\(\renewcommand{\AA}{\text{Å}}\)

# compute sna/atom command

# compute snad/atom command

# compute snap command

# compute sna/grid command

# compute sna/grid/local command

## Syntax

```
compute ID group-ID sna/atom rcutfac rfac0 twojmax R_1 R_2 ... w_1 w_2 ... keyword values ...
compute ID group-ID snad/atom rcutfac rfac0 twojmax R_1 R_2 ... w_1 w_2 ... keyword values ...
compute ID group-ID snav/atom rcutfac rfac0 twojmax R_1 R_2 ... w_1 w_2 ... keyword values ...
compute ID group-ID snap rcutfac rfac0 twojmax R_1 R_2 ... w_1 w_2 ... keyword values ...
compute ID group-ID snap rcutfac rfac0 twojmax R_1 R_2 ... w_1 w_2 ... keyword values ...
compute ID group-ID sna/grid nx ny nz rcutfac rfac0 twojmax R_1 R_2 ... w_1 w_2 ... keyword values ...
compute ID group-ID sna/grid/local nx ny nz rcutfac rfac0 twojmax R_1 R_2 ... w_1 w_2 ... keyword values ...
```

ID, group-ID are documented in compute command

sna/atom = style name of this compute command

rcutfac = scale factor applied to all cutoff radii (positive real)

rfac0 = parameter in distance to angle conversion (0 < rcutfac < 1)

twojmax = band limit for bispectrum components (non-negative integer)

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

nx, ny, nz = number of grid points in x, y, and z directions (positive integer)

zero or more keyword/value pairs may be appended

keyword =

*rmin0*or*switchflag*or*bzeroflag*or*quadraticflag*or*chem*or*bnormflag*or*wselfallflag*or*bikflag*or*switchinnerflag*or*sinner*or*dinner*or*dgradflag*or*nnn*or*wmode*or*delta**rmin0*value = parameter in distance to angle conversion (distance units)*switchflag*value =*0*or*1**0*= do not use switching function*1*= use switching function*bzeroflag*value =*0*or*1**0*= do not subtract B0*1*= subtract B0*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*switchinnerflag*value =*0*or*1**0*= do not use inner switching function*1*= use inner switching function*sinner*values =*sinnerlist**sinnerlist*=*ntypes*values of*Sinner*(distance units)*dinner*values =*dinnerlist**dinnerlist*=*ntypes*values of*Dinner*(distance units)*bikflag*value =*0*or*1*(only implemented for compute snap)*0*= descriptors are summed over atoms of each type*1*= descriptors are listed separately for each atom*dgradflag*value =*0*or*1*(only implemented for compute snap)*0*= descriptor gradients are summed over atoms of each type*1*= descriptor gradients are listed separately for each atom pairadditional keyword =

*nnn*or*wmode*or*delta**nnn*value = number of considered nearest neighbors to compute the bispectrum over a target specific number of neighbors (only implemented for compute sna/atom)*wmode*value = weight function for finding optimal cutoff to match the target number of neighbors (required if nnn used, only implemented for compute sna/atom)*0*= heavyside weight function*1*= hyperbolic tangent weight function*delta*value = transition interval centered at cutoff distance for hyperbolic tangent weight function (ignored if wmode=0, required if wmode=1, only implemented for compute sna/atom)

## Examples

```
compute b all sna/atom 1.4 0.99363 6 2.0 2.4 0.75 1.0 rmin0 0.0
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
compute snap all snap 1.0 0.99363 6 3.81 3.83 1.0 0.93 switchinnerflag 1 sinner 1.35 1.6 dinner 0.25 0.3
compute bgrid all sna/grid/local 200 200 200 1.4 0.95 6 2.0 1.0
compute bnnn all sna/atom 9.0 0.99363 8 0.5 1.0 rmin0 0.0 nnn 24 wmode 1 delta 0.2
```

## Description

Define a computation that calculates a set of quantities related to the bispectrum components of the atoms in a group. These computes are used primarily for calculating the dependence of energy, force, and stress components on the linear coefficients in the snap pair_style, which is useful when training a SNAP potential to match target data.

