diff --git a/examples/mliap/ace/Ta_example/Ta06A.ace.pytorch.mliap b/examples/mliap/ace/Ta_example/Ta06A.ace.pytorch.mliap
new file mode 100644
index 0000000000..d2f6c4fce4
--- /dev/null
+++ b/examples/mliap/ace/Ta_example/Ta06A.ace.pytorch.mliap
@@ -0,0 +1,18 @@
+# DATE: 2014-09-05 UNITS: metal CONTRIBUTOR: Aidan Thompson athomps@sandia.gov CITATION: Thompson, Swiler, Trott, Foiles and Tucker, arxiv.org, 1409.3880 (2014) 
+
+# Definition of SNAP potential Ta_Cand06A
+# Assumes 1 LAMMPS atom type
+ 
+variable zblcutinner equal 4
+variable zblcutouter equal 4.8
+variable zblz equal 73
+
+# Specify hybrid with SNAP, ZBL
+
+pair_style hybrid/overlay &
+zbl ${zblcutinner} ${zblcutouter} &
+mliap model mliappy Ta_Pytorch.pt &
+descriptor ace coupling_coefficients.yace
+pair_coeff 1 1 zbl ${zblz} ${zblz}
+pair_coeff * * mliap Ta
+
diff --git a/examples/mliap/ace/Ta_example/Ta_Pytorch.pt b/examples/mliap/ace/Ta_example/Ta_Pytorch.pt
new file mode 100644
index 0000000000..79acf6e724
Binary files /dev/null and b/examples/mliap/ace/Ta_example/Ta_Pytorch.pt differ
diff --git a/examples/mliap/ace/Ta_example/coupling_coefficients.yace b/examples/mliap/ace/Ta_example/coupling_coefficients.yace
new file mode 100644
index 0000000000..a1310efd6d
--- /dev/null
+++ b/examples/mliap/ace/Ta_example/coupling_coefficients.yace
@@ -0,0 +1,41 @@
+elements: [Ta] 
+E0: [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}
+bonds:
+  [0, 0]: {nradmax: 12, lmax: 2, nradbasemax: 12, radbasename: ChebExpCos, radparameters: [3.059235105], radcoefficients: [[[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [[0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [[0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0]], [[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0]], [[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0]], [[0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0]], [[0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0]], [[0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0]], [[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0]], [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0]], [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]]], prehc: 0, lambdahc: 3.059235105, rcut: 4.604694451, dcut: 0.01, rcut_in: 0.0, 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: [0], ns: [3], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 0, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [0], ns: [4], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 0, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [0], ns: [5], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 0, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [0], ns: [6], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 0, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [0], ns: [7], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 0, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [0], ns: [8], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 0, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [0], ns: [9], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 0, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [0], ns: [10], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 0, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [0], ns: [11], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 0, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [0], ns: [12], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 0, rank: 2, ndensity: 1, num_ms_combs: 1, mus: [0, 0], ns: [1, 1], ls: [0, 0], ms_combs: [0, 0], ctildes: [1.0]}
+    - {mu0: 0, rank: 2, ndensity: 1, num_ms_combs: 1, mus: [0, 0], ns: [1, 2], ls: [0, 0], ms_combs: [0, 0], ctildes: [1.0]}
+    - {mu0: 0, rank: 2, ndensity: 1, num_ms_combs: 1, mus: [0, 0], ns: [2, 2], ls: [0, 0], ms_combs: [0, 0], ctildes: [1.0]}
+    - {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: [0.5773502588272095, -0.5773502588272095, 0.5773502588272095]}
+    - {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: [0.5773502588272095, -0.5773502588272095, 0.5773502588272095]}
+    - {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: [0.5773502588272095, -0.5773502588272095, 0.5773502588272095]}
+    - {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: [0.4472135901451111, -0.4472135901451111, 0.4472135901451111, -0.4472135901451111, 0.4472135901451111]}
+    - {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: [0.4472135901451111, -0.4472135901451111, 0.4472135901451111, -0.4472135901451111, 0.4472135901451111]}
+    - {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: [0.4472135901451111, -0.4472135901451111, 0.4472135901451111, -0.4472135901451111, 0.4472135901451111]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 0, 0], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.1999999952104794, -0.14142135359335306, 0.08164966239889981, -0.14142135359335306, 0.16329932479779963, -0.14142135359335306, 0.08164966239889981, -0.14142135359335306, 0.1999999952104794]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 0, 0], ns: [1, 1, 2], 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: [0.1999999952104794, -0.14142135359335306, 0.08164966239889981, -0.14142135359335306, 0.16329932479779963, -0.14142135359335306, 0.08164966239889981, -0.14142135359335306, 0.1999999952104794]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 0, 0], ns: [1, 2, 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: [0.1999999952104794, -0.14142135359335306, 0.08164966239889981, -0.14142135359335306, 0.16329932479779963, -0.14142135359335306, 0.08164966239889981, -0.14142135359335306, 0.1999999952104794]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 0, 0], ns: [1, 2, 2], 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: [0.1999999952104794, -0.14142135359335306, 0.08164966239889981, -0.14142135359335306, 0.16329932479779963, -0.14142135359335306, 0.08164966239889981, -0.14142135359335306, 0.1999999952104794]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 0, 0], ns: [2, 2, 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: [0.1999999952104794, -0.14142135359335306, 0.08164966239889981, -0.14142135359335306, 0.16329932479779963, -0.14142135359335306, 0.08164966239889981, -0.14142135359335306, 0.1999999952104794]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 0, 0], ns: [2, 2, 2], 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: [0.1999999952104794, -0.14142135359335306, 0.08164966239889981, -0.14142135359335306, 0.16329932479779963, -0.14142135359335306, 0.08164966239889981, -0.14142135359335306, 0.1999999952104794]}
+    - {mu0: 0, 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: [0.106904496560845, -0.13093073535988964, 0.106904496560845, -0.13093073535988964, 0.0534522482804225, 0.0534522482804225, -0.13093073535988964, 0.106904496560845, 0.0534522482804225, -0.106904496560845, 0.0534522482804225, 0.106904496560845, -0.13093073535988964, 0.0534522482804225, 0.0534522482804225, -0.13093073535988964, 0.106904496560845, -0.13093073535988964, 0.106904496560845]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 0], ns: [1, 1, 2], 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: [0.106904496560845, -0.13093073535988964, 0.106904496560845, -0.13093073535988964, 0.0534522482804225, 0.0534522482804225, -0.13093073535988964, 0.106904496560845, 0.0534522482804225, -0.106904496560845, 0.0534522482804225, 0.106904496560845, -0.13093073535988964, 0.0534522482804225, 0.0534522482804225, -0.13093073535988964, 0.106904496560845, -0.13093073535988964, 0.106904496560845]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 0], ns: [1, 2, 2], 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: [0.106904496560845, -0.13093073535988964, 0.106904496560845, -0.13093073535988964, 0.0534522482804225, 0.0534522482804225, -0.13093073535988964, 0.106904496560845, 0.0534522482804225, -0.106904496560845, 0.0534522482804225, 0.106904496560845, -0.13093073535988964, 0.0534522482804225, 0.0534522482804225, -0.13093073535988964, 0.106904496560845, -0.13093073535988964, 0.106904496560845]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 0], ns: [2, 2, 2], 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: [0.106904496560845, -0.13093073535988964, 0.106904496560845, -0.13093073535988964, 0.0534522482804225, 0.0534522482804225, -0.13093073535988964, 0.106904496560845, 0.0534522482804225, -0.106904496560845, 0.0534522482804225, 0.106904496560845, -0.13093073535988964, 0.0534522482804225, 0.0534522482804225, -0.13093073535988964, 0.106904496560845, -0.13093073535988964, 0.106904496560845]}
+    - {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: [0.08944271588708351, -0.0447213584132956, -0.0447213584132956, 0.014907121600868895, 0.02981424320173779, 0.014907121600868895, -0.0447213584132956, -0.0447213584132956, 0.02981424320173779, 0.05962848640347558, 0.02981424320173779, -0.0447213584132956, -0.0447213584132956, 0.014907121600868895, 0.02981424320173779, 0.014907121600868895, -0.0447213584132956, -0.0447213584132956, 0.08944271588708351]}
diff --git a/examples/mliap/ace/Ta_example/in.mliap.pytorch.ace.Ta06A b/examples/mliap/ace/Ta_example/in.mliap.pytorch.ace.Ta06A
new file mode 100644
index 0000000000..1f70422745
--- /dev/null
+++ b/examples/mliap/ace/Ta_example/in.mliap.pytorch.ace.Ta06A
@@ -0,0 +1,53 @@
+# Demonstrate MLIAP/PyTorch interface to linear SNAP potential
+
+# Initialize simulation
+
+variable nsteps index 100
+variable nrep equal 4
+variable a equal 3.316
+units           metal
+
+# generate the box and atom positions using a BCC lattice
+
+variable nx equal ${nrep}
+variable ny equal ${nrep}
+variable nz equal ${nrep}
+
+boundary        p p p
+
+lattice         bcc $a
+region          box block 0 ${nx} 0 ${ny} 0 ${nz}
+create_box      1 box
+create_atoms    1 box
+
+mass 1 180.88
+
+# choose potential
+
+include Ta06A.ace.pytorch.mliap
+
+# Setup output
+
+compute  eatom all pe/atom
+compute  energy all reduce sum c_eatom
+
+compute  satom all stress/atom NULL
+compute  str all reduce sum c_satom[1] c_satom[2] c_satom[3]
+variable press equal (c_str[1]+c_str[2]+c_str[3])/(3*vol)
+
+thermo_style    custom step temp epair c_energy etotal press v_press
+thermo          10
+thermo_modify norm yes
+
+# Set up NVE run
+
+timestep 0.5e-3
+neighbor 1.0 bin
+neigh_modify once no every 1 delay 0 check yes
+
+# Run MD
+
+velocity all create 300.0 4928459 loop geom
+fix 1 all nve
+run             ${nsteps}
+
diff --git a/examples/mliap/ace/Ta_example/in.numdiff b/examples/mliap/ace/Ta_example/in.numdiff
new file mode 100644
index 0000000000..05c47d8547
--- /dev/null
+++ b/examples/mliap/ace/Ta_example/in.numdiff
@@ -0,0 +1,72 @@
+# Demonstrate MLIAP/PyTorch interface to linear SNAP potential
+
+# Initialize simulation
+
+# Initialize simulation
+variable    nthermo index 10    # thermo output interval 
+variable    ndump index 10     # dump output interval
+variable    fdelta index 1.0e-4 # displacement size
+variable    vdelta index 1.0e-6 # strain size for numdiff/virial
+variable    bdelta index 1.0e-8 # strain size for numdiff Born matrix
+variable    temp index 10.0     # temperature
+variable    nugget equal 1.0e-6 # regularization for relerr
+
+variable nsteps index 100
+units           metal
+atom_modify     map yes
+
+variable nrep equal 1
+variable a equal 3.316
+
+# generate the box and atom positions using a BCC lattice
+
+variable nx equal ${nrep}
+variable ny equal ${nrep}
+variable nz equal ${nrep}
+
+boundary        p p p
+
+lattice         bcc $a
+region          box block 0 ${nx} 0 ${ny} 0 ${nz}
+create_box      1 box
+create_atoms    1 box
+
+mass 1 180.88
+
+# choose potential
+
+include Ta06A.ace.pytorch.mliap
+
+# Set up NVE run
+
+velocity all create 300.0 4928459 loop geom
+fix 1 all nve
+timestep 0.5e-3
+neighbor 1.0 bin
+neigh_modify once no every 1 delay 0 check yes
+
+# Run MD
+compute  eatom all pe/atom
+compute  energy all reduce sum c_eatom
+
+compute  satom all stress/atom NULL
+compute  str all reduce sum c_satom[1] c_satom[2] c_satom[3]
+variable press equal (c_str[1]+c_str[2]+c_str[3])/(3*vol)
+
+
+fix          numforce all numdiff ${nthermo} ${fdelta}
+variable     ferrx atom f_numforce[1]-fx
+variable     ferry atom f_numforce[2]-fy
+variable     ferrz atom f_numforce[3]-fz
+variable     ferrsq atom v_ferrx^2+v_ferry^2+v_ferrz^2
+compute      faverrsq all reduce ave v_ferrsq
+variable     fsq atom fx^2+fy^2+fz^2
+compute      favsq all reduce ave v_fsq
+variable     frelerr equal sqrt(c_faverrsq/(c_favsq+${nugget}))
+dump errors  all custom ${ndump} force_error.dump v_ferrx v_ferry v_ferrz
+
+thermo_style    custom step temp v_frelerr
+
+run             ${nsteps}
+
+
diff --git a/examples/mliap/ace/Ta_example/numerical_diff_reference/force_error.dump b/examples/mliap/ace/Ta_example/numerical_diff_reference/force_error.dump
new file mode 100644
index 0000000000..b097c187da
--- /dev/null
+++ b/examples/mliap/ace/Ta_example/numerical_diff_reference/force_error.dump
@@ -0,0 +1,234 @@
+ITEM: TIMESTEP
+0
+ITEM: NUMBER OF ATOMS
+108
+ITEM: BOX BOUNDS pp pp pp
+0.0000000000000000e+00 1.6073999999999998e+01
+0.0000000000000000e+00 1.6073999999999998e+01
+0.0000000000000000e+00 1.6073999999999998e+01
+ITEM: ATOMS v_ferrx v_ferry v_ferrz
+-1.70003e-16 -1.70003e-16 4.44072e-12
+6.93889e-17 -1.5786e-16 4.44074e-12
+0 -1.85615e-16 4.44071e-12
+0 0 4.44075e-12
+-1.9082e-16 -4.33681e-19 4.44073e-12
+2.80158e-16 0 4.44072e-12
+-1.5786e-16 6.93889e-17 4.44073e-12
+4.98733e-17 5.0307e-17 4.44075e-12
+0 1.05384e-16 4.44072e-12
+0 -1.9082e-16 4.44073e-12
+-1.39212e-16 -1.39645e-16 4.44072e-12
+2.43295e-16 -1.62197e-16 4.4407e-12
+-1.85615e-16 0 4.44071e-12
+1.04951e-16 0 4.44072e-12
+0 0 4.44069e-12
+0 2.7929e-16 4.44072e-12
+-1.6263e-16 2.43295e-16 -1.89085e-16
+2.60209e-16 2.59341e-16 -2.01228e-16
+0 -4.44103e-12 -4.44089e-12
+-1.9082e-16 -4.44105e-12 -4.44089e-12
+2.80158e-16 -4.44106e-12 -4.44089e-12
+-1.38778e-16 0 -4.44089e-12
+6.245e-17 4.33681e-19 -4.44089e-12
+0 0 -4.44089e-12
+0 6.245e-17 -4.44089e-12
+-1.67401e-16 6.54858e-17 -4.44089e-12
+2.63678e-16 8.54351e-17 -4.44089e-12
+-1.5786e-16 -1.9082e-16 -4.44089e-12
+6.59195e-17 -1.66967e-16 -4.44089e-12
+0 -1.71304e-16 -4.44089e-12
+0 0 -4.44089e-12
+-1.7087e-16 0 -4.44089e-12
+4.44113e-12 4.44089e-12 4.44089e-12
+4.44072e-12 4.44117e-12 4.44089e-12
+-4.44081e-12 4.44116e-12 4.44089e-12
+0 4.44113e-12 4.44089e-12
+-1.5786e-16 -1.5786e-16 4.44096e-12
+4.98733e-17 -1.45717e-16 4.44094e-12
+0 -1.73472e-16 4.441e-12
+0 0 8.88185e-12
+-1.67401e-16 -4.33681e-19 8.88185e-12
+2.63678e-16 4.33681e-19 8.88187e-12
+-1.45717e-16 4.98733e-17 8.88183e-12
+3.03577e-17 3.07913e-17 8.88181e-12
+0 8.54351e-17 8.88187e-12
+0 -1.66533e-16 8.88185e-12
+-1.15359e-16 -1.14492e-16 8.88185e-12
+2.27249e-16 -1.39645e-16 8.88187e-12
+-1.73472e-16 4.44089e-12 1.04951e-16
+8.54351e-17 4.44089e-12 4.44098e-12
+0 4.44089e-12 4.44103e-12
+0 2.62811e-16 8.54351e-17
+-1.39212e-16 2.28116e-16 8.84709e-17
+2.43729e-16 2.43295e-16 1.0842e-16
+0 4.44073e-12 -1.91253e-16
+-1.39212e-16 4.44072e-12 -1.40079e-16
+2.43295e-16 4.4407e-12 -1.6263e-16
+-1.5786e-16 4.33681e-19 -1.90386e-16
+6.59195e-17 0 -1.68268e-16
+0 -4.33681e-19 -1.72605e-16
+4.33681e-19 6.63532e-17 -1.68268e-16
+-1.15359e-16 6.93889e-17 -1.16226e-16
+2.27249e-16 8.89046e-17 -1.4138e-16
+-1.76942e-16 -1.39212e-16 -1.38344e-16
+6.93889e-17 -1.15359e-16 -1.16226e-16
+0 -1.19696e-16 -1.18829e-16
+-4.33681e-19 0 -1.72605e-16
+-1.19696e-16 -8.88178e-12 -4.44101e-12
+2.02963e-16 -8.88178e-12 -4.44104e-12
+-1.89085e-16 -4.44065e-12 -4.44106e-12
+8.89046e-17 -4.44067e-12 -8.88193e-12
+-4.44089e-12 -4.44069e-12 -4.44104e-12
+-1.85615e-16 4.44071e-12 -8.88178e-12
+1.04951e-16 4.44072e-12 -8.88178e-12
+0 4.44069e-12 -8.88178e-12
+0 0 -8.88179e-12
+-1.7087e-16 -4.33681e-19 -8.88178e-12
+2.38524e-16 0 -8.88179e-12
+-1.73472e-16 1.04951e-16 -8.88178e-12
+8.54351e-17 8.54351e-17 -8.88179e-12
+0 1.39645e-16 -8.88179e-12
+0 -1.7087e-16 -8.88178e-12
+-1.19262e-16 -4.44101e-12 -8.88178e-12
+2.02963e-16 -4.44103e-12 -8.88178e-12
+4.44069e-12 -4.44089e-12 -4.44089e-12
+-4.44075e-12 -4.44089e-12 -4.44089e-12
+0 -4.44089e-12 -4.44089e-12
+0 4.44113e-12 -4.44089e-12
+-1.42681e-16 4.4411e-12 -4.44089e-12
+2.2031e-16 4.44111e-12 -4.44089e-12
+0 -1.70003e-16 2.78857e-16
+-1.6263e-16 -1.89085e-16 2.43729e-16
+2.58474e-16 -2.01228e-16 2.59341e-16
+-1.70003e-16 0 2.78857e-16
+8.54351e-17 4.33681e-19 2.64112e-16
+0 0 2.39392e-16
+0 8.50015e-17 2.62377e-16
+-1.39212e-16 8.89046e-17 2.26815e-16
+2.43729e-16 1.0842e-16 2.43729e-16
+-1.89085e-16 -1.6263e-16 2.43729e-16
+8.89046e-17 -1.39212e-16 2.26815e-16
+0 -1.42681e-16 2.04697e-16
+0 0 2.39392e-16
+-1.42681e-16 0 2.04697e-16
+2.18575e-16 0 2.19009e-16
+-2.01228e-16 2.59341e-16 2.59341e-16
+1.0842e-16 2.43729e-16 2.43729e-16
+0 2.18575e-16 2.19009e-16
+ITEM: TIMESTEP
+500
+ITEM: NUMBER OF ATOMS
+108
+ITEM: BOX BOUNDS pp pp pp
+0.0000000000000000e+00 1.6073999999999998e+01
+0.0000000000000000e+00 1.6073999999999998e+01
+0.0000000000000000e+00 1.6073999999999998e+01
+ITEM: ATOMS v_ferrx v_ferry v_ferrz
+-1.3776e-10 2.47272e-10 -3.28864e-10
+-2.41697e-10 1.01804e-10 5.23161e-11
+1.05049e-10 -2.15415e-10 -8.92249e-11
+-5.67845e-11 -1.42956e-10 2.87984e-10
+-1.89563e-10 2.33915e-11 1.77797e-10
+5.64386e-12 -1.40725e-10 2.11666e-10
+1.79474e-10 5.26194e-11 7.01598e-11
+-1.26354e-10 1.60715e-11 -1.16155e-10
+1.88433e-10 7.20246e-11 6.80476e-11
+9.5471e-11 -2.07041e-10 5.19479e-11
+2.38244e-11 1.825e-10 -2.83878e-10
+-1.5185e-10 1.84394e-10 -7.86389e-11
+-8.81272e-11 1.62343e-11 9.85176e-11
+8.94604e-11 -1.65739e-12 1.28668e-10
+1.54379e-10 -1.79534e-10 2.66559e-10
+9.45847e-11 -5.87442e-11 1.55227e-10
+1.15658e-10 -1.60218e-10 1.21751e-11
+-2.80168e-10 9.90372e-11 -1.27114e-10
+1.11145e-10 5.18136e-11 -3.0418e-11
+-9.83397e-11 1.73196e-10 1.15687e-10
+-5.74285e-11 2.51531e-11 9.7697e-12
+1.42362e-10 -9.5646e-11 -1.73654e-10
+-7.39821e-11 -3.20981e-11 -1.78449e-10
+-2.51951e-10 -1.15197e-10 8.82936e-11
+3.35573e-10 -2.75032e-10 -3.1707e-10
+1.22688e-10 -2.27996e-10 1.47766e-10
+-1.37771e-10 5.33214e-12 -5.85577e-11
+-1.70622e-10 3.12829e-10 -1.08447e-11
+-1.73536e-10 -1.02976e-10 7.83582e-11
+-6.68286e-12 -8.49568e-11 6.82726e-11
+-1.7133e-11 3.19051e-10 -2.68584e-10
+2.23857e-10 8.36696e-11 -5.43273e-11
+-6.24681e-12 7.01483e-12 -1.35879e-10
+2.33202e-10 -2.00896e-10 2.73521e-10
+-1.31892e-10 6.05411e-11 3.54368e-11
+1.6424e-10 -2.71671e-10 -5.86501e-11
+-1.39463e-11 -5.81513e-11 -3.01565e-10
+-6.34934e-12 -2.72284e-11 1.14104e-10
+1.12429e-10 -2.01987e-10 -2.18914e-10
+8.71607e-11 5.52618e-11 -4.93286e-11
+-2.18783e-10 -3.32576e-11 -2.89466e-10
+-1.67884e-11 -8.16377e-11 9.96883e-11
+-3.33057e-10 -1.66266e-10 2.67318e-10
+-5.82787e-11 6.04089e-11 6.6292e-11
+1.91093e-11 2.3181e-10 -1.96098e-10
+5.11125e-11 1.12249e-10 3.87786e-10
+9.09899e-11 -1.57395e-10 -3.16673e-10
+-1.46917e-10 -8.38336e-11 1.12849e-10
+9.48542e-12 -9.77006e-11 -1.52892e-10
+-8.43689e-11 1.94004e-10 1.03202e-10
+1.38461e-10 1.19281e-10 -5.49165e-11
+4.62478e-11 9.47571e-11 -2.00772e-11
+-2.93004e-11 2.99581e-11 1.06484e-10
+-2.49249e-10 1.32561e-11 -9.25232e-11
+1.68231e-10 -2.38684e-11 -1.64172e-10
+-2.3154e-10 -1.6209e-10 -1.54156e-10
+-7.44595e-11 8.23192e-11 1.09241e-10
+-8.66415e-11 1.92223e-10 1.68339e-10
+-7.12903e-12 -3.91225e-11 -1.46808e-10
+2.81078e-10 2.09221e-10 5.19054e-10
+1.33327e-10 2.87604e-10 1.85848e-10
+-1.77755e-10 -6.877e-11 1.23268e-10
+6.54644e-13 -1.84329e-11 -6.09743e-11
+3.02108e-10 5.19683e-11 3.13886e-10
+-1.66554e-10 -2.36886e-10 2.83899e-10
+4.30115e-10 6.74615e-11 2.07986e-10
+2.76612e-11 -2.24249e-10 -2.59811e-10
+-3.65799e-12 4.70878e-11 7.68043e-11
+9.82275e-11 -3.51711e-10 6.80762e-11
+-3.68015e-11 3.30873e-12 1.40977e-10
+-2.32046e-10 2.97306e-11 -2.27689e-10
+-2.04805e-10 1.72586e-10 -1.67954e-10
+-3.9955e-10 2.92143e-10 4.21952e-11
+8.50465e-11 -8.00231e-11 1.11889e-12
+1.96812e-10 1.92032e-10 1.05057e-10
+-1.12763e-10 3.7697e-10 -3.3687e-11
+1.98275e-10 3.73163e-10 3.62537e-11
+-5.24916e-11 -1.05259e-10 -2.60468e-10
+1.17582e-11 3.79021e-11 -1.22134e-10
+-1.22277e-11 1.49988e-10 8.18378e-11
+1.09992e-10 -2.9566e-10 -1.14248e-10
+-1.50336e-10 -9.98117e-11 -5.42109e-10
+-2.48108e-10 3.93239e-11 -2.61267e-10
+-1.91012e-10 -3.2863e-11 -2.8852e-10
+-1.54055e-10 -9.63702e-12 1.39615e-10
+8.58463e-11 1.00663e-10 1.28437e-10
+9.69413e-11 -4.0911e-11 1.61141e-10
+2.3482e-10 -7.52639e-11 -3.10965e-11
+6.72632e-11 3.32637e-11 1.44226e-10
+2.09652e-10 -9.75756e-12 4.6282e-11
+-2.18177e-10 -4.11436e-11 3.03962e-10
+8.21879e-12 -7.75314e-11 4.52972e-11
+1.85027e-11 -1.42285e-10 8.0357e-11
+1.8043e-10 -3.69802e-10 -1.85688e-10
+1.51044e-10 1.85886e-10 -1.21116e-10
+-3.83219e-11 -6.44858e-11 -2.28935e-10
+2.43517e-10 -8.0455e-11 -1.83719e-11
+-2.93897e-10 -9.12731e-11 -1.7552e-10
+-1.85963e-10 2.3358e-10 -1.2909e-10
+3.62023e-11 9.03903e-11 5.6199e-11
+3.06953e-10 -2.85687e-10 2.32101e-10
+3.24893e-10 3.94381e-11 1.91749e-10
+-4.38448e-11 1.8545e-10 -2.72547e-11
+-1.38376e-10 6.95532e-11 1.42392e-10
+7.23818e-11 1.65909e-12 -3.54841e-11
+-1.47706e-10 -6.05487e-11 8.23716e-11
+8.60403e-11 -1.08309e-10 -1.21542e-10
+2.60305e-11 6.84718e-11 -2.40918e-11
diff --git a/examples/mliap/ace/Ta_example/numerical_diff_reference/in.numdiff b/examples/mliap/ace/Ta_example/numerical_diff_reference/in.numdiff
new file mode 100644
index 0000000000..05c47d8547
--- /dev/null
+++ b/examples/mliap/ace/Ta_example/numerical_diff_reference/in.numdiff
@@ -0,0 +1,72 @@
+# Demonstrate MLIAP/PyTorch interface to linear SNAP potential
+
+# Initialize simulation
+
+# Initialize simulation
+variable    nthermo index 10    # thermo output interval 
+variable    ndump index 10     # dump output interval
+variable    fdelta index 1.0e-4 # displacement size
+variable    vdelta index 1.0e-6 # strain size for numdiff/virial
+variable    bdelta index 1.0e-8 # strain size for numdiff Born matrix
+variable    temp index 10.0     # temperature
+variable    nugget equal 1.0e-6 # regularization for relerr
+
+variable nsteps index 100
+units           metal
+atom_modify     map yes
+
+variable nrep equal 1
+variable a equal 3.316
+
+# generate the box and atom positions using a BCC lattice
+
+variable nx equal ${nrep}
+variable ny equal ${nrep}
+variable nz equal ${nrep}
+
+boundary        p p p
+
+lattice         bcc $a
+region          box block 0 ${nx} 0 ${ny} 0 ${nz}
+create_box      1 box
+create_atoms    1 box
+
+mass 1 180.88
+
+# choose potential
+
+include Ta06A.ace.pytorch.mliap
+
+# Set up NVE run
+
+velocity all create 300.0 4928459 loop geom
+fix 1 all nve
+timestep 0.5e-3
+neighbor 1.0 bin
+neigh_modify once no every 1 delay 0 check yes
+
+# Run MD
+compute  eatom all pe/atom
+compute  energy all reduce sum c_eatom
+
+compute  satom all stress/atom NULL
+compute  str all reduce sum c_satom[1] c_satom[2] c_satom[3]
+variable press equal (c_str[1]+c_str[2]+c_str[3])/(3*vol)
+
+
+fix          numforce all numdiff ${nthermo} ${fdelta}
+variable     ferrx atom f_numforce[1]-fx
+variable     ferry atom f_numforce[2]-fy
+variable     ferrz atom f_numforce[3]-fz
+variable     ferrsq atom v_ferrx^2+v_ferry^2+v_ferrz^2
+compute      faverrsq all reduce ave v_ferrsq
+variable     fsq atom fx^2+fy^2+fz^2
+compute      favsq all reduce ave v_fsq
+variable     frelerr equal sqrt(c_faverrsq/(c_favsq+${nugget}))
+dump errors  all custom ${ndump} force_error.dump v_ferrx v_ferry v_ferrz
+
+thermo_style    custom step temp v_frelerr
+
+run             ${nsteps}
+
+
diff --git a/examples/mliap/ace/multielement_compute/coupling_coefficients.yace b/examples/mliap/ace/multielement_compute/coupling_coefficients.yace
new file mode 100644
index 0000000000..2953222216
--- /dev/null
+++ b/examples/mliap/ace/multielement_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]}
+    - {mu0: 0, 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: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 2, 2], ns: [1, 1, 1], ls: [2, 1, 1], ms_combs: [-2, 1, 1, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 2, -1, -1], ctildes: [0.25819888974716115, -0.18257418583505536, -0.18257418583505536, 0.10540925533894599, 0.21081851067789198, 0.10540925533894599, -0.18257418583505536, -0.18257418583505536, 0.25819888974716115]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 1, 2], ns: [1, 1, 1], ls: [1, 2, 1], ms_combs: [-1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -2, 1, 1, -1, 0, 1, 0, -1], ctildes: [0.10540925533894599, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.21081851067789198, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.10540925533894599]}
+    - {mu0: 0, 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: 0, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [1, 1, 2], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [1, 1, 2], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [1, 2, 2], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
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+    - {mu0: 0, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [0, 1, 2], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
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+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 1, 1], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 1, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 1, 1, 1], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 1, 1, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 1, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 2, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [1, 1, 1, 1], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [1, 1, 1, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [1, 1, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [1, 2, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 0, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [2, 2, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+  1:
+    - {mu0: 1, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [1], ns: [1], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 1, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [1], ns: [2], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 1, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [2], ns: [1], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 1, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [2], ns: [2], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 1, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [0], ns: [2], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 1, rank: 1, ndensity: 1, num_ms_combs: 1, mus: [0], ns: [1], ls: [0], ms_combs: [0], ctildes: [1.0]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 1], ns: [1, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [2, 2], ns: [1, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [1, 2], ns: [1, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [1, 1], ns: [2, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [2, 2], ns: [2, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 1], ns: [2, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [1, 2], ns: [1, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 2], ns: [1, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [1, 1], ns: [1, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [1, 1], ns: [1, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 1], ns: [1, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 1], ns: [2, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 2], ns: [2, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [1, 1], ns: [1, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 2], ns: [2, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 0], ns: [2, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 2], ns: [1, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [1, 1], ns: [2, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [2, 2], ns: [1, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 2], ns: [1, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 1], ns: [2, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [2, 2], ns: [2, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 0], ns: [2, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 0], ns: [1, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [2, 2], ns: [1, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [1, 2], ns: [2, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 2], ns: [1, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 0], ns: [1, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [1, 2], ns: [1, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [0, 0], ns: [1, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [1, 2], ns: [2, 1], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
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+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [2, 2], ns: [1, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [1, 2], ns: [1, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 5, mus: [0, 2], ns: [2, 2], ls: [2, 2], ms_combs: [-2, 2, -1, 1, 0, 0, 1, -1, 2, -2], ctildes: [0.4472135954999579, -0.4472135954999579, 0.447213595499958, -0.4472135954999579, 0.4472135954999579]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [1, 1], ns: [1, 1], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
+    - {mu0: 1, rank: 2, ndensity: 1, num_ms_combs: 3, mus: [1, 2], ns: [2, 2], ls: [1, 1], ms_combs: [-1, 1, 0, 0, 1, -1], ctildes: [0.5773502691896257, -0.5773502691896257, 0.5773502691896257]}
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+    - {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]}
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+    - {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]}
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+    - {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]}
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+    - {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]}
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+    - {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]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 0, 2], ns: [1, 1, 1], ls: [1, 2, 1], ms_combs: [-1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -2, 1, 1, -1, 0, 1, 0, -1], ctildes: [0.10540925533894599, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.21081851067789198, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.10540925533894599]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 1, 1], ns: [1, 1, 1], ls: [2, 1, 1], ms_combs: [-2, 1, 1, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 2, -1, -1], ctildes: [0.25819888974716115, -0.18257418583505536, -0.18257418583505536, 0.10540925533894599, 0.21081851067789198, 0.10540925533894599, -0.18257418583505536, -0.18257418583505536, 0.25819888974716115]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 1, 2], ns: [1, 1, 1], ls: [2, 1, 1], ms_combs: [-2, 1, 1, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 2, -1, -1], ctildes: [0.25819888974716115, -0.18257418583505536, -0.18257418583505536, 0.10540925533894599, 0.21081851067789198, 0.10540925533894599, -0.18257418583505536, -0.18257418583505536, 0.25819888974716115]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 0, 1], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 2, 2], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 0, 2], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 0, 1], ns: [1, 1, 1], ls: [1, 2, 1], ms_combs: [-1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -2, 1, 1, -1, 0, 1, 0, -1], ctildes: [0.10540925533894599, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.21081851067789198, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.10540925533894599]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 1, 2], ns: [1, 1, 1], ls: [1, 2, 1], ms_combs: [-1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -2, 1, 1, -1, 0, 1, 0, -1], ctildes: [0.10540925533894599, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.21081851067789198, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.10540925533894599]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 1], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 0], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 1, 2], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [2, 2, 2], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [1, 1, 2], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [2, 2, 2], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [0, 1, 2], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [0, 1, 1], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [1, 1, 2], ns: [1, 1, 1], ls: [1, 2, 1], ms_combs: [-1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -2, 1, 1, -1, 0, 1, 0, -1], ctildes: [0.10540925533894599, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.21081851067789198, -0.18257418583505536, 0.25819888974716115, -0.18257418583505536, 0.10540925533894599]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [1, 1, 1], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [1, 2, 2], ns: [1, 1, 1], ls: [2, 1, 1], ms_combs: [-2, 1, 1, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 2, -1, -1], ctildes: [0.25819888974716115, -0.18257418583505536, -0.18257418583505536, 0.10540925533894599, 0.21081851067789198, 0.10540925533894599, -0.18257418583505536, -0.18257418583505536, 0.25819888974716115]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 9, mus: [1, 2, 2], ns: [1, 1, 1], ls: [1, 1, 2], ms_combs: [-1, -1, 2, -1, 0, 1, -1, 1, 0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, -2], ctildes: [0.19999999999999998, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.16329931618554522, -0.1414213562373095, 0.08164965809277261, -0.1414213562373095, 0.19999999999999998]}
+    - {mu0: 2, rank: 3, ndensity: 1, num_ms_combs: 19, mus: [0, 1, 1], ns: [1, 1, 1], ls: [2, 2, 2], ms_combs: [-2, 0, 2, -2, 1, 1, -2, 2, 0, -1, -1, 2, -1, 0, 1, -1, 1, 0, -1, 2, -1, 0, -2, 2, 0, -1, 1, 0, 0, 0, 0, 1, -1, 0, 2, -2, 1, -2, 1, 1, -1, 0, 1, 0, -1, 1, 1, -2, 2, -2, 0, 2, -1, -1, 2, 0, -2], ctildes: [0.10690449676496976, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, 0.05345224838248488, -0.10690449676496976, 0.05345224838248488, 0.10690449676496976, -0.1309307341415954, 0.05345224838248488, 0.05345224838248488, -0.1309307341415954, 0.10690449676496976, -0.1309307341415954, 0.10690449676496976]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 0, 0], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 0, 1], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 0, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 1, 1], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 1, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 0, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 1, 1, 1], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 1, 1, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 1, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [0, 2, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [1, 1, 1, 1], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [1, 1, 1, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [1, 1, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [1, 2, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
+    - {mu0: 2, rank: 4, ndensity: 1, num_ms_combs: 19, mus: [2, 2, 2, 2], ns: [1, 1, 1, 1], ls: [1, 1, 1, 1], ms_combs: [-1, -1, 1, 1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 1, -1, 1, -1, 1, 0, 0, -1, 1, 1, -1, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 1, 0, -1, 1, -1, -1, 1, 1, -1, 0, 0, 1, -1, 1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 1, -1, -1], ctildes: [0.0, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, -0.3333333333333333, 0.3333333333333333, -0.3333333333333333, 0.0, 0.0, 0.3333333333333333, -0.3333333333333333, 0.3333333333333333, 0.0, 0.0, 0.0]}
diff --git a/examples/mliap/ace/multielement_compute/in.mliap.ace.compute b/examples/mliap/ace/multielement_compute/in.mliap.ace.compute
new file mode 100644
index 0000000000..40000b5493
--- /dev/null
+++ b/examples/mliap/ace/multielement_compute/in.mliap.ace.compute
@@ -0,0 +1,25 @@
+#info all out log
+units  metal
+atom_style  atomic
+boundary    p p p
+atom_modify    map hash
+read_data  latte_cell_0.data
+mass  1 1.00
+mass  2 14.00
+mass  3 15.999
+
+pair_style     zero 5.7
+pair_coeff     * *
+
+
+compute   	ace all mliap descriptor ace coupling_coefficients.yace model linear gradgradflag 1
+
+thermo 1
+thermo_style  custom step c_ace[1][1]
+thermo_modify   norm no
+
+fix             pc all ave/time 1 1 1 c_ace[*] file desc_out.dat  mode vector
+
+
+run 0
+
diff --git a/src/ML-IAP/compute_mliap.cpp b/src/ML-IAP/compute_mliap.cpp
index 4fa88abad1..1f858b608b 100644
--- a/src/ML-IAP/compute_mliap.cpp
+++ b/src/ML-IAP/compute_mliap.cpp
@@ -23,6 +23,7 @@
 #include "mliap_model_quadratic.h"
 #include "mliap_descriptor_snap.h"
 #include "mliap_descriptor_so3.h"
+#include "mliap_descriptor_ace.h"
 #ifdef MLIAP_PYTHON
 #include "mliap_model_python.h"
 #endif
@@ -94,6 +95,10 @@ ComputeMLIAP::ComputeMLIAP(LAMMPS *lmp, int narg, char **arg) :
         if (iarg+3 > narg) error->all(FLERR,"Illegal pair_style mliap command");
         descriptor = new MLIAPDescriptorSO3(lmp,arg[iarg+2]);
         iarg += 3;
+      } else if (strcmp(arg[iarg+1],"ace") == 0) {
+        if (iarg+3 > narg) error->all(FLERR,"Illegal pair_style mliap command");
+        descriptor = new MLIAPDescriptorACE(lmp,arg[iarg+2]);
+        iarg += 3;
       } else error->all(FLERR,"Illegal compute mliap command");
       descriptorflag = 1;
     } else if (strcmp(arg[iarg],"gradgradflag") == 0) {
diff --git a/src/ML-IAP/mliap_data.cpp b/src/ML-IAP/mliap_data.cpp
index b40fefd3ba..bc29ad9004 100644
--- a/src/ML-IAP/mliap_data.cpp
+++ b/src/ML-IAP/mliap_data.cpp
@@ -33,7 +33,7 @@ MLIAPData::MLIAPData(LAMMPS *lmp, int gradgradflag_in, int *map_in, class MLIAPM
     gamma(nullptr), gamma_row_index(nullptr), gamma_col_index(nullptr), egradient(nullptr),
     numneighs(nullptr), iatoms(nullptr), ielems(nullptr), pair_i(nullptr), jatoms(nullptr),
     jelems(nullptr), elems(nullptr), rij(nullptr), graddesc(nullptr), model(nullptr),
-    descriptor(nullptr), list(nullptr)
+    descriptor(nullptr), list(nullptr), itypes(nullptr), ix(nullptr), cpy_frstngh(nullptr)
 {
   gradgradflag = gradgradflag_in;
   map = map_in;
@@ -84,8 +84,11 @@ MLIAPData::~MLIAPData()
   memory->destroy(iatoms);
   memory->destroy(pair_i);
   memory->destroy(ielems);
+  memory->destroy(itypes);
+  memory->destroy(ix);
   memory->destroy(numneighs);
   memory->destroy(jatoms);
+  memory->destroy(cpy_frstngh);
   memory->destroy(jelems);
   memory->destroy(elems);
   memory->destroy(rij);
@@ -183,6 +186,7 @@ void MLIAPData::generate_neighdata(NeighList *list_in, int eflag_in, int vflag_i
       int jtype = type[j];
       const int jelem = map[jtype];
 
+      cpy_frstngh[ii][jj] = firstneigh[i][jj];
       if (rsq < descriptor->cutsq[ielem][jelem]) {
         pair_i[ij] = i;
         jatoms[ij] = j;
@@ -196,6 +200,7 @@ void MLIAPData::generate_neighdata(NeighList *list_in, int eflag_in, int vflag_i
     }
     iatoms[ii] = i;
     ielems[ii] = ielem;
+    itypes[ii] = itype;
     numneighs[ii] = ninside;
     npairs += ninside;
   }
@@ -222,6 +227,8 @@ void MLIAPData::grow_neigharrays()
   if (natomneigh_max < natomneigh) {
     memory->grow(iatoms, natomneigh, "MLIAPData:iatoms");
     memory->grow(ielems, natomneigh, "MLIAPData:ielems");
+    memory->grow(itypes, natomneigh, "MLIAPData:itypes");
+    memory->grow(ix, natomneigh, 3, "MLIAPData:ix");
     memory->grow(numneighs, natomneigh, "MLIAPData:numneighs");
     natomneigh_max = natomneigh;
   }
@@ -241,6 +248,9 @@ void MLIAPData::grow_neigharrays()
     const double xtmp = x[i][0];
     const double ytmp = x[i][1];
     const double ztmp = x[i][2];
+    ix[i][0] = x[i][0];
+    ix[i][1] = x[i][1];
+    ix[i][2] = x[i][2];
     const int itype = type[i];
     const int ielem = map[itype];
 
@@ -265,6 +275,7 @@ void MLIAPData::grow_neigharrays()
   if (nneigh_max < nneigh) {
     memory->grow(pair_i, nneigh, "MLIAPData:pair_i");
     memory->grow(jatoms, nneigh, "MLIAPData:jatoms");
+    memory->grow(cpy_frstngh, natomneigh, nneigh, "MLIAPData:cpy_frstngh");
     memory->grow(jelems, nneigh, "MLIAPData:jelems");
     memory->grow(rij, nneigh, 3, "MLIAPData:rij");
     if (gradgradflag == 0) memory->grow(graddesc, nneigh, ndescriptors, 3, "MLIAPData:graddesc");
@@ -292,12 +303,15 @@ double MLIAPData::memory_usage()
 
   bytes += (double) natomneigh_max * sizeof(int);    // iatoms
   bytes += (double) natomneigh_max * sizeof(int);    // ielems
+  bytes += (double) natomneigh_max * sizeof(int);    // itypes
+  bytes += (double) natomneigh_max * 3 * sizeof(int);// ix
   bytes += (double) natomneigh_max * sizeof(int);    // numneighs
 
-  bytes += (double) nneigh_max * sizeof(int);           // pair_i
-  bytes += (double) nneigh_max * sizeof(int);           // jatoms
-  bytes += (double) nneigh_max * sizeof(int);           // jelems
-  bytes += (double) nneigh_max * 3 * sizeof(double);    // rij"
+  bytes += (double) nneigh_max * sizeof(int);                 // pair_i
+  bytes += (double) nneigh_max * sizeof(int);                 // jatoms
+  bytes += (double) nneigh_max * sizeof(int);                 // jelems
+  bytes += (double) nneigh_max * natomneigh_max * sizeof(int);// cpy_frstngh
+  bytes += (double) nneigh_max * 3 * sizeof(double);          // rij"
 
   if (gradgradflag == 0)
     bytes += (double) nneigh_max * ndescriptors * 3 * sizeof(double);    // graddesc
diff --git a/src/ML-IAP/mliap_data.h b/src/ML-IAP/mliap_data.h
index 9746b4da5e..6dfc0ec9b6 100644
--- a/src/ML-IAP/mliap_data.h
+++ b/src/ML-IAP/mliap_data.h
@@ -67,12 +67,15 @@ class MLIAPData : protected Pointers {
   int *numneighs;                // neighbors count for each atom
   int *iatoms;                   // index of each atom
   int *ielems;                   // element of each atom
+  int *itypes;                   // LAMMPS type of each atom for external evaluators
   int nneigh_max;                // number of ij neighbors allocated
   int npairs;                    // number of ij neighbor pairs
   int *pair_i;                   // index of each i atom for each ij pair
   int *jatoms;                   // index of each neighbor
   int *jelems;                   // element of each neighbor
   int *elems;                    // element of each atom in or not in the neighborlist
+  int **cpy_frstngh;             // index of each neighbor for external evaluators
+  double **ix;                   // neighbor positions for external evaluators
   double **rij;                  // distance vector of each neighbor
   double ***graddesc;            // descriptor gradient w.r.t. each neighbor
   int eflag;                     // indicates if energy is needed
diff --git a/src/ML-IAP/mliap_descriptor.h b/src/ML-IAP/mliap_descriptor.h
index 7f05c6804a..6d66356960 100644
--- a/src/ML-IAP/mliap_descriptor.h
+++ b/src/ML-IAP/mliap_descriptor.h
@@ -31,6 +31,7 @@ class MLIAPDescriptor : virtual protected Pointers {
 
   int ndescriptors;     // number of descriptors
   int nelements;        // # of unique elements
+  int allocated_elements; // is the element array allocated
   char **elements;      // names of unique elements
   double **cutsq;       // nelem x nelem rcutsq values
   double **cutghost;    // cutoff for each ghost pair
diff --git a/src/ML-IAP/mliap_descriptor_ace.cpp b/src/ML-IAP/mliap_descriptor_ace.cpp
new file mode 100644
index 0000000000..f89b5c2fd6
--- /dev/null
+++ b/src/ML-IAP/mliap_descriptor_ace.cpp
@@ -0,0 +1,447 @@
+/* ----------------------------------------------------------------------
+   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.
+------------------------------------------------------------------------- */
+
+/* ----------------------------------------------------------------------
+   Contributing author: James Goff (SNL)
+------------------------------------------------------------------------- */
+
+#include "mliap_descriptor_ace.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 "atom.h"
+#include "comm.h"
+#include "error.h"
+#include "memory.h"
+#include "mliap_data.h"
+#include "neigh_list.h"
+#include "pair_mliap.h"
+#include "tokenizer.h"
+
+#include <cmath>
+#include <cstring>
+#include <typeinfo>
+//#include <map>
+
+namespace LAMMPS_NS {
+struct ACE_ML_impl {
+  ACE_ML_impl() : basis_set(nullptr), ace(nullptr) {}
+  ~ACE_ML_impl()
+  {
+    delete basis_set;
+    delete ace;
+  }
+  ACECTildeBasisSet *basis_set;
+  ACECTildeEvaluator *ace;
+};
+}
+
+
+using namespace LAMMPS_NS;
+
+#define MAXLINE 1024
+#define MAXWORD 3
+
+/* ---------------------------------------------------------------------- */
+
+MLIAPDescriptorACE::MLIAPDescriptorACE(LAMMPS *_lmp, char *yacefilename) : Pointers(_lmp),  MLIAPDescriptor(_lmp)
+{
+
+  acemlimpl = new ACE_ML_impl;
+  int ntypes = atom->ntypes;
+  allocated_elements = 0;
+  //read in file with CG coefficients or c_tilde coefficients
+  //auto coeff_file_name = utils::get_potential_file_path(yacefilename);
+  //char*  ctilde_file_name = yacefilename;
+  //ctilde_file = *ctilde_file_name
+  delete acemlimpl -> basis_set;
+  //acemlimpl -> basis_set = new ACECTildeBasisSet(ctilde_file);
+  //acemlimpl -> basis_set = new ACECTildeBasisSet(&ctilde_file_name);
+  acemlimpl -> basis_set = new ACECTildeBasisSet("coupling_coefficients.yace");
+  //ntypes = acemlimpl -> basis_set->nelements;
+  int tot_num = 0;
+  for (int mu = 0; mu <  ntypes; mu++) {
+    if ( max_num < acemlimpl -> basis_set->total_basis_size_rank1[mu] + acemlimpl ->basis_set->total_basis_size[mu]) {
+      max_num = acemlimpl -> basis_set->total_basis_size_rank1[mu] + acemlimpl ->basis_set->total_basis_size[mu];
+    }
+    tot_num += acemlimpl -> basis_set->total_basis_size_rank1[mu] + acemlimpl ->basis_set->total_basis_size[mu];
+  }
+
+  ndescriptors = max_num; //n_r1 + n_rp;
+  nelements = acemlimpl -> basis_set ->nelements;
+
+  memory->destroy(cutsq);
+  
+  if (allocated_elements) {
+    for (int iielem = 0; iielem < nelements; iielem++) delete[] elements[iielem];
+    delete[] elements;
+    allocated_elements = 0;
+  } else if (not allocated_elements) {
+    elements = new char * [nelements];
+    for (int iielem = 0; iielem < nelements; iielem ++){
+      elements[iielem] = utils::strdup(acemlimpl->basis_set->elements_name[iielem]);
+    }
+    allocated_elements = 1;
+  }
+  
+  memory->create(cutsq,ntypes+1,ntypes+1,"mliap/descriptor/ace:cutsq");
+  float icmax = 0.0;
+  float icuti, icutj;
+  for (int mui = 0; mui < acemlimpl -> basis_set ->nelements; mui++) {
+    icuti = acemlimpl -> basis_set->radial_functions->cut(mui, mui); 
+    if (icuti > icmax) icmax = icuti;
+    for (int muj = mui+1; muj < acemlimpl -> basis_set ->nelements; muj++) {
+      icutj = acemlimpl -> basis_set->radial_functions->cut(mui, muj);
+      if (icutj > icmax) icmax = icutj;
+    }
+  }
+  float cutmax = 0.0;
+  float cuti,cutj;
+  float cutfac = 1.0;
+  for (int mui = 0; mui < acemlimpl -> basis_set ->nelements; mui++) {
+    cuti = acemlimpl -> basis_set->radial_functions->cut(mui, mui);
+    if (cuti > cutmax) cutmax = cuti;
+    
+    cutsq[mui][mui] = ((2*cuti*cutfac)*(2*cuti*cutfac));
+    for (int muj = mui + 1; muj < ntypes; muj++) {
+      cutj = acemlimpl -> basis_set->radial_functions->cut(mui, muj);
+      cutsq[mui][muj] = cutsq[muj][mui] = ((2*cuti *cutfac)*(2*cutj*cutfac));
+    }
+  }
+
+}
+
+
+void MLIAPDescriptorACE::allocate()
+{
+}
+
+/* ---------------------------------------------------------------------- */
+
+MLIAPDescriptorACE::~MLIAPDescriptorACE()
+{
+  delete acemlimpl;
+  if (allocated) {
+    memory->destroy(cutsq);
+  }
+}
+
+/* ----------------------------------------------------------------------
+   compute descriptors for each atom
+   ---------------------------------------------------------------------- */
+
+void MLIAPDescriptorACE::compute_descriptors(class MLIAPData *data)
+{
+  int max_jnum = -1;
+  int nei = 0;
+  int jtmp =0;
+  int ntypes = atom->ntypes;
+  for (int iitmp = 0; iitmp < data->nlistatoms; iitmp++) {
+    int itmp = data->iatoms[iitmp];
+    jtmp = data->numneighs[iitmp];
+    nei = nei + jtmp;
+    if (jtmp > max_jnum){
+      max_jnum = jtmp;
+    }
+  }
+  delete acemlimpl -> basis_set;
+  acemlimpl -> basis_set = new ACECTildeBasisSet("coupling_coefficients.yace");
+
+
+  for (int ii = 0; ii < data->nlistatoms; ii++) {
+    const int i = data->iatoms[ii];
+    const int ielemx = data->ielems[ii];
+    const int jnum = data->numneighs[ii];
+    int elem_offset = ndescriptors * ielemx; //data->ielems[ii];
+
+    delete acemlimpl -> ace;
+    acemlimpl -> ace = new ACECTildeEvaluator(*acemlimpl -> basis_set);
+    acemlimpl -> ace->compute_projections = 1;
+    acemlimpl -> ace->compute_b_grad = 1;
+
+    acemlimpl -> ace->element_type_mapping.init(ntypes+1);
+    for (int ik = 1; ik <= ntypes; ik++) {
+      for(int mu = 0; mu < ntypes; mu++){
+        if (mu != -1) {
+          if (mu == ik - 1) {
+            acemlimpl -> ace->element_type_mapping(ik) = mu;
+          }
+        }
+      }
+    }
+    
+    
+    acemlimpl -> ace->resize_neighbours_cache(jnum);
+    acemlimpl -> ace->compute_atom(ii, atom->x, atom->type, data->numneighs[ii], data->cpy_frstngh[ii]);
+    for (int icoeff = 0; icoeff < data->ndescriptors; icoeff++){
+      data->descriptors[ii][icoeff] = acemlimpl -> ace -> projections(icoeff);
+    }
+  
+  }
+}
+
+/* ----------------------------------------------------------------------
+   compute forces for each atom
+   ---------------------------------------------------------------------- */
+
+void MLIAPDescriptorACE::compute_forces(class MLIAPData *data)
+{
+  double fij[3];
+  double **f = atom->f;
+  int ij = 0;
+
+  int max_jnum = -1;
+  int nei = 0;
+  int jtmp =0;
+  int ntypes = atom->ntypes;
+  for (int iitmp = 0; iitmp < data->nlistatoms; iitmp++) {
+    //int itmp = ilist[iitmp];
+    int itmp = data->iatoms[iitmp];
+    jtmp = data->numneighs[iitmp];
+    nei = nei + jtmp;
+    if (jtmp > max_jnum){
+      max_jnum = jtmp;
+    }
+  }
+
+  // BEGIN force loop
+  for (int ii = 0; ii < data->nlistatoms; ii++) {
+    const int i = data->iatoms[ii];
+    const int ielem = data->ielems[ii];
+    int elem_offset = max_num * ielem;
+    delete acemlimpl -> ace;
+    acemlimpl -> ace = new ACECTildeEvaluator(*acemlimpl -> basis_set);
+    acemlimpl -> ace->compute_projections = 1;
+    acemlimpl -> ace->compute_b_grad = 1;
+    acemlimpl -> ace->element_type_mapping.init(ntypes+1);
+    for (int ik = 1; ik <= ntypes; ik++) {
+      for(int mu = 0; mu < acemlimpl -> basis_set ->nelements; mu++){
+        if (mu != -1) {
+          if (mu == ik - 1) {
+            acemlimpl -> ace->element_type_mapping(ik) = mu;
+          }
+        }
+      }
+    }
+
+
+    const int jnum = data->numneighs[ii];
+    acemlimpl -> ace->resize_neighbours_cache(jnum);
+    acemlimpl -> ace->compute_atom(ii, atom->x, atom->type, data->numneighs[ii], data->cpy_frstngh[ii]);
+    int ij0 = ij;
+    int ninside = 0;
+    for (int jj = 0; jj < jnum; jj++) {
+      ninside++;
+      ij++;
+    }
+
+    ij = ij0;
+    const int* const jlist = data->cpy_frstngh[ii];
+    double **x = atom->x;
+    const double xtmp = x[i][0];
+    const double ytmp = x[i][1];
+    const double ztmp = x[i][2];
+
+    for (int jj = 0; jj < jnum; jj++) {
+      const int j = jlist[jj];
+      for (int iicoeff = 0; iicoeff < ndescriptors; iicoeff++) {
+        DOUBLE_TYPE fx_dB = acemlimpl -> ace -> neighbours_dB(iicoeff,jj,0)*data->betas[ii][iicoeff];
+        DOUBLE_TYPE fy_dB = acemlimpl -> ace -> neighbours_dB(iicoeff,jj,1)*data->betas[ii][iicoeff];
+        DOUBLE_TYPE fz_dB = acemlimpl -> ace -> neighbours_dB(iicoeff,jj,2)*data->betas[ii][iicoeff];
+        // add force contribution from each descriptor
+        f[i][0] += fx_dB;
+        f[i][1] += fy_dB;
+        f[i][2] += fz_dB;
+        f[j][0] -= fx_dB;
+        f[j][1] -= fy_dB;
+        f[j][2] -= fz_dB;
+      }
+      //xi = atom-
+      const double delx = x[j][0] - xtmp;
+      const double dely = x[j][1] - ytmp;
+      const double delz = x[j][2] - ztmp;
+      double rij_tmp[3] = {delx,dely,delz};
+      if (data->vflag) data->pairmliap->v_tally(i, j, fij, rij_tmp);
+      ij++;
+    }
+  }
+}
+
+/* ----------------------------------------------------------------------
+   calculate gradients of forces w.r.t. parameters
+   ---------------------------------------------------------------------- */
+
+void MLIAPDescriptorACE::compute_force_gradients(class MLIAPData *data)
+{
+  int ij = 0;
+  int ntypes = atom->ntypes;
+
+  int max_jnum = -1;
+  int nei = 0;
+  int jtmp =0;
+  for (int iitmp = 0; iitmp < data->nlistatoms; iitmp++) {
+    int itmp = data->iatoms[iitmp];
+    jtmp = data->numneighs[iitmp];
+    nei = nei + jtmp;
+    if (jtmp > max_jnum){
+      max_jnum = jtmp;
+    }
+  }
+
+  for (int ii = 0; ii < data->nlistatoms; ii++) {
+    const int i = data->iatoms[ii];
+    const int ielem = data->ielems[ii];
+    delete acemlimpl -> ace;
+    acemlimpl -> ace = new ACECTildeEvaluator(*acemlimpl -> basis_set);
+    acemlimpl -> ace->compute_projections = 1;
+    acemlimpl -> ace->compute_b_grad = 1;
+    int n_r1, n_rp = 0;
+    n_r1 = acemlimpl -> basis_set->total_basis_size_rank1[0];
+    n_rp = acemlimpl -> basis_set->total_basis_size[0];
+
+    acemlimpl -> ace->element_type_mapping.init(ntypes+1);
+    for (int ik = 1; ik <= ntypes; ik++) {
+      for(int mu = 0; mu < acemlimpl -> basis_set ->nelements; mu++){
+        if (mu != -1) {
+          if (mu == ik - 1) {
+            acemlimpl -> ace->element_type_mapping(ik) = mu;
+          }
+        }
+      }
+    }
+
+
+
+    const int jnum = data->numneighs[ii];
+    const int* const jlist = data->cpy_frstngh[ii];
+    acemlimpl -> ace->resize_neighbours_cache(jnum);
+    acemlimpl -> ace->compute_atom(ii, atom->x, atom->type, data->numneighs[ii], data->cpy_frstngh[ii]);
+    for (int jj = 0; jj < jnum; jj++) {
+      const int jt = data->jatoms[ij];
+      const int jtt = data->jatoms[jj];
+      const int j = jlist[jj];
+        for (int inz = 0; inz < data->gamma_nnz; inz++) {
+          const int l = data->gamma_row_index[ii][inz];
+          const int k = data->gamma_col_index[ii][inz];
+          DOUBLE_TYPE fx_dB = acemlimpl -> ace -> neighbours_dB(k,jj,0);
+          DOUBLE_TYPE fy_dB = acemlimpl -> ace -> neighbours_dB(k,jj,1);
+          DOUBLE_TYPE fz_dB = acemlimpl -> ace -> neighbours_dB(k,jj,2);
+          data->gradforce[i][l] += data->gamma[ii][inz] * fx_dB;
+          data->gradforce[i][l + data->yoffset] += data->gamma[ii][inz] * fy_dB;
+          data->gradforce[i][l + data->zoffset] += data->gamma[ii][inz] * fz_dB;
+          data->gradforce[j][l] -= data->gamma[ii][inz] * fx_dB;
+          data->gradforce[j][l + data->yoffset] -= data->gamma[ii][inz] * fy_dB;
+          data->gradforce[j][l + data->zoffset] -= data->gamma[ii][inz] * fz_dB;
+
+        }
+      ij ++;
+    }
+  }
+}
+
+/* ----------------------------------------------------------------------
+   compute descriptor gradients for each neighbor atom
+   ---------------------------------------------------------------------- */
+
+void MLIAPDescriptorACE::compute_descriptor_gradients(class MLIAPData *data)
+{
+  int ij = 0;
+  int max_jnum = -1;
+  int nei = 0;
+  int jtmp =0;
+  int ntypes = atom->ntypes;
+  for (int iitmp = 0; iitmp < data->nlistatoms; iitmp++) {
+    int itmp = data->iatoms[iitmp];
+    jtmp = data->numneighs[iitmp];
+    nei = nei + jtmp;
+    if (jtmp > max_jnum){
+      max_jnum = jtmp;
+    }
+  }
+  for (int ii = 0; ii < data->nlistatoms; ii++) {
+    const int i = data->iatoms[ii];
+    const int ielem = data->ielems[ii];
+    int elem_offset = max_num * ielem;
+    // ensure rij, inside, wj, and rcutij are of size jnum
+    delete acemlimpl -> ace;
+    acemlimpl -> ace = new ACECTildeEvaluator(*acemlimpl -> basis_set);
+    acemlimpl -> ace->compute_projections = 1;
+    acemlimpl -> ace->compute_b_grad = 1;
+    int n_r1, n_rp = 0;
+    n_r1 = acemlimpl -> basis_set->total_basis_size_rank1[0];
+    n_rp = acemlimpl -> basis_set->total_basis_size[0];
+
+    acemlimpl -> ace->element_type_mapping.init(ntypes+1);
+    for (int ik = 1; ik <= ntypes; ik++) {
+      for(int mu = 0; mu < acemlimpl -> basis_set ->nelements; mu++){
+        if (mu != -1) {
+          if (mu == ik - 1) {
+            acemlimpl -> ace->element_type_mapping(ik) = mu;
+          }
+        }
+      }
+    }
+
+    const int* const jlist = data->cpy_frstngh[ii];
+    const int jnum = data->numneighs[ii];
+    acemlimpl -> ace->resize_neighbours_cache(jnum);
+    acemlimpl -> ace->compute_atom(i, atom->x, atom->type, data->numneighs[ii], data->cpy_frstngh[ii]);
+    int ij0 = ij;
+    int ninside = 0;
+    for (int jj = 0; jj < jnum; jj++) {
+      ninside++;
+      ij++;
+    }
+
+    ij = ij0;
+    for (int jj = 0; jj < data->numneighs[ii]; jj++) {
+      const int jt = data->jatoms[ij];
+      
+      const int j = jlist[jj];
+      int yoffset = ndescriptors;
+      int zoffset = ndescriptors*2;
+      for (int iicoeff = 0; iicoeff < ndescriptors; iicoeff++) {
+        DOUBLE_TYPE fx_dB = acemlimpl -> ace -> neighbours_dB(iicoeff,jj,0);
+        DOUBLE_TYPE fy_dB = acemlimpl -> ace -> neighbours_dB(iicoeff,jj,1);
+        DOUBLE_TYPE fz_dB = acemlimpl -> ace -> neighbours_dB(iicoeff,jj,2);
+      // Accumulate dB_k^i/dRi, dB_k^i/dRj
+        data->graddesc[ij][iicoeff][0] = fx_dB;
+        data->graddesc[ij][iicoeff][1] = fy_dB;
+        data->graddesc[ij][iicoeff][2] = fz_dB; 
+
+      }
+      ij++;
+    }
+  }
+}
+
+void MLIAPDescriptorACE::init()
+{
+}
+
+/* ---------------------------------------------------------------------- */
+
+
+/* ----------------------------------------------------------------------
+   memory usage
+------------------------------------------------------------------------- */
+
+double MLIAPDescriptorACE::memory_usage()
+{
+  double bytes = MLIAPDescriptor::memory_usage();
+
+  return bytes;
+}
diff --git a/src/ML-IAP/mliap_descriptor_ace.h b/src/ML-IAP/mliap_descriptor_ace.h
new file mode 100644
index 0000000000..435746becf
--- /dev/null
+++ b/src/ML-IAP/mliap_descriptor_ace.h
@@ -0,0 +1,53 @@
+/* -*- 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.
+------------------------------------------------------------------------- */
+
+#ifndef LMP_MLIAP_DESCRIPTOR_ACE_H
+#define LMP_MLIAP_DESCRIPTOR_ACE_H
+
+#include "mliap_descriptor.h"
+
+namespace LAMMPS_NS {
+
+class MLIAPDescriptorACE : public MLIAPDescriptor {
+ public:
+  MLIAPDescriptorACE(LAMMPS *, char *);
+  ~MLIAPDescriptorACE() override;
+  void compute_descriptors(class MLIAPData *) override;
+  void compute_forces(class MLIAPData *) override;
+  void compute_force_gradients(class MLIAPData *) override;
+  void compute_descriptor_gradients(class MLIAPData *) override;
+  void init() override;
+  double memory_usage() override;
+
+  double rcutfac;
+  int allocated=0;
+  int max_num=0;
+  //char* ctilde_file_name = nullptr;
+  //char* ctilde_file ="coupling_coefficients.yace";
+  //char *const ctilde_file = nullptr;
+
+ protected:
+  virtual void allocate();
+  int natoms, nmax, size_peratom, lastcol;
+  int ncoeff, nvalues, nperdim, yoffset, zoffset;
+  int ndims_peratom, ndims_force, ndims_virial;
+  int n_r1, n_rp;
+  int chemflag;
+  int bikflag, bik_rows, dgradflag, dgrad_rows;
+  double cutmax;
+  struct ACE_ML_impl *acemlimpl;
+};
+
+}    // namespace LAMMPS_NS
+
+#endif
diff --git a/src/ML-IAP/pair_mliap.cpp b/src/ML-IAP/pair_mliap.cpp
index 66e8b848cd..f68e9c0b56 100644
--- a/src/ML-IAP/pair_mliap.cpp
+++ b/src/ML-IAP/pair_mliap.cpp
@@ -21,6 +21,7 @@
 #include "mliap_data.h"
 #include "mliap_descriptor_snap.h"
 #include "mliap_descriptor_so3.h"
+#include "mliap_descriptor_ace.h"
 #include "mliap_model_linear.h"
 #include "mliap_model_nn.h"
 #include "mliap_model_quadratic.h"
@@ -181,7 +182,9 @@ void PairMLIAP::settings(int narg, char ** arg)
         if (iarg+3 > narg) utils::missing_cmd_args(FLERR, "pair_style mliap descriptor so3", error);
         descriptor = new MLIAPDescriptorSO3(lmp,arg[iarg+2]);
         iarg += 3;
-
+      } else if (strcmp(arg[iarg+1],"ace") == 0) {
+        descriptor = new MLIAPDescriptorACE(lmp,arg[iarg+2]);
+        iarg += 3;
       } else error->all(FLERR,"Illegal pair_style mliap command");
     } else if (strcmp(arg[iarg], "unified") == 0) {
 #ifdef MLIAP_PYTHON