Update pair_agni docs

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\begin{eqnarray*}

  F_i^u & = & \sum_t^{N_t}\alpha_t \cdot \exp\left[-\frac{\left(d_{i,t}^u\right)^2}{2l^2}\right] \\

  d_{i,t}^u & = & \left|\left| V_i^u(\eta) - V_t^u(\eta) \right|\right| \\

  V_i^u(\eta) & = & \sum_{j \neq i}\frac{r^u_{ij}}{r_{ij}} \cdot e^{-\left(\frac{r_{ij}}{\eta} \right)^2} \cdot f_d\left(r_{ij}\right) \\

  f_d\left(r_{ij}\right) & = & \frac{1}{2} \left[\cos\left(\frac{\pi r_{ij}}{R_c}\right) + 1 \right]

\end{eqnarray*}

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.. index:: pair\_style agni

.. index:: pair_style agni

pair\_style agni command

pair_style agni command

========================

pair\_style agni/omp command

pair_style agni/omp command

============================

Syntax

""""""

.. parsed-literal::

.. code-block:: LAMMPS

   pair_style agni

Examples

""""""""

pair\_style      agni

pair\_coeff      \* \* Al.agni Al

.. code-block:: LAMMPS

   pair_style      agni

   pair_coeff      * * Al.agni Al

Description

"""""""""""

Style *agni* style computes the many-body vectorial force components for

an atom as

.. image:: Eqs/pair_agni.jpg

   :align: center

.. math::

   F_i^u                  & = \sum_t^{N_t}\alpha_t \cdot \exp\left[-\frac{\left(d_{i,t}^u\right)^2}{2l^2}\right] \\

   d_{i,t}^u              & = \left|\left| V_i^u(\eta) - V_t^u(\eta) \right|\right| \\

   V_i^u(\eta)            & = \sum_{j \neq i}\frac{r^u_{ij}}{r_{ij}} \cdot e^{-\left(\frac{r_{ij}}{\eta} \right)^2} \cdot f_d\left(r_{ij}\right) \\

   f_d\left(r_{ij}\right) & = \frac{1}{2} \left[\cos\left(\frac{\pi r_{ij}}{R_c}\right) + 1 \right]

*u* labels the individual components, i.e. x, y or z, and *V* is the

corresponding atomic fingerprint. *d* is the Euclidean distance between

any two atomic fingerprints. A total of N\_t reference atomic

environments are considered to construct the force field file. *alpha\_t*

and *l* are the weight coefficients and length scale parameter of the

:math:`u` labels the individual components, i.e. :math:`x`, :math:`y` or :math:`z`, and :math:`V` is the

corresponding atomic fingerprint. :math:`d` is the Euclidean distance between

any two atomic fingerprints. A total of :math:`N_t` reference atomic

environments are considered to construct the force field file. :math:`\alpha_t`

and :math:`l` are the weight coefficients and length scale parameter of the

non-linear regression model.

The method implements the recently proposed machine learning access to

Only a single pair\_coeff command is used with the *agni* style which

specifies an AGNI potential file containing the parameters of the

force field for the needed elements. These are mapped to LAMMPS atom

types by specifying N additional arguments after the filename in the

pair\_coeff command, where N is the number of LAMMPS atom types:

types by specifying :math:`N` additional arguments after the filename in the

pair\_coeff command, where :math:`N` is the number of LAMMPS atom types:

* filename

* N element names = mapping of AGNI elements to atom types

* :math:`N` element names = mapping of AGNI elements to atom types

See the :doc:`pair_coeff <pair_coeff>` doc page for alternate ways

to specify the path for the force field file.