lecture-2.ipynb

{
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  {
   "cell_type": "markdown",
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   "source": [
    "<p style=\"font-size:32px; font-weight: bolder; text-align: center\"> Atom-centered density correlation representations </p>\n",
    "<p style=\"text-align: center\"><i> authored by: <a href=\"mailto:michele.ceriotti@gmail.com\"> Michele Ceriotti </a></i></p>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This notebook demonstrates some of the concepts discussed in the second lecture in the \"Introduction to Atomic-Scale Machine Learning\" mini-course. \n",
    "It demonstrates the construction of descriptors of an atom-centered environment based on an expansion of the neighbor density in radial functions \n",
    "and spherical harmonics, and how these can be combined to build higher-order invariant and equivariant descriptors. \n",
    "\n",
    "The notebook contains examples for a water-molecule dataset, and introduces the concepts of density-correlation descriptors, as well as how their significance in terms of SO(3) equivariance. \n",
    "\n",
    "1. [Water dataset](#loading)\n",
    "1. [Density expansion coefficients](#density-expansion)\n",
    "2. [Compute the SOAP representation](#soap)\n",
    "3. [Relation with atom-centered symmetry functions](#variations)\n",
    "4. [Equivariance and equivariant features](#equivariance)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import ase, ase.io\n",
    "import matplotlib as mpl\n",
    "import matplotlib.pyplot as plt \n",
    "import itertools\n",
    "\n",
    "import chemiscope\n",
    "\n",
    "import rascaline\n",
    "import sphericart as sc\n",
    "import metatensor as mt\n",
    "from metatensor import mean_over_samples, Labels\n",
    "\n",
    "from skmatter.metrics import global_reconstruction_error, local_reconstruction_error\n",
    "\n",
    "# loads some utilities from a helper file\n",
    "from utils import wigner_d_matrix, wigner_d_real, rotation_matrix, xyz_to_spherical, spherical_to_xyz, ClebschGordanReal, _real_clebsch_gordan_matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id=\"loading\"> </a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Load and prepare the water dataset"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This dataset contains a collection of configurations for a single water molecule. The coinfigurations are generated by distorting an equilibrium configuration along the bending mode, and the asymmetric stretching coordinate. Each frame contains also the energy and dipole moment, computed with the Partridge-Schwenke monomer potential ([Partridge, Schwenke, J. Chem. Phys. (1997)](http://doi.org/10.1063/1.473987)). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "h2o_frames = ase.io.read(\"data/water_energy-dipole.xyz\", \":\")\n",
    "\n",
    "h2o_energy = np.zeros(len(h2o_frames))\n",
    "h2o_dipole = np.zeros((len(h2o_frames),3))\n",
    "h2o_force = np.zeros((len(h2o_frames),3,3))\n",
    "for fi, f in enumerate(h2o_frames):\n",
    "    h2o_energy[fi] = f.info['energy']\n",
    "    h2o_dipole[fi] = f.info['dipole']\n",
    "    h2o_force[fi] = f.arrays['force']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "94915a1511ed4df2bd496b539693ae52",
       "version_major": 2,
       "version_minor": 0
      },
      "text/html": [
       "<ChemiscopeWidget(meta={'name': ' '}, settings={'map': {'x': {'property': 'HOH'}, 'y': {'property': 'OH1'}, 'co...}, structures=[{'size': 3, 'names': ['O', 'H', 'H'], 'x': [0.0, -0.35, 0.6], '...}, shapes={'dipole': {'kind': 'arrow', 'parameters': {'global': {'color': ...}, properties={'OH2': {'target': 'structure', 'values': [1.2000000000000002, 1...}, )>"
      ],
      "text/plain": [
       "<ChemiscopeWidget(meta={'name': ' '}, settings={'map': {'x': {'property': 'HOH'}, 'y': {'property': 'OH1'}, 'c…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dipole_arrows = chemiscope.ase_vectors_to_arrows(h2o_frames, \"dipole\", scale=4, head_length_scale=3);\n",
    "dipole_arrows[\"parameters\"][\"global\"].update({ \"color\": 0x60A0FF })\n",
    "cs = chemiscope.show(h2o_frames, properties = chemiscope.extract_properties(h2o_frames),\n",
    "        shapes={ \"dipole\" : dipole_arrows},\n",
    "        settings={\n",
    "            'map' : { 'x':{'property' : \"HOH\"},  'y':{'property' : \"OH1\"}, 'color' : {'property' : 'energy'} },\n",
    "            'structure': [{'axes': 'off','keepOrientation': True, 'shape': ['dipole']}]\n",
    "        })\n",
    "\n",
    "if chemiscope.jupyter._is_running_in_notebook():\n",
    "    from IPython.display import display\n",
    "    display(cs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id=\"density-expansion\"> </a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Compute the density expansion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now move on to discuss how to describe an atomic environment using expansion coefficients of the atomic density. \n",
    "In short, the idea is to start describing a molecule $A$ in terms of localized functions (e.g. Gaussians) centered on each atom $i$, \"labelled\" by their chemical nature $a$\n",
    "\n",
    "$$\n",
    "\\langle a \\mathbf{x} | A; \\rho\\rangle = \\sum_{i \\in A} \\delta_{a a_i} \\langle \\mathbf{x} | \\mathbf{r}_i \\rangle.\n",
    "$$\n",
    "\n",
    "We use  the notation $\\langle \\mathbf{x} | \\mathbf{r}_i \\rangle = g(\\mathbf{x}-\\mathbf{r}_i)$ to emphasize how the full structure is built as a sum of terms that describe individual atoms, and how this description can be implemented in any continuous or discrete basis. \n",
    "In general terms, in analogy with the Dirac notation used to describe a quantum state, we use the notation $\\langle q | A\\rangle$ to indicate a descriptor $| A\\rangle$ for an entity $A$, discretized in a basis that is enumerated by the index $q$. \n",
    "See Section 3.1 of [this review](https://doi.org/10.1021/acs.chemrev.1c00021) for a gentler introduction. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This density is then symmetrized with respect to translations (reflecting the fact that atomic properties are invariant to rigid translations of a molecule) which leads to expressing the structure descriptors as a sum of descriptors of _atom centered environments_ $A_i$,\n",