Bispectrum components of an atom are order parameters characterizing the radial and angular distribution of neighbor atoms. The detailed mathematical definition is given in the paper by Thompson et al. (Thompson)

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*
\((R_i + R_i')\)

Bartok et al. (Bartok), 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 \(\theta_0\) defined by,

In this way, all possible neighbor positions are mapped on to a subset
of the 3-sphere. Points south of the latitude \(\theta_0\) =
*rfac0* \(\pi\) are excluded.

The natural basis for functions on the 3-sphere is formed by the
representatives of *SU(2)*, the matrices \(U^j_{m,m'}(\theta, \phi,
\theta_0)\). These functions are better known as \(D^j_{m,m'}\), the
elements of the Wigner *D*-matrices (Meremianin, Varshalovich, Mason) 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 Fourier series in the basis functions, we can write each
Fourier coefficient as

The \(w_{\mu_{i'}}\) neighbor weights are dimensionless numbers that
depend on \(\mu_{i'}\), the SNAP element of atom *i’*, while the
central atom is arbitrarily assigned a unit weight. The function
\(f_c(r)\) ensures that the contribution of each neighbor atom goes
smoothly to zero at \(R_{ii'}\):

The expansion coefficients \(u^j_{m,m'}\) 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 real-valued and invariant under rotation (Bartok).

The constants \(H^{jmm'}_{j_1 m_1 m_{1'},j_2 m_ 2m_{2'}}\) 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 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. 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 LAMMPS atom type:

The sum is over all atoms *i’* of atom type *I*. For each atom *i*,
this compute evaluates the above expression for each direction, each
atom type, and each bispectrum component. See section below on output
for a detailed explanation.

Compute *snav/atom* calculates the virial contribution due to the
derivatives:

Again, the sum is over all atoms *i’* of atom type *I*. For each atom
*i*, this compute evaluates the above expression for each of the six
virial components, each atom type, and each bispectrum component. See
section below on output for a detailed explanation.

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 of
the array contain the summation of *snav/atom* over all atoms, broken
out by type. In between these are 3**N* rows containing the same
values computed by *snad/atom* (these are already summed over all atoms
and broken out by type). The element in the last column of each row
contains the potential energy, force, or stress, according to the row.
These quantities correspond to the user-specified reference potential
that must be subtracted from the target data when fitting SNAP. The
potential energy calculation uses the built in compute *thermo_pe*. The
stress calculation uses a compute called *snap_press* that is
automatically created behind the scenes, according to the following
command:

```
compute snap_press all pressure NULL virial
```

See section below on output for a detailed explanation of the data layout in the global array.

Added in version 3Aug2022.

The compute *sna/grid* and *sna/grid/local* commands calculate
bispectrum components for a regular grid of points. These are
calculated from the local density of nearby atoms *i’* around each grid
point, as if there was a central atom *i* at the grid point. This is
useful for characterizing fine-scale structure in a configuration of
atoms, and it is used in the MALA package to build machine-learning surrogates
for finite-temperature Kohn-Sham density functional theory (Ellis
et al.) Neighbor atoms not in the group do not contribute
to the bispectrum components of the grid points. The distance cutoff
\(R_{ii'}\) assumes that *i* has the same type as the neighbor atom
*i’*.

Compute *sna/grid* calculates a global array containing bispectrum
components for a regular grid of points.
The grid is aligned with the current box dimensions, with the
first point at the box origin, and forming a regular 3d array with
*nx*, *ny*, and *nz* points in the x, y, and z directions. For triclinic
boxes, the array is congruent with the periodic lattice vectors
a, b, and c. The array contains one row for each of the
\(nx \times ny \times nz\) grid points, looping over the index for *ix* fastest,
then *iy*, and *iz* slowest. Each row of the array contains the *x*, *y*,
and *z* coordinates of the grid point, followed by the bispectrum
components. See section below on output for a detailed explanation of the data
layout in the global array.

Compute *sna/grid/local* calculates bispectrum components of a regular
grid of points similarly to compute *sna/grid* described above.
However, because the array is local, it contains only rows for grid points
that are local to the processor subdomain. The global grid
of \(nx \times ny \times nz\) points is still laid out in space the same as for *sna/grid*,
but grid points are strictly partitioned, so that every grid point appears in
one and only one local array. The array contains one row for each of the
local grid points, looping over the global index *ix* fastest,
then *iy*, and *iz* slowest. Each row of the array contains
the global indexes *ix*, *iy*, and *iz* first, followed by the *x*, *y*,
and *z* coordinates of the grid point, followed by the bispectrum
components. See section below on output for a detailed explanation of the data
layout in the global array.

