diff --git a/cmake/CMakeLists.txt b/cmake/CMakeLists.txt
index 28e02bbee7..f7e9b314bd 100644
--- a/cmake/CMakeLists.txt
+++ b/cmake/CMakeLists.txt
@@ -971,33 +971,53 @@ if(PKG_KOKKOS)
   endif()
 endif()
 if(PKG_KSPACE)
-  message(STATUS "<<< FFT settings >>>
--- Primary FFT lib:  ${FFT}")
-  if(FFT_SINGLE)
-    message(STATUS "Using single precision FFTs")
-  else()
-    message(STATUS "Using double precision FFTs")
-  endif()
-  if(FFT_FFTW_THREADS OR FFT_MKL_THREADS)
-    message(STATUS "Using threaded FFTs")
-  else()
-    message(STATUS "Using non-threaded FFTs")
-  endif()
-  if(PKG_KOKKOS)
-    if(Kokkos_ENABLE_CUDA)
-      if(FFT STREQUAL "KISS")
-        message(STATUS "Kokkos FFT: KISS")
-      else()
-        message(STATUS "Kokkos FFT: cuFFT")
-      endif()
-    elseif(Kokkos_ENABLE_HIP)
-      if(FFT STREQUAL "KISS")
-        message(STATUS "Kokkos FFT: KISS")
-      else()
-        message(STATUS "Kokkos FFT: hipFFT")
-      endif()
+  if (LMP_HEFFTE)
+    message(STATUS "<<< FFT settings >>>
+-- Primary FFT lib:  heFFTe")
+    if (HEFFTE_BACKEND)
+      message(STATUS "heFFTe backend:  ${HEFFTE_BACKEND}")
     else()
-      message(STATUS "Kokkos FFT: ${FFT}")
+      message(STATUS "heFFTe backend:  stock (builtin FFT implementation, tested for corrected but not optimized for production)")
+    endif()
+    if(FFT_SINGLE)
+      message(STATUS "Using single precision FFTs")
+    else()
+      message(STATUS "Using double precision FFTs")
+    endif()
+  else()
+    message(STATUS "<<< FFT settings >>>
+-- Primary FFT lib:  ${FFT}")
+    if(FFT_SINGLE)
+      message(STATUS "Using single precision FFTs")
+    else()
+      message(STATUS "Using double precision FFTs")
+    endif()
+    if(FFT_FFTW_THREADS OR FFT_MKL_THREADS)
+      message(STATUS "Using threaded FFTs")
+    else()
+      message(STATUS "Using non-threaded FFTs")
+    endif()
+    if (FFT_HEFFTE)
+      message(STATUS "Using distributed algorithms from heFTTe")
+    else()
+      message(STATUS "Using builtin distributed algorithms")
+    endif()
+    if(PKG_KOKKOS)
+      if(Kokkos_ENABLE_CUDA)
+        if(FFT STREQUAL "KISS")
+          message(STATUS "Kokkos FFT: KISS")
+        else()
+          message(STATUS "Kokkos FFT: cuFFT")
+        endif()
+      elseif(Kokkos_ENABLE_HIP)
+        if(FFT STREQUAL "KISS")
+          message(STATUS "Kokkos FFT: KISS")
+        else()
+          message(STATUS "Kokkos FFT: hipFFT")
+        endif()
+      else()
+        message(STATUS "Kokkos FFT: ${FFT}")
+      endif()
     endif()
   endif()
 endif()
diff --git a/cmake/Modules/Packages/KSPACE.cmake b/cmake/Modules/Packages/KSPACE.cmake
index de7e7e5b20..9c9c879cd4 100644
--- a/cmake/Modules/Packages/KSPACE.cmake
+++ b/cmake/Modules/Packages/KSPACE.cmake
@@ -46,6 +46,42 @@ else()
   target_compile_definitions(lammps PRIVATE -DFFT_KISS)
 endif()
 
+option(FFT_USE_HEFFTE  "Use heFFTe as the distributed FFT engine, overrides the FFT option."  OFF)
+if(FFT_USE_HEFFTE)
+  # if FFT_HEFFTE is enabled, switch the builtin FFT engine with Heffte
+  set(FFT_HEFFTE_BACKEND_VALUES FFTW MKL)
+  set(FFT_HEFFTE_BACKEND "" CACHE STRING "Select heFFTe backend, e.g., FFTW or MKL")
+  set_property(CACHE FFT_HEFFTE_BACKEND PROPERTY STRINGS ${FFT_HEFFTE_BACKEND_VALUES})
+
+  if(FFT_HEFFTE_BACKEND STREQUAL "FFTW") # respect the backend choice, FFTW or MKL
+    set(HEFFTE_COMPONENTS "FFTW")
+    set(Heffte_ENABLE_FFTW "ON" CACHE BOOL "Enables FFTW backend for heFFTe")
+  elseif(FFT_HEFFTE_BACKEND STREQUAL "MKL")
+    set(HEFFTE_COMPONENTS "MKL")
+    set(Heffte_ENABLE_MKL "ON" CACHE BOOL "Enables MKL backend for heFFTe")
+  else()
+    message(WARNING "FFT_HEFFTE_BACKEND not selected, defaulting to the builtin 'stock' backend, which is intended for testing and is not optimized for production runs")
+  endif()
+
+  find_package(Heffte 2.4.0 QUIET COMPONENTS ${HEFFTE_COMPONENTS})
+  if (NOT Heffte_FOUND) # download and build
+    include(FetchContent)
+    FetchContent_Declare(HEFFTE_PROJECT # using v2.4.0
+      URL  "https://github.com/icl-utk-edu/heffte/archive/refs/tags/v2.4.0.tar.gz"
+      URL_HASH SHA256=02310fb4f9688df02f7181667e61c3adb7e38baf79611d80919d47452ff7881d
+      )
+    FetchContent_Populate(HEFFTE_PROJECT)
+    add_subdirectory(${heffte_project_SOURCE_DIR} ${heffte_project_BINARY_DIR})
+    set_target_properties(lmp PROPERTIES INSTALL_RPATH "${CMAKE_INSTALL_PREFIX}/lib")
+    set_target_properties(lammps PROPERTIES INSTALL_RPATH "${CMAKE_INSTALL_PREFIX}/lib")
+    add_library(Heffte::Heffte INTERFACE IMPORTED GLOBAL)
+    target_link_libraries(Heffte::Heffte INTERFACE Heffte)
+  endif()
+
+  target_compile_definitions(lammps PRIVATE -DFFT_HEFFTE "-DFFT_HEFFTE_${FFT_HEFFTE_BACKEND}")
+  target_link_libraries(lammps PRIVATE Heffte::Heffte)
+endif()
+
 set(FFT_PACK "array" CACHE STRING "Optimization for FFT")
 set(FFT_PACK_VALUES array pointer memcpy)
 set_property(CACHE FFT_PACK PROPERTY STRINGS ${FFT_PACK_VALUES})
diff --git a/doc/src/Build_settings.rst b/doc/src/Build_settings.rst
index 75112cce42..39608a5c32 100644
--- a/doc/src/Build_settings.rst
+++ b/doc/src/Build_settings.rst
@@ -43,6 +43,12 @@ When the KSPACE package is included in a LAMMPS build, the
 require use of an FFT library to compute 1d FFTs.  The KISS FFT
 library is included with LAMMPS, but other libraries can be faster.
 LAMMPS can use them if they are available on your system.
+Alternatively, LAMMPS can use the
+`heFFTe <https://icl-utk-edu.github.io/heffte/>`_
+library for the MPI communication algorithms,
+which comes with many optimizations for special cases,
+e.g., leveraging 2D and 3D backend transforms and
+better pipelining for packing and communication.
 
 .. tabs::
 
@@ -53,6 +59,7 @@ LAMMPS can use them if they are available on your system.
          -D FFT=value              # FFTW3 or MKL or KISS, default is FFTW3 if found, else KISS
          -D FFT_SINGLE=value       # yes or no (default), no = double precision
          -D FFT_PACK=value         # array (default) or pointer or memcpy
+         -D FFT_USE_HEFFTE=value   # yes or no (default), yes links to heFFTe
 
       .. note::
 
@@ -76,6 +83,15 @@ LAMMPS can use them if they are available on your system.
          -D MKL_INCLUDE_DIR=path     # ditto for Intel MKL library
          -D FFT_MKL_THREADS=on       # enable using threaded FFTs with MKL libraries
          -D MKL_LIBRARY=path         # path to MKL libraries
+         -D FFT_HEFFTE_BACKEND=value # FFTW or MKL or empty/undefined for the stock backend
+         -D Heffte_ROOT=path         # path to an existing heFFTe installation
+
+      .. note::
+
+         heFFTe comes with a builtin stock backend for FFTs; however, the backend
+         is intended for testing purposes and is not performance optimized
+         for large scale production runs.
+
 
    .. tab:: Traditional make
 
@@ -111,6 +127,24 @@ LAMMPS can use them if they are available on your system.
       files in its default search path.  You must specify ``FFT_LIB``
       with the appropriate FFT libraries to include in the link.
 
+      Traditional make can also link to heFFTe using an existing installation
+
+      .. code-block:: make
+
+         include <path-to-heffte-installation>/share/heffte/HeffteMakefile.in
+         FFT_INC = -DFFT_HEFFTE -DFFT_HEFFTE_FFTW $(heffte_include)
+         FFT_PATH =
+         FFT_LIB = $(heffte_link) $(heffte_libs)
+
+      The heFFTe install path will contain `HeffteMakefile.in`.
+      which will define the `heffte_` include variables needed to link to heFFTe from
+      an external project using traditional make.
+      The `-DFFT_HEFFTE` is required to switch to using heFFTe, while the optional `-DFFT_HEFFTE_FFTW`
+      selects the desired heFFTe backend, e.g., `-DFFT_HEFFTE_FFTW` or `-DFFT_HEFFTE_MKL`,
+      omitting the variable will default to the `stock` backend.
+      The heFFTe `stock` backend is intended to be used for testing and debugging,
+      but is not performance optimized for large scale production runs.
+
 The `KISS FFT library <https://github.com/mborgerding/kissfft>`_ is
 included in the LAMMPS distribution.  It is portable across all
 platforms.  Depending on the size of the FFTs and the number of
@@ -170,6 +204,16 @@ Depending on the machine, the size of the FFT grid, the number of
 processors used, one option may be slightly faster.  The default is
 ARRAY mode.
 
+When using ``-DFFT_HEFFTE`` CMake will first look for an existing install
+with hints provided by ``-DHeffte_ROOT``, as recommended by the CMake
+standard and note that the name is case sensitive. If CMake cannot find
+a heFFTe installation with the correct backend (e.g., FFTW or MKL),
+it will attempt to download and build the library automatically.
+In this case, LAMMPS CMake will also accept all heFFTe specific variables
+listed in the
+`heFFTe documentation <https://mkstoyanov.bitbucket.io/heffte/md_doxygen_installation.html>`_
+and those variables will be passed into the heFFTe build.
+
 ----------
 
 .. _size:
diff --git a/doc/src/Commands_compute.rst b/doc/src/Commands_compute.rst
index 07f5100ea4..b7e2156d88 100644
--- a/doc/src/Commands_compute.rst
+++ b/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>`
diff --git a/doc/src/Commands_fix.rst b/doc/src/Commands_fix.rst
index 3283561c41..e89e302673 100644
--- a/doc/src/Commands_fix.rst
+++ b/doc/src/Commands_fix.rst
@@ -239,10 +239,10 @@ OPT.
    * :doc:`store/force <fix_store_force>`
    * :doc:`store/state <fix_store_state>`
    * :doc:`tdpd/source <fix_dpd_source>`
-   * :doc:`temp/berendsen <fix_temp_berendsen>`
+   * :doc:`temp/berendsen (k) <fix_temp_berendsen>`
    * :doc:`temp/csld <fix_temp_csvr>`
    * :doc:`temp/csvr <fix_temp_csvr>`
-   * :doc:`temp/rescale <fix_temp_rescale>`
+   * :doc:`temp/rescale (k) <fix_temp_rescale>`
    * :doc:`temp/rescale/eff <fix_temp_rescale_eff>`
    * :doc:`tfmc <fix_tfmc>`
    * :doc:`tgnpt/drude <fix_tgnh_drude>`
diff --git a/doc/src/compute.rst b/doc/src/compute.rst
index a99d7d7c9c..6491696271 100644
--- a/doc/src/compute.rst
+++ b/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
diff --git a/doc/src/compute_pace.rst b/doc/src/compute_pace.rst
new file mode 100644
index 0000000000..0c4a295f1b
--- /dev/null
+++ b/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).
diff --git a/doc/src/fix_nh_uef.rst b/doc/src/fix_nh_uef.rst
index 2d4cc32c76..60d47994f0 100644
--- a/doc/src/fix_nh_uef.rst
+++ b/doc/src/fix_nh_uef.rst
@@ -23,7 +23,7 @@ Syntax
   .. parsed-literal::
 
      keyword = *erate* or *ext* or *strain* or *temp* or *iso* or *x* or *y* or *z* or *tchain* or *pchain* or *tloop* or *ploop* or *mtk*
-       *erate* values = e_x e_y = engineering strain rates (required)
+       *erate* values = e_x e_y = true strain rates (required)
        *ext* value = *x* or *y* or *z* or *xy* or *yz* or *xz* = external dimensions
          sets the external dimensions used to calculate the scalar pressure
        *strain* values = e_x e_y = initial strain
diff --git a/doc/src/fix_temp_berendsen.rst b/doc/src/fix_temp_berendsen.rst
index 67e496e6c5..541f3191d5 100644
--- a/doc/src/fix_temp_berendsen.rst
+++ b/doc/src/fix_temp_berendsen.rst
@@ -1,8 +1,11 @@
 .. index:: fix temp/berendsen
+.. index:: fix temp/berendsen/kk
 
 fix temp/berendsen command
 ==========================
 
+Accelerator Variants: *temp/berendsen/kk*
+
 Syntax
 """"""
 
@@ -118,6 +121,10 @@ remaining thermal degrees of freedom, and the bias is added back in.
 
 ----------
 
+.. include:: accel_styles.rst
+
+----------
+
 Restart, fix_modify, output, run start/stop, minimize info
 """""""""""""""""""""""""""""""""""""""""""""""""""""""""""
 
diff --git a/doc/src/fix_temp_rescale.rst b/doc/src/fix_temp_rescale.rst
index bfdcaa90f8..2dd2178346 100644
--- a/doc/src/fix_temp_rescale.rst
+++ b/doc/src/fix_temp_rescale.rst
@@ -1,8 +1,11 @@
 .. index:: fix temp/rescale
+.. index:: fix temp/rescale/kk
 
 fix temp/rescale command
 ========================
 
+Accelerator Variants: *temp/rescale/kk*
+
 Syntax
 """"""
 
@@ -125,6 +128,10 @@ remaining thermal degrees of freedom, and the bias is added back in.
 
 ----------
 
+.. include:: accel_styles.rst
+
+----------
+
 Restart, fix_modify, output, run start/stop, minimize info
 """""""""""""""""""""""""""""""""""""""""""""""""""""""""""
 
diff --git a/examples/PACKAGES/pace/README.md b/examples/PACKAGES/pace/README.md
new file mode 100644
index 0000000000..66254b5241
--- /dev/null
+++ b/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`
+
diff --git a/examples/PACKAGES/pace/compute/coupling_coefficients.yace b/examples/PACKAGES/pace/compute/coupling_coefficients.yace
new file mode 100644
index 0000000000..2953222216
--- /dev/null
+++ b/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]}
+    - {mu0: 0, 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: 0, 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: 0, 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: 0, 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: 0, 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: 0, 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: 0, 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: 0, 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: 0, 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: 0, 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: 0, 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: 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]}
+    - {mu0: 0, 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: 0, 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: 0, 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: 0, 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: 0, 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: 0, 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: 0, 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: 0, 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: 0, 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: 0, 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: 0, 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: 0, 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: 0, 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: 0, 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]}
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+    - {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]}
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+    - {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]}
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+    - {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]}
+    - {mu0: 1, 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: 1, 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: 1, 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: 1, 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: 1, 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: 1, 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: 1, 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: 1, 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: 1, 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: 1, 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: 1, 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: 1, 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: 1, 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: 1, 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: 1, 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: 1, 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: 1, 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: 1, 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: 1, 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: 1, 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: 1, 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: 1, 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: 1, 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: 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]}
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+    - {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]}
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+    - {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]}
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+    - {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]}
diff --git a/examples/PACKAGES/pace/compute/in.compute b/examples/PACKAGES/pace/compute/in.compute
new file mode 100644
index 0000000000..a0ef25f606
--- /dev/null
+++ b/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
diff --git a/examples/PACKAGES/pace/compute/latte_cell_0.data b/examples/PACKAGES/pace/compute/latte_cell_0.data
new file mode 100644
index 0000000000..b18ec13a4d
--- /dev/null
+++ b/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
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+   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
diff --git a/examples/PACKAGES/pace/compute/latte_cell_0.xyz b/examples/PACKAGES/pace/compute/latte_cell_0.xyz
new file mode 100644
index 0000000000..afe0a27f35
--- /dev/null
+++ b/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
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+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
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+H        4.79808795       1.14034947       9.67399628
+H        5.44558456       2.01021000      10.62077351
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+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
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+H        0.86490722      11.83879228       3.63899745
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+H        3.89904877       6.42834904       8.88324096
+H        5.95997296       9.93297232       9.47466547
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+H       10.98366636       9.51571998       9.14787573
+O        1.52232797       5.33273311       0.57537606
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+H       10.79234019       2.97552251       5.90000184
+H       10.75124206       2.18961560       7.18074015
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+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
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+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
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+H        1.47666664       8.10878664      10.87290333
+H        2.10220669       8.19476202       0.29510553
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+H        5.50353016      10.96568825       0.73651277
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+H        4.19055349       1.77303760       2.79630494
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+H        1.15184937       0.17494341      11.99928285
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+H        3.41660269      10.82155767       7.32965639
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+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
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+H        9.61319804      11.88341918       7.16809240
+H        9.97840503      11.90238635       8.63894187
+O        3.94241531       6.96502967      11.60258943
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+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
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+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
diff --git a/examples/PACKAGES/pace/compute/log.5Dec23.compute.g++.1 b/examples/PACKAGES/pace/compute/log.5Dec23.compute.g++.1
new file mode 100644
index 0000000000..e5036cfe5b
--- /dev/null
+++ b/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
diff --git a/examples/PACKAGES/pace/compute/log.5Dec23.compute.g++.4 b/examples/PACKAGES/pace/compute/log.5Dec23.compute.g++.4
new file mode 100644
index 0000000000..49ca6129b6
--- /dev/null
+++ b/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
diff --git a/src/.gitignore b/src/.gitignore
index e2c9fa6d5b..74164574c2 100644
--- a/src/.gitignore
+++ b/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
diff --git a/src/KOKKOS/Install.sh b/src/KOKKOS/Install.sh
index 489efc55a0..4be74334c9 100755
--- a/src/KOKKOS/Install.sh
+++ b/src/KOKKOS/Install.sh
@@ -177,6 +177,10 @@ action fix_shardlow_kokkos.cpp fix_shardlow.cpp
 action fix_shardlow_kokkos.h fix_shardlow.h
 action fix_spring_self_kokkos.cpp
 action fix_spring_self_kokkos.h
+action fix_temp_berendsen_kokkos.cpp
+action fix_temp_berendsen_kokkos.h
+action fix_temp_rescale_kokkos.cpp
+action fix_temp_rescale_kokkos.h
 action fix_viscous_kokkos.cpp
 action fix_viscous_kokkos.h
 action fix_wall_gran_kokkos.cpp fix_wall_gran.cpp
diff --git a/src/KOKKOS/fix_temp_berendsen_kokkos.cpp b/src/KOKKOS/fix_temp_berendsen_kokkos.cpp
new file mode 100644
index 0000000000..b986b3189a
--- /dev/null
+++ b/src/KOKKOS/fix_temp_berendsen_kokkos.cpp
@@ -0,0 +1,135 @@
+// 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 "fix_temp_berendsen_kokkos.h"
+
+#include "atom_kokkos.h"
+#include "comm.h"
+#include "compute.h"
+#include "error.h"
+#include "force.h"
+#include "group.h"
+#include "input.h"
+#include "modify.h"
+#include "update.h"
+#include "variable.h"
+#include "atom_masks.h"
+
+#include <cmath>
+#include <cstring>
+
+using namespace LAMMPS_NS;
+using namespace FixConst;
+
+enum{NOBIAS,BIAS};
+enum{CONSTANT,EQUAL};
+
+/* ---------------------------------------------------------------------- */
+
+template<class DeviceType>
+FixTempBerendsenKokkos<DeviceType>::FixTempBerendsenKokkos(LAMMPS *lmp, int narg, char **arg) :
+  FixTempBerendsen(lmp, narg, arg)
+{
+  kokkosable = 1;
+  atomKK = (AtomKokkos *)atom;
+  execution_space = ExecutionSpaceFromDevice<DeviceType>::space;
+
+  datamask_read = EMPTY_MASK;
+  datamask_modify = EMPTY_MASK;
+}
+
+/* ---------------------------------------------------------------------- */
+
+template<class DeviceType>
+void FixTempBerendsenKokkos<DeviceType>::end_of_step()
+{
+  atomKK->sync(temperature->execution_space,temperature->datamask_read);
+  double t_current = temperature->compute_scalar();
+  atomKK->modified(temperature->execution_space,temperature->datamask_modify);
+  atomKK->sync(execution_space,temperature->datamask_modify);
+
+  double tdof = temperature->dof;
+
+  // there is nothing to do, if there are no degrees of freedom
+
+  if (tdof < 1) return;
+
+  if (t_current == 0.0)
+    error->all(FLERR, "Computed current temperature for fix temp/berendsen must not be 0.0");
+
+  double delta = update->ntimestep - update->beginstep;
+  if (delta != 0.0) delta /= update->endstep - update->beginstep;
+
+  // set current t_target
+  // if variable temp, evaluate variable, wrap with clear/add
+
+  if (tstyle == CONSTANT)
+    t_target = t_start + delta * (t_stop-t_start);
+  else {
+    modify->clearstep_compute();
+    t_target = input->variable->compute_equal(tvar);
+    if (t_target < 0.0)
+      error->one(FLERR, "Fix temp/berendsen variable {} returned negative temperature",
+                 input->variable->names[tvar]);
+    modify->addstep_compute(update->ntimestep + nevery);
+  }
+
+  // rescale velocities by lamda
+  // for BIAS:
+  //   temperature is current, so do not need to re-compute
+  //   OK to not test returned v = 0, since lamda is multiplied by v
+
+  double lamda = sqrt(1.0 + update->dt/t_period*(t_target/t_current - 1.0));
+  double efactor = 0.5 * force->boltz * tdof;
+  energy += t_current * (1.0-lamda*lamda) * efactor;
+
+  auto v = atomKK->k_v.view<DeviceType>();
+  auto mask = atomKK->k_mask.view<DeviceType>();
+  int nlocal = atom->nlocal;
+  auto groupbit = this->groupbit;
+
+  if (which == NOBIAS) {
+    atomKK->sync(temperature->execution_space,temperature->datamask_read);
+    temperature->remove_bias_all();
+    atomKK->modified(temperature->execution_space,temperature->datamask_modify);
+    atomKK->sync(execution_space,temperature->datamask_modify);
+  }
+
+  atomKK->sync(execution_space,V_MASK|MASK_MASK);
+
+  Kokkos::parallel_for(Kokkos::RangePolicy<DeviceType>(0,nlocal), LAMMPS_LAMBDA(int i) {
+    if (mask[i] & groupbit) {
+      v(i,0) *= lamda;
+      v(i,1) *= lamda;
+      v(i,2) *= lamda;
+    }
+  });
+
+  atomKK->modified(execution_space,V_MASK);
+
+  if (which == NOBIAS) {
+    atomKK->sync(temperature->execution_space,temperature->datamask_read);
+    temperature->restore_bias_all();
+    atomKK->modified(temperature->execution_space,temperature->datamask_modify);
+    atomKK->sync(execution_space,temperature->datamask_modify);
+  }
+}
+/* ---------------------------------------------------------------------- */
+
+namespace LAMMPS_NS {
+template class FixTempBerendsenKokkos<LMPDeviceType>;
+#ifdef LMP_KOKKOS_GPU
+template class FixTempBerendsenKokkos<LMPHostType>;
+#endif
+}
diff --git a/src/KOKKOS/fix_temp_berendsen_kokkos.h b/src/KOKKOS/fix_temp_berendsen_kokkos.h
new file mode 100644
index 0000000000..6a0aa5ce98
--- /dev/null
+++ b/src/KOKKOS/fix_temp_berendsen_kokkos.h
@@ -0,0 +1,44 @@
+/* -*- 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 FIX_CLASS
+// clang-format off
+FixStyle(temp/berendsen/kk,FixTempBerendsenKokkos<LMPDeviceType>);
+FixStyle(temp/berendsen/kk/device,FixTempBerendsenKokkos<LMPDeviceType>);
+FixStyle(temp/berendsen/kk/host,FixTempBerendsenKokkos<LMPHostType>);
+// clang-format on
+#else
+
+// clang-format off
+#ifndef LMP_FIX_TEMP_BERENDSEN_KOKKOS_H
+#define LMP_FIX_TEMP_BERENDSEN_KOKKOS_H
+
+#include "fix_temp_berendsen.h"
+#include "kokkos_type.h"
+
+namespace LAMMPS_NS {
+
+template<class DeviceType>
+class FixTempBerendsenKokkos : public FixTempBerendsen {
+ public:
+  typedef DeviceType device_type;
+
+  FixTempBerendsenKokkos(class LAMMPS *, int, char **);
+  ~FixTempBerendsenKokkos() override {}
+  void end_of_step() override;
+};
+
+}    // namespace LAMMPS_NS
+
+#endif
+#endif
diff --git a/src/KOKKOS/fix_temp_rescale_kokkos.cpp b/src/KOKKOS/fix_temp_rescale_kokkos.cpp
new file mode 100644
index 0000000000..3a1c6ddd26
--- /dev/null
+++ b/src/KOKKOS/fix_temp_rescale_kokkos.cpp
@@ -0,0 +1,140 @@
+// 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 "fix_temp_rescale_kokkos.h"
+
+#include "atom_kokkos.h"
+#include "comm.h"
+#include "compute.h"
+#include "error.h"
+#include "force.h"
+#include "group.h"
+#include "input.h"
+#include "modify.h"
+#include "update.h"
+#include "variable.h"
+#include "atom_masks.h"
+
+#include <cmath>
+#include <cstring>
+
+using namespace LAMMPS_NS;
+using namespace FixConst;
+
+enum{NOBIAS,BIAS};
+enum{CONSTANT,EQUAL};
+
+/* ---------------------------------------------------------------------- */
+
+template<class DeviceType>
+FixTempRescaleKokkos<DeviceType>::FixTempRescaleKokkos(LAMMPS *lmp, int narg, char **arg) :
+  FixTempRescale(lmp, narg, arg)
+{
+  kokkosable = 1;
+  atomKK = (AtomKokkos *)atom;
+  execution_space = ExecutionSpaceFromDevice<DeviceType>::space;
+
+  datamask_read = EMPTY_MASK;
+  datamask_modify = EMPTY_MASK;
+}
+
+/* ---------------------------------------------------------------------- */
+
+template<class DeviceType>
+void FixTempRescaleKokkos<DeviceType>::end_of_step()
+{
+  atomKK->sync(temperature->execution_space,temperature->datamask_read);
+  double t_current = temperature->compute_scalar();
+  atomKK->modified(temperature->execution_space,temperature->datamask_modify);
+  atomKK->sync(execution_space,temperature->datamask_modify);
+
+  // there is nothing to do, if there are no degrees of freedom
+
+  if (temperature->dof < 1) return;
+
+  // protect against division by zero
+
+  if (t_current == 0.0)
+    error->all(FLERR,"Computed temperature for fix temp/rescale cannot be 0.0");
+
+  double delta = update->ntimestep - update->beginstep;
+  if (delta != 0.0) delta /= update->endstep - update->beginstep;
+
+  // set current t_target
+  // if variable temp, evaluate variable, wrap with clear/add
+
+  if (tstyle == CONSTANT)
+    t_target = t_start + delta * (t_stop-t_start);
+  else {
+    modify->clearstep_compute();
+    t_target = input->variable->compute_equal(tvar);
+    if (t_target < 0.0)
+      error->one(FLERR, "Fix temp/rescale variable returned negative temperature");
+    modify->addstep_compute(update->ntimestep + nevery);
+  }
+
+  // rescale velocity of appropriate atoms if outside window
+  // for BIAS:
+  //   temperature is current, so do not need to re-compute
+  //   OK to not test returned v = 0, since factor is multiplied by v
+
+  if (fabs(t_current-t_target) > t_window) {
+    t_target = t_current - fraction*(t_current-t_target);
+    double factor = sqrt(t_target/t_current);
+    double efactor = 0.5 * force->boltz * temperature->dof;
+
+    energy += (t_current-t_target) * efactor;
+
+    auto v = atomKK->k_v.view<DeviceType>();
+    auto mask = atomKK->k_mask.view<DeviceType>();
+    int nlocal = atom->nlocal;
+    auto groupbit = this->groupbit;
+
+    if (which == NOBIAS) {
+      atomKK->sync(temperature->execution_space,temperature->datamask_read);
+      temperature->remove_bias_all();
+      atomKK->modified(temperature->execution_space,temperature->datamask_modify);
+      atomKK->sync(execution_space,temperature->datamask_modify);
+    }
+
+    atomKK->sync(execution_space,V_MASK|MASK_MASK);
+
+    Kokkos::parallel_for(Kokkos::RangePolicy<DeviceType>(0,nlocal), LAMMPS_LAMBDA(int i) {
+      if (mask[i] & groupbit) {
+        v(i,0) *= factor;
+        v(i,1) *= factor;
+        v(i,2) *= factor;
+      }
+    });
+
+    atomKK->modified(execution_space,V_MASK);
+
+    if (which == NOBIAS) {
+      atomKK->sync(temperature->execution_space,temperature->datamask_read);
+      temperature->restore_bias_all();
+      atomKK->modified(temperature->execution_space,temperature->datamask_modify);
+      atomKK->sync(execution_space,temperature->datamask_modify);
+
+    }
+  }
+}
+
+/* ---------------------------------------------------------------------- */
+
+namespace LAMMPS_NS {
+template class FixTempRescaleKokkos<LMPDeviceType>;
+#ifdef LMP_KOKKOS_GPU
+template class FixTempRescaleKokkos<LMPHostType>;
+#endif
+}
diff --git a/src/KOKKOS/fix_temp_rescale_kokkos.h b/src/KOKKOS/fix_temp_rescale_kokkos.h
new file mode 100644
index 0000000000..7dd3111325
--- /dev/null
+++ b/src/KOKKOS/fix_temp_rescale_kokkos.h
@@ -0,0 +1,44 @@
+/* -*- 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 FIX_CLASS
+// clang-format off
+FixStyle(temp/rescale/kk,FixTempRescaleKokkos<LMPDeviceType>);
+FixStyle(temp/rescale/kk/device,FixTempRescaleKokkos<LMPDeviceType>);
+FixStyle(temp/rescale/kk/host,FixTempRescaleKokkos<LMPHostType>);
+// clang-format on
+#else
+
+// clang-format off
+#ifndef LMP_FIX_TEMP_RESCALE_KOKKOS_H
+#define LMP_FIX_TEMP_RESCALE_KOKKOS_H
+
+#include "fix_temp_rescale.h"
+#include "kokkos_type.h"
+
+namespace LAMMPS_NS {
+
+template<class DeviceType>
+class FixTempRescaleKokkos : public FixTempRescale {
+ public:
+  typedef DeviceType device_type;
+
+  FixTempRescaleKokkos(class LAMMPS *, int, char **);
+  ~FixTempRescaleKokkos() override {}
+  void end_of_step() override;
+};
+
+}    // namespace LAMMPS_NS
+
+#endif
+#endif
diff --git a/src/KSPACE/fft3d_wrap.cpp b/src/KSPACE/fft3d_wrap.cpp
index 478cf6fc9d..7b00543eea 100644
--- a/src/KSPACE/fft3d_wrap.cpp
+++ b/src/KSPACE/fft3d_wrap.cpp
@@ -27,30 +27,66 @@ FFT3d::FFT3d(LAMMPS *lmp, MPI_Comm comm, int nfast, int nmid, int nslow,
              int out_klo, int out_khi,
              int scaled, int permute, int *nbuf, int usecollective) : Pointers(lmp)
 {
+  #ifndef FFT_HEFFTE
   plan = fft_3d_create_plan(comm,nfast,nmid,nslow,
                             in_ilo,in_ihi,in_jlo,in_jhi,in_klo,in_khi,
                             out_ilo,out_ihi,out_jlo,out_jhi,out_klo,out_khi,
                             scaled,permute,nbuf,usecollective);
   if (plan == nullptr) error->one(FLERR,"Could not create 3d FFT plan");
+  #else
+  heffte::plan_options options = heffte::default_options<heffte_backend>();
+  options.algorithm = (usecollective == 0) ?
+                          heffte::reshape_algorithm::p2p_plined
+                        : heffte::reshape_algorithm::alltoallv;
+  options.use_reorder = (permute != 0);
+  hscale = (scaled == 0) ? heffte::scale::none : heffte::scale::full;
+
+  heffte_plan = std::unique_ptr<heffte::fft3d<heffte_backend>>(
+        new heffte::fft3d<heffte_backend>(
+                heffte::box3d<>({in_ilo,in_jlo,in_klo}, {in_ihi, in_jhi, in_khi}),
+                heffte::box3d<>({out_ilo,out_jlo,out_klo}, {out_ihi, out_jhi, out_khi}),
+                comm, options)
+      );
+  *nbuf = heffte_plan->size_workspace();
+  heffte_workspace.resize(heffte_plan->size_workspace());
+  #endif
 }
 
 /* ---------------------------------------------------------------------- */
 
 FFT3d::~FFT3d()
 {
+  #ifndef FFT_HEFFTE
   fft_3d_destroy_plan(plan);
+  #endif
 }
 
 /* ---------------------------------------------------------------------- */
 
 void FFT3d::compute(FFT_SCALAR *in, FFT_SCALAR *out, int flag)
 {
+  #ifndef FFT_HEFFTE
   fft_3d((FFT_DATA *) in,(FFT_DATA *) out,flag,plan);
+  #else
+  if (flag == 1)
+      heffte_plan->forward(reinterpret_cast<std::complex<FFT_SCALAR>*>(in),
+                           reinterpret_cast<std::complex<FFT_SCALAR>*>(out),
+                           reinterpret_cast<std::complex<FFT_SCALAR>*>(heffte_workspace.data())
+                          );
+  else
+      heffte_plan->backward(reinterpret_cast<std::complex<FFT_SCALAR>*>(in),
+                            reinterpret_cast<std::complex<FFT_SCALAR>*>(out),
+                            reinterpret_cast<std::complex<FFT_SCALAR>*>(heffte_workspace.data()),
+                            hscale
+                           );
+  #endif
 }
 
 /* ---------------------------------------------------------------------- */
 
 void FFT3d::timing1d(FFT_SCALAR *in, int nsize, int flag)
 {
+  #ifndef FFT_HEFFTE
   fft_1d_only((FFT_DATA *) in,nsize,flag,plan);
+  #endif
 }
diff --git a/src/KSPACE/fft3d_wrap.h b/src/KSPACE/fft3d_wrap.h
index f72cfd4622..04b828b7de 100644
--- a/src/KSPACE/fft3d_wrap.h
+++ b/src/KSPACE/fft3d_wrap.h
@@ -17,6 +17,19 @@
 #include "fft3d.h"    // IWYU pragma: export
 #include "pointers.h"
 
+#ifdef FFT_HEFFTE
+#include "heffte.h"
+// select the backend
+#if defined(FFT_HEFFTE_FFTW)
+using heffte_backend = heffte::backend::fftw;
+#elif defined(FFT_HEFFTE_MKL)
+using heffte_backend = heffte::backend::mkl;
+#else
+using heffte_backend = heffte::backend::stock;
+#endif
+
+#endif // FFT_HEFFTE
+
 namespace LAMMPS_NS {
 
 class FFT3d : protected Pointers {
@@ -30,7 +43,14 @@ class FFT3d : protected Pointers {
   void timing1d(FFT_SCALAR *, int, int);
 
  private:
+  #ifdef FFT_HEFFTE
+  // the heFFTe plan supersedes the internal fft_plan_3d
+  std::unique_ptr<heffte::fft3d<heffte_backend>> heffte_plan;
+  std::vector<std::complex<FFT_SCALAR>> heffte_workspace;
+  heffte::scale hscale;
+  #else
   struct fft_plan_3d *plan;
+  #endif
 };
 
 }    // namespace LAMMPS_NS
diff --git a/src/ML-PACE/compute_pace.cpp b/src/ML-PACE/compute_pace.cpp
new file mode 100644
index 0000000000..cf3be8f47f
--- /dev/null
+++ b/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;
+}
diff --git a/src/ML-PACE/compute_pace.h b/src/ML-PACE/compute_pace.h
new file mode 100644
index 0000000000..2f2f8d7f21
--- /dev/null
+++ b/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
diff --git a/src/PYTHON/python_impl.cpp b/src/PYTHON/python_impl.cpp
index 57f8ea1cf0..6dd5175621 100644
--- a/src/PYTHON/python_impl.cpp
+++ b/src/PYTHON/python_impl.cpp
@@ -61,16 +61,19 @@ PythonImpl::PythonImpl(LAMMPS *lmp) : Pointers(lmp)
   nfunc = 0;
   pfuncs = nullptr;
 
-#if PY_MAJOR_VERSION >= 3
-#ifndef Py_LIMITED_API
+#if PY_MAJOR_VERSION >= 3 && !defined(Py_LIMITED_API)
   // check for PYTHONUNBUFFERED environment variable
   const char *PYTHONUNBUFFERED = getenv("PYTHONUNBUFFERED");
+  // Force the stdout and stderr streams to be unbuffered.
+  bool unbuffered = PYTHONUNBUFFERED != nullptr && strcmp(PYTHONUNBUFFERED, "1") == 0;
 
-  if (PYTHONUNBUFFERED != nullptr && strcmp(PYTHONUNBUFFERED, "1") == 0) {
-    // Python Global configuration variable
-    // Force the stdout and stderr streams to be unbuffered.
-    Py_UnbufferedStdioFlag = 1;
-  }
+#if PY_VERSION_HEX >= 0x030800f0
+  PyConfig config;
+  PyConfig_InitPythonConfig(&config);
+  config.buffered_stdio = !unbuffered;
+#else
+  // Python Global configuration variable
+  Py_UnbufferedStdioFlag = unbuffered;
 #endif
 #endif
 
@@ -94,12 +97,17 @@ PythonImpl::PythonImpl(LAMMPS *lmp) : Pointers(lmp)
 #endif
 #endif
 
+#if PY_VERSION_HEX >= 0x030800f0 && !defined(Py_LIMITED_API)
+  Py_InitializeFromConfig(&config);
+  PyConfig_Clear(&config);
+#else
   Py_Initialize();
+#endif
 
   // only needed for Python 2.x and Python 3 < 3.7
   // With Python 3.7 this function is now called by Py_Initialize()
   // Deprecated since version 3.9, will be removed in version 3.11
-#if PY_MAJOR_VERSION < 3 || PY_MINOR_VERSION < 7
+#if PY_VERSION_HEX < 0x030700f0
   if (!PyEval_ThreadsInitialized()) { PyEval_InitThreads(); }
 #endif
 
diff --git a/src/fix_temp_berendsen.h b/src/fix_temp_berendsen.h
index 78ece2af22..f137830508 100644
--- a/src/fix_temp_berendsen.h
+++ b/src/fix_temp_berendsen.h
@@ -38,7 +38,7 @@ class FixTempBerendsen : public Fix {
   void restart(char *buf) override;
   void *extract(const char *, int &) override;
 
- private:
+ protected:
   int which;
   double t_start, t_stop, t_period, t_target;
   double energy;