Merge remote-tracking branch 'refs/remotes/origin/chem_snap' into chem_snap

* R_1, R_2,... = list of cutoff radii, one for each type (distance units)

* w_1, w_2,... = list of neighbor weights, one for each type

* zero or more keyword/value pairs may be appended

* keyword = *rmin0* or *switchflag* or *bzeroflag* or *quadraticflag*

* keyword = *rmin0* or *switchflag* or *bzeroflag* or *quadraticflag* or *chem* or *bnormflag* or *wselfallflag*

  .. parsed-literal::

       *quadraticflag* value = *0* or *1*

          *0* = do not generate quadratic terms

          *1* = generate quadratic terms

       *chem* values = *nelements* *elementlist*

          *nelements* = number of SNAP elements

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

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

          *0* = do not normalize

          *1* = normalize bispectrum components

       *wselfallflag* value = *0* or *1*

          *0* = self-contribution only for element of central atom 

          *1* = self-contribution for all elements

Examples

""""""""

   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 *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 = {\tt 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 *theta0max=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 4D

hyperspherical harmonics *U\^j_m,m'(theta, phi, theta0).*  These

functions are better known as *D\^j_m,m',* the elements of the Wigner

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

representatives of *SU(2)*, the matrices :math:`U^j_{m,m'}(\theta, \phi, \theta_0)`.  

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:`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

.. math::

  u^j_{m,m'} = U^j_{m,m'}(0,0,0) + \sum_{r_{ii'} < R_{ii'}}{f_c(r_{ii'}) w_{i'} U^j_{m,m'}(\theta_0,\theta,\phi)}

  u^j_{m,m'} = U^j_{m,m'}(0,0,0) + \sum_{r_{ii'} < R_{ii'}}{f_c(r_{ii'}) w_{\mu_{i'}} U^j_{m,m'}(\theta_0,\theta,\phi)}

The *w_i'* neighbor weights are dimensionless numbers that are chosen

to distinguish atoms of different types, while the central atom is

arbitrarily assigned a unit weight.  The function *fc(r)* ensures that

The :math:`w_{\mu_{i'}}` neighbor weights are dimensionless numbers that depend on 

:math:`\mu_{i'}`, the SNAP element of atom *i'*, while the central atom is

arbitrarily assigned a unit weight.  The function :math:`f_c(r)` ensures that

the contribution of each neighbor atom goes smoothly to zero at

*R_ii'*:

:math:`R_{ii'}`:

.. math::

  f_c(r)   = & \frac{1}{2}(\cos(\pi \frac{r-r_{min0}}{R_{ii'}-r_{min0}}) + 1), r \leq R_{ii'} \\

           = & 0,  r > R_{ii'}

The expansion coefficients *u\^j_m,m'* are complex-valued and they are

The expansion coefficients :math:`u^j_{m,m'}` are complex-valued and they are

not directly useful as descriptors, because they are not invariant

under rotation of the polar coordinate frame. However, the following

scalar triple products of expansion coefficients can be shown to be

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

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

The constants *H\^jmm'_j1m1m1'_j2m2m2'* are coupling coefficients,

The constants :math:`H^{jmm'}_{j_1 m_1 m_{1'},j_2 m_ 2m_{2'}}` 

are coupling coefficients,

analogous to Clebsch-Gordan coefficients for rotations on the

2-sphere. These invariants are the components of the bispectrum and

these are the quantities calculated by the compute *sna/atom*\ . They

the 3-sphere. The j2=0 subset form the power spectrum, which

characterizes the correlations of two points. The lowest-order

components describe the coarsest features of the density function,

while higher-order components reflect finer detail.  Note that the

central atom is included in the expansion, so three point-correlations

can be either due to three neighbors, or two neighbors and the central

atom.

while higher-order components reflect finer detail. Each bispectrum

component contains terms that depend on the positions of up to 4

atoms (3 neighbors and the central atom).

Compute *snad/atom* calculates the derivative of the bispectrum components

summed separately for each atom type:

summed separately for each LAMMPS atom type:

.. math::

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

3-sphere.

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

3-sphere, given above.

The argument *twojmax* defines which

bispectrum components are generated. See section below on output for a

ordered in which they are listed.

The keyword *switchflag* can be used to turn off the switching

function.

function :math:`f_c(r)`.

The keyword *bzeroflag* determines whether or not *B0*\ , the bispectrum

components of an atom with no neighbors, are subtracted from

*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

detailed explanation of the number of quadratic terms and the

ordered in which they are listed.

The keyword *chem* activates the explicit multi-element variant

of the SNAP bispectrum components. The argument *nelements*

specifies the number of SNAP elements that will be handled.

This is followed by *elementlist*, a list of integers of

length *ntypes*, with values in the range [0, *nelements* ),

which maps each LAMMPS type to one of the SNAP elements.

Note that multiple LAMMPS types can be mapped to the same element,

and some elements may be mapped by no LAMMPS type. However, in typical

use cases (training SNAP potentials) the mapping from LAMMPS types 

to elements is one-to-one. 

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>`

The bispectrum components are indexed on 

ordered triplets of elements:

.. math::

   B_{j_1,j_2,j}^{\kappa\lambda\mu} =

   \sum_{m_1,m'_1=-j_1}^{j_1}\sum_{m_2,m'_2=-j_2}^{j_2}\sum_{m,m'=-j}^{j} (u^{\mu}_{j,m,m'})^*

   H {\scriptscriptstyle \begin{array}{l} {j} {m} {m'} \\

        {j_1} {m_1} {m'_1} \\

        {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 

of element :math:`\mu`

.. math::

  u^{\mu}_{j,m,m'} =  w^{self}_{\mu_{i}\mu} U^{j,m,m'}(0,0,0) + \sum_{r_{ii'} < R_{ii'}}{\delta_{\mu\mu_{i'}}f_c(r_{ii'}) w_{\mu_{i'}} U^{j,m,m'}(\theta_0,\theta,\phi)}

where :math:`w^{self}_{\mu_{i}\mu}` is the self-conribution, which is either 1 or 0 

(see keyword *wselfallflag* below), :math:`\delta_{\mu\mu_{i'}}` indicates 

that the sum is only over neighbor atoms of element :math:`\mu`,

and all other quantities are the same as those appearing in the 

original equation for :math:`u^j_{m,m'}` given above.  

The keyword *wselfallflag* defines the rule used for the self-contribution.

If *wselfallflag* is on, then :math:`w^{self}_{\mu_{i}\mu}` = 1. If it is

off then :math:`w^{self}_{\mu_{i}\mu}` = 0, except in the case

of :math:`{\mu_{i}=\mu}`, when :math:`w^{self}_{\mu_{i}\mu}` = 1.

When the *chem* keyword is not used, this keyword has no effect.

The keyword *bnormflag* determines whether or not the bispectrum

component :math:`B_{j_1,j_2,j}` is divided by a factor of :math:`2j+1`.

This normalization simplifies force calculations because of the

following symmetry relation

.. math::

 \frac{B_{j_1,j_2,j}}{2j+1} = \frac{B_{j,j_2,j_1}}{2j_1+1} = \frac{B_{j_1,j,j_2}}{2j_2+1}

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 not both unset.

.. note::

   If you have a bonded system, then the settings of

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

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

For even twojmax = 2(*m*\ -1), :math:`K = m(m+1)(2m+1)/6`, the *m*\ -th pyramidal number. For odd twojmax = 2 *m*\ -1, :math:`K = m(m+1)(m+2)/3`, twice the *m*\ -th tetrahedral number.

.. note::

   the *diagonal* keyword allowing other possible choices

Compute *snad/atom* evaluates a per-atom array. The columns are

arranged into *ntypes* blocks, listed in order of atom type *I*\ .  Each

block contains three sub-blocks corresponding to the *x*\ , *y*\ , and *z*

components of the atom position.  Each of these sub-blocks contains

one column for each bispectrum component, the same as for compute

*sna/atom*

components of the atom position.  Each of these sub-blocks contains *K*

columns for the *K* bispectrum components, the same as for compute *sna/atom*

Compute *snav/atom* evaluates a per-atom array. The columns are

arranged into *ntypes* blocks, listed in order of atom type *I*\ .  Each

block contains six sub-blocks corresponding to the *xx*\ , *yy*\ , *zz*\ ,

*yz*\ , *xz*\ , and *xy* components of the virial tensor in Voigt

notation.  Each of these sub-blocks contains one column for each

bispectrum component, the same as for compute *sna/atom*

notation.  Each of these sub-blocks contains *K*

columns for the *K* bispectrum components, the same as for compute *sna/atom*

Compute *snap* evaluates a global array.