The value of all bispectrum components will be zero for atoms not in the group. Neighbor atoms not in the group do not contribute to the bispectrum of atoms in the group.

The neighbor list needed to compute this quantity is constructed each time the calculation is performed (i.e. each time a snapshot of atoms is dumped). Thus it can be inefficient to compute/dump this quantity too frequently.

The argument *rcutfac* is a scale factor that controls the ratio of
atomic radius to radial cutoff distance.

The argument *rfac0* and the optional keyword *rmin0* define the
linear mapping from radial distance to polar angle \(theta_0\) on the
3-sphere, given above.

The argument *twojmax* defines which
bispectrum components are generated. See section below on output for a
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 \(f_c(r)\).

The keyword *bzeroflag* determines whether or not *B0*, the bispectrum
components of an atom with no neighbors, are subtracted from the
calculated bispectrum components. This optional keyword normally only
affects compute *sna/atom*. However, when *quadraticflag* is on, it
also affects *snad/atom* and *snav/atom*.

The keyword *quadraticflag* determines whether or not the 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
Cusentino et al. The bispectrum components are
indexed on ordered triplets of elements:

where \(u^{\mu}_{j,m,m'}\) is an expansion coefficient for the partial density of neighbors of element \(\mu\)

where \(w^{self}_{\mu_{i}\mu}\) is the self-contribution, which is
either 1 or 0 (see keyword *wselfallflag* below),
\(\delta_{\mu\mu_{i'}}\) indicates that the sum is only over
neighbor atoms of element \(\mu\), and all other quantities are the
same as those appearing in the original equation for \(u^j_{m,m'}\)
given above.

The keyword *wselfallflag* defines the rule used for the
self-contribution. If *wselfallflag* is on, then
\(w^{self}_{\mu_{i}\mu}\) = 1. If it is off then
\(w^{self}_{\mu_{i}\mu}\) = 0, except in the case of
\({\mu_{i}=\mu}\), when \(w^{self}_{\mu_{i}\mu}\) = 1. When the
*chem* keyword is not used, this keyword has no effect.

The keyword *bnormflag* determines whether or not the bispectrum
component \(B_{j_1,j_2,j}\) is divided by a factor of \(2j+1\).
This normalization simplifies force calculations because of the
following symmetry relation

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.

The keyword *switchinnerflag* with value 1
activates an additional radial switching
function similar to \(f_c(r)\) above, but acting to switch off
smoothly contributions from neighbor atoms at short separation distances.
This is useful when SNAP is used in combination with a simple
repulsive potential. For a neighbor atom at
distance \(r\), its contribution is scaled by a multiplicative
factor \(f_{inner}(r)\) defined as follows:

where the switching region is centered at \(S_{inner}\) and it extends a distance \(D_{inner}\)
to the left and to the right of this.
With this option, additional keywords *sinner* and *dinner* must be used,
each followed by *ntypes*
values for \(S_{inner}\) and \(D_{inner}\), respectively.
When the central atom and the neighbor atom have different types,
the values of \(S_{inner}\) and \(D_{inner}\) are
the arithmetic means of the values for both types.

The keywords *bikflag* and *dgradflag* are only used by compute *snap*.
The keyword *bikflag* determines whether or not to list the descriptors
of each atom separately, or sum them together and list in a single row.
If *bikflag* is set
to *0* then a single bispectrum row is used, which contains the per-atom bispectrum
descriptors \(B_{i,k}\) summed over all atoms *i* to produce
\(B_k\). If *bikflag* is set
to *1* this is replaced by a separate per-atom bispectrum row for each atom.
In this case, the entries in the final column for these rows
are set to zero.

The keyword *dgradflag* determines whether to sum atom gradients or list
them separately. If *dgradflag* is set to 0, the bispectrum
descriptor gradients w.r.t. atom *j* are summed over all atoms *i’*
of type *I* (similar to *snad/atom* above).
If *dgradflag* is set to 1, gradients are listed separately for each pair of atoms.
Each row corresponds
to a single term \(\frac{\partial {B_{i,k} }}{\partial {r}^a_j}\)
where \({r}^a_j\) is the *a-th* position coordinate of the atom with global
index *j*. This also changes
the number of columns to be equal to the number of bispectrum components, with 3
additional columns representing the indices \(i\), \(j\), and \(a\),
as explained more in the Output info section below. The option *dgradflag=1*
requires that *bikflag=1*.

Note

Using *dgradflag* = 1 produces a global array with \(N + 3N^2 + 1\) rows
which becomes expensive for systems with more than 1000 atoms.

Note

If you have a bonded system, then the settings of special_bonds command can remove pairwise interactions between atoms in the same bond, angle, or dihedral. This is the default setting for the special_bonds command, and means those pairwise interactions do not appear in the neighbor list. Because this fix uses the neighbor list, it also means those pairs will not be included in the calculation. One way to get around this, is to write a dump file, and use the rerun command to compute the bispectrum components for snapshots in the dump file. The rerun script can use a special_bonds command that includes all pairs in the neighbor list.

The keyword *nnn* allows for the calculation of the bispectrum over a
specific target number of neighbors. This option is only implemented for
the compute *sna/atom*. An optimal cutoff radius for defining the
neighborhood of the central atom is calculated by means of a dichotomy
algorithm. This iterative process allows to assign weights to
neighboring atoms in order to match the total sum of weights with the
target number of neighbors. Depending on the radial weight function
used in that process, the cutoff radius can fluctuate a lot in the
presence of thermal noise. Therefore, in addition to the *nnn* keyword,
the keyword *wmode* allows to choose whether a Heaviside (*wmode* = 0)
function or a Hyperbolic tangent function (*wmode* = 1) should be used.
If the Heaviside function is used, the cutoff radius exactly matches the
distance between the central atom an its *nnn*’th neighbor. However, in
the case of the hyperbolic tangent function, the dichotomy algorithm
allows to span the weights over a distance *delta* in order to reduce
fluctuations in the resulting local atomic environment fingerprint. The
detailed formalism is given in the paper by Lafourcade et
al. (Lafourcade).

## Output info

Compute *sna/atom* calculates a per-atom array, each column
corresponding to a particular bispectrum component. The total number of
columns and the identity of the bispectrum component contained in each
column depend of the value of *twojmax*, as described by the following
piece of python code:

```
for j1 in range(0,twojmax+1):
for j2 in range(0,j1+1):
for j in range(j1-j2,min(twojmax,j1+j2)+1,2):
if (j>=j1): print j1/2.,j2/2.,j/2.
```

There are \(m(m+1)/2\) descriptors with last index *j*,
where *m* = \(\lfloor j \rfloor + 1\).
Hence, for even *twojmax* = 2(*m*-1), \(K = m(m+1)(2m+1)/6\), the *m*-th pyramidal number,
and for odd *twojmax* = 2 *m*-1, \(K = m(m+1)(m+2)/3\), twice the *m*-th tetrahedral number.

Note

the *diagonal* keyword allowing other possible choices
for the number of bispectrum components was removed in 2019,
since all potentials use the value of 3, corresponding to the
above set of bispectrum components.

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 *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 *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
*ntypes* blocks, listed in order of atom type *I*. Each block contains
one column for each bispectrum component, the same as for compute
*sna/atom*. A final column contains the corresponding energy, force
component on an atom, or virial stress component. The rows of the array
appear in the following order:

1 row:

*sna/atom*quantities summed for all atoms of type*I*3*

*N*rows:*snad/atom*quantities, with derivatives w.r.t. x, y, and z coordinate of atom*i*appearing in consecutive rows. The atoms are sorted based on atom ID.6 rows:

*snav/atom*quantities summed for all atoms of type*I*

For example, if *K* =30 and ntypes=1, the number of columns in the
per-atom arrays generated by *sna/atom*, *snad/atom*, and
*snav/atom* are 30, 90, and 180, respectively. With *quadratic* value=1,
the numbers of columns are 930, 2790, and 5580, respectively. The
number of columns in the global array generated by *snap* are 31, and
931, respectively, while the number of rows is 1+3**N*+6, where *N*
is the total number of atoms.

Compute *sna/grid* evaluates a global array.
The array contains one row for each of the
\(nx \times ny \times nz\) grid points, looping over the index for *ix* fastest,
then *iy*, and *iz* slowest. Each row of the array contains the *x*, *y*,
and *z* coordinates of the grid point, followed by the bispectrum
components.

Compute *sna/grid/local* evaluates a local array.
The array contains one row for each of the
local grid points, looping over the global index *ix* fastest,
then *iy*, and *iz* slowest. Each row of the array contains
the global indexes *ix*, *iy*, and *iz* first, followed by the *x*, *y*,
and *z* coordinates of the grid point, followed by the bispectrum
components.

If the *quadratic* keyword value is set to 1, then additional columns
are generated, corresponding to the products of all distinct pairs of
bispectrum components. If the number of bispectrum components is *K*,
then the number of distinct pairs is *K*(*K*+1)/2. For compute
*sna/atom* these columns are appended to existing *K* columns. The
ordering of quadratic terms is upper-triangular, (1,1),(1,2)…(1,*K*),(2,1)…(*K*-1,*K*-1),(*K*-1,*K*),(*K*,*K*).
For computes *snad/atom* and *snav/atom* each set of *K*(*K*+1)/2
additional columns is inserted directly after each of sub-block 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
\(N_{elem}^3\) sub-blocks, each sub-block corresponding to a
particular chemical labeling \(\kappa\lambda\mu\) 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 \(N_{elem}^3\) sub-blocks are
treated as a single block of \(K N_{elem}^3\) columns.

If the *bik* keyword is set to 1, the structure of the snap array is expanded.
The first \(N\) rows of the snap array
correspond to \(B_{i,k}\) instead of a single row summed over atoms \(i\).
In this case, the entries in the final column for these rows
are set to zero. Also, each row contains only non-zero entries for the
columns corresponding to the type of that atom. This is not true in the case
of *dgradflag* keyword = 1 (see below).

If the *dgradflag* keyword is set to 1, this changes the structure of the
global array completely.
Here the *snad/atom* quantities are replaced with rows corresponding to
descriptor gradient components on single atoms:

where \({r}^a_j\) is the *a-th* position coordinate of the atom with global
index *j*. The rows are
organized in chunks, where each chunk corresponds to an atom with global index
\(j\). The rows in an atom \(j\) chunk correspond to
atoms with global index \(i\). The total number of rows for
these descriptor gradients is therefore \(3N^2\).
The number of columns is equal to the number of bispectrum components,
plus 3 additional left-most columns representing the global atom indices
\(i\), \(j\),
and Cartesian direction \(a\) (0, 1, 2, for x, y, z).
The first 3 columns of the first \(N\) rows belong to the reference
potential force components. The remaining K columns contain the
\(B_{i,k}\) per-atom descriptors corresponding to the non-zero entries
obtained when *bikflag* = 1.
The first column of the last row, after the first
\(N + 3N^2\) rows, contains the reference potential
energy. The virial components are not used with this option. The total number of
rows is therefore \(N + 3N^2 + 1\) and the number of columns is \(K + 3\).

These values can be accessed by any command that uses per-atom values from a compute as input. See the Howto output doc page for an overview of LAMMPS output options. To see how this command can be used within a Python workflow to train SNAP potentials, see the examples in FitSNAP.

## Restrictions

These computes are part of the ML-SNAP package. They are only enabled if LAMMPS was built with that package. See the Build package page for more info.

## Default

The optional keyword defaults are *rmin0* = 0,
*switchflag* = 1, *bzeroflag* = 1, *quadraticflag* = 0,
*bnormflag* = 0, *wselfallflag* = 0, *switchinnerflag* = 0,
*nnn* = -1, *wmode* = 0, *delta* = 1.e-3

**(Thompson)** Thompson, Swiler, Trott, Foiles, Tucker, J Comp Phys, 285, 316, (2015).

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

**(Meremianin)** Meremianin, J. Phys. A, 39, 3099 (2006).

**(Varshalovich)** Varshalovich, Moskalev, Khersonskii, Quantum Theory
of Angular Momentum, World Scientific, Singapore (1987).

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

**(Cusentino)** Cusentino, Wood, Thompson, J Phys Chem A, 124, 5456, (2020)

**(Ellis)** Ellis, Fiedler, Popoola, Modine, Stephens, Thompson, Cangi, Rajamanickam, Phys Rev B, 104, 035120, (2021)

**(Lafourcade)** Lafourcade, Maillet, Denoual, Duval, Allera, Goryaeva, and Marinica,
Comp. Mat. Science, 230, 112534 (2023)