Merge pull request #3869 from jmgoff/compute-pace-update

Compute pace update

doc/src/Commands_compute.rst

@@ -100,6 +100,7 @@ KOKKOS, o = OPENMP, t = OPT.
    * :doc:`nbond/atom <compute_nbond_atom>`
    * :doc:`omega/chunk <compute_omega_chunk>`
    * :doc:`orientorder/atom (k) <compute_orientorder_atom>`
+   * :doc:`pace <compute_pace>`
    * :doc:`pair <compute_pair>`
    * :doc:`pair/local <compute_pair_local>`
    * :doc:`pe <compute_pe>`

doc/src/compute.rst

@@ -264,6 +264,7 @@ The individual style names on the :doc:`Commands compute <Commands_compute>` pag
 * :doc:`nbond/atom <compute_nbond_atom>` - calculates number of bonds per atom
 * :doc:`omega/chunk <compute_omega_chunk>` - angular velocity for each chunk
 * :doc:`orientorder/atom <compute_orientorder_atom>` - Steinhardt bond orientational order parameters Ql
+* :doc:`pace <compute_pace>` - atomic cluster expansion descriptors and related quantities
 * :doc:`pair <compute_pair>` - values computed by a pair style
 * :doc:`pair/local <compute_pair_local>` - distance/energy/force of each pairwise interaction
 * :doc:`pe <compute_pe>` - potential energy

doc/src/compute_pace.rst

@@ -0,0 +1,251 @@
+.. index:: compute pace
+
+compute pace command
+========================
+
+Syntax
+""""""
+
+.. code-block:: LAMMPS
+
+   compute ID group-ID pace ace_potential_filename ... keyword values ...
+
+* ID, group-ID are documented in :doc:`compute <compute>` command
+* pace = style name of this compute command
+* ace_potential_filename = file name (in the .yace or .ace format from :doc:`pace pair_style <pair_pace>`) including ACE hyperparameters, bonds, and generalized coupling coefficients
+* keyword = *bikflag* or *dgradflag*
+
+  .. parsed-literal::
+
+       *bikflag* value = *0* or *1*
+          *0* = descriptors are summed over atoms of each type
+          *1* = descriptors are listed separately for each atom
+       *dgradflag* value = *0* or *1*
+          *0* = descriptor gradients are summed over atoms of each type
+          *1* = descriptor gradients are listed separately for each atom pair
+
+Examples
+""""""""
+
+.. code-block:: LAMMPS
+
+   compute pace all pace coupling_coefficients.yace
+   compute pace all pace coupling_coefficients.yace 0 1
+   compute pace all pace coupling_coefficients.yace 1 1
+
+Description
+"""""""""""
+
+.. versionadded:: TBD
+
+This compute calculates a set of quantities related to the atomic cluster
+expansion (ACE) descriptors of the atoms in a group. ACE descriptors are
+a highly generalizable atomic descriptor, encoding the radial and angular
+distribution of neighbor atoms, up to arbitrary bond order (rank). The
+detailed mathematical definition is given in the paper by
+:ref:`(Drautz) <Drautz19>`. These descriptors are used in the
+:doc:`pace pair_style <pair_pace>`. Quantities obtained from `compute pace`
+are related to those used in :doc:`pace pair_style <pair_pace>` to
+evaluate atomic energies, forces, and stresses for linear ACE models.
+For example, the energy for a linear ACE model is calculated as:
+:math:`E=\sum_i^{N\_atoms} \sum_{\boldsymbol{\nu}} c_{\boldsymbol{\nu}}  B_{i,\boldsymbol{\boldsymbol{\nu}}}`.
+The ACE descriptors for atom `i` :math:`B_{i,\boldsymbol{\nu}}`, and
+:math:`c_{\nu}` are linear model parameters. The detailed definition
+and indexing convention for ACE descriptors is given in :ref:`(Drautz) <Drautz19>`.
+In short, body order :math:`N`, angular character, radial character,
+and chemical elements in the *N-body* descriptor are encoded by :math:`\nu`.
+In the :doc:`pace pair_style <pair_pace>`, the linear model parameters
+and the ACE descriptors are combined for efficient evaluation of energies
+and forces. The details and benefits of this efficient implementation are
+given in :ref:`(Lysogorskiy) <Lysogorskiy21>`. et. al, but the combined
+descriptors and linear model parameters for the purposes of `compute pace`
+may be expressed in terms of the ACE descriptors mentioned above.
+
+:math:`c_{\boldsymbol{\nu}} B_{i,\boldsymbol{\nu}}= \sum_{\boldsymbol{\nu}' \in \boldsymbol{\nu} } \big[ c_{\boldsymbol{\nu}} C(\boldsymbol{\nu}') \big] A_{i,\boldsymbol{\nu}'}`
+
+where the bracketed terms on the right-hand side are the combined functions
+with linear model parameters typically provided in the `<name>.yace` potential
+file for `pace pair_style`. When these bracketed terms are multiplied by the
+products of the atomic base from :ref:`(Drautz) <Drautz19>`,
+:math:`A_{i,\boldsymbol{\nu'}}`, the ACE descriptors are recovered but they
+are also scaled by linear model parameters. The generalized coupling coefficients,
+written in short-hand here as :math:`C(\boldsymbol{\nu}')`, are the generalized
+Clebsch-Gordan or generalized Wigner symbols. It may be desirable to reverse the
+combination of these descriptors and the linear model parameters so that the
+ACE descriptors themselves may be used. The ACE descriptors and their gradients
+are often used when training ACE models, performing custom data analysis,
+generalizing ACE model forms, and other tasks that involve direct computation of
+descriptors. The key utility of `compute pace` is that it can compute the ACE
+descriptors and gradients so that these tasks can be performed during a LAMMPS
+simulation or so that LAMMPS can be used as a driver for tasks like ACE model
+parameterization. To see how this command can be used within a Python workflow
+to train ACE potentials, see the examples in
+`FitSNAP <https://github.com/FitSNAP/FitSNAP>`_. Examples on using outputs from
+this compute to construct general ACE potential forms are demonstrated in
+:ref:`(Goff) <Goff23>`. The various keywords and inputs to `compute pace`
+determine what ACE descriptors and related quantities are returned in a compute
+array.
+
+The coefficient file, `<name>.yace`, ultimately defines the number of ACE
+descriptors to be computed, their maximum body-order, the degree of angular
+character they have, the degree of radial character they have, the chemical
+character (which element-element interactions are encoded by descriptors),
+and other hyperparameters defined in :ref:`(Drautz) <Drautz19>`. These may
+be modeled after the potential files in :doc:`pace pair_style <pair_pace>`,
+and have the same format. Details on how to generate the coefficient files
+to train ACE models may be found in `FitSNAP <https://github.com/FitSNAP/FitSNAP>`_.
+
+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 descriptor row is used, which contains
+the per-atom ACE descriptors :math:`B_{i,\boldsymbol{\nu}}` summed over all
+atoms *i* to produce :math:`B_{\boldsymbol{\nu}}`. If *bikflag* is set to
+*1* this is replaced by a separate per-atom ACE descriptor 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 ACE
+descriptor gradients w.r.t. atom *j* are summed over all atoms *i'*
+of, which may be useful when training linear ACE models on atomic forces.
+If *dgradflag* is set to 1, gradients are listed separately for each pair of atoms.
+Each row corresponds
+to a single term :math:`\frac{\partial {B_{i,\boldsymbol{\nu}}}}{\partial {r}^a_j}`
+where :math:`{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
+ACE descriptors, with 3 additional columns representing the indices :math:`i`,
+:math:`j`, and :math:`a`, as explained more in the Output info section below.
+The option *dgradflag=1* requires that *bikflag=1*.
+
+.. note::
+
+    It is noted here that in contrast to :doc:`pace pair_style <pair_pace>`,
+    the *.yace* file for `compute pace` typically should not contain linear
+    parameters for an ACE potential. If :math:`c_{\nu}` are included,
+    the value of the descriptor will not be returned in the `compute` array,
+    but instead, the energy contribution from that descriptor will be returned.
+    Do not do this unless it is the desired behavior.
+    *In short, you should not plug in a '.yace' for a pace potential into this
+    compute to evaluate descriptors.*
+
+.. note::
+
+    *Generalized Clebsch-Gordan or Generalized Wigner symbols (with appropriate
+    factors) must be used to evaluate ACE descriptors with this compute.* There
+    are multiple ways to define the generalized coupling coefficients. Because
+    of this, this compute will not revert your potential file to a coupling
+    coefficient file. Instead this compute allows the user to supply coupling
+    coefficients that follow any convention.
+
+.. note::
+
+   Using *dgradflag* = 1 produces a global array with :math:`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 :doc:`special_bonds
+   <special_bonds>` command can remove pairwise interactions between
+   atoms in the same bond, angle, or dihedral.  This is the default
+   setting for the :doc:`special_bonds <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 :doc:`rerun <rerun>` command to
+   compute the ACE descriptors for snapshots in the dump file.
+   The rerun script can use a :doc:`special_bonds <special_bonds>`
+   command that includes all pairs in the neighbor list.
+
+----------
+
+Output info
+"""""""""""
+
+Compute *pace* 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 ACE descriptor, the same as for compute
+*sna/atom*\ in :doc:`compute snap <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: *pace* average descriptor values for all atoms of type *I*
+* 3\*\ *n* force rows: 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 and run up to the total number of atoms, *n*.
+* 6 rows: *virial* quantities summed for all atoms of type *I*
+
+For example, if :math:`\# \; B_{i, \boldsymbol{\nu}}` =30 and ntypes=1, the number of columns in the
+The number of columns in the global array generated by *pace* are 31, and
+931, respectively, while the number of rows is 1+3\*\ *n*\ +6, where *n*
+is the total number of atoms.
+
+If the *bik* keyword is set to 1, the structure of the pace array is expanded.
+The first :math:`N` rows of the pace array
+correspond to :math:`\# \; B_{i,\boldsymbol{\nu}}` instead of a single row summed over atoms :math:`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 per-atom quantities are replaced with rows corresponding to
+descriptor gradient components on single atoms:
+
+.. math::
+
+  \frac{\partial {B_{i,\boldsymbol{\nu}}  }}{\partial {r}^a_j}
+
+where :math:`{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
+:math:`j`. The rows in an atom :math:`j` chunk correspond to
+atoms with global index :math:`i`. The total number of rows for
+these descriptor gradients is therefore :math:`3N^2`.
+The number of columns is equal to the number of ACE descriptors,
+plus 3 additional left-most columns representing the global atom indices
+:math:`i`, :math:`j`,
+and Cartesian direction :math:`a`  (0, 1, 2, for x, y, z).
+The first 3 columns of the first :math:`N` rows belong to the reference
+potential force components. The remaining K columns contain the
+:math:`B_{i,\boldsymbol{\nu}}` per-atom descriptors corresponding to the non-zero entries
+obtained when *bikflag* = 1.
+The first column of the last row, after the first
+:math:`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 :math:`N + 3N^2 + 1` and the number of columns is :math:`K + 3`.
+
+These values can be accessed by any command that uses global values
+from a compute as input.  See the :doc:`Howto output <Howto_output>` doc
+page for an overview of LAMMPS output options.
+
+Restrictions
+""""""""""""
+
+These computes are part of the ML-PACE package.  They are only enabled
+if LAMMPS was built with that package.  See the :doc:`Build package
+<Build_package>` page for more info.
+
+Related commands
+""""""""""""""""
+
+:doc:`pair_style pace <pair_pace>`
+:doc:`pair_style snap <pair_snap>`
+:doc:`compute snap <compute_sna_atom>`
+
+Default
+"""""""
+
+The optional keyword defaults are *bikflag* = 0,
+*dgradflag* = 0
+
+----------
+
+.. _Drautz19:
+
+**(Drautz)** Drautz, Phys Rev B, 99, 014104 (2019).
+
+.. _Lysogorskiy21:
+
+**(Lysogorskiy)** Lysogorskiy, van der Oord, Bochkarev, Menon, Rinaldi, Hammerschmidt, Mrovec, Thompson, Csanyi, Ortner, Drautz, npj Comp Mat, 7, 97 (2021).
+
+.. _Goff23:
+
+**(Goff)** Goff, Zhang, Negre, Rohskopf, Niklasson, Journal of Chemical Theory and Computation 19, no. 13 (2023).

examples/PACKAGES/pace/README.md

@@ -0,0 +1,9 @@
+# This folder contains examples for pace in LAMMPS
+
+
+## Compute pace usage
+compute/latte_cell_0.data            # lammps data file with C-H-O structure
+compute/latte_cell_0.xyz             # xyz file with C-H-O structure 
+compute/coupling_coefficients.yace   # .yace file containing coupling coefficients (or ACE potential parameters)
+compute/in.compute                   # input file for calling `compute pace`
+

examples/PACKAGES/pace/compute/coupling_coefficients.yace

@@ -0,0 +1,294 @@
+elements: [H, N, O] 
+E0: [0.000000, 0.000000, 0.000000] 
+deltaSplineBins: 0.001000 
+embeddings:
+  0: {ndensity: 1, FS_parameters: [1.0, 1.0], npoti: FinnisSinclair, rho_core_cutoff: 100000, drho_core_cutoff: 250}
+  1: {ndensity: 1, FS_parameters: [1.0, 1.0], npoti: FinnisSinclair, rho_core_cutoff: 100000, drho_core_cutoff: 250}
+  2: {ndensity: 1, FS_parameters: [1.0, 1.0], npoti: FinnisSinclair, rho_core_cutoff: 100000, drho_core_cutoff: 250}
+bonds:
+  [0, 0]: {nradmax: 2, lmax: 2, nradbasemax: 2, radbasename: ChebExpCos, radparameters: [3.3], radcoefficients: [[[1, 0], [1, 0], [1, 0]], [[0, 1], [0, 1], [0, 1]]], prehc: 0, lambdahc: 3.3, rcut: 5.0, dcut: 0.01, rcut_in: 0.1, dcut_in: 0.01, inner_cutoff_type: distance}
+  [0, 1]: {nradmax: 2, lmax: 2, nradbasemax: 2, radbasename: ChebExpCos, radparameters: [3.3], radcoefficients: [[[1, 0], [1, 0], [1, 0]], [[0, 1], [0, 1], [0, 1]]], prehc: 0, lambdahc: 3.3, rcut: 5.5, dcut: 0.01, rcut_in: 0.1, dcut_in: 0.01, inner_cutoff_type: distance}
+  [0, 2]: {nradmax: 2, lmax: 2, nradbasemax: 2, radbasename: ChebExpCos, radparameters: [3.3], radcoefficients: [[[1, 0], [1, 0], [1, 0]], [[0, 1], [0, 1], [0, 1]]], prehc: 0, lambdahc: 3.3, rcut: 5.7, dcut: 0.01, rcut_in: 0.1, dcut_in: 0.01, inner_cutoff_type: distance}
+  [1, 0]: {nradmax: 2, lmax: 2, nradbasemax: 2, radbasename: ChebExpCos, radparameters: [3.3], radcoefficients: [[[1, 0], [1, 0], [1, 0]], [[0, 1], [0, 1], [0, 1]]], prehc: 0, lambdahc: 3.3, rcut: 5.5, dcut: 0.01, rcut_in: 0.1, dcut_in: 0.01, inner_cutoff_type: distance}
+  [1, 1]: {nradmax: 2, lmax: 2, nradbasemax: 2, radbasename: ChebExpCos, radparameters: [3.3], radcoefficients: [[[1, 0], [1, 0], [1, 0]], [[0, 1], [0, 1], [0, 1]]], prehc: 0, lambdahc: 3.3, rcut: 4.4, dcut: 0.01, rcut_in: 0.1, dcut_in: 0.01, inner_cutoff_type: distance}
+  [1, 2]: {nradmax: 2, lmax: 2, nradbasemax: 2, radbasename: ChebExpCos, radparameters: [3.3], radcoefficients: [[[1, 0], [1, 0], [1, 0]], [[0, 1], [0, 1], [0, 1]]], prehc: 0, lambdahc: 3.3, rcut: 5.7, dcut: 0.01, rcut_in: 0.1, dcut_in: 0.01, inner_cutoff_type: distance}
+  [2, 0]: {nradmax: 2, lmax: 2, nradbasemax: 2, radbasename: ChebExpCos, radparameters: [3.3], radcoefficients: [[[1, 0], [1, 0], [1, 0]], [[0, 1], [0, 1], [0, 1]]], prehc: 0, lambdahc: 3.3, rcut: 5.7, dcut: 0.01, rcut_in: 0.1, dcut_in: 0.01, inner_cutoff_type: distance}
+  [2, 1]: {nradmax: 2, lmax: 2, nradbasemax: 2, radbasename: ChebExpCos, radparameters: [3.3], radcoefficients: [[[1, 0], [1, 0], [1, 0]], [[0, 1], [0, 1], [0, 1]]], prehc: 0, lambdahc: 3.3, rcut: 5.7, dcut: 0.01, rcut_in: 0.1, dcut_in: 0.01, inner_cutoff_type: distance}
+  [2, 2]: {nradmax: 2, lmax: 2, nradbasemax: 2, radbasename: ChebExpCos, radparameters: [3.3], radcoefficients: [[[1, 0], [1, 0], [1, 0]], [[0, 1], [0, 1], [0, 1]]], prehc: 0, lambdahc: 3.3, rcut: 5.5, dcut: 0.01, rcut_in: 0.1, dcut_in: 0.01, inner_cutoff_type: distance}
+functions:
+  0:
+    - {mu0: 0, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [0], ns: [1], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 0, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [0], ns: [2], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 0, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [1], ns: [2], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 0, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [1], ns: [1], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 0, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [2], ns: [1], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 0, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [2], ns: [2], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 0, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 2], ns: [1, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 0, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 2], ns: [1, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 0, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [1, 1], ns: [1, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 0, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 1], ns: [1, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 0, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [1, 2], ns: [1, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 0, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 2], ns: [2, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 0, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [1, 1], ns: [2, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 0, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 2], ns: [1, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 0, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [1, 2], ns: [2, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 0, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 0], ns: [1, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 0, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 0], ns: [1, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 0, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 1], ns: [2, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 0, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [1, 2], ns: [2, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 0, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [2, 2], ns: [1, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 0, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 1], ns: [2, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 0, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 0], ns: [2, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
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+    - {mu0: 0, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 1], ns: [2, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
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+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 2, 2], ns: [1, 1, 1], ls: [2, 1, 1], ms_combs: [-2, 1, 1, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 2, -1, -1], ctildes: [0.25819888974716115, -0.18257418583505536, -0.18257418583505536, 0.10540925533894599, 0.21081851067789198, 0.10540925533894599, -0.18257418583505536, -0.18257418583505536, 0.25819888974716115]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 1, 2], ns: [1, 1, 1], ls: [1, 2, 1], ms_combs: [-1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -2, 1, 1, -1, 0, 1, 0, -1], ctildes: [0.10540925533894599, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.21081851067789198, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.10540925533894599]}
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+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [1, 1, 2], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [1, 1, 2], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [1, 2, 2], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 2], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [0, 1, 2], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [1, 2, 2], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 1, 1], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [1, 1, 1], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 0, 2], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [2, 2, 2], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 0, 0], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 0, 2], ns: [1, 1, 1], ls: [1, 2, 1], ms_combs: [-1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -2, 1, 1, -1, 0, 1, 0, -1], ctildes: [0.10540925533894599, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.21081851067789198, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.10540925533894599]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 1, 2], ns: [1, 1, 1], ls: [2, 1, 1], ms_combs: [-2, 1, 1, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 2, -1, -1], ctildes: [0.25819888974716115, -0.18257418583505536, -0.18257418583505536, 0.10540925533894599, 0.21081851067789198, 0.10540925533894599, -0.18257418583505536, -0.18257418583505536, 0.25819888974716115]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 2, 2], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 0, 1], ns: [1, 1, 1], ls: [1, 2, 1], ms_combs: [-1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -2, 1, 1, -1, 0, 1, 0, -1], ctildes: [0.10540925533894599, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.21081851067789198, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.10540925533894599]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 1, 1], ns: [1, 1, 1], ls: [2, 1, 1], ms_combs: [-2, 1, 1, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 2, -1, -1], ctildes: [0.25819888974716115, -0.18257418583505536, -0.18257418583505536, 0.10540925533894599, 0.21081851067789198, 0.10540925533894599, -0.18257418583505536, -0.18257418583505536, 0.25819888974716115]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [1, 1, 1], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 0, 1], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 1, 2], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [1, 2, 2], ns: [1, 1, 1], ls: [2, 1, 1], ms_combs: [-2, 1, 1, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 2, -1, -1], ctildes: [0.25819888974716115, -0.18257418583505536, -0.18257418583505536, 0.10540925533894599, 0.21081851067789198, 0.10540925533894599, -0.18257418583505536, -0.18257418583505536, 0.25819888974716115]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [0, 1, 1], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [0, 2, 2], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [2, 2, 2], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 1], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 0, 0], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 0, 1], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 0, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 1, 1], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 1, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 1, 1, 1], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 1, 1, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 1, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 2, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [1, 1, 1, 1], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [1, 1, 1, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [1, 1, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [1, 2, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [2, 2, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+  1:
+    - {mu0: 1, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [1], ns: [1], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 1, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [1], ns: [2], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 1, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [2], ns: [1], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 1, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [2], ns: [2], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 1, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [0], ns: [2], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 1, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [0], ns: [1], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 1], ns: [1, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [2, 2], ns: [1, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [1, 2], ns: [1, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [1, 1], ns: [2, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [2, 2], ns: [2, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 1], ns: [2, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [1, 2], ns: [1, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 2], ns: [1, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [1, 1], ns: [1, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [1, 1], ns: [1, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 1], ns: [1, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 1], ns: [2, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 2], ns: [2, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [1, 1], ns: [1, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 2], ns: [2, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 0], ns: [2, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 2], ns: [1, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [1, 1], ns: [2, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [2, 2], ns: [1, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 2], ns: [1, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 1], ns: [2, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [2, 2], ns: [2, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 0], ns: [2, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 0], ns: [1, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [2, 2], ns: [1, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [1, 2], ns: [2, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 2], ns: [1, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 0], ns: [1, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [1, 2], ns: [1, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 0], ns: [1, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [1, 2], ns: [2, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 2], ns: [2, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [2, 2], ns: [1, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [1, 2], ns: [1, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 2], ns: [2, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [1, 1], ns: [1, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [1, 2], ns: [2, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 0], ns: [1, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 1], ns: [2, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 1], ns: [1, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [1, 2], ns: [2, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 1], ns: [1, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 0, 1], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 1, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [1, 1, 2], ns: [1, 1, 1], ls: [1, 2, 1], ms_combs: [-1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -2, 1, 1, -1, 0, 1, 0, -1], ctildes: [0.10540925533894599, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.21081851067789198, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.10540925533894599]}
+    - {mu0: 1, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [1, 1, 2], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 1, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 0, 2], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
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+    - {mu0: 1, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [1, 1, 1], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 1, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 0, 0], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 1, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 0, 1], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 1, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 0, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 1, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 1, 1], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 1, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 1, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 1, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 1, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 1, 1, 1], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 1, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 1, 1, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 1, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 1, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 1, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 2, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 1, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [1, 1, 1, 1], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 1, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [1, 1, 1, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 1, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [1, 1, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 1, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [1, 2, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 1, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [2, 2, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+  2:
+    - {mu0: 2, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [0], ns: [1], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 2, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [1], ns: [1], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 2, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [0], ns: [2], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 2, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [1], ns: [2], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 2, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [2], ns: [1], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 2, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [2], ns: [2], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [1, 1], ns: [1, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 2], ns: [2, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 1], ns: [1, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [1, 1], ns: [1, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [1, 2], ns: [1, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 1], ns: [1, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [1, 1], ns: [2, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 1], ns: [1, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [1, 1], ns: [1, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 1], ns: [2, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [1, 2], ns: [2, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [1, 1], ns: [1, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 2], ns: [2, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [2, 2], ns: [1, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 0], ns: [2, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 2], ns: [1, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 1], ns: [2, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 2], ns: [2, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [1, 2], ns: [1, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [2, 2], ns: [1, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 1], ns: [2, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [2, 2], ns: [2, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 0], ns: [1, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [1, 2], ns: [1, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 0], ns: [1, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 0], ns: [2, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [1, 1], ns: [2, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 2], ns: [1, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 0], ns: [1, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 1], ns: [2, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [1, 2], ns: [2, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 2], ns: [2, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [1, 2], ns: [1, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [2, 2], ns: [1, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 1], ns: [1, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [2, 2], ns: [2, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 0], ns: [1, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 2], ns: [1, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [1, 2], ns: [2, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [1, 2], ns: [2, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 2], ns: [1, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 2, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [2, 2], ns: [1, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [1, 2, 2], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [0, 2, 2], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 0, 0], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 2, 2], ns: [1, 1, 1], ls: [2, 1, 1], ms_combs: [-2, 1, 1, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 2, -1, -1], ctildes: [0.25819888974716115, -0.18257418583505536, -0.18257418583505536, 0.10540925533894599, 0.21081851067789198, 0.10540925533894599, -0.18257418583505536, -0.18257418583505536, 0.25819888974716115]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [1, 1, 2], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 2], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [1, 1, 1], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 0, 2], ns: [1, 1, 1], ls: [1, 2, 1], ms_combs: [-1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -2, 1, 1, -1, 0, 1, 0, -1], ctildes: [0.10540925533894599, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.21081851067789198, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.10540925533894599]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 1, 1], ns: [1, 1, 1], ls: [2, 1, 1], ms_combs: [-2, 1, 1, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 2, -1, -1], ctildes: [0.25819888974716115, -0.18257418583505536, -0.18257418583505536, 0.10540925533894599, 0.21081851067789198, 0.10540925533894599, -0.18257418583505536, -0.18257418583505536, 0.25819888974716115]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 1, 2], ns: [1, 1, 1], ls: [2, 1, 1], ms_combs: [-2, 1, 1, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 2, -1, -1], ctildes: [0.25819888974716115, -0.18257418583505536, -0.18257418583505536, 0.10540925533894599, 0.21081851067789198, 0.10540925533894599, -0.18257418583505536, -0.18257418583505536, 0.25819888974716115]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 0, 1], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 2, 2], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 0, 2], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 0, 1], ns: [1, 1, 1], ls: [1, 2, 1], ms_combs: [-1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -2, 1, 1, -1, 0, 1, 0, -1], ctildes: [0.10540925533894599, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.21081851067789198, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.10540925533894599]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 1, 2], ns: [1, 1, 1], ls: [1, 2, 1], ms_combs: [-1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -2, 1, 1, -1, 0, 1, 0, -1], ctildes: [0.10540925533894599, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.21081851067789198, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.10540925533894599]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 1], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 0], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 1, 2], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [2, 2, 2], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [1, 1, 2], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [2, 2, 2], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [0, 1, 2], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 1, 1], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [1, 1, 2], ns: [1, 1, 1], ls: [1, 2, 1], ms_combs: [-1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -2, 1, 1, -1, 0, 1, 0, -1], ctildes: [0.10540925533894599, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.21081851067789198, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.10540925533894599]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [1, 1, 1], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [1, 2, 2], ns: [1, 1, 1], ls: [2, 1, 1], ms_combs: [-2, 1, 1, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 2, -1, -1], ctildes: [0.25819888974716115, -0.18257418583505536, -0.18257418583505536, 0.10540925533894599, 0.21081851067789198, 0.10540925533894599, -0.18257418583505536, -0.18257418583505536, 0.25819888974716115]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [1, 2, 2], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [0, 1, 1], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 0, 0], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 0, 1], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 0, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 1, 1], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 1, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 1, 1, 1], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 1, 1, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 1, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 2, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [1, 1, 1, 1], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [1, 1, 1, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [1, 1, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [1, 2, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [2, 2, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}

examples/PACKAGES/pace/compute/in.compute

@@ -0,0 +1,22 @@
+#info all out log
+units  metal
+atom_style  atomic
+boundary    p p p
+atom_modify    map hash
+boundary  p p p
+read_data  latte_cell_0.data
+mass  1 1.00
+mass  2 14.00
+mass  3 15.999
+
+        # potential settings
+
+pair_style     zero 5.7
+pair_coeff     * *
+
+compute     pace all pace coupling_coefficients.yace 1 0
+
+thermo 1 
+thermo_style  custom step temp c_pace[1][183]
+
+run 0

examples/PACKAGES/pace/compute/latte_cell_0.data

@@ -0,0 +1,172 @@
+latte_cell_0.data (written by ASE) 
+
+161 	 atoms 
+3  atom types
+0.0                      12  xlo xhi
+0.0                      12  ylo yhi
+0.0                      12  zlo zhi
+
+
+Atoms 
+
+     1   3      1.2688096799999999      2.0079938400000001      2.7446829899999998
+     2   1      1.5343068200000001      2.0638766500000001      3.7105626900000002
+     3   1      1.7848279600000001      2.6755003400000001      2.2268847200000002
+     4   1              1.56251195      1.1089126899999999      2.3978115199999999
+     5   1             11.61728216              5.71881094      2.4732045999999999
+     6   2      6.5501865600000002      4.7439566800000001      3.6526025500000001
+     7   1      6.4564895299999998      4.1571673000000002      2.6975267999999999
+     8   2               2.0835561              1.59406078      8.5498047600000007
+     9   1      1.1041162499999999      1.4971771599999999      8.1507879200000009
+    10   1              2.60115534      2.2945960400000001              7.95374187
+    11   1      1.9817723300000001      2.0194066400000001      9.5128239400000005
+    12   1     0.99333338000000004      3.6983907299999998      8.1903947899999991
+    13   3      4.9484070999999998      5.3645501400000004              9.16152503
+    14   1      9.0716170599999995      9.3748453999999999      4.2276462400000003
+    15   2     0.30864418999999998      7.7136657499999997      2.9274995599999998
+    16   1              0.47661671              10.1807211              3.71160091
+    17   1              1.07465334      7.8226921999999997      3.5771466900000002
+    18   1     0.38402249999999999      8.3770493300000002      2.1748437100000002
+    19   1      11.435413410000001      7.7903735999999997      3.4040245499999999
+    20   3      6.1570384599999999             10.25988474              3.50899568
+    21   1      5.5932224399999999      9.5632944700000007      3.1446559000000001
+    22   2      1.7785569000000001      7.6312579300000003      9.1488452299999992
+    23   1      2.5594048599999999              6.96832838      9.3069700199999996
+    24   1              2.12441551      8.4547986999999996      8.6428622900000001
+    25   1              1.04552782      7.1697722800000001      8.5894244999999998
+    26   1     0.34824445999999998             10.17844028      9.1629463799999993
+    27   3      5.9638830399999998      10.723709400000001      9.4568803900000002
+    28   1      6.5890835699999997      10.926486110000001      8.7981925800000003
+    29   2      7.1065890400000002              1.83029753      3.3452543600000002
+    30   1      6.9229304999999997      1.8465022099999999      4.3089037100000001
+    31   1      8.0780433600000006      1.9303052199999999      3.2089521400000001
+    32   1      5.6795373600000003      10.471831630000001      4.3244390499999996
+    33   1              6.82999417     0.95850113000000003      2.9815288199999999
+    34   2      11.383805349999999      4.6301225199999996      2.5393688399999998
+    35   1     0.37927047000000003      4.1943216300000001              2.59073807
+    36   3      5.2376410099999999              1.91523463      9.7240636400000007
+    37   1      4.7887202499999999      2.7036936499999999      9.5698142300000004
+    38   1      9.8129906699999996      9.2075140700000002              4.08265499
+    39   1      4.7980879500000002      1.1403494700000001      9.6739962800000008
+    40   1      5.4455845600000004      2.0102099999999998      10.620773509999999
+    41   3     0.90954338999999995      4.6240093199999999      8.3108110600000007
+    42   1      11.909735319999999      4.7483814000000004      8.2500624600000005
+    43   2      7.3223424499999998               7.5866457      3.0245226500000002
+    44   1      7.4470362200000002      8.3169646700000008      3.7148003300000001
+    45   1      6.9073805300000002               7.9385021      2.1723768699999999
+    46   1      5.5542868500000004      5.1176065800000003      3.7655251999999999
+    47   1      6.8124309500000004      6.7778811599999997      3.3973232499999999
+    48   2     0.29575823000000001             11.04303794      3.1016142499999999
+    49   1     0.86490721999999998             11.83879228      3.6389974500000002
+    50   3              6.85201686      8.0846369300000003      8.8762878799999996
+    51   1      7.3351430100000004      7.4263498700000001      9.3821674799999997
+    52   1      6.7919613300000004      7.7595477199999996      7.9716174799999999
+    53   1      3.8990487699999998      6.4283490399999996      8.8832409600000002
+    54   1              5.95997296      9.9329723199999993      9.4746654699999997
+    55   3      11.403658979999999      10.371960359999999      9.2766092199999992
+    56   1      10.983666360000001      9.5157199800000001      9.1478757300000009
+    57   3      1.5223279700000001      5.3327331100000004     0.57537605999999997
+    58   1      2.3815113999999999      5.7251991200000001     0.77945295999999997
+    59   1     0.92079957000000001      6.0931282299999996     0.62203253000000003
+    60   3             11.23490924      2.9153355200000002      6.7585064099999999
+    61   1      10.792340190000001      2.9755225099999998      5.9000018399999998
+    62   1      10.751242059999999      2.1896156000000002      7.1807401500000001
+    63   3             11.39027944      7.3462855600000001      6.7258299499999996
+    64   1             10.92025679              6.69831954      7.2776696599999999
+    65   1             11.12238028      7.0632020999999998      5.8394107799999997
+    66   3      8.4684319499999994             10.71736286             10.60018556
+    67   1      8.5672201599999998      11.420466080000001             11.25794033
+    68   1      9.0803109800000001             10.04804949              10.9406517
+    69   3      6.5851757299999996      9.9940623399999993      6.5574614899999997
+    70   1      7.0276325799999997             10.76096604      6.1723333699999996
+    71   1      5.7419327400000002      10.384583920000001      6.8228822999999998
+    72   3      1.7600546399999999              1.01771919      5.4926787700000004
+    73   1      1.9704209100000001              1.01748419      6.4429703700000003
+    74   1              1.42973007              0.11076352      5.3470644900000002
+    75   1              2.61130613      9.7034123700000006      10.450306830000001
+    76   3              3.05086908             10.48131334      10.085189310000001
+    77   1      3.0032693199999998             10.93357295      9.3652121000000008
+    78   3      6.4631532199999997      8.7652058299999993      11.967847969999999
+    79   1      6.4506808400000004      9.6596595300000008      11.588956019999999
+    80   1      5.6611629700000003      8.3535737700000006      11.638443329999999
+    81   3      1.1745999300000001      5.2420690800000003      5.1001449699999997
+    82   1              1.31932881      5.5236392399999996      6.0128966300000002
+    83   1     0.58053834999999998      4.4898134599999997      5.2325565000000003
+    84   3      6.7275549699999999     0.78840874999999999      7.3817280900000002
+    85   1      6.3887965600000003              1.54670982      6.8634520400000003
+    86   1      7.6791783999999996     0.94039024000000004      7.2649461000000004
+    87   3      8.5476657199999995   0.0064750299999999997      5.0450514100000001
+    88   1      8.8736290899999997             11.10484108      4.8601807900000003
+    89   1      8.0477597599999999     0.20198361000000001      4.2357399400000002
+    90   3      1.2895030000000001      8.4280097900000008             11.82038504
+    91   1      1.4766666399999999      8.1087866399999999             10.87290333
+    92   1              2.10220669      8.1947620200000006              0.29510553
+    93   3      9.6797907599999995      6.4207335499999996      4.3469150599999997
+    94   1      8.9271530099999996              6.72940235      3.7974122399999999
+    95   1             10.20024126      5.9167739199999998              3.66976111
+    96   3              3.57411616      6.7041021699999996      3.8825478499999999
+    97   1      2.8894899500000002      6.1560529800000001      4.2980848099999998
+    98   1      4.3613707699999997      6.4304732400000004      4.3804965400000002
+    99   3               4.7506556      11.441853350000001              1.12537088
+   100   1      4.0861192800000001      10.748523670000001      1.1923347099999999
+   101   1      5.5035301600000004      10.965688249999999     0.73651277000000004
+   102   3      9.5254526399999992      4.8994443900000002      8.3732284099999994
+   103   1      8.7885959800000002      4.3508043900000004      8.6632831400000008
+   104   1      9.6149067499999994      4.6084911499999999      7.4540068699999997
+   105   3      4.1970746700000001              1.34592128              3.67401439
+   106   1      4.9437011999999996     0.74406280999999996                3.514068
+   107   1      4.1905534900000001      1.7730376000000001      2.7963049400000002
+   108   3              1.88232618             11.95451227     0.60024434000000004
+   109   1      2.0464587299999999             11.02454723              0.38329541
+   110   1      1.1518493700000001     0.17494340999999999             11.99928285
+   111   3      3.7593842199999998             11.01685511      6.4562050800000002
+   112   1      3.2125414299999999              10.4553747      5.8894917099999997
+   113   1      3.4166026899999999      10.821557670000001      7.3296563900000002
+   114   3      9.7039841399999993              3.95001545      11.894743249999999
+   115   1      10.461666060000001      3.9163117999999999      11.285435229999999
+   116   1             10.09834695      4.4026997400000001     0.68193007999999999
+   117   3      8.5639596400000002      3.5169507499999999      5.6224104199999996
+   118   1      8.3966650299999994      2.6262214699999999      5.2638164300000003
+   119   1      7.9695371399999999      4.0825059799999996      5.0049407400000003
+   120   3      9.6736245000000007     0.48030482000000002      7.9257577799999996
+   121   1      9.6131980400000003      11.883419180000001      7.1680923999999999
+   122   1      9.9784050299999993             11.90238635              8.63894187
+   123   3      3.9424153099999999      6.9650296699999998             11.60258943
+   124   1      4.2767152700000004      6.8460048999999996      10.670225220000001
+   125   1      4.6570638500000001      6.5129461500000003    0.091159879999999999
+   126   3      3.0570173199999999      9.6631958499999993      3.6611250599999998
+   127   1      2.5400490100000002      9.5743355000000001      2.8444047600000002
+   128   1      2.9314874400000002      8.7809807200000005      4.0425234200000002
+   129   3      7.4549612700000001      5.8430850799999998      11.011384720000001
+   130   1      8.1675884100000005      5.4639182799999997             10.47644287
+   131   1      6.7135573700000002      5.8393818399999997      10.361099749999999
+   132   3      9.8029139300000008      7.9578901699999998             10.21404942
+   133   1             10.38910242      8.3400641400000008             10.87949429
+   134   1      9.0637612000000001      7.6392374099999998      10.756928869999999
+   135   3      4.4963435599999997      4.1067935799999997             11.73387805
+   136   1      4.5473727899999998      4.9577970899999997             11.19223377
+   137   1      5.3588818399999996      4.1756111699999998     0.20355936999999999
+   138   3      9.5923448100000002      7.3418014600000001              1.34856172
+   139   1      8.8715593300000002      7.4776837199999999              2.05040471
+   140   1      9.0443221699999992      7.2732200799999998              0.54011714
+   141   3      7.0350963100000001              3.22348773               0.7070824
+   142   1      7.1784470499999999      4.1340314300000003      1.0184109699999999
+   143   1      7.7787854400000001      2.7888888399999998              1.15838887
+   144   3      9.2124107800000008     0.48085899999999998              1.21751966
+   145   1      9.6620436499999993      11.657271079999999              1.45318397
+   146   1      9.9404883900000005              1.11619136              1.18684594
+   147   3              1.19704207      9.5859959200000002      6.6190888899999996
+   148   1              0.25606413      9.6737366500000004      6.8319340899999998
+   149   1      1.2690051899999999      8.6249354900000004      6.5480112500000001
+   150   3     0.78256133999999999      2.6040609300000002      11.453408359999999
+   151   1              0.61502181      3.5607405999999999             11.40300991
+   152   1              1.55655312      2.5457368800000002      10.866733030000001
+   153   3      5.8627936099999998      7.1217054800000001              5.89173203
+   154   1      6.3432410700000004      7.9400136699999999      6.0855840299999997
+   155   1      5.5077296699999998      6.8468306800000001      6.7436875799999996
+   156   3      10.887828150000001      9.9637482500000001     0.51092815999999996
+   157   1             11.78841776      10.322043069999999     0.44704989000000001
+   158   1             11.02688182      9.2051906700000004      1.0976661299999999
+   159   3              3.93073389      4.1645674499999998      5.7137877000000001
+   160   1      4.6884062999999996      3.5788913299999998      5.5644605800000004
+   161   1      4.2956948500000003      4.7644888099999996      6.3801669700000003

examples/PACKAGES/pace/compute/latte_cell_0.xyz

@@ -0,0 +1,163 @@
+161
+Lattice="12.0 0.0 0.0 0.0 12.0 0.0 0.0 0.0 12.0" Properties=species:S:1:pos:R:3 pbc="T T T"
+O        1.26880968       2.00799384       2.74468299
+H        1.53430682       2.06387665       3.71056269
+H        1.78482796       2.67550034       2.22688472
+H        1.56251195       1.10891269       2.39781152
+H       11.61728216       5.71881094       2.47320460
+N        6.55018656       4.74395668       3.65260255
+H        6.45648953       4.15716730       2.69752680
+N        2.08355610       1.59406078       8.54980476
+H        1.10411625       1.49717716       8.15078792
+H        2.60115534       2.29459604       7.95374187
+H        1.98177233       2.01940664       9.51282394
+H        0.99333338       3.69839073       8.19039479
+O        4.94840710       5.36455014       9.16152503
+H        9.07161706       9.37484540       4.22764624
+N        0.30864419       7.71366575       2.92749956
+H        0.47661671      10.18072110       3.71160091
+H        1.07465334       7.82269220       3.57714669
+H        0.38402250       8.37704933       2.17484371
+H       11.43541341       7.79037360       3.40402455
+O        6.15703846      10.25988474       3.50899568
+H        5.59322244       9.56329447       3.14465590
+N        1.77855690       7.63125793       9.14884523
+H        2.55940486       6.96832838       9.30697002
+H        2.12441551       8.45479870       8.64286229
+H        1.04552782       7.16977228       8.58942450
+H        0.34824446      10.17844028       9.16294638
+O        5.96388304      10.72370940       9.45688039
+H        6.58908357      10.92648611       8.79819258
+N        7.10658904       1.83029753       3.34525436
+H        6.92293050       1.84650221       4.30890371
+H        8.07804336       1.93030522       3.20895214
+H        5.67953736      10.47183163       4.32443905
+H        6.82999417       0.95850113       2.98152882
+N       11.38380535       4.63012252       2.53936884
+H        0.37927047       4.19432163       2.59073807
+O        5.23764101       1.91523463       9.72406364
+H        4.78872025       2.70369365       9.56981423
+H        9.81299067       9.20751407       4.08265499
+H        4.79808795       1.14034947       9.67399628
+H        5.44558456       2.01021000      10.62077351
+O        0.90954339       4.62400932       8.31081106
+H       11.90973532       4.74838140       8.25006246
+N        7.32234245       7.58664570       3.02452265
+H        7.44703622       8.31696467       3.71480033
+H        6.90738053       7.93850210       2.17237687
+H        5.55428685       5.11760658       3.76552520
+H        6.81243095       6.77788116       3.39732325
+N        0.29575823      11.04303794       3.10161425
+H        0.86490722      11.83879228       3.63899745
+O        6.85201686       8.08463693       8.87628788
+H        7.33514301       7.42634987       9.38216748
+H        6.79196133       7.75954772       7.97161748
+H        3.89904877       6.42834904       8.88324096
+H        5.95997296       9.93297232       9.47466547
+O       11.40365898      10.37196036       9.27660922
+H       10.98366636       9.51571998       9.14787573
+O        1.52232797       5.33273311       0.57537606
+H        2.38151140       5.72519912       0.77945296
+H        0.92079957       6.09312823       0.62203253
+O       11.23490924       2.91533552       6.75850641
+H       10.79234019       2.97552251       5.90000184
+H       10.75124206       2.18961560       7.18074015
+O       11.39027944       7.34628556       6.72582995
+H       10.92025679       6.69831954       7.27766966
+H       11.12238028       7.06320210       5.83941078
+O        8.46843195      10.71736286      10.60018556
+H        8.56722016      11.42046608      11.25794033
+H        9.08031098      10.04804949      10.94065170
+O        6.58517573       9.99406234       6.55746149
+H        7.02763258      10.76096604       6.17233337
+H        5.74193274      10.38458392       6.82288230
+O        1.76005464       1.01771919       5.49267877
+H        1.97042091       1.01748419       6.44297037
+H        1.42973007       0.11076352       5.34706449
+H        2.61130613       9.70341237      10.45030683
+O        3.05086908      10.48131334      10.08518931
+H        3.00326932      10.93357295       9.36521210
+O        6.46315322       8.76520583      11.96784797
+H        6.45068084       9.65965953      11.58895602
+H        5.66116297       8.35357377      11.63844333
+O        1.17459993       5.24206908       5.10014497
+H        1.31932881       5.52363924       6.01289663
+H        0.58053835       4.48981346       5.23255650
+O        6.72755497       0.78840875       7.38172809
+H        6.38879656       1.54670982       6.86345204
+H        7.67917840       0.94039024       7.26494610
+O        8.54766572       0.00647503       5.04505141
+H        8.87362909      11.10484108       4.86018079
+H        8.04775976       0.20198361       4.23573994
+O        1.28950300       8.42800979      11.82038504
+H        1.47666664       8.10878664      10.87290333
+H        2.10220669       8.19476202       0.29510553
+O        9.67979076       6.42073355       4.34691506
+H        8.92715301       6.72940235       3.79741224
+H       10.20024126       5.91677392       3.66976111
+O        3.57411616       6.70410217       3.88254785
+H        2.88948995       6.15605298       4.29808481
+H        4.36137077       6.43047324       4.38049654
+O        4.75065560      11.44185335       1.12537088
+H        4.08611928      10.74852367       1.19233471
+H        5.50353016      10.96568825       0.73651277
+O        9.52545264       4.89944439       8.37322841
+H        8.78859598       4.35080439       8.66328314
+H        9.61490675       4.60849115       7.45400687
+O        4.19707467       1.34592128       3.67401439
+H        4.94370120       0.74406281       3.51406800
+H        4.19055349       1.77303760       2.79630494
+O        1.88232618      11.95451227       0.60024434
+H        2.04645873      11.02454723       0.38329541
+H        1.15184937       0.17494341      11.99928285
+O        3.75938422      11.01685511       6.45620508
+H        3.21254143      10.45537470       5.88949171
+H        3.41660269      10.82155767       7.32965639
+O        9.70398414       3.95001545      11.89474325
+H       10.46166606       3.91631180      11.28543523
+H       10.09834695       4.40269974       0.68193008
+O        8.56395964       3.51695075       5.62241042
+H        8.39666503       2.62622147       5.26381643
+H        7.96953714       4.08250598       5.00494074
+O        9.67362450       0.48030482       7.92575778
+H        9.61319804      11.88341918       7.16809240
+H        9.97840503      11.90238635       8.63894187
+O        3.94241531       6.96502967      11.60258943
+H        4.27671527       6.84600490      10.67022522
+H        4.65706385       6.51294615       0.09115988
+O        3.05701732       9.66319585       3.66112506
+H        2.54004901       9.57433550       2.84440476
+H        2.93148744       8.78098072       4.04252342
+O        7.45496127       5.84308508      11.01138472
+H        8.16758841       5.46391828      10.47644287
+H        6.71355737       5.83938184      10.36109975
+O        9.80291393       7.95789017      10.21404942
+H       10.38910242       8.34006414      10.87949429
+H        9.06376120       7.63923741      10.75692887
+O        4.49634356       4.10679358      11.73387805
+H        4.54737279       4.95779709      11.19223377
+H        5.35888184       4.17561117       0.20355937
+O        9.59234481       7.34180146       1.34856172
+H        8.87155933       7.47768372       2.05040471
+H        9.04432217       7.27322008       0.54011714
+O        7.03509631       3.22348773       0.70708240
+H        7.17844705       4.13403143       1.01841097
+H        7.77878544       2.78888884       1.15838887
+O        9.21241078       0.48085900       1.21751966
+H        9.66204365      11.65727108       1.45318397
+H        9.94048839       1.11619136       1.18684594
+O        1.19704207       9.58599592       6.61908889
+H        0.25606413       9.67373665       6.83193409
+H        1.26900519       8.62493549       6.54801125
+O        0.78256134       2.60406093      11.45340836
+H        0.61502181       3.56074060      11.40300991
+H        1.55655312       2.54573688      10.86673303
+O        5.86279361       7.12170548       5.89173203
+H        6.34324107       7.94001367       6.08558403
+H        5.50772967       6.84683068       6.74368758
+O       10.88782815       9.96374825       0.51092816
+H       11.78841776      10.32204307       0.44704989
+H       11.02688182       9.20519067       1.09766613
+O        3.93073389       4.16456745       5.71378770
+H        4.68840630       3.57889133       5.56446058
+H        4.29569485       4.76448881       6.38016697

examples/PACKAGES/pace/compute/log.5Dec23.compute.g++.1

@@ -0,0 +1,81 @@
+LAMMPS (21 Nov 2023)
+OMP_NUM_THREADS environment is not set. Defaulting to 1 thread. (src/comm.cpp:98)
+  using 1 OpenMP thread(s) per MPI task
+#info all out log
+units  metal
+atom_style  atomic
+boundary    p p p
+atom_modify    map hash
+boundary  p p p
+read_data  latte_cell_0.data
+Reading data file ...
+  orthogonal box = (0 0 0) to (12 12 12)
+  1 by 1 by 1 MPI processor grid
+  reading atoms ...
+  161 atoms
+  read_data CPU = 0.001 seconds
+mass  1 1.00
+mass  2 14.00
+mass  3 15.999
+
+        # potential settings
+
+pair_style     zero 5.7
+pair_coeff     * *
+
+compute     pace all pace coupling_coefficients.yace 1 0
+
+thermo 1
+thermo_style  custom step temp c_pace[1][183]
+
+run 0
+WARNING: No fixes with time integration, atoms won't move (src/verlet.cpp:60)
+Generated 0 of 3 mixed pair_coeff terms from geometric mixing rule
+Neighbor list info ...
+  update: every = 1 steps, delay = 0 steps, check = yes
+  max neighbors/atom: 2000, page size: 100000
+  master list distance cutoff = 7.7
+  ghost atom cutoff = 7.7
+  binsize = 3.85, bins = 4 4 4
+  2 neighbor lists, perpetual/occasional/extra = 1 1 0
+  (1) pair zero, perpetual
+      attributes: half, newton on
+      pair build: half/bin/atomonly/newton
+      stencil: half/bin/3d
+      bin: standard
+  (2) compute pace, occasional
+      attributes: full, newton on
+      pair build: full/bin/atomonly
+      stencil: full/bin/3d
+      bin: standard
+Per MPI rank memory allocation (min/avg/max) = 6.993 | 6.993 | 6.993 Mbytes
+   Step          Temp      c_pace[1][183]
+         0   0              8.6885642    
+Loop time of 1.217e-06 on 1 procs for 0 steps with 161 atoms
+
+164.3% CPU use with 1 MPI tasks x 1 OpenMP threads
+
+MPI task timing breakdown:
+Section |  min time  |  avg time  |  max time  |%varavg| %total
+---------------------------------------------------------------
+Pair    | 0          | 0          | 0          |   0.0 |  0.00
+Neigh   | 0          | 0          | 0          |   0.0 |  0.00
+Comm    | 0          | 0          | 0          |   0.0 |  0.00
+Output  | 0          | 0          | 0          |   0.0 |  0.00
+Modify  | 0          | 0          | 0          |   0.0 |  0.00
+Other   |            | 1.217e-06  |            |       |100.00
+
+Nlocal:            161 ave         161 max         161 min
+Histogram: 1 0 0 0 0 0 0 0 0 0
+Nghost:           1754 ave        1754 max        1754 min
+Histogram: 1 0 0 0 0 0 0 0 0 0
+Neighs:          14230 ave       14230 max       14230 min
+Histogram: 1 0 0 0 0 0 0 0 0 0
+FullNghs:        28460 ave       28460 max       28460 min
+Histogram: 1 0 0 0 0 0 0 0 0 0
+
+Total # of neighbors = 28460
+Ave neighs/atom = 176.77019
+Neighbor list builds = 0
+Dangerous builds = 0
+Total wall time: 0:00:00

examples/PACKAGES/pace/compute/log.5Dec23.compute.g++.4

@@ -0,0 +1,81 @@
+LAMMPS (21 Nov 2023)
+OMP_NUM_THREADS environment is not set. Defaulting to 1 thread. (src/comm.cpp:98)
+  using 1 OpenMP thread(s) per MPI task
+#info all out log
+units  metal
+atom_style  atomic
+boundary    p p p
+atom_modify    map hash
+boundary  p p p
+read_data  latte_cell_0.data
+Reading data file ...
+  orthogonal box = (0 0 0) to (12 12 12)
+  1 by 2 by 2 MPI processor grid
+  reading atoms ...
+  161 atoms
+  read_data CPU = 0.001 seconds
+mass  1 1.00
+mass  2 14.00
+mass  3 15.999
+
+        # potential settings
+
+pair_style     zero 5.7
+pair_coeff     * *
+
+compute     pace all pace coupling_coefficients.yace 1 0
+
+thermo 1
+thermo_style  custom step temp c_pace[1][183]
+
+run 0
+WARNING: No fixes with time integration, atoms won't move (src/verlet.cpp:60)
+Generated 0 of 3 mixed pair_coeff terms from geometric mixing rule
+Neighbor list info ...
+  update: every = 1 steps, delay = 0 steps, check = yes
+  max neighbors/atom: 2000, page size: 100000
+  master list distance cutoff = 7.7
+  ghost atom cutoff = 7.7
+  binsize = 3.85, bins = 4 4 4
+  2 neighbor lists, perpetual/occasional/extra = 1 1 0
+  (1) pair zero, perpetual
+      attributes: half, newton on
+      pair build: half/bin/atomonly/newton
+      stencil: half/bin/3d
+      bin: standard
+  (2) compute pace, occasional
+      attributes: full, newton on
+      pair build: full/bin/atomonly
+      stencil: full/bin/3d
+      bin: standard
+Per MPI rank memory allocation (min/avg/max) = 6.97 | 6.97 | 6.971 Mbytes
+   Step          Temp      c_pace[1][183]
+         0   0              8.6885642    
+Loop time of 1.979e-06 on 4 procs for 0 steps with 161 atoms
+
+164.2% CPU use with 4 MPI tasks x 1 OpenMP threads
+
+MPI task timing breakdown:
+Section |  min time  |  avg time  |  max time  |%varavg| %total
+---------------------------------------------------------------
+Pair    | 0          | 0          | 0          |   0.0 |  0.00
+Neigh   | 0          | 0          | 0          |   0.0 |  0.00
+Comm    | 0          | 0          | 0          |   0.0 |  0.00
+Output  | 0          | 0          | 0          |   0.0 |  0.00
+Modify  | 0          | 0          | 0          |   0.0 |  0.00
+Other   |            | 1.979e-06  |            |       |100.00
+
+Nlocal:          40.25 ave          44 max          35 min
+Histogram: 1 0 0 0 1 0 0 0 1 1
+Nghost:         1134.5 ave        1159 max        1117 min
+Histogram: 1 1 0 0 1 0 0 0 0 1
+Neighs:         3557.5 ave        4115 max        3189 min
+Histogram: 2 0 0 0 0 1 0 0 0 1
+FullNghs:         7115 ave        7755 max        6158 min
+Histogram: 1 0 0 0 1 0 0 0 0 2
+
+Total # of neighbors = 28460
+Ave neighs/atom = 176.77019
+Neighbor list builds = 0
+Dangerous builds = 0
+Total wall time: 0:00:00

src/.gitignore

@@ -100,6 +100,8 @@
 /lepton_utils.cpp
 /lepton_utils.h
 
+/compute_pace.cpp
+/compute_pace.h
 /pair_pace.cpp
 /pair_pace.h
 /pair_pace_extrapolation.cpp

src/ML-PACE/compute_pace.cpp

@@ -0,0 +1,496 @@
+// clang-format off
+/* ----------------------------------------------------------------------
+   LAMMPS - Large-scale Atomic/Molecular Massively Parallel Simulator
+   https://www.lammps.org/, Sandia National Laboratories
+   LAMMPS Development team: developers@lammps.org
+   Copyright (2003) Sandia Corporation.  Under the terms of Contract
+   DE-AC04-94AL85000 with Sandia Corporation, the U.S. Government retains
+   certain rights in this software.  This software is distributed under
+   the GNU General Public License.
+   See the README file in the top-level LAMMPS directory.
+------------------------------------------------------------------------- */
+
+#include "compute_pace.h"
+#include "ace-evaluator/ace_evaluator.h"
+#include "ace-evaluator/ace_c_basis.h"
+#include "ace-evaluator/ace_abstract_basis.h"
+#include "ace-evaluator/ace_types.h"
+#include <cstring>
+#include <map>
+
+#include "atom.h"
+#include "update.h"
+#include "modify.h"
+#include "neighbor.h"
+#include "neigh_list.h"
+#include "neigh_request.h"
+#include "force.h"
+#include "pair.h"
+#include "comm.h"
+#include "memory.h"
+#include "error.h"
+
+namespace LAMMPS_NS {
+struct ACECimpl {
+  ACECimpl() : basis_set(nullptr), ace(nullptr) {}
+  ~ACECimpl()
+  {
+    delete basis_set;
+    delete ace;
+  }
+  ACECTildeBasisSet *basis_set;
+  ACECTildeEvaluator *ace;
+};
+}
+
+using namespace LAMMPS_NS;
+
+enum{SCALAR,VECTOR,ARRAY};
+ComputePACE::ComputePACE(LAMMPS *lmp, int narg, char **arg) :
+  Compute(lmp, narg, arg), cutsq(nullptr), list(nullptr), pace(nullptr),
+  paceall(nullptr), pace_peratom(nullptr), map(nullptr), cg(nullptr)
+{
+  array_flag = 1;
+  extarray = 0;
+  bikflag = 0;
+  dgradflag = 0;
+
+  int ntypes = atom->ntypes;
+  int nargmin = 4;
+
+  acecimpl = new ACECimpl;
+  if (narg < nargmin) error->all(FLERR,"Illegal compute pace command");
+
+  bikflag = utils::inumeric(FLERR, arg[4], false, lmp);
+  dgradflag = utils::inumeric(FLERR, arg[5], false, lmp);
+  if (dgradflag && !bikflag)
+    error->all(FLERR,"Illegal compute pace command: dgradflag=1 requires bikflag=1");
+
+  memory->create(map,ntypes+1,"pace:map");
+
+  //read in file with CG coefficients or c_tilde coefficients
+
+  auto potential_file_name = utils::get_potential_file_path(arg[3]);
+  delete acecimpl -> basis_set;
+  acecimpl -> basis_set = new ACECTildeBasisSet(potential_file_name);
+  double cut = acecimpl -> basis_set->cutoffmax;
+  cutmax = acecimpl -> basis_set->cutoffmax;
+  double cuti;
+  double radelemall = 0.5;
+
+  //# of rank 1, rank > 1 functions
+
+  int n_r1, n_rp = 0;
+  n_r1 = acecimpl -> basis_set->total_basis_size_rank1[0];
+  n_rp = acecimpl -> basis_set->total_basis_size[0];
+
+  int ncoeff = n_r1 + n_rp;
+
+  //int nvalues = ncoeff;
+
+  nvalues = ncoeff;
+
+  //-----------------------------------------------------------
+  //nperdim = ncoeff;
+
+  ndims_force = 3;
+  ndims_virial = 6;
+  bik_rows = 1;
+  yoffset = nvalues; //nperdim;
+  zoffset = 2*nvalues; //nperdim;
+  natoms = atom->natoms;
+  if (bikflag) bik_rows = natoms;
+    dgrad_rows = ndims_force*natoms;
+  size_array_rows = bik_rows+dgrad_rows + ndims_virial;
+  if (dgradflag) {
+    size_array_rows = bik_rows + 3*natoms*natoms + 1;
+    size_array_cols = nvalues + 3;
+    if (comm->me == 0)
+      error->warning(FLERR,"dgradflag=1 creates a N^2 array, beware of large systems.");
+  } else size_array_cols = nvalues*atom->ntypes + 1;
+  lastcol = size_array_cols-1;
+
+  ndims_peratom = ndims_force;
+  size_peratom = ndims_peratom*nvalues*atom->ntypes;
+
+  nmax = 0;
+}
+
+/* ---------------------------------------------------------------------- */
+
+ComputePACE::~ComputePACE()
+{
+  delete acecimpl;
+  memory->destroy(pace);
+  memory->destroy(paceall);
+  memory->destroy(cutsq);
+  memory->destroy(pace_peratom);
+  memory->destroy(map);
+}
+
+/* ---------------------------------------------------------------------- */
+
+void ComputePACE::init()
+{
+  if (force->pair == nullptr)
+    error->all(FLERR,"Compute pace requires a pair style be defined");
+
+  if (cutmax > force->pair->cutforce)
+    error->all(FLERR,"Compute pace cutoff is longer than pairwise cutoff");
+
+  // need an occasional full neighbor list
+  neighbor->add_request(this, NeighConst::REQ_FULL | NeighConst::REQ_OCCASIONAL);
+
+  int count = 0;
+  for (int i = 0; i < modify->ncompute; i++)
+    if (strcmp(modify->compute[i]->style,"pace") == 0) count++;
+  if (count > 1 && comm->me == 0)
+    error->warning(FLERR,"More than one compute pace");
+
+  // allocate memory for global array
+  memory->create(pace,size_array_rows,size_array_cols, "pace:pace");
+  memory->create(paceall,size_array_rows,size_array_cols, "pace:paceall");
+  array = paceall;
+
+  // find compute for reference energy
+
+  std::string id_pe = std::string("thermo_pe");
+  int ipe = modify->find_compute(id_pe);
+  if (ipe == -1)
+    error->all(FLERR,"compute thermo_pe does not exist.");
+  c_pe = modify->compute[ipe];
+
+  // add compute for reference virial tensor
+
+  std::string id_virial = std::string("pace_press");
+  std::string pcmd = id_virial + " all pressure NULL virial";
+  modify->add_compute(pcmd);
+
+  int ivirial = modify->find_compute(id_virial);
+  if (ivirial == -1)
+    error->all(FLERR,"compute pace_press does not exist.");
+  c_virial = modify->compute[ivirial];
+}
+
+/* ---------------------------------------------------------------------- */
+
+void ComputePACE::init_list(int /*id*/, NeighList *ptr)
+{
+  list = ptr;
+}
+
+/* ---------------------------------------------------------------------- */
+
+void ComputePACE::compute_array()
+{
+  int ntotal = atom->nlocal + atom->nghost;
+  double **f = atom->f;
+  invoked_array = update->ntimestep;
+
+  // grow pace_peratom array if necessary
+
+  if (atom->nmax > nmax) {
+    memory->destroy(pace_peratom);
+    nmax = atom->nmax;
+    memory->create(pace_peratom,nmax,size_peratom,"pace:pace_peratom");
+  }
+
+  // clear global array
+
+  for (int irow = 0; irow < size_array_rows; irow++){
+    for (int icoeff = 0; icoeff < size_array_cols; icoeff++){
+      pace[irow][icoeff] = 0.0;
+    }
+  }
+
+  // clear local peratom array
+
+  for (int i = 0; i < ntotal; i++){
+    for (int icoeff = 0; icoeff < size_peratom; icoeff++) {
+      pace_peratom[i][icoeff] = 0.0;
+    }
+  }
+
+  // invoke full neighbor list (will copy or build if necessary)
+
+  neighbor->build_one(list);
+  SPECIES_TYPE *mus;
+  NS_TYPE *ns;
+  LS_TYPE *ls;
+
+  const int inum = list->inum;
+  const int* const ilist = list->ilist;
+  const int* const numneigh = list->numneigh;
+  int** const firstneigh = list->firstneigh;
+  int * const type = atom->type;
+
+  //determine the maximum number of neighbours
+  int max_jnum = -1;
+  int nei = 0;
+  int jtmp =0;
+  for (int iitmp = 0; iitmp < list->inum; iitmp++) {
+    int itmp = ilist[iitmp];
+    jtmp = numneigh[itmp];
+    nei = nei + jtmp;
+    if (jtmp > max_jnum){
+      max_jnum = jtmp;
+    }
+  }
+
+  // compute pace derivatives for each atom in group
+  // use full neighbor list to count atoms less than cutoff
+
+  double** const x = atom->x;
+  const int* const mask = atom->mask;
+  const int ntypes = atom->ntypes;
+
+  for (int ii = 0; ii < inum; ii++) {
+    int irow = 0;
+    if (bikflag) irow = atom->tag[ilist[ii] & NEIGHMASK]-1;
+    const int i = ilist[ii];
+    if (mask[i] & groupbit) {
+      const int itype = type[i];
+      const int* const jlist = firstneigh[i];
+      const int jnum = numneigh[i];
+      const int typeoffset_local = ndims_peratom*nvalues*(itype-1);
+      const int typeoffset_global = nvalues*(itype-1);
+
+      delete acecimpl -> ace;
+      acecimpl -> ace = new ACECTildeEvaluator(*acecimpl -> basis_set);
+      acecimpl -> ace->compute_projections = 1;
+      acecimpl -> ace->compute_b_grad = 1;
+      int n_r1, n_rp = 0;
+      n_r1 = acecimpl -> basis_set->total_basis_size_rank1[0];
+      n_rp = acecimpl -> basis_set->total_basis_size[0];
+
+      int ncoeff = n_r1 + n_rp;
+      acecimpl -> ace->element_type_mapping.init(ntypes+1);
+      for (int ik = 1; ik <= ntypes; ik++) {
+        for(int mu = 0; mu < acecimpl -> basis_set ->nelements; mu++){
+          if (mu != -1) {
+            if (mu == ik - 1) {
+              map[ik] = mu;
+              acecimpl -> ace->element_type_mapping(ik) = mu;
+            }
+          }
+        }
+      }
+
+
+      if (dgradflag) {
+
+        // dBi/dRi tags
+
+        pace[bik_rows + ((atom->tag[i]-1)*3*natoms) + 3*(atom->tag[i]-1) + 0][0] = atom->tag[i]-1;
+        pace[bik_rows + ((atom->tag[i]-1)*3*natoms) + 3*(atom->tag[i]-1) + 0][1] = atom->tag[i]-1;
+        pace[bik_rows + ((atom->tag[i]-1)*3*natoms) + 3*(atom->tag[i]-1) + 0][2] = 0;
+        pace[bik_rows + ((atom->tag[i]-1)*3*natoms) + 3*(atom->tag[i]-1) + 1][0] = atom->tag[i]-1;
+        pace[bik_rows + ((atom->tag[i]-1)*3*natoms) + 3*(atom->tag[i]-1) + 1][1] = atom->tag[i]-1;
+        pace[bik_rows + ((atom->tag[i]-1)*3*natoms) + 3*(atom->tag[i]-1) + 1][2] = 1;
+        pace[bik_rows + ((atom->tag[i]-1)*3*natoms) + 3*(atom->tag[i]-1) + 2][0] = atom->tag[i]-1;
+        pace[bik_rows + ((atom->tag[i]-1)*3*natoms) + 3*(atom->tag[i]-1) + 2][1] = atom->tag[i]-1;
+        pace[bik_rows + ((atom->tag[i]-1)*3*natoms) + 3*(atom->tag[i]-1) + 2][2] = 2;
+
+        // dBi/dRj tags
+
+        for (int j=0; j<natoms; j++) {
+          pace[bik_rows + ((j)*3*natoms) + 3*(atom->tag[i]-1) + 0][0] = atom->tag[i]-1;
+          pace[bik_rows + ((j)*3*natoms) + 3*(atom->tag[i]-1) + 0][1] = j;
+          pace[bik_rows + ((j)*3*natoms) + 3*(atom->tag[i]-1) + 0][2] = 0;
+          pace[bik_rows + ((j)*3*natoms) + 3*(atom->tag[i]-1) + 1][0] = atom->tag[i]-1;
+          pace[bik_rows + ((j)*3*natoms) + 3*(atom->tag[i]-1) + 1][1] = j;
+          pace[bik_rows + ((j)*3*natoms) + 3*(atom->tag[i]-1) + 1][2] = 1;
+          pace[bik_rows + ((j)*3*natoms) + 3*(atom->tag[i]-1) + 2][0] = atom->tag[i]-1;
+          pace[bik_rows + ((j)*3*natoms) + 3*(atom->tag[i]-1) + 2][1] = j;
+          pace[bik_rows + ((j)*3*natoms) + 3*(atom->tag[i]-1) + 2][2] = 2;
+        }
+      }
+
+      // resize the neighbor cache after setting the basis
+      acecimpl -> ace->resize_neighbours_cache(max_jnum);
+      acecimpl -> ace->compute_atom(i, atom->x, atom->type, list->numneigh[i], list->firstneigh[i]);
+      Array1D<DOUBLE_TYPE> Bs = acecimpl -> ace -> projections;
+
+      for (int jj = 0; jj < jnum; jj++) {
+        const int j = jlist[jj];
+        //replace mapping of jj to j
+        if (!dgradflag) {
+          double *pacedi = pace_peratom[i]+typeoffset_local;
+          double *pacedj = pace_peratom[j]+typeoffset_local;
+
+          //force array in (func_ind,neighbour_ind,xyz_ind) format
+          // dimension: (n_descriptors,max_jnum,3)
+          //example to access entries for neighbour jj after running compute_atom for atom i:
+          for (int func_ind =0; func_ind < n_r1 + n_rp; func_ind++){
+            DOUBLE_TYPE fx_dB = acecimpl -> ace -> neighbours_dB(func_ind,jj,0);
+            DOUBLE_TYPE fy_dB = acecimpl -> ace -> neighbours_dB(func_ind,jj,1);
+            DOUBLE_TYPE fz_dB = acecimpl -> ace -> neighbours_dB(func_ind,jj,2);
+            pacedi[func_ind] += fx_dB;
+            pacedi[func_ind+yoffset] += fy_dB;
+            pacedi[func_ind+zoffset] += fz_dB;
+            pacedj[func_ind] -= fx_dB;
+            pacedj[func_ind+yoffset] -= fy_dB;
+            pacedj[func_ind+zoffset] -= fz_dB;
+            }
+         } else {
+            //printf("inside dBi/dRj logical : ncoeff = %d \n", ncoeff);
+            for (int iicoeff = 0; iicoeff < ncoeff; iicoeff++) {
+
+              // add to pace array for this proc
+              //printf("inside dBi/dRj loop\n");
+              // dBi/dRj
+              DOUBLE_TYPE fx_dB = acecimpl -> ace -> neighbours_dB(iicoeff,jj,0);
+              DOUBLE_TYPE fy_dB = acecimpl -> ace -> neighbours_dB(iicoeff,jj,1);
+              DOUBLE_TYPE fz_dB = acecimpl -> ace -> neighbours_dB(iicoeff,jj,2);
+              pace[bik_rows + ((atom->tag[j]-1)*3*natoms) + 3*(atom->tag[i]-1) + 0][iicoeff+3] -= fx_dB;
+              pace[bik_rows + ((atom->tag[j]-1)*3*natoms) + 3*(atom->tag[i]-1) + 1][iicoeff+3] -= fy_dB;
+              pace[bik_rows + ((atom->tag[j]-1)*3*natoms) + 3*(atom->tag[i]-1) + 2][iicoeff+3] -= fz_dB;
+
+              // dBi/dRi
+              pace[bik_rows + ((atom->tag[i]-1)*3*natoms) + 3*(atom->tag[i]-1) + 0][iicoeff+3] += fx_dB;
+              pace[bik_rows + ((atom->tag[i]-1)*3*natoms) + 3*(atom->tag[i]-1) + 1][iicoeff+3] += fy_dB;
+              pace[bik_rows + ((atom->tag[i]-1)*3*natoms) + 3*(atom->tag[i]-1) + 2][iicoeff+3] += fz_dB;
+            }
+          }
+        } // loop over jj inside
+      if (!dgradflag) {
+
+        int k = typeoffset_global;
+
+        for (int icoeff = 0; icoeff < ncoeff; icoeff++){
+          pace[irow][k++] += Bs(icoeff);
+        }
+      } else {
+        int k = 3;
+        for (int icoeff = 0; icoeff < ncoeff; icoeff++){
+          pace[irow][k++] += Bs(icoeff);
+        }
+      }
+    } //group bit
+  } // for ii loop
+  // accumulate force contributions to global array
+  if (!dgradflag){
+    for (int itype = 0; itype < atom->ntypes; itype++) {
+      const int typeoffset_local = ndims_peratom*nvalues*itype;
+      const int typeoffset_global = nvalues*itype;
+      for (int icoeff = 0; icoeff < nvalues; icoeff++) {
+        for (int i = 0; i < ntotal; i++) {
+          double *pacedi = pace_peratom[i]+typeoffset_local;
+          int iglobal = atom->tag[i];
+          int irow = 3*(iglobal-1)+1;
+          pace[irow++][icoeff+typeoffset_global] += pacedi[icoeff];
+          pace[irow++][icoeff+typeoffset_global] += pacedi[icoeff+yoffset];
+          pace[irow][icoeff+typeoffset_global] += pacedi[icoeff+zoffset];
+        }
+      }
+    }
+  }
+
+  if (!dgradflag) {
+    // accumulate forces to global array
+    for (int i = 0; i < atom->nlocal; i++) {
+      int iglobal = atom->tag[i];
+      int irow = 3*(iglobal-1)+1;
+      pace[irow++][lastcol] = atom->f[i][0];
+      pace[irow++][lastcol] = atom->f[i][1];
+      pace[irow][lastcol] = atom->f[i][2];
+    }
+  } else {
+
+    // for dgradflag=1, put forces at first 3 columns of bik rows
+
+    for (int i=0; i<atom->nlocal; i++) {
+      int iglobal = atom->tag[i];
+      pace[iglobal-1][0+0] = atom->f[i][0];
+      pace[iglobal-1][0+1] = atom->f[i][1];
+      pace[iglobal-1][0+2] = atom->f[i][2];
+    }
+  }
+
+  dbdotr_compute();
+
+  // sum up over all processes
+  MPI_Allreduce(&pace[0][0],&paceall[0][0],size_array_rows*size_array_cols,MPI_DOUBLE,MPI_SUM,world);
+
+  // assign energy to last column
+
+  if (!dgradflag) {
+    for (int i = 0; i < bik_rows; i++) paceall[i][lastcol] = 0;
+    int irow = 0;
+    double reference_energy = c_pe->compute_scalar();
+    paceall[irow][lastcol] = reference_energy;
+  } else {
+
+    // assign reference energy right after the dgrad rows, first column
+
+    int irow = bik_rows + 3*natoms*natoms;
+    double reference_energy = c_pe->compute_scalar();
+    paceall[irow][0] = reference_energy;
+  }
+
+  // assign virial stress to last column
+  // switch to Voigt notation
+
+  if (!dgradflag) {
+    c_virial->compute_vector();
+    int irow = 3*natoms+bik_rows;
+    paceall[irow++][lastcol] = c_virial->vector[0];
+    paceall[irow++][lastcol] = c_virial->vector[1];
+    paceall[irow++][lastcol] = c_virial->vector[2];
+    paceall[irow++][lastcol] = c_virial->vector[5];
+    paceall[irow++][lastcol] = c_virial->vector[4];
+    paceall[irow++][lastcol] = c_virial->vector[3];
+  }
+}
+
+/* ----------------------------------------------------------------------
+   compute global virial contributions via summing r_i.dB^j/dr_i over
+   own & ghost atoms
+------------------------------------------------------------------------- */
+void ComputePACE::dbdotr_compute()
+{
+
+  if (dgradflag) return;
+
+  double **x = atom->x;
+  int irow0 = bik_rows+ndims_force*natoms;
+
+  // sum over ace contributions to forces
+  // on all particles including ghosts
+
+  int nall = atom->nlocal + atom->nghost;
+  for (int i = 0; i < nall; i++)
+    for (int itype = 0; itype < atom->ntypes; itype++) {
+      const int typeoffset_local = ndims_peratom*nvalues*itype;
+      const int typeoffset_global = nvalues*itype;
+      double *pacedi = pace_peratom[i]+typeoffset_local;
+      for (int icoeff = 0; icoeff < nvalues; icoeff++) {
+        double dbdx = pacedi[icoeff];
+        double dbdy = pacedi[icoeff+yoffset];
+        double dbdz = pacedi[icoeff+zoffset];
+        int irow = irow0;
+        pace[irow++][icoeff+typeoffset_global] += dbdx*x[i][0];
+        pace[irow++][icoeff+typeoffset_global] += dbdy*x[i][1];
+        pace[irow++][icoeff+typeoffset_global] += dbdz*x[i][2];
+        pace[irow++][icoeff+typeoffset_global] += dbdz*x[i][1];
+        pace[irow++][icoeff+typeoffset_global] += dbdz*x[i][0];
+        pace[irow++][icoeff+typeoffset_global] += dbdy*x[i][0];
+      }
+    }
+}
+
+/* ----------------------------------------------------------------------
+   memory usage
+------------------------------------------------------------------------- */
+
+double ComputePACE::memory_usage()
+{
+
+  double bytes = (double)size_array_rows*size_array_cols*sizeof(double); // pace
+  bytes += (double)size_array_rows*size_array_cols*sizeof(double);       // paceall
+  bytes += (double)nmax*size_peratom * sizeof(double);                   // pace_peratom
+  int n = atom->ntypes+1;
+  bytes += (double)n*sizeof(int);        // map
+
+  return bytes;
+}

src/ML-PACE/compute_pace.h

@@ -0,0 +1,60 @@
+/* -*- c++ -*- ----------------------------------------------------------
+   LAMMPS - Large-scale Atomic/Molecular Massively Parallel Simulator
+   https://www.lammps.org/, Sandia National Laboratories
+   LAMMPS Development team: developers@lammps.org
+
+   Copyright (2003) Sandia Corporation.  Under the terms of Contract
+   DE-AC04-94AL85000 with Sandia Corporation, the U.S. Government retains
+   certain rights in this software.  This software is distributed under
+   the GNU General Public License.
+
+   See the README file in the top-level LAMMPS directory.
+------------------------------------------------------------------------- */
+
+#ifdef COMPUTE_CLASS
+// clang-format off
+ComputeStyle(pace,ComputePACE);
+// clang-format on
+#else
+
+#ifndef LMP_COMPUTE_PACE_H
+#define LMP_COMPUTE_PACE_H
+
+#include "compute.h"
+
+namespace LAMMPS_NS {
+
+class ComputePACE : public Compute {
+ public:
+  ComputePACE(class LAMMPS *, int, char **);
+  ~ComputePACE();
+  void init();
+  void init_list(int, class NeighList *);
+  void compute_array();
+  double memory_usage();
+
+ private:
+  int natoms, nmax, size_peratom, lastcol;
+  int ncoeff, nvalues, nperdim, yoffset, zoffset;
+  int ndims_peratom, ndims_force, ndims_virial;
+  double **cutsq;
+  class NeighList *list;
+  double **pace, **paceall;
+  double **pace_peratom;
+  double rcutfac;
+  int *map;    // map types to [0,nelements)
+  int nelements, chemflag;
+  int bikflag, bik_rows, dgradflag, dgrad_rows;
+  double *cg;
+  double cutmax;
+  Compute *c_pe;
+  Compute *c_virial;
+
+  void dbdotr_compute();
+  struct ACECimpl *acecimpl;
+};
+
+}    // namespace LAMMPS_NS
+
+#endif
+#endif