    "\n",
    "\n",
    "$$\n",
    "\\langle a \\mathbf{x} | A; \\rho\\rangle = \\sum_i \\langle a \\mathbf{x} | \\rho_i\\rangle\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\langle a \\mathbf{x} | \\rho_i\\rangle = \\sum_{j \\in A_i} \\delta_{a a_j} \\langle \\mathbf{x} | \\mathbf{r}_{ji} \\rangle.\n",
    "$$\n",
    "\n",
    "where the Gaussians are evaluated at the interatomic distance vectors $\\mathbf{r}_{ji}=\\mathbf{r}_j-\\mathbf{r}_i$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To manipulate this atom-centered density, it is more convenient to express it on a discrete basis. Guided by symmetry considerations that will become apparent, and in analogy with what is done routinely in quantum chemistry for the electron wafefunction (or density) we use a basis of radial functions $R_{nl}(x) \\equiv \\langle x||nl\\rangle$ and spherical harmonics $Y^m_{l}(\\hat{\\mathbf{x}}) \\equiv \\langle \\hat{\\mathbf{x}}|lm\\rangle$\n",
    "\n",
    "$$\n",
    "\\langle a nlm | \\rho_i\\rangle = \\int \\mathrm{d}\\mathbf{x} \n",
    " \\langle nl| x\\rangle  \\langle lm| \\hat{\\mathbf{x}} \\rangle\n",
    "\\langle a \\mathbf{x} | \\rho_i\\rangle  \n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "hypers = {\n",
    "    \"cutoff\": 3,\n",
    "    \"max_radial\": 4,\n",
    "    \"max_angular\": 2,\n",
    "    \"atomic_gaussian_width\": 0.1,\n",
    "    \"cutoff_function\": {\"ShiftedCosine\": {\"width\": 0.5}},\n",
    "    \"radial_basis\": {\"Gto\":  {\"spline_accuracy\": 1e-6 }},\n",
    "    \"center_atom_weight\": 1.0\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "calculator = rascaline.SphericalExpansion(**hypers)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "rhoi = calculator.compute(h2o_frames)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Metatensor structured feature storage"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`rascaline` calculators return a `metatensor.TensorMap` object, that works as a container that holds blocks of data. The pattern is reminiscent of a `dict`, but with some more structure and metadata: each block is associated with a _key_, which consists in a tuple of ints. The set of keys is a `metatensor.Labels` object, that also keeps track of the _names_ that describe each index in the key. \n",
    "\n",
    "The expansion keys hold the `spherical_harmonics_l` index, and two indices corresponding to the atomic number of the central atom `species_center` and of the neighbors `species_neighbor`. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "TensorMap with 12 blocks\n",
       "keys: spherical_harmonics_l  species_center  species_neighbor\n",
       "                0                  1                1\n",
       "                1                  1                1\n",
       "                               ...\n",
       "                1                  8                8\n",
       "                2                  8                8"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rhoi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Each block is associated to a `metatensor.TensorBlock` object. Each block is associated with a dense tensor with a `samples` direction (enumerating the items that are described) a `properties` direction (enumerating actual properties, or descriptors) and zero or more `components` - typical examples would be the Cartesian coordinates of a vector, or the $m$ index in spherical harmonics. Each is associated with a `Labels` object that stores metadata that describe the entries."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "TensorBlock\n",
       "    samples (242): ['structure', 'center']\n",
       "    components (3): ['spherical_harmonics_m']\n",
       "    properties (4): ['n']\n",
       "    gradients: None"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rhoi.block(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['structure', 'center']\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "LabelsValues([[0, 1],\n",
       "              [0, 2],\n",
       "              [1, 1],\n",
       "              [1, 2],\n",
       "              [2, 1],\n",
       "              [2, 2],\n",
       "              [3, 1],\n",
       "              [3, 2],\n",
       "              [4, 1],\n",
       "              [4, 2]], dtype=int32)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(rhoi.block(1).samples.names)\n",
    "rhoi.block(1).samples.values[:10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['spherical_harmonics_m']\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "LabelsValues([[-1],\n",
       "              [ 0],\n",
       "              [ 1]], dtype=int32)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(rhoi.block(1).components[0].names)\n",
    "rhoi.block(1).components[0].values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['n']\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "LabelsValues([[0],\n",
       "              [1],\n",
       "              [2],\n",
       "              [3]], dtype=int32)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(rhoi.block(1).properties.names)\n",
    "rhoi.block(1).properties.values[:10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(242, 1, 4)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rhoi.block(0).values.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$l=0$ components are rotationally invariant, so e.g. O-centred features will be independent on the H-O-H angle. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f44ecf693a0>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 600x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1,1,figsize=(6,4))\n",
    "block = rhoi.block(spherical_harmonics_l=0, species_center=8, species_neighbor=1)\n",
    "y = block.values\n",
    "ax.plot(y[0,0], 'rx', label=f\"O-H1={h2o_frames[0].info['OH1']}Å, OHO={h2o_frames[0].info['HOH']}\")\n",
    "ax.plot(y[4,0], 'r--', label=f\"O-H1={h2o_frames[4].info['OH1']}Å, OHO={h2o_frames[4].info['HOH']}\")\n",
    "ax.plot(y[11,0], 'bo', label=f\"O-H1={h2o_frames[11].info['OH1']}Å, OHO={h2o_frames[11].info['HOH']}\")\n",
    "ax.plot(y[15,0], 'b--', label=f\"O-H1={h2o_frames[15].info['OH1']}Å, OHO={h2o_frames[15].info['HOH']}\")\n",
    "ax.set_xlabel('feature index')\n",
    "ax.set_ylabel('feature value')\n",
