :math:`\epsilon_{ij}`, and :math:`\sigma_{ij}` are, respectively, the separation, minimum potential, and collision diameter for the pair of interaction sites :math:`i` and :math:`j`. The constant :math:`C_n` is a normalization factor according to Eq. 3, such that the minimum of the potential remains at :math:`-\epsilon_{ij}` for all :math:`n_{ij}`. In the 12-6 potential, :math:`C_n` reduces to the familiar value of 4.

  The factor :math:`\varphi_E` is defined as:

  .. math::

    \varphi_E(r_{ij}) = 
    \begin{cases}
      1 & r_{ij} \leq r_{switch} \\
      \frac{\big({r_{cut}}^2 - {r_{ij}}^2 \big)^2 \times \big({r_{cut}}^2 - 3{r_{switch}}^2 + 2{r_{ij}}^2 \big)}{\big({r_{cut}}^2 - {r_{switch}}^2 \big)^3} & r_{switch} < r_{ij} < r_{cut} \\
      0 & r_{ij} \geq r_{cut}
    \end{cases}

``Virial Calculation``
  Virial is basically the negative derivative of energy with respect to distance, multiplied by distance.

  .. math:: 

    W_{\texttt{VDW}}(r_{ij}) = -\frac{dE_{\texttt{VDW}}(r_{ij})}{r_{ij}}\times \frac{\overrightarrow{r_{ij}}}{{r_{ij}}} = F_{\texttt{VDW}}(r_{ij}) \times \frac{\overrightarrow{r_{ij}}}{{r_{ij}}}

  Using SWITCH potential function defined above:

  .. math::

    F_{\texttt{VDW}}(r_{ij}) = \Bigg[6 C_{n_{ij}} \epsilon_{ij} \bigg[\frac{n_{ij}}{6} \times \bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^{n_{ij}} - \bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^6\bigg]\times \frac{\varphi_E(r_{ij})}{{r_{ij}}}  - 
    
    C_{n_{ij}} \epsilon_{ij} \bigg[\bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^{n_{ij}} - \bigg(\frac{\sigma_{ij}}{r_{ij}}\bigg)^6\bigg] \times \varphi_F(r_{ij}) \Bigg]

  The factor :math:`\varphi_F` is defined as:

  .. math::

    \varphi_F(r_{ij}) = 
    \begin{cases}
      0 & r_{ij} \leq r_{switch} \\
      \frac{12r_{ij}\big({r_{cut}}^2 - {r_{ij}}^2 \big) \times \big({r_{switch}}^2 - {r_{ij}}^2 \big)}{\big({r_{cut}}^2 - {r_{switch}}^2 \big)^3} & r_{switch} < r_{ij} < r_{cut} \\
      0 & r_{ij} \geq r_{cut}
    \end{cases}

  .. figure:: static/SWITCH.png

    Graph of Van der Waals potential with and without the application of the ``SWITCH`` function. With the ``SWITCH`` function active, the potential is smoothly reduced to 0.0 at the ``Rcut`` distance.

SWITCH (MARTINI)
----------------

This option in ``MARTINI`` force field smoothly forces the potential energy to be zero at Rcut distance and starts modifying the potential at ``Rswitch`` distance.

``Potential Calculation``
  Potential Calculation: Interactions between atoms can be modeled with an n-6 potential. In standard MARTINI, :math:`n` is equal to 12,

  .. math:: 

    E_{\texttt{VDW}}(r_{ij}) = C_{n_{ij}}\epsilon_{ij} \Bigg[ {\sigma_{ij}}^{n} \bigg(\frac{1}{{r_{ij}}^{n}} + \varphi_{E, n} (r_{ij}) \bigg) - {\sigma_{ij}}^{6} \bigg(\frac{1}{{r_{ij}}^{6}} + \varphi_{E, 6} (r_{ij}) \bigg) \Bigg]
	
  where :math:`r_{ij}`, :math:`\epsilon_{ij}`, and :math:`\sigma_{ij}` are, respectively, the separation, minimum potential, and collision diameter for the pair of interaction sites :math:`i` and :math:`j`. The constant :math:`C_n` is a normalization factor according to Eq. 3, such that the minimum of the potential remains at :math:`-\epsilon_{ij}` for all :math:`n_{ij}`. In the 12-6 potential, :math:`C_n` reduces to the familiar value of 4.

  The factor :math:`\varphi_{E, \alpha}` and constants are defined as:

  .. math::

    \varphi_{E, \alpha}(r_{ij}) = 
    \begin{cases}
      -C_{\alpha} & r_{ij} \leq r_{switch} \\
      -\frac{A_{\alpha}}{3} (r_{ij} - r_{switch})^3 -\frac{B_{\alpha}}{4} (r_{ij} - r_{switch})^4 - C_{\alpha} & r_{switch} < r_{ij} < r_{cut} \\
      0 & r_{ij} \geq r_{cut}
    \end{cases}

  .. math::

    A_{\alpha} = \alpha \frac{(\alpha + 1) r_{switch} - (\alpha +4) r_{cut}} {{r_{cut}}^{(\alpha + 2)} {(r_{cut} - r_{switch})}^2}

  .. math::

    B_{\alpha} = \alpha \frac{(\alpha + 1) r_{switch} - (\alpha +3) r_{cut}} {{r_{cut}}^{(\alpha + 2)} {(r_{cut} - r_{switch})}^3}

  .. math::

    C_{\alpha} =  \frac{1}{{r_{cut}}^{\alpha}} -\frac{A_{\alpha}}{3} (r_{cut} - r_{switch})^3 -\frac{B_{\alpha}}{4} (r_{cut} - r_{switch})^4

``Virial Calculation``
  Virial is basically the negative derivative of energy with respect to distance, multiplied by distance.

  .. math:: 

    W_{\texttt{VDW}}(r_{ij}) = -\frac{dE_{\texttt{VDW}}(r_{ij})}{r_{ij}}\times \frac{\overrightarrow{r_{ij}}}{{r_{ij}}} = F_{\texttt{VDW}}(r_{ij}) \times \frac{\overrightarrow{r_{ij}}}{{r_{ij}}}

  Using the ``SWITCH`` potential function defined for ``MARTINI`` force field:

  .. math::

    F_{\texttt{VDW}}(r_{ij}) = C_{n_{ij}}\epsilon_{ij} \Bigg[ {\sigma_{ij}}^{n} \bigg(\frac{n}{{r_{ij}}^{(n+1)}} + \varphi_{F, n} (r_{ij}) \bigg) - {\sigma_{ij}}^{6} \bigg(\frac{6}{{r_{ij}}^{(6+1)}} + \varphi_{F, 6} (r_{ij}) \bigg) \Bigg]
	
  The constants defined in Eq. 14-16 and the factor :math:`\varphi_{F, \alpha}` defined as:

  .. math::

    \varphi_{F, \alpha}(r_{ij}) = 
    \begin{cases}
      0 & r_{ij} \leq r_{switch} \\
      A_{\alpha} (r_{ij} - r_{switch})^2 + B_{\alpha} (r_{ij} - r_{switch})^3 & r_{switch} < r_{ij} < r_{cut} \\
      0 & r_{ij} \geq r_{cut}
    \end{cases}

  .. figure:: static/MARTINI.png

    Graph of Van der Waals potential with and without the application of the ``SWITCH`` function in ``MARTINI`` force field. With the ``SWITCH`` function active, the potential is smoothly reduced to 0.0 at the ``Rcut`` distance.