The columns are arranged into

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:`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 *K* bispectrum components. For the purposes

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

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

page for an overview of LAMMPS output options.

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

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

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

The optional keyword defaults are *rmin0* = 0,

*switchflag* = 1, *bzeroflag* = 1, *quadraticflag* = 0,

*bnormflag* = 0, *wselfallflag* = 0

----------

**(Varshalovich)** Varshalovich, Moskalev, Khersonskii, Quantum Theory

of Angular Momentum, World Scientific, Singapore (1987).

.. _Varshalovich1987:

.. _Mason2009:

**(Mason)** J. K. Mason, Acta Cryst A65, 259 (2009).

.. _Cusentino2020:

**(Cusentino)** Cusentino, Wood, and Thompson, J Phys Chem A, xxx, xxxxx, (2020)

Description

"""""""""""

Pair style *snap* computes interactions using the spectral

neighbor analysis potential (SNAP) :ref:`(Thompson) <Thompson20142>`.

Pair style *snap* defines the spectral

neighbor analysis potential (SNAP), a machine-learning 

interatomic potential :ref:`(Thompson) <Thompson20142>`.

Like the GAP framework of Bartok et al. :ref:`(Bartok2010) <Bartok20102>`, 

:ref:`(Bartok2013) <Bartok2013>` which uses bispectrum components

SNAP uses bispectrum components

to characterize the local neighborhood of each atom

in a very general way. The mathematical definition of the

bispectrum calculation used by SNAP is identical

to that used by :doc:`compute sna/atom <compute_sna_atom>`.

bispectrum calculation and its derivatives w.r.t. atom positions

is identical to that used by :doc:`compute snap <compute_sna_atom>`,

which is used to fit SNAP potentials to *ab initio* energy, force,

and stress data.

In SNAP, the total energy is decomposed into a sum over

atom energies. The energy of atom *i* is

expressed as a weighted sum over bispectrum components.

.. math::

   E^i_{SNAP}(B_1^i,...,B_K^i) = \beta^{\alpha_i}_0 + \sum_{k=1}^K \beta_k^{\alpha_i} B_k^i

   E^i_{SNAP}(B_1^i,...,B_K^i) = \beta^{\mu_i}_0 + \sum_{k=1}^K \beta_k^{\mu_i} B_k^i

where :math:`B_k^i` is the *k*\ -th bispectrum component of atom *i*\ ,

and :math:`\beta_k^{\alpha_i}` is the corresponding linear coefficient

that depends on :math:\alpha_i`, the SNAP element of atom *i*\ . The

and :math:`\beta_k^{\mu_i}` is the corresponding linear coefficient

that depends on :math:`\mu_i`, the SNAP element of atom *i*\ . The

number of bispectrum components used and their definitions

depend on the value of *twojmax*

depend on the value of *twojmax* and other parameters

defined in the SNAP parameter file described below.

The bispectrum calculation is described in more detail

in :doc:`compute sna/atom <compute_sna_atom>`.

with #) anywhere. Each non-blank non-comment line must contain one

keyword/value pair. The required keywords are *rcutfac* and

*twojmax*\ . Optional keywords are *rfac0*\ , *rmin0*\ ,

*switchflag*\ , *bzeroflag*\, and *chunksize*\.

*switchflag*\ , *bzeroflag*\ , *quadraticflag*\ , *chemflag*\ , 

*bnormflag*\ , *wselfallflag*\ , and *chunksize*\ .

The default values for these keywords are

* *rfac0* = 0.99363

* *rmin0* = 0.0

* *switchflag* = 0

* *switchflag* = 1

* *bzeroflag* = 1

* *quadraticflag* = 1

* *quadraticflag* = 0

* *chemflag* = 0

* *bnormflag* = 0

* *wselfallflag* = 0

* *chunksize* = 2000

If *quadraticflag* is set to 1, then the SNAP energy expression includes additional quadratic terms 

that have been shown to increase the overall accuracy of the potential without much increase

in computational cost :ref:`(Wood) <Wood20182>`. 

.. math::

   E^i_{SNAP}(\mathbf{B}^i) = \beta^{\mu_i}_0 + \boldsymbol{\beta}^{\mu_i} \cdot \mathbf{B}_i + \frac{1}{2}\mathbf{B}^t_i \cdot \boldsymbol{\alpha}^{\mu_i} \cdot \mathbf{B}_i

where :math:`\mathbf{B}_i` is the *K*-vector of bispectrum components, 

:math:`\boldsymbol{\beta}^{\mu_i}` is the *K*-vector of linear coefficients 

for element :math:`\mu_i`, and :math:`\boldsymbol{\alpha}^{\mu_i}` 

is the symmetric *K* by *K* matrix of quadratic coefficients.

The SNAP element file should contain *K*\ (\ *K*\ +1)/2 additional coefficients

for each element, the upper-triangular elements of :math:`\boldsymbol{\alpha}^{\mu_i}`.

If *chemflag* is set to 1, then the energy expression is written in terms of explicit multi-element bispectrum

components indexed on ordered triplets of elements, which has been shown to increase the ability of the SNAP

potential to capture energy differences in chemically complex systems, 

at the expense of a significant increase in computational cost :ref:`(Cusentino) <Cusentino20202>`.

.. math::

   E^i_{SNAP}(\mathbf{B}^i) = \beta^{\mu_i}_0 + \sum_{\kappa,\lambda,\mu} \boldsymbol{\beta}^{\kappa\lambda\mu}_{\mu_i} \cdot \mathbf{B}^{\kappa\lambda\mu}_i 

where :math:`\mathbf{B}^{\kappa\lambda\mu}_i` is the *K*-vector of bispectrum components 

for neighbors of elements :math:`\kappa`, :math:`\lambda`, and :math:`\mu` and 

:math:`\boldsymbol{\beta}^{\kappa\lambda\mu}_{\mu_i}` is the corresponding *K*-vector 

of linear coefficients for element :math:`\mu_i`. The SNAP element file should contain 

a total of :math:`K N_{elem}^3` coefficients for each of the :math:`N_{elem}` elements.

The keyword *chunksize* is only applicable when using the

pair style *snap* with the KOKKOS package and is ignored otherwise.

This keyword controls

Detailed definitions for all the other keywords

are given on the :doc:`compute sna/atom <compute_sna_atom>` doc page.

If *quadraticflag* is set to 1, then the SNAP energy expression includes the quadratic term, 0.5\*B\^t.alpha.B, where alpha is a symmetric *K* by *K* matrix.

The SNAP element file should contain *K*\ (\ *K*\ +1)/2 additional coefficients

for each element, the upper-triangular elements of alpha.

.. note::

   The previously used *diagonalstyle* keyword was removed in 2019,

:doc:`compute sna/atom <compute_sna_atom>`,

:doc:`compute snad/atom <compute_sna_atom>`,

:doc:`compute snav/atom <compute_sna_atom>`

:doc:`compute snav/atom <compute_sna_atom>`,

:doc:`compute snap <compute_sna_atom>`

**Default:** none

**(Bartok2010)** Bartok, Payne, Risi, Csanyi, Phys Rev Lett, 104, 136403 (2010).

.. _Bartok2013:

.. _Wood20182:

**(Wood)** Wood and Thompson, J Chem Phys, 148, 241721, (2018)

.. _Cusentino20202:

**(Bartok2013)** Bartok, Gillan, Manby, Csanyi, Phys Rev B 87, 184115 (2013).

**(Cusentino)** Cusentino, Wood, and Thompson, J Phys Chem A, xxx, xxxxx, (2020)

../../potentials/InP_JCPA2020.snap

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../../potentials/InP_JCPA2020.snapcoeff

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../../potentials/InP_JCPA2020.snapparam

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