update dynamical_matrix docs

Calculate the dynamical matrix by finite difference of the selected group,

.. image:: JPG/dynamical_matrix_dynmat.jpg

   :align: center

.. math::

where D is the dynamical matrix and Phi is the force constant matrix defined by

   D = \frac{\Phi_{ij}^{\alpha\beta}}{\sqrt{M_i M_j}}

.. image:: JPG/dynamical_matrix_force_constant.jpg

   :align: center

where D is the dynamical matrix and :math:`\Phi` is the force constant

matrix defined by

The output for the dynamical matrix is printed three elements at a time. The

three elements are the three beta elements for a respective i/alpha/j combination. 

Each line is printed in order of j increasing first, alpha second, and i last.

.. math::

If the style eskm is selected, the dynamical matrix will be in units of inverse squared

femtoseconds. These units will then conveniently leave frequencies in THz, where

frequencies, represented as omega, can be calculated from

   \Phi_{ij}^{\alpha\beta} = \frac{\partial^2 U}{\partial x_{i,\alpha} \partial x_{j,\beta}}

:c, image(Eqs/dynamical\_matrix\_phonons.jpg)

The output for the dynamical matrix is printed three elements at a time.

The three elements are the three :math:`\beta` elements for a respective

i/:math:`\alpha`/j combination.  Each line is printed in order of j

increasing first, :math:`\alpha` second, and i last.

If the style eskm is selected, the dynamical matrix will be in units of

inverse squared femtoseconds. These units will then conveniently leave

frequencies in THz.

Restrictions

""""""""""""

The command collects an array of nine times the number of atoms in a group

on every single MPI rank, so the memory requirements can be very significant

for large systems.

This command is part of the USER-PHONON package.  It is only enabled if

LAMMPS was built with that package.  See the :doc:`Build package <Build_package>` doc page for more info.

LAMMPS was built with that package.

See the :doc:`Build package <Build_package>` doc page for more info.

Related commands

""""""""""""""""

:doc:`fix phonon <fix_phonon>`

:doc:`compute hma <compute_hma>` uses an analytic formulation of the hessian

provided by Pair's single\_hessian.

:doc:`compute hma <compute_hma>` uses an analytic formulation of the

Hessian provided by a pair_style's Pair::single\_hessian() funcion,

if implemented.

Default

"""""""