    "ax.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$l=1$ components on the other hand are sensitive to the position of neighbors.\n",
    "Overall, for a sufficiently fine-grained discretization of the density, the expansion coefficients contain full information on the position of all neighbors in 3D space. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f44ecdf8d60>"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 600x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1,1,figsize=(6,4))\n",
    "block = rhoi.block(spherical_harmonics_l=1, species_center=8, species_neighbor=1)\n",
    "y = block.values\n",
    "ax.plot(y[0,0], 'rx', label=f\"O-H1={h2o_frames[0].info['OH1']:.1}Å, OHO={h2o_frames[0].info['HOH']}\")\n",
    "ax.plot(y[4,0], 'r--', label=f\"O-H1={h2o_frames[4].info['OH1']:.1}Å, OHO={h2o_frames[4].info['HOH']}\")\n",
    "ax.plot(y[11,0], 'bo', label=f\"O-H1={h2o_frames[11].info['OH1']:.1}Å, OHO={h2o_frames[11].info['HOH']}\")\n",
    "ax.plot(y[15,0], 'b--', label=f\"O-H1={h2o_frames[15].info['OH1']:.1}Å, OHO={h2o_frames[15].info['HOH']}\")\n",
    "ax.set_xlabel('feature index')\n",
    "ax.set_ylabel('feature value')\n",
    "ax.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Computing information content with the GFRE"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given the possibility of having different types of descriptors, an obvious question is which choice of features provides the \"best\" representation of a set of structures. \n",
    "One possible answer is given by the \"feature reconstruction error\" metrics ([Goscinski et al., MLST (2021)](http://doi.org/10.1088/2632-2153/abdaf7)), that measures the error made by a linear model that attempts to predict features $\\mathbf{X}'$ starting from an alternative featurization $\\mathbf{X}$. \n",
    "Features are standardized so that complete failure in reconstructing the target representation yields a reconstruction error around 1. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "# we restrict the analysis to O-centered features, so there is one environment per structure. \n",
    "# we use `metatensor` utilities to consolidate l,m indices in the property block, and select exclusively \n",
    "# the blocks associated with O centers and H neighbors\n",
    "block = (rhoi.components_to_properties([\"spherical_harmonics_m\"])\n",
    "             .keys_to_properties([\"spherical_harmonics_l\"])\n",
    "             .block(species_center=8, species_neighbor=1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, we look how density-expansion coefficients compare with the naive descriptors based on the Cartesian coordinates of the H atoms \n",
    "relative to the O (which sits at the origin). \n",
    "\n",
    "Surprisingly (given that the density descriptors are computed from the positions!) the reconstruction error is larger when going from the \n",
    "positions to the coefficients than vice versa. This happens because the relationship between the two is highly non-linear: in fact, using a \n",
    "local version of the algorithm that only tries to perform a locally linear mapping shows that the two representations contain similar \n",
    "amounts of information. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "positions = np.array([f.positions.flatten() for f in h2o_frames])\n",
    "h_positions = positions[3:] "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Global reconstruction error positions -> density coefficients: 0.207318\n",
      "Global reconstruction error density coefficients -> positions: 0.055457\n",
      "\n",
      "Local reconstruction error positions -> density coefficients: 0.039321\n",
      "Local reconstruction error density coefficients -> positions: 0.033520\n",
      "\n"
     ]
    }
   ],
   "source": [
    "print(f\"\"\"\n",
    "Global reconstruction error positions -> density coefficients: {global_reconstruction_error(positions, block.values):4f}\n",
    "Global reconstruction error density coefficients -> positions: {global_reconstruction_error(block.values, positions):4f}\n",
    "\n",
    "Local reconstruction error positions -> density coefficients: {local_reconstruction_error(positions, block.values, n_local_points=3):4f}\n",
    "Local reconstruction error density coefficients -> positions: {local_reconstruction_error(block.values, positions, n_local_points=3):4f}\n",
    "\"\"\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another useful case is looking for convergence of the density expansion. Here a linear framework is natural, even though one should keep in mind that the error is an indication of how well the discretization encodes the neighbor density information, which is redundant relative to the simple encoding of the _atomic positions_.\n",
    "\n",
    "NB: the high->low resolution reconstruction error is _not_ zero because even the larger basis is not fully converged and the GTO basis changes with the `max_radial` parameter so the larger basis does not contain explicitly the small-basis functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "# update the hypers to increase the number of channels\n",
    "fine_calculator = rascaline.SphericalExpansion(**{**hypers, \"max_radial\":12, \"max_angular\":8})\n",
    "fine_rhoi = fine_calculator.compute(h2o_frames)\n",
    "fine_block = fine_rhoi.components_to_properties([\"spherical_harmonics_m\"]).keys_to_properties([\"spherical_harmonics_l\"]).block(species_center=8, species_neighbor=1)\n",
    "\n",
    "mid_calculator = rascaline.SphericalExpansion(**{**hypers, \"max_radial\":6, \"max_angular\":4})\n",
    "mid_rhoi = mid_calculator.compute(h2o_frames)\n",
    "mid_block = mid_rhoi.components_to_properties([\"spherical_harmonics_m\"]).keys_to_properties([\"spherical_harmonics_l\"]).block(species_center=8, species_neighbor=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Global reconstruction error high-> low resolution: 0.001055\n",
      "Global reconstruction error low -> high resolution: 0.547158\n",
      "Global reconstruction error mid -> high resolution: 0.154662\n",
      "\n"
     ]
    }
   ],
   "source": [
    "print(f\"\"\"\n",
    "Global reconstruction error high-> low resolution: {global_reconstruction_error(fine_block.values, block.values):4f}\n",
    "Global reconstruction error low -> high resolution: {global_reconstruction_error(block.values, fine_block.values):4f}\n",
    "Global reconstruction error mid -> high resolution: {global_reconstruction_error(mid_block.values, fine_block.values):4f}\n",
    "\"\"\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id=\"soap\"> </a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Compute the SOAP representation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The $l=0$ part of the density-expansion coefficients $\\langle an00|\\rho_i\\rangle$ corresponds to a discretization of the pair correlation function: using a real-space basis,\n",
    "\n",
    "$$\n",
    "    \\langle ax00|\\rho_i\\rangle \\approx \\sum_{j\\in A_i} \\delta_{a a_j} \\langle x | r_{ji} \\rangle \n",
    "$$\n",
    "\n",
    "where $ \\langle x | r_{ji} \\rangle $ is a localized function centered on $r_{ji}$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to obtain a richer description of the atomic environment it is possible to combine several copies of $\\langle a\\mathbf{x} | \\rho_i \\rangle$, to build $\\nu$-neighbors atom-centered density correlations (ACDCs). \n",
    "The formalism we use was introduced by [Willatt et al.](https://doi.org/10.1063/1.5090481), and is explained in detail, discussing its relation with the leading frameworks for atomistic machine learning, in a [review by Musil et al.](https://doi.org/10.1021/acs.chemrev.1c00021)\n",
    "\n",
    "Essentially, the idea is that considering tensor products of the atom density provides simultaneous information on the mutual position of several neighbors\n",
    "\n",
    "$$\n",
    "\\langle \\mathbf{x} |  \\rho_i \\rangle \\langle \\mathbf{x}' |  \\rho_i \\rangle =\n",
    "\\sum_{jj'\\in A_i}\n",
    "\\langle \\mathbf{x} |\\mathbf{r}_{ji} \\rangle \\langle \\mathbf{x}' |\\mathbf{r}_{j'i} \\rangle. \n",
    "$$\n",
    "\n",
    "The invariant part of this two-neighbor correlation function can be extracted by taking a symmetrized product of the density coefficients,\n",
    "$$\n",
    "\\langle aa'nn'l|\\overline{\\rho_i^{\\otimes 2}}\\rangle \\propto \n",
    "\\sum_m  \\langle anlm|\\rho_i\\rangle \\langle a'n'lm|\\rho_i\\rangle\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`rascaline` provides the infrastructure to evaluate SOAP features using the `SoapPowerSpectrum` calculator. We can compare the results with those obtained by explicitly computing the symmetrized products. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "soap_calculator = rascaline.SoapPowerSpectrum(**hypers)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "rho2i = soap_calculator.compute(h2o_frames)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The calculator returns only invariant features, with separate blocks depending on the nature of the central atom and the neighbors. \n",
    "Select the block corresponding to O-centered $\\langle HHnn'(l=2)|$ features"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "TensorMap with 6 blocks\n",
       "keys: species_center  species_neighbor_1  species_neighbor_2\n",
       "            1                 1                   1\n",
       "            1                 1                   8\n",
       "                               ...\n",
       "            8                 1                   8\n",
       "            8                 8                   8"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rho2i"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "soap_OHH = rho2i.block(species_center=8, species_neighbor_1=1, species_neighbor_2=1)\n",
    "soap_OHH_l2 = soap_OHH.values[:,np.where(soap_OHH.properties[\"l\"]==2)[0]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "... and now compute the same manually. Note the $1/\\sqrt{2l+1}$ factor."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "rhoi_OH = rhoi.blocks(species_center=8, species_neighbor=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "manual_OHH_l2 = np.einsum(\"imn, imp-> inp\", rhoi_OH[2].values, rhoi_OH[2].values).reshape(len(h2o_frames),-1)/np.sqrt(5.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 600x600 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1,2,figsize=(6,6), sharex=True, sharey=True, constrained_layout=True)\n",
    "ax[0].matshow(soap_OHH_l2[:40])\n",
    "ax[0].set_ylabel(\"structure index\")\n",
    "cbvals=ax[1].matshow(manual_OHH_l2[:40])\n",
    "fig.colorbar(cbvals, label=\"feature value\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id=\"variations\"> </a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Relation between density correlation features, real-space neighbor-density and atom-centered symmetry functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The purpose of this (rather contrieved) demonstration is to show explicitly the relation between density expansion coefficients and a smooth neighbor density, and between SOAP descriptors and atom-centered symmetry functions. \n",
    "We use `SphericalBesselBasis` (the Laplacian Eigenvalues basis from [Bigi et al., JCP (2022)](http://doi.org/10.1063/5.0124363)) because it's orthonormal and makes it easier to post-process based on the utility functions in `rascaline`. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Density reconstruction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can explicitly plot the real-space neighbor density by computing it from the coefficients:\n",
    "\n",
    "$$\n",
    "\\rho_i(\\mathbf{x}) = \\sum_{nlm} R_{nl}(x) Y_l^m(\\hat{\\mathbf{x}}) \\langle nlm | \\rho_i \\rangle\n",
    "$$\n",
    "\n",
    "We use `SoapSpliner` to compute the radial integrals that enter the calculation of the density expansion. As explained e.g. in [Musil et al., JCP (2021)](http://doi.org/10.1063/5.0044689), the coefficients for the expansion when using a Gaussian smearing with standard deviation $\\sigma$ can be computed by evaluating at the neighbor positions a suitably-computed radial integral function $R_nl^{(\\sigma)}$. \n",
    "\n",
    "$$\n",
    " \\langle nlm | \\rho_i \\rangle = \\sum_{j\\in A_i} R_{nl}^{(\\sigma)}(r_{ji}) Y_l^m(\\hat{\\mathbf{r}}_{ji}),\n",
    "$$\n",
    "\n",
    "However, when reconstructing the density one needs to use the bare (non-smoothed) functions "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "hypers = {\n",
    "    \"cutoff\": 4,\n",
    "    \"max_radial\": 4,\n",
    "    \"max_angular\": 16,\n",
    "    \"atomic_gaussian_width\": 0.3,\n",
    "    \"cutoff_function\": {\"ShiftedCosine\": {\"width\": 0.5}},\n",
    "    \"center_atom_weight\": 1.0\n",
    "}\n",
    "\n",
    "basis_functions = rascaline.utils.SphericalBesselBasis(hypers[\"cutoff\"], hypers[\"max_radial\"], hypers[\"max_angular\"])\n",
    "\n",
    "basis_integrals = rascaline.utils.SoapSpliner(\n",
    "    cutoff=hypers[\"cutoff\"],\n",
    "    max_radial=hypers[\"max_radial\"],\n",
    "    max_angular=hypers[\"max_angular\"],\n",
    "    basis=basis_functions,\n",
    "    density=rascaline.utils.GaussianDensity(atomic_gaussian_width=hypers[\"atomic_gaussian_width\"]),\n",
    "    accuracy=1e-4\n",
    ").compute()\n",
    "\n",
    "hypers[\"radial_basis\"] = basis_integrals"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nice twist: we use `selected_keys` to only compute one set of element-element density coefficients, between O and H"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "calculator = rascaline.SphericalExpansion(**hypers)\n",
    "rhoi_oh = calculator.compute(h2o_frames, \n",
    "                      selected_keys=mt.Labels([\"spherical_harmonics_l\", \"species_center\", \"species_neighbor\"],\n",
    "                                              np.array([[l,8,1] for l in range(hypers[\"max_angular\"])]  ) )\n",
    "                             )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "# a utility function to compute the density projected on a list of 3D points\n",
    "def real_space_rho(x, cnlm, hypers):\n",
    "    sph = sc.SphericalHarmonics(l_max=hypers[\"max_angular\"], normalized=True).compute(x)    \n",
    "    values = np.zeros(x.shape[0])\n",
    "    for key, block in cnlm.items():\n",
    "        l = key[\"spherical_harmonics_l\"]\n",
    "        for n in range(hypers[\"max_radial\"]):\n",
    "            rbf = basis_functions.compute(n, l, np.sqrt(np.sum(x**2,axis=1)) )\n",
    "            sph_l = (sph[:, l*l:(l+1)*(l+1)] @ block.values[0, :, n])\n",
    "        values += rbf*sph_l\n",
    "    return values"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we compute the reconstructed density. \n",
    "NB: instabilities in the orthogonalization procedure for the basis make it difficult to converge the expansion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "ngrid = 100\n",
    "iframe = 15\n",
    "grid = np.array(list(itertools.product(np.linspace(-1.5,1.5,ngrid), np.linspace(-1.5,1.5,ngrid), [0])) \n",
    "               ).reshape(-1,3)\n",
    "# this selects the coefficients for a single structure \n",
    "cnlm = mt.slice(rhoi_oh, axis=\"samples\", labels=Labels(\"structure\", np.array([[iframe]])))\n",
    "rbf = real_space_rho(grid, cnlm, hypers)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'y / Å')"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 600x600 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1,1,figsize=(6,6), constrained_layout=True)\n",
    "ctf = ax.contourf(grid[:,0].reshape(ngrid,ngrid), grid[:,1].reshape(ngrid,ngrid), \n",
    "             rbf.reshape(ngrid,ngrid))\n",
    "fig.colorbar(ctf, label=r'$\\rho_i$')\n",
    "ax.plot(h2o_frames[iframe].positions[1:,0], h2o_frames[iframe].positions[1:,1], 'kx')\n",
    "ax.plot(h2o_frames[iframe].positions[:1,0], h2o_frames[iframe].positions[:1,1], 'ro')\n",
    "ax.set_xlabel('x / Å')\n",
    "ax.set_ylabel('y / Å')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Density coefficients and atom-centered symmetry functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now demonstrate another link between density correlations and a widely-used family of descriptors, atom-centered symmetry functions. \n",
    "\n",
    "In the Dirac-$\\delta$ limit of Gaussian smearing, the expansion coefficients are just the radial functions evaluated at the atomic positions, and one can show that\n",
    "\n",
    "$$\n",
    "\\langle nn'l|\\overline{\\rho_i^{\\otimes 2}}\\rangle \\propto\n",
    "\\sum_{jj'\\in A_i} R_{nl}(r_{ji}) R_{nl}(r_{j'i}) P_l(\\hat{\\mathbf{r}}_{ji}\\cdot \\hat{\\mathbf{r}}_{j'i})\n",
    "$$\n",
    "\n",
    "let's see this in action"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "hypers = {\n",
    "    \"cutoff\": 4,\n",
    "    \"max_radial\": 4,\n",
    "    \"max_angular\": 16,\n",
    "    \"atomic_gaussian_width\": 0.3,\n",
    "    \"cutoff_function\": {\"ShiftedCosine\": {\"width\": 0.5}},\n",
    "    \"center_atom_weight\": 1.0\n",
    "}\n",
    "\n",
    "basis_functions = rascaline.utils.SphericalBesselBasis(hypers[\"cutoff\"], hypers[\"max_radial\"], hypers[\"max_angular\"])\n",
    "\n",
    "basis_integrals = rascaline.utils.SoapSpliner(\n",
    "    cutoff=hypers[\"cutoff\"],\n",
    "    max_radial=hypers[\"max_radial\"],\n",
    "    max_angular=hypers[\"max_angular\"],\n",
    "    basis=basis_functions,\n",
    "    density=rascaline.utils.DeltaDensity(),  # <<< we evaluate with a delta density\n",
    "    accuracy=1e-5\n",
    ").compute()\n",
    "\n",
    "hypers[\"radial_basis\"] = basis_integrals"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We compute only the OHH component of the SOAP power spectrum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "soap_calculator = rascaline.SoapPowerSpectrum(**hypers)\n",
    "rhoi_ohh = soap_calculator.compute(h2o_frames, \n",
    "                    selected_keys=mt.Labels([\"species_center\", \"species_neighbor_1\", \"species_neighbor_2\"],\n",
    "                                           np.array([[8,1,1]]))\n",
    "                                   )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now compute explicitly the symmetry functions, summing over all pairs of neighbors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "def acsf(r_list, l, n, n1, Rnl):\n",
    "    \"\"\" Compute ACSF based on the Rnl radial functions \n",
    "    \n",
    "    r_list: list of neighbor positions (n_neigbors, 3)\n",
    "    l: angular momentum channel\n",
    "    n, n1: the two radial channels\n",
    "    Rnl: the basis function from rascaline.utils\n",
    "    \"\"\"\n",
    "    \n",
    "    value = 0.0\n",
    "    Rnl_list = []\n",
    "    Rn1l_list = []    \n",
    "    for rij in r_list:\n",
    "        Rnl_list.append(Rnl.compute(n=n, ell=l, integrand_positions=np.sqrt(rij@rij)))\n",
    "        Rn1l_list.append(Rnl.compute(n=n1, ell=l, integrand_positions=np.sqrt(rij@rij)))\n",
    "    for rij, R in zip(r_list, Rnl_list):    \n",
    "        for rijp, Rp in zip(r_list, Rn1l_list):\n",
    "            cosjjp = rij@rijp/np.sqrt((rij@rij) *(rijp@rijp))\n",
    "            value += R*Rp*np.polynomial.legendre.legval(cosjjp, [0]*l+[1])\n",
    "    return value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [],
   "source": [
    "# accumulate the reference (SOAP-based) and manual (computed ACSF) values for all structures and channels\n",
    "ref_manual = np.zeros((\n",
    "    len(h2o_frames), \n",
    "    len(rhoi_ohh.block(0).properties),\n",
    "    2))\n",
    "\n",
    "for i in range(len(h2o_frames)):    \n",
    "    for j, p in enumerate(rhoi_ohh.block(0).properties):\n",
    "        # Note how we use the properties keys to keep track of what each property is \n",
    "        ref_manual[i,j] =  (\n",
    "          rhoi_ohh.block(0).values[i,j],\n",
    "          acsf(h2o_frames[i].positions[1:],p[\"l\"],p[\"n1\"],p[\"n2\"],basis_functions), \n",
    "        )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We disregard a bunch of normalization and scaling coefficients, so the density correlations and ACSF descriptors are only proportional and not identical"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'ACSF')"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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AABhEGAMAADCIMAYAAGAQYQwAAMAgwhgAAIBBhDEAAACDCGMAAAAGEcYAAAAMIowBAAAYRBgDAAAwiDAGBGG1WKpqqJLVYpkuBQCQwNgoHAjAXeNW9aZq32NXnkueUo+5ggAACYuZMeAIVovlF8QkqXpTNTNkAICIIIwBR6jbWRdwvL6pPsqVAACSAWEMOEJxbnHA8SJHUZQrAQAkA8IYcARnjlOuPJffmCvPJWeO01BFAIBElmLbtm26iGjyer3KyspSe3u7MjMzTZeDGGa1WKpvqleRo4ggBgA4ZqFmDu6mBIJw5jgJYQCAiOMyJQAAgEGEMQAAAIMIYwAAAAYRxgAAAAwijAEAABhEGAP6qNnbrHXb16nZ22y6FABAHKK1BdAHSzcs1XWrr1On3anUlFQtnrpYZeeVmS4LABBHmBkDeqnZ2+wLYpLUaXfqulXXMUMGADgmhDGgl7bu2eoLYod0qlOznp1lqCIAQDwijAG9NPqk0UpRSrfxP//jz7JaLAMVAQDiEWEM6KXhmcP17bO+HfBYfVN9lKsBAMQrwhjQBz+Z9JOA40WOoihXAgCIV4QxoA+cOU658lx+Y648FxuMAwBClmLbtm26iGjyer3KyspSe3u7MjMzTZeDBGG1WKpvqleRo4ggBgCQFHrmoM8YEAbOHCchDADQK4QxIIKsFku/3fxbyZa+P+77BDYAQDeEMSBC3DVuVW+q9j1+sPFBufJc8pR6zBUFAIg5LOAHIsBqsfyC2CHVm6rpQQYA8EMYAyKgbmdd0GP0IAMAHI4wBkRAcW5x0GP0IAMAHI4wBkRAoP5jEj3IAADd0WcMiCCrxdKTm5+UJF19ztUEMQBIIvQZA2IA/ccAAEfDZUoAAACDCGMAAAAGEcYAAAAMIowBAAAYRBgDDLNaLFU1VNGZHwCSFHdTAgYduX8le1cCQPJhZgwwJND+lexdCQDJhzAGGBJs/0r2rgSA5EIYAwwJtn8le1cCQHIhjAGGBNq/kr0rASD5sDclYJjVYqm+qV5FjiKCGAAkEPamBOIE+1cCQHLjMiUAAIBBhDEAAACDCGMAAAAGEcYAAAAMIowBAAAYRBgDAAAwiDAGAABgEGEMAADAIMIYEIesFktVDVWyWizTpQAA+ogO/ECccde4Vb2p2vfYleeSp9RjriAAQJ8wMwbEEavF8gtiklS9qZoZMgCIY4QxII7U7awLOF7fVB/lSgAA4UIYA+JIcW5xwPEiR1GUKwEAhAthDIgjzhynXHkuvzFXnkvOHKehigAAfZVi27Ztuoho8nq9ysrKUnt7uzIzM02XA/SK1WKpvqleRY4ighgAxKhQM0dMzIwtWrRII0aMUEZGhgoLC9XY2Bj03D/84Q+aMGGChgwZouOPP175+fl6/PHHo1gtYJ4zx6k5588hiAFAAjAexlasWKHy8nLNnz9fGzZsUF5eniZPnqz33nsv4PknnniibrvtNjU0NOj111/XzJkzNXPmTP35z3+OcuUAAAB9Z/wyZWFhoZxOpxYuXChJ6uzslMPh0I033qi5c+eG9BrnnXeepkyZorvuuuuo53KZEgAARENcXKY8ePCg1q9fr5KSEt9YamqqSkpK1NDQcNTn27at2tpabdmyRV/72tcCnnPgwAF5vV6/HwAAgFhhNIzt3r1bHR0dys7O9hvPzs5Wa2tr0Oe1t7dr0KBBSktL05QpU/SrX/1K3/jGNwKeW1lZqaysLN+Pw+EI62cAAADoC+Nrxnpj8ODB2rhxoyzL0j333KPy8nK98MILAc+dN2+e2tvbfT9NTU3RLRYAAKAHRvemHDp0qPr166e2tja/8ba2Ng0bNizo81JTUzVq1ChJUn5+vt58801VVlbqwgsv7HZuenq60tPTw1o3AABAuBidGUtLS9P48eNVW1vrG+vs7FRtba0mTpwY8ut0dnbqwIEDkSgRAAAgoozOjElSeXm5XC6XJkyYoIKCAi1YsED79+/XzJkzJUkzZsxQTk6OKisrJXWtAZswYYJGjhypAwcOaM2aNXr88cf10EMPmfwYAAAAvWI8jE2fPl3vv/++Kioq1Nraqvz8fK1du9a3qH/nzp1KTf1iAm///v264YYb1NzcrOOOO05jx47VE088oenTp5v6CAAAAL1mvM9YtNFnDAAARENc9BkDAABIdoQxIIk0e5u1bvs6NXubTZcCAPgX42vGAETH0g1Ldd3q69Rpdyo1JVWLpy5W2XllpssCgKTHzBiQBJq9zb4gJkmddqdmrZrFDBkAxADCGJAEtu7Z6gtihytcUmigGgDA4QhjQBIYfdLogOPvfvSuPBs90S0GAOCHMAYkgeGZw5WXnRfw2LNvPRvlagAAhyOMAUlizvlzAo5PGzstuoUAAPwQxoAk4c53a+QJI/3GRp4wUu58t5mCAACSaG0BJJVtN22TZ6NHz771rKaNnUYQA4AYEPLMWG5urvbs2eN7vHDhQnm93ogUBSBy3PluPfPdZ/yCmNViqaqhSlaLZa4wAEhSIc+MNTc3q6Ojw/f41ltv1WWXXcb+jkCcc9e4Vb2p2vfYleeSp9RjriAASDK9XjOWZPuLAwnJarH8gpgkVW+qZoYMAKKIBfxAEqvbWRdwvL6pPsqVAEDyOqYF/I8++qgGDRokSfr888/l8Xg0dOhQv3Nuuumm8FUHIKKKc4sDjhc5iqJcCQAkrxQ7xOuNI0aMUEpKSs8vlpKif/zjH2EpLFK8Xq+ysrLU3t7OejdArBkDgEgJNXOEHMYSBWEM6M5qsVTfVK8iR5GcOU7T5QBAQgg1c9BnDICcOU5CGAAYEvIC/oaGBq1evdpv7De/+Y2+9KUv6ZRTTtF1112nAwcOhL1AAACARBZyGLvzzjv1xhtv+B5v3rxZZWVlKikp0dy5c7Vq1SpVVlZGpEgAAIBEFXIY27hxoy6++GLf4+XLl6uwsFBLlixReXm5fvnLX+qpp56KSJEAzKJDPwBETshrxj788ENlZ2f7Hv/f//2fvvnNb/oeO51ONTU1hbc6AMZxtyUARFbIM2PZ2dnavn27JOngwYPasGGDzj//fN/xffv2acCAAeGvEIAxdOgHgMgLOYxddtllmjt3rurq6jRv3jwNHDhQxcVfNIx8/fXXNXLkyIgUCcAMOvQDQOSFfJnyrrvu0pVXXqkLLrhAgwYNUnV1tdLS0nzHly1bpksuuSQiRQIwgw79ABB5x9z0tb29XYMGDVK/fv38xj/44AMNGjTIL6DFIpq+AseGNWMA0Dth78Df0dGhN954Q6NHj9Zxxx3nd+zjjz/Wtm3bdPbZZys1Nbb3HieMAceODv0AcOxCzRwhJ6fHH39c11xzTcCZr7S0NF1zzTX63e9+17tqAcQ0Z45Tc86fQxADgAgIOYwtXbpUP/nJT7pdnpSk/v376z//8z+1ePHisBYHAACQ6EIOY1u2bPFrZXEkp9OpN998MyxFAQAAJIuQw9j+/fvl9XqDHt+3b58+/vjjsBQFAACQLEIOY6NHj9ZLL70U9PiLL76o0aNHh6UoAACAZBFyGPve976n22+/Xa+//nq3Y5s2bVJFRYW+973vhbU4AACARBdya4vPPvtMl1xyiV588UWVlJRo7NixkqS33npLzz//vCZNmqTnn38+5rdEorUFAACIhrC3thgwYICee+453XPPPdq1a5cWL16sRx55RLt27dI999yj559/Xlu2bAlL8QDik9Viqaqhir0rAeAYHHMH/iN5vV4tX75cS5cu1auvvqqOjo5w1RYRzIwBkUGnfgDwF/aZsSP99a9/lcvl0mmnnab7779fX//61/Xyyy/39uUAxDGrxfILYpJUvamaGTIACEHIG4VLUmtrqzwej5YuXSqv16vvfOc7OnDggGpqanTWWWdFqkYAMa5uZ13A8fqmerr2A8BRhDwzdvnll2vMmDF6/fXXtWDBAr377rv61a9+FcnaAMSJ4tzigONFjqIoVwIA8SfkMPanP/1JZWVl+u///m9NmTIl4LZIAJKTM8cpV57Lb8yV52JWDABCEHIYe/HFF7Vv3z6NHz9ehYWFWrhwoXbv3h3J2gDEEU+pR42zGvXA5AfUOKuRxfsAEKJjvpty//79WrFihZYtW6bGxkZ1dHSoqqpK11xzjQYPHhypOsOGuykBAEA0hJo5+tTaYsuWLVq6dKkef/xx7d27V9/4xje0cuXK3r5cVBDGAABANES8tYUkjRkzRj//+c/V3NysJ598si8vBQAAkJT63PQ13jAzBgAAoiEqM2MAAADoG8IYAACAQYQxAAAAgwhjAAAABhHGAAAADCKMAQAAGEQYAwAAMIgwBsAoq8VSVUOVrBbLdCkAYER/0wUASF7uGreqN1X7HrvyXGwwDiDpMDMGwAirxfILYpJUvamaGTIASYcwBsCIup11Acfrm+qjXAkAmEUYA2BEcW5xwPEiR1GUKwEAswhjAIxw5jjlynP5jbnyXDp18Klat32dmr3NhioDgOhiAT8AYzylHs12zlZ9U72KHEV6ve11nb7gdHXanUpNSdXiqYtVdl6Z6TIBIKJSbNu2TRcRTV6vV1lZWWpvb1dmZqbpcgD8S7O32RfEDklRilZevVJTz5xqsDIA6J1QMweXKQHEhK17tvoFMUmyZevyJy+Xu8ZtpigAiALCGICYMPqk0UpNCfyvJFpeAEhkhDEAMWF45nAtnrpYKUoJeJyWFwASFWEMQMwoO69MK69eGfAYLS8AJCrCGICYMvXMqd1aXkwZPUV1O+u4VAkgIXE3JYCYZLVYqm+q1/++879as22Nb5z9KwHEC+6mBBDXnDlOFTmK/IKYxGJ+AImHMAYgZrF/JYBkQBgDELPYvxJAMiCMAYhZwfavdOY4DVUEAOHHAn4AMe/QYv4iRxFBDEDcCDVzsFE4gJjnzHESwgAkLC5TAgAAGEQYAxDXrBZLVQ1VtLsAELe4TAkgbrlr3KreVO17TENYAPGImTEAcclqsfyCmERDWADxiTAGIC7REBZAoiCMAYhLNIQFkCgIYwDiEg1hASQKmr4CiGs0hAUQq2j6CiAp0BAWQLzjMiUAAIBBhDEAAACDCGMAAAAGxUQYW7RokUaMGKGMjAwVFhaqsbEx6LlLlixRcXGxTjjhBJ1wwgkqKSnp8XwAYMskALHMeBhbsWKFysvLNX/+fG3YsEF5eXmaPHmy3nvvvYDnv/DCC7r66qu1bt06NTQ0yOFw6JJLLlFLS0uUKwcQD9w1bhU8WqBbnrtFBY8WyF3jNl0SAPgx3tqisLBQTqdTCxculCR1dnbK4XDoxhtv1Ny5c4/6/I6ODp1wwglauHChZsyYcdTzaW0BJA+rxVLBowXdxhtnNXIHJoCICzVzGJ0ZO3jwoNavX6+SkhLfWGpqqkpKStTQ0BDSa3z88cf67LPPdOKJJwY8fuDAAXm9Xr8fAMmBLZMAxAOjYWz37t3q6OhQdna233h2drZaW1tDeo3/+q//0mmnneYX6A5XWVmprKws34/D4ehz3QDiA1smAYgHxteM9cW9996r5cuX65lnnlFGRkbAc+bNm6f29nbfT1NTU5SrBGAKWyYBiAdGO/APHTpU/fr1U1tbm994W1ubhg0b1uNz77//ft177716/vnnNW7cuKDnpaenKz09PSz1Aog/nlKPZjtns2USgJhldGYsLS1N48ePV21trW+ss7NTtbW1mjhxYtDn/fznP9ddd92ltWvXasKECdEoFUAcc+Y4Nef8OQQxADHJ+N6U5eXlcrlcmjBhggoKCrRgwQLt379fM2fOlCTNmDFDOTk5qqyslCT97Gc/U0VFhX73u99pxIgRvrVlgwYN0qBBg4x9DgAAgN4wHsamT5+u999/XxUVFWptbVV+fr7Wrl3rW9S/c+dOpaZ+MYH30EMP6eDBg7rqqqv8Xmf+/Pn66U9/Gs3SAQAA+sx4n7Foo88YAACIhrjoMwYAAJDsCGMAAAAGEcYAAAAMIowBAAAYRBgDgH+xWixVNVTJarFMlwIgiRhvbQEAscBd41b1pmrfY1eeS55Sj7mCACQNZsYAJD2rxfILYpJUvamaGTIAUUEYA5D06nbWBRyvb6qPciUAkhFhDEDSK84tDjhe5CiKciUAkhFhDEDSc+Y45cpz+Y258lxsLA4gKtgOCQD+xWqxVN9UryJHEUEMQJ+Fmjm4mxIA/sWZ4ySEAYg6LlMCAAAYRBgDAAAwiDAGACFq9jZr3fZ1avY2my4FQAJhzRgAhGDphqW6bvV16rQ7lZqSqsVTF6vsvDLTZQFIAMyMAcBRNHubfUFMkjrtTl236jpmyACEBWEMAI5i656tviB2SKc6NevZWYYqApBICGMAcBSjTxqtFKV0G//zP/7M/pUA+owwBgBHMTxzuL591rcDHmP/SgB9RRgDgBD8ZNJPAo7v+HAHs2MA+oQwBgAhCLR/pSQ92PigCh4tkLvGHf2iACQEwhgAhMhT6lHjrEbdXHhzt2PVm6qZIQPQK4QxADgGzhynhmcND3iM9WMAeoMwBgDHqDi3OOB4kaMoypUASASEMQA4RoHWj7nyXHLmOA1VBCCepdi2bZsuIpq8Xq+ysrLU3t6uzMxM0+UAiGNWi6X6pnoVOYoIYgC6CTVzsDclAPSSM8cZMIRZLZbqdtapOLeYkAbgqAhjABBG7hq3qjdV+x678lzylHrMFQQg5rFmDADCxGqx/IKYRMsLAEdHGAOAMKnbWRdwnJYXAHpCGAOAMKHlBYDeIIwBQJjQ8gJAb9DaAgDCjJYXACRaWwCAMcFaXgBAIFymBAAAMIgwBgBRZLVYqmqoot0FAB8uUwJAlNAQFkAgzIwBQBTQEBZAMIQxAIgCGsICCIYwBgBRQENYAMEQxgAgCmgICyAYmr4CQBTREBZIHjR9BYAYRENYAEfiMiUAAIBBhDEAAACDCGMAAAAGEcYAAAAMIowBQAxh70og+XA3JQDECPauBJITM2MAEAPYuxJIXoQxAIgB7F0JJC/CGADEAPauBJIXYQwAYgB7VwLJi70pASCGsHclkDjYmxIA4hB7VwLJh8uUAAAABhHGAAAADCKMAQAAGEQYAwAAMIgwBgAAYBBhDAAAwCDCGADEqWZvs9ZtX6dmb7PpUgD0AX3GACAOLd2wVNeuula2bKUoRUsuX6Ky88pMlwWgF5gZA4A40+xt1qxVs2SrawMVW7auXXUtM2RAnCKMAUCcWbVlVbcxW7ZWb1ltoBoAfUUYA4A4s2vfroDjz7z1jKwWK8rVAOgrwhgAxJnLx1wecPy5fzyngkcL5K5xR7cgIA41N0vr1nX9aRphDADijDPHKVeeK+jx6k3VzJABPbj9dik3V7roIun006WlS83WQxgDgDjkKfWocVajpp05LeDx+qb6KFcExIdvf1u65x7J7rr/RZ2d0vXXm50hI4wBQJxy5jh129duC3isyFEU5WqA2GdZ0u9/3328o0Pati369RxCGAOAOBbokqUrzyVnjtNQRUBssizp7rsDH0tJkUaNim49h6PpKwDEOU+pR7Ods1XfVK8iR5GcOU5ZLZbqdtapOLeYYIak53ZL1dXBj996qzR8eNTK6SbFtg9dNU0OXq9XWVlZam9vV2ZmpulyACDs3DVuVW/64r88rjyXPKUecwUBBlmWVFAQ/PhVV0lPPx2Z9w41c3CZEgASiNVi+QUxibsrkdzq6gKPl5ZKjY2RC2LHgjAGAAmkbmfg//JwdyWSVXFx4PFbb5WcMXIFnzAGAAmkODfwf3m4uxLJxLKkqqquP51OyXVEWz6XK3aCmMQCfgBIKIfurjxyzRiL+JEsjlys73JJHo80e7ZUXy8VFcVWEJNYwG+6HACICKvF8ru7EkgGwRbrNzaaCWChZg5mxgAgATlznN1CGO0ukOiCLdavr4+92bDDEcYAIAnQ7gKJrLlZ2rpVOvPMwMeLYnzJJAv4ASDB0e4CiezwTb+nTZMmTfI/HmuL9QNhZgwAElxP7S64XIl4dvHF0l/+8sXjzk7plVekVau69pqMxcX6gRDGACDB0e4Cicayuma83nyz+7GODmnQIGnOnKiX1WvGL1MuWrRII0aMUEZGhgoLC9XY2Bj03DfeeEPf+ta3NGLECKWkpGjBggXRKxQA4hSbiSORuN1dd0wGCmKS+U2/e8NoGFuxYoXKy8s1f/58bdiwQXl5eZo8ebLee++9gOd//PHHOuOMM3Tvvfdq2LBhUa4WAOKXp9SjxlmNemDyA2qc1cjifcQly+p5w2/J/KbfvWG0z1hhYaGcTqcWLlwoSers7JTD4dCNN96ouXPn9vjcESNGaM6cOZpzjPOQ9BkDACA+VVVJt9wS/PhFF0m1tdGr52hivs/YwYMHtX79es2bN883lpqaqpKSEjU0NITtfQ4cOKADBw74Hnu93rC9NgAAiDzL6uohduKJwc+57DLpj3+MXk3hZOwy5e7du9XR0aHs7Gy/8ezsbLW2tobtfSorK5WVleX7cTgcYXttAEgUVoulqoYq2l0g5hxaI3bLLdLMmdLIkf7Hnc6uDvvxGsSkJLibct68eSovL/c99nq9BDIAOAwNYRGrPJ7ua8TeeUd67DFp7974aV1xNMZmxoYOHap+/fqpra3Nb7ytrS2si/PT09OVmZnp9wMA6EJDWMQqt7trJiyQvXu7WlckQhCTDIaxtLQ0jR8/XrWHrbTr7OxUbW2tJk6caKosAEgqPTWEBUw52l2Tsb690bEyepmyvLxcLpdLEyZMUEFBgRYsWKD9+/dr5r+i8IwZM5STk6PKykpJXYv+//73v/v+uaWlRRs3btSgQYM0Kt6aigBADKAhLGJRsA2/pfjY3uhYGQ1j06dP1/vvv6+Kigq1trYqPz9fa9eu9S3q37lzp1JTv5i8e/fdd3Xuuef6Ht9///26//77dcEFF+iFF16IdvkAEPcONYQ9cs0YDWFhwtHumnzssa7Ll4nGaJ8xE+gzBgDdWS2W6pvqVeQoIojBCLfb/9LkyJFdi/UPcbm6FvTHk1AzB2EMAAAYtXq1dPnl3cfj/a7JmG/6CgAAcPvt0j33BD526K7JRGd8o3AAABAFFRVSbq503nldU1Ex4NvfDh7EpMS7azIYZsYAAEh0gwdLH33U9c9NTV3XBCdNkurNtTDxeKTf/z748US8azIYZsYAAEhkFRVfBLHDvfSSsRmynhq6StKqVfG3WL8vCGMAACSyJ54Ifmzt2ujVoa7WFXPm9NzQ9bbbpKlTo1ZSTOAyJQAAiWzUKGn79sDHLr00amUc2boikKuuku6+OyrlxBRmxgAASGTBVsiPGxe1KahAG34f6bHHpKefjko5MYcwBgA4Js3eZj1kPaSKv1SwoXg8cDq7VsMf7qKLpE2bovL2R1sfJnWVl4id9UNF01cAQMiWbliqWatm+Y258lzylHrMFITQWVbX3ZNR7KBqWVJBQfDjN98sXX114t41GWrmYGYMABCSZm+zrl11bbfx6k3VzJDFGo9HmjbN/5ZEp7Nr9XwUk8/RNvyuqkrcIHYsCGMAgJBs3bNVtgJfTHly85NRrgZBjRrVdV1w5cquP0eNMlZKcXHg8cceS67WFUdDGAMAhGT0SaOVopSAxx545QG5a9zRLQjdeTz+u2tLXY+jmHwsq2vGy7ICL1dL9vVhgbBmDAAQsqUbluraVdcGnSFrnNUoZw7XnYz50pekHTu6j59xRveQFgFHtq9wubpyoIHlajGBNWMAgLArO69MO2/eqW+N/VbA4/VN5rbXgSSvN/B4e3vE39qyurevqK7+YoYsysvV4gphDABwTIZnDtd/ffW/Ah7b8eEOFvOb9IMfBB7/f/8v4m8dbLG+we0v4waXKQEAveKucat6U+BOnrS7MCgjQzpw4IvH6enSp59G7O2am6WtW6X9+7v2Hz9SY2PyzohxmRIAEFGeUo8aZzXq5sKbux2j3YVBn34q/fjH0ogRXX9GMIjdfruUm9vVQ3baNGnSJP/jLlfyBrFjwd6UAIBec+Y4Vbcz8PWp+qZ6FvObsmBB108Effvb0u9//8Xjzk7plVekVaukbduSb7F+XxDGAAB9UpwbuJlUkaMoypUkKcvqWrBVXBy19FNR4R/EDunokAYN6lqsj9ARxgAAfeLMccqV5/JbP+bKczErFg1Tpkhr1nzx+FAviQgaNSp4l4yUFKM9ZuMWC/gBAGFhtViqb6pXkaNIzhynrBZLdTvrVJxbTDCLhKIi6aWXuo9HcMW8x9Pzpt+33SbdfXdE3jouhZo5mBkDAISFM8fpC11H3mnJ3ZVhNnVq4CAmdfWSCHMYO3QltKYm+DlXXUUQ6y3upgQAhJXVYnVrecHdlWFkWdIf/xj8eFF41+q53VJBgXTLLcF7id1xh/T002F926RCGAMAhFVPd1ciDHqafrrssrDOigXqqn+kkSOlO+8M21smJS5TAgDCirsrI2z79sDjgwb1PGPWC8Fmwq6+Wvrkk67eYmz63XfMjAEAwurQ3ZWH4+7KMAqWfv77v8P+VsWBc7Vuvll65hmCWLhwNyUAICKOvLvy0Bh3WIbBKadI77//xeOTT5beey8sL31k2zK32/9SZRS6ZySMUDMHYQwAEBXcYRlmVVXSH/4gXXmlVF4elpcMFrwsq+smTbrqHxvCWBCEMQCIPqvFUsGjBd3GG2c1MkMWiih02besrrsmj5TMG333FRuFAwBiBndY9sHhvSUKCiK2UCvYYv16fkURRxgDAEQcd1j2UqDeEtXVXeNhfIuqKunEEwMfD3PbMgRAawsAQMSxf2UvzZgReDxMXfaP3FFp5Ej/fSddLi5RRgNhDAAQFZ5Sj2Y7Z3e7wxJBTJ0qvfVW4GNhmK468oZMqSuIPfaYtHcvi/WjiTAGAIiaw/evRA962vJowoQ+p6RRo7oHsUP27pXmzOnTy+MYEcYAAIg1V1wR/Nivf92nlx4+XGppCX6cNWLRxwJ+AABiyerVUmtr4GNf/nKfZsU8np6D2KRJXJo0gTAGAEAsWbMm+LGj7drdA49Hmjs3+PGTT6aNhSmEMQAAYslllwUev+iiXk9bjRolzZwptbUFPj5yZNh2U0IvEMYAAIgVHo+0ZElXOjrcuHFSbW2vX/LwdhVHysmRtm3r1UsjTFjADwBALBg1yj81ZWdLV10lXXppV5uLXnrmmcDj2dnSvfdGrKE/jgEzYwCAmLP67dW64Y83aPXbq02XEh2Bpq/a2rraWPQiiHk80rRpXX/+278FPocgFjvYKBwAEFOKlhbppeYv2sJPGj5J9WUJvrJ84EDpk0+6j5eWBp/aCuLICbZDVzyPHOPSZOSxUTgAIO6sfnu1XxCTpJeaX1JVQ5WhiqLA4wkcxKSu6a0QWZb03e92n2B75x3p9tu7OuuXlnb9SRCLLcyMAQBixg1/vEEPvfpQwGOuPJc8pZ7oFhQNaWnSZ591Hx84UNq/P6SXcLt77nrRiwk2hAEzYwCAuHPZ6CBtHSRVb6qW1WJFsZooOO64wEFMkhYtCuklLOvo7ceOYYINBhDGAAAxY+qZUzVp+KSgx5/c/GQUq4mwjAzp008DHxs4MOTV9d/5Ts/HR45koX6sI4wBAGJKfVm9fnHJLwIee+CVB+SucUe3oEioqJAOHAh+PMRZsQEDpB07Ah876STWh8ULwhgAIOaUTyyXK88V8FhCXK589NHgx4YOPepUlscjORzS558HP+f++5kRixc0fQUAxCRPqUdD0ofowcYHux2rb6qXMyeOd7Tu6Ag8npIivf9+j089snVFIFyajC/MjAEAYtb3x30/4PiOD3fE7+zYqFHBN4JctqzHpx5tayNJ+upXuTQZbwhjAICY5cxxBrxc+WDjgyp4tCD+1o8VFwdPU0eZzrKsrkuPPenfX6qr6315MIMwBgCIaZ5SjxpnNermwpu7HYur9WMDBkgvvhj4WH5+j9NZo0ZJBQXSG28EPp6ZKV1zTfAuGYhtrBkDAMQ8Z45TdTsDT/nExfqxsrKeV9v/+MdBD/XvH3yJmcTWRomAmTEAQFwozi0OOF7kKIpyJb3Q01qwHi5Pfve7wYPY2WfTuiJREMYAAHEh0PqxQ4+rGqpi93JlWlrwYz2str/9dmnFiuBPXbaMOyYTBZcpAQBxw1Pq0WznbNU31avIUaRF1iIVPFrgOx5z+1dWVPS8kCvIavsvf1l6663gTxs5UnLG+JVZhI6NwgEAcclqsfyC2CGNsxpjZw3ZkCFSe3vgY6NHS2+/3W042L7hh0uu/3LHLzYKBwAktJ4W9McEywoexCTp1lu7DQ0ZcvQg9thjfSsLsYcwBgCISzG/oP8XgffXlBRw0X5FRc/ZLcjTkAAIYwCAuBRsQX/MXKIMtvp+zJhui/aLiqS77gr+Uscfz52TiYwF/ACAuHXkgv6YCWL9e/jP69y5fg+HD5daWoKfPmCA9NFHYaoLMYkwBgCIa84cZ+yEMKnrVshgzcH69/e7zlhR0XMQy8qS9u4Na3WIQVymBAAgXG6/veeeFPPm+f7R7e750uQddxDEkgWtLQAACIfmZsnhCH58wADp4EFJXTNiPQWxSZOk+hi5KRS9F2rm4DIlACBhNXubtXXPVo0+abSGZw6P7Jtt3drz8X8FsVGjpHfeCX5aTg5BLNlwmRIAkJCWbliq3AdyddFvLlLuA7laumFpZN9w9GgpJSXwscZGWZZ09dU9B7E77uiaYENyIYwBABJOs7dZs1bNkq2ulTi2bM1aNUvN3ggmneHDpSVLugcyl0vuRU4VFEjLlwd/ussl3Xln5MpD7OIyJQAg4azasirg+FUrrtLL174cuTcuK5MmT5ZWr5ZaW6UpU2TJqeruuzb5ueMOglgyI4wBABLOrn27Ao6/8u4r8mz0yJ3vjtybDx8u/fCHvod1VT2fPnIkQSzZcZkSAJBwLh9zedBjM5+dKXeNO2q1FAfetUlf+xpd9dGFMAYASDjOHKemjJ4S9Hj1pmpZLVZ0anF2rQc7nMsl/d//sc8kuhDGAAAJafX3VuuyUZcFPf7k5iejVovHIzU2Sg880PWnxxO1t0YcoOkrACCheTZ6NPPZmQGPufJc8pR6olsQkkaomYOZMQBAQnPnu+XKcwU8Fs3LlUAwhDEAQMLzlHr044IfBzxW30S7e5hFGAMAJIXvj/t+wPEdH+5gdgxGEcYAAEnBmeMMeLnywcYHVfBoQVTbXQCHI4wBAJKGp9SjxlmNurnw5m7HWD8GUwhjAICk4sxxanjW8IDHWD8GEwhjAICkU5wbuC1+kaMoypUAhDEAQBIKtH7MleeSM8dpqCIkM5q+AgCSltViqb6pXkWOIoIYwi7UzNE/ijUBABBTnDlOQhiMi4nLlIsWLdKIESOUkZGhwsJCNTY29nj+008/rbFjxyojI0PnnHOO1qxZE6VKAQAAwst4GFuxYoXKy8s1f/58bdiwQXl5eZo8ebLee++9gOe/9NJLuvrqq1VWVqbXXntNpaWlKi0t1d/+9rcoVw4AANB3xteMFRYWyul0auHChZKkzs5OORwO3XjjjZo7d26386dPn679+/dr9erVvrHzzz9f+fn5evjhh4/6fqwZAwAA0RAXG4UfPHhQ69evV0lJiW8sNTVVJSUlamhoCPichoYGv/MlafLkyUHPP3DggLxer98PAABArDAaxnbv3q2Ojg5lZ2f7jWdnZ6u1tTXgc1pbW4/p/MrKSmVlZfl+HA5HeIoHAAAIA+NrxiJt3rx5am9v9/00NTWZLgkAAMDHaGuLoUOHql+/fmpra/Mbb2tr07BhwwI+Z9iwYcd0fnp6utLT08NTMAAAQJgZnRlLS0vT+PHjVVtb6xvr7OxUbW2tJk6cGPA5EydO9Dtfkv73f/836PkAAACxzHjT1/LycrlcLk2YMEEFBQVasGCB9u/fr5kzZ0qSZsyYoZycHFVWVkqSfvzjH+uCCy7QL37xC02ZMkXLly/Xq6++qsWLF5v8GAAAAL1iPIxNnz5d77//vioqKtTa2qr8/HytXbvWt0h/586dSk39YgJv0qRJ+t3vfqfbb79dt956q0aPHq2amhqdffbZpj4CAABArxnvMxZt9BkDAADREBd9xgAAAJIdYQwAAMAgwhgAAIBBhDEAAACDCGMAAAAGEcYAAAAMIowBAAAYZLzpa7Qdaqvm9XoNVwIAABLZoaxxtJauSRfG9u3bJ0lyOByGKwEAAMlg3759ysrKCno86Trwd3Z26t1339XgwYOVkpIS9Dyv1yuHw6GmpiY69RvC78Asvn/z+B2YxfdvViJ8/7Zta9++fTrttNP8tnY8UtLNjKWmpmr48OEhn5+ZmRm3fwkSBb8Ds/j+zeN3YBbfv1nx/v33NCN2CAv4AQAADCKMAQAAGEQYCyI9PV3z589Xenq66VKSFr8Ds/j+zeN3YBbfv1nJ9P0n3QJ+AACAWMLMGAAAgEGEMQAAAIMIYwAAAAYRxgAAAAwijAEAABiU1GHsgw8+0Pe//31lZmZqyJAhKisr00cffdTj+TfeeKPGjBmj4447Trm5ubrpppvU3t7ud97OnTs1ZcoUDRw4UKeccor+4z/+Q59//nmkP07cOdbvX5IWL16sCy+8UJmZmUpJSdHevXu7nTNixAilpKT4/dx7770R+hTxK1Lff29eN1n15rv69NNPNXv2bJ100kkaNGiQvvWtb6mtrc3vnCP//qekpGj58uWR/ChxY9GiRRoxYoQyMjJUWFioxsbGHs9/+umnNXbsWGVkZOicc87RmjVr/I7btq2KigqdeuqpOu6441RSUqKtW7dG8iPEtXB//263u9vf9UsvvTSSHyEy7CR26aWX2nl5efbLL79s19XV2aNGjbKvvvrqoOdv3rzZvvLKK+2VK1fa27Zts2tra+3Ro0fb3/rWt3znfP755/bZZ59tl5SU2K+99pq9Zs0ae+jQofa8efOi8ZHiyrF+/7Zt2w888IBdWVlpV1ZW2pLsDz/8sNs5p59+un3nnXfau3bt8v189NFHEfoU8StS339vXjdZ9ea7+uEPf2g7HA67trbWfvXVV+3zzz/fnjRpkt85kuzHHnvM738Dn3zySSQ/SlxYvny5nZaWZi9btsx+44037GuvvdYeMmSI3dbWFvD8+vp6u1+/fvbPf/5z++9//7t9++232wMGDLA3b97sO+fee++1s7Ky7JqaGnvTpk32FVdcYX/pS1/i+w4gEt+/y+WyL730Ur+/6x988EG0PlLYJG0Y+/vf/25Lsi3L8o396U9/slNSUuyWlpaQX+epp56y09LS7M8++8y2bdtes2aNnZqaare2tvrOeeihh+zMzEz7wIED4fsAca6v3/+6det6DGMPPPBAGKtNPJH6/sP1v6tk0Jvvau/evfaAAQPsp59+2jf25ptv2pLshoYG35gk+5lnnolY7fGqoKDAnj17tu9xR0eHfdppp9mVlZUBz//Od75jT5kyxW+ssLDQvv76623btu3Ozk572LBh9n333ec7vnfvXjs9Pd1+8sknI/AJ4lu4v3/b7gpj06ZNi0i90ZS0lykbGho0ZMgQTZgwwTdWUlKi1NRUvfLKKyG/Tnt7uzIzM9W/f3/f655zzjnKzs72nTN58mR5vV698cYb4fsAcS5c338w9957r0466SSde+65uu+++7hMfIRIff+R/r0mkt58V+vXr9dnn32mkpIS39jYsWOVm5urhoYGv3Nnz56toUOHqqCgQMuWLZOd5P29Dx48qPXr1/t9d6mpqSopKen23R3S0NDgd77U9e/zQ+dv375dra2tfudkZWWpsLAw6Gsmq0h8/4e88MILOuWUUzRmzBj96Ec/0p49e8L/ASKsv+kCTGltbdUpp5ziN9a/f3+deOKJam1tDek1du/erbvuukvXXXed3+seHsQk+R6H+rrJIBzffzA33XSTzjvvPJ144ol66aWXNG/ePO3atUtVVVV9et1EEqnvP5K/10TTm++qtbVVaWlpGjJkiN94dna233PuvPNOXXTRRRo4cKCee+453XDDDfroo4900003hf1zxIvdu3ero6Mj4L+f33rrrYDPCfbv80Pf9aE/ezoHXSLx/UvSpZdeqiuvvFJf+tKX9M477+jWW2/VN7/5TTU0NKhfv37h/yARknBhbO7cufrZz37W4zlvvvlmn9/H6/VqypQpOuuss/TTn/60z6+XKKL1/fekvLzc98/jxo1TWlqarr/+elVWVib8Hmex8P0nu1j4Hdxxxx2+fz733HO1f/9+3XfffUkdxpCYvvvd7/r++ZxzztG4ceM0cuRIvfDCC7r44osNVnZsEi6M3XLLLXK73T2ec8YZZ2jYsGF67733/MY///xzffDBBxo2bFiPz9+3b58uvfRSDR48WM8884wGDBjgOzZs2LBud4ccutPpaK+bCKLx/R+rwsJCff7559qxY4fGjBkT1teONaa//2j+XmNVJH8Hw4YN08GDB7V3716/2bG2trYev9/CwkLdddddOnDgQML/H5Jghg4dqn79+nW787Sn727YsGE9nn/oz7a2Np166ql+5+Tn54ex+vgXie8/kDPOOENDhw7Vtm3b4iqMJf0C/ldffdU39uc///moC43b29vt888/377gggvs/fv3dzt+aAH/4XeHPPLII3ZmZqb96aefhvdDxLHefv+H9LSA/0hPPPGEnZqaGpd32ERKpL7/vr5uMunNd3VoAf/vf/9739hbb73VbQH/ke6++277hBNOCF/xcaqgoMD+93//d9/jjo4OOycnp8cF5FOnTvUbmzhxYrcF/Pfff7/veHt7Owv4gwj39x9IU1OTnZKSYj/77LPhKTpKkjaM2XbXbeXnnnuu/corr9gvvviiPXr0aL/bypubm+0xY8bYr7zyim3bXf8jKywstM855xx727ZtfrfSfv7557Ztf9Ha4pJLLrE3btxor1271j755JNpbRHAsX7/tm3bu3btsl977TV7yZIltiT7r3/9q/3aa6/Ze/bssW3btl966SX7gQcesDdu3Gi/88479hNPPGGffPLJ9owZM6L++WJdJL7/UF4XX+jN7+CHP/yhnZuba//lL3+xX331VXvixIn2xIkTfcdXrlxpL1myxN68ebO9detW+9e//rU9cOBAu6KiIqqfLRYtX77cTk9Ptz0ej/33v//dvu666+whQ4b47n7/wQ9+YM+dO9d3fn19vd2/f3/7/vvvt9988017/vz5AVtbDBkyxH722Wft119/3Z42bRqtLYII9/e/b98++yc/+Ynd0NBgb9++3X7++eft8847zx49enTcTX4kdRjbs2ePffXVV9uDBg2yMzMz7ZkzZ9r79u3zHd++fbstyV63bp1t21/MBgT62b59u+95O3bssL/5zW/axx13nD106FD7lltu8bW+wBeO9fu3bdueP39+wO//scces23bttevX28XFhbaWVlZdkZGhv3lL3/Z/p//+Z+4+x9mNETi+w/ldfGF3vwOPvnkE/uGG26wTzjhBHvgwIH2v/3bv9m7du3yHf/Tn/5k5+fn24MGDbKPP/54Oy8vz3744Yftjo6OaH60mPWrX/3Kzs3NtdPS0uyCggL75Zdf9h274IILbJfL5Xf+U089ZZ955pl2Wlqa/ZWvfMX+4x//6He8s7PTvuOOO+zs7Gw7PT3dvvjii+0tW7ZE46PEpXB+/x9//LF9ySWX2CeffLI9YMAA+/TTT7evvfZav9ZS8SLFtpP8fmcAAACDkrbPGAAAQCwgjAEAABhEGAMAADCIMAYAAGAQYQwