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Added chemsnap keywords and upgraded equations to mathjax

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Aidan Thompson
2020-06-13 17:55:45 -06:00
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145
doc/src/compute_sna_atom.rst
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@ -30,7 +30,7 @@ Syntax
* 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::
@ -44,6 +44,15 @@ Syntax
*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* element indices [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
""""""""
@ -71,7 +80,7 @@ mathematical definition is given in the paper by Thompson et
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 (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
@ -80,17 +89,17 @@ polar angle *theta0* defined by,
.. math::
\theta_0 = {\tt rfac0} \frac{r-r_{min0}}{R_{ii'}-r_{min0}} \pi
\theta_0 = 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 \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
@ -100,20 +109,20 @@ coefficient as
.. 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
@ -128,7 +137,8 @@ real-valued and invariant under rotation :ref:`(Bartok) <Bartok20101>`.
{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
@ -136,13 +146,12 @@ characterize the strength of density correlations at three points on
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::
@ -202,7 +211,7 @@ 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.
3-sphere, given above.
The argument *twojmax* defines which
bispectrum components are generated. See section below on output for a
@ -210,7 +219,7 @@ detailed explanation of the number of bispectrum components and the
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
@ -226,6 +235,65 @@ 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 both unset.
.. note::
If you have a bonded system, then the settings of
@ -257,6 +325,12 @@ described by the following piece of python code:
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}
.. note::
the *diagonal* keyword allowing other possible choices
@ -267,16 +341,15 @@ described by the following piece of python code:
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
@ -312,6 +385,14 @@ of linear terms i.e. linear and quadratic terms are contiguous.
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`
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
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 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.
@ -332,6 +413,7 @@ Default
The optional keyword defaults are *rmin0* = 0,
*switchflag* = 1, *bzeroflag* = 1, *quadraticflag* = 0,
*bnormflag* = 0, *wselfallflag* = 0
----------
@ -352,3 +434,12 @@ available at `arXiv:1409.3880 <http://arxiv.org/abs/1409.3880>`_
**(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)

78
doc/src/pair_snap.rst
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@ -24,27 +24,30 @@ Examples
Description
"""""""""""
Pair style *snap* computes interactions using the spectral
neighbor analysis potential (SNAP) :ref:`(Thompson) <Thompson20142>`.
Like the GAP framework of Bartok et al. :ref:`(Bartok2010) <Bartok20102>`,
:ref:`(Bartok2013) <Bartok2013>` which uses bispectrum components
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>`,
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>`.
@ -136,17 +139,51 @@ The SNAP parameter file can contain blank and comment lines (start
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
@ -159,10 +196,6 @@ into two passes.
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,
@ -221,7 +254,8 @@ Related commands
: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
@ -235,6 +269,10 @@ Related commands
**(Bartok2010)** Bartok, Payne, Risi, Csanyi, Phys Rev Lett, 104, 136403 (2010).
.. _Bartok2013:
.. _Wood20182:
**(Bartok2013)** Bartok, Gillan, Manby, Csanyi, Phys Rev B 87, 184115 (2013).
**(Wood)** Wood and Thompson, J Chem Phys, 148, 241721, (2018)
.. _Cusentino20202:
**(Cusentino)** Cusentino, Wood, and Thompson, J Phys Chem A, xxx, xxxxx, (2020)