Polished the doc files a bit

          *1* = generate quadratic terms

       *chem* values = *nelements* *elementlist*

          *nelements* = number of SNAP elements

          *elementlist* = *ntypes* element indices [0, *nelements* ) 

          *elementlist* = *ntypes* integers in range [0, *nelements*) 

       *bnormflag* value = *0* or *1*

          *0* = do not normalize

          *1* = normalize bispectrum components

   compute db all sna/atom 1.4 0.95 6 2.0 1.0

   compute vb all sna/atom 1.4 0.95 6 2.0 1.0

   compute snap all snap 1.4 0.95 6 2.0 1.0

   compute snap all snap 1.0 0.99363 6 3.81 3.83 1.0 0.93 chem 2 0 1

Description

"""""""""""

al. :ref:`(Thompson) <Thompson20141>`

The position of a neighbor atom *i'* relative to a central atom *i* is

a point within the 3D ball of radius :math:`R_{ii'} = rcutfac (R_i + R_i')`

a point within the 3D ball of radius :math:`R_{ii'}` = *rcutfac* :math:`(R_i + R_i')`

Bartok et al. :ref:`(Bartok) <Bartok20101>`, proposed mapping this 3D ball

onto the 3-sphere, the surface of the unit ball in a four-dimensional

space.  The radial distance *r*  within *R_ii'* is mapped on to a third

polar angle *theta0* defined by,

polar angle :math:`\theta_0` defined by,

.. math::

  \theta_0 = rfac0 \frac{r-r_{min0}}{R_{ii'}-r_{min0}} \pi

  \theta_0 = {\sf rfac0} \frac{r-r_{min0}}{R_{ii'}-r_{min0}} \pi

In this way, all possible neighbor positions are mapped on to a subset

of the 3-sphere.  Points south of the latitude :math:`\theta_0=rfac0 \pi`

of the 3-sphere.  Points south of the latitude :math:`\theta_0` = *rfac0* :math:`\pi`

are excluded.

The natural basis for functions on the 3-sphere is formed by the 

These functions are better known as :math:`D^j_{m,m'}`, the elements of the Wigner

*D*\ -matrices :ref:`(Meremianin <Meremianin2006>`,

:ref:`Varshalovich <Varshalovich1987>`, :ref:`Mason) <Mason2009>`

The density of neighbors on the 3-sphere can be written as a sum of

Dirac-delta functions, one for each neighbor, weighted by species and

radial distance. Expanding this density function as a generalized

virial components, each atom type, and each bispectrum component.  See

section below on output for a detailed explanation.

Compute *snap* calculates a global array contains information related

Compute *snap* calculates a global array containing information related

to all three of the above per-atom computes *sna/atom*\ , *snad/atom*\ ,

and *snav/atom*\ . The first row of the array contains the summation of

*sna/atom* over all atoms, but broken out by type. The last six rows

atomic radius to radial cutoff distance.

The argument *rfac0* and the optional keyword *rmin0* define the

linear mapping from radial distance to polar angle *theta0* on the

linear mapping from radial distance to polar angle :math:`theta_0` on the

3-sphere, given above.

The argument *twojmax* defines which

*quadraticflag* is on, it also affects *snad/atom* and *snav/atom*\ .

The keyword *quadraticflag* determines whether or not the

quadratic analogs to the bispectrum quantities are generated.

quadratic combinations of bispectrum quantities are generated.

These are formed by taking the outer product of the vector

of bispectrum components with itself.

See section below on output for a

The explicit multi-element variant invoked by the *chem* keyword

partitions the density of neighbors into partial densities

for each chemical element.  This is described in detail in the 

paper by :ref:`(Cusentino et al.) <Cusentino2020>`.

paper by :ref:`Cusentino et al. <Cusentino2020>`

The bispectrum components are indexed on 

ordered triplets of elements.

ordered triplets of elements:

.. math::

        {j_2} {m_2} {m'_2} \end{array}}

        u^{\kappa}_{j_1,m_1,m'_1} u^{\lambda}_{j_2,m_2,m'_2}

Where :math:`u^{\mu}_{j,m,m'}` is an expansion coefficient for the partial density of neighbors 

where :math:`u^{\mu}_{j,m,m'}` is an expansion coefficient for the partial density of neighbors 

of element :math:`\mu`

.. math::

This option is typically used in conjunction with the *chem* keyword, 

and LAMMPS will generate a warning if both *chem* and *bnormflag*

are not both set or both unset.

are not both set or not both unset.

.. note::

           for j in range(j1-j2,min(twojmax,j1+j2)+1,2):

               if (j>=j1): print j1/2.,j2/2.,j/2.

The total number of bispectrum components is given by the following expression

.. math::

  K = \frac{(twojmax+2)(twojmax+3)(twojmax+4)}{24}

The total number of bispectrum components is :math:`K = (J+2)(J+3)(J+4)/24`

where *J* is equal to *twojmax*\ .

.. note::

So the nesting order from inside to outside is bispectrum component,

linear then quadratic, vector/tensor component, type.

if the *chem* keyword is used, then the data is arranged into :math:`nelements^3`

If the *chem* keyword is used, then the data is arranged into :math:`N_{elem}^3`

sub-blocks, each sub-block corresponding to a particular chemical labelling 

:math:`\kappa\lambda\mu` with the last label changing fastest.

Each sub-block contains the *K* bispectrum components. For the purposes

Each sub-block contains *K* bispectrum components. For the purposes

of handling contributions to force, virial, and quadratic combinations,

these :math:`nelements^3` sub-blocks are treated as a single block

of :math:`nelements^3 K` columns.

these :math:`N_{elem}^3` sub-blocks are treated as a single block

of :math:`K N_{elem}^3` columns.

These values can be accessed by any command that uses per-atom values

from a compute as input.  See the :doc:`Howto output <Howto_output>` doc

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

These computes are part of the SNAP package.  They are 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

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

   pair_coeff * * snap InP.snapcoeff InP.snapparam In In P P

The 1st 2 arguments must be \* \* so as to span all LAMMPS atom types.

The first 2 arguments must be \* \* so as to span all LAMMPS atom types.

The two filenames are for the coefficient and parameter files, respectively.

The two trailing 'In' arguments map LAMMPS atom types 1 and 2 to the

SNAP 'In' element. The two trailing 'P' arguments map LAMMPS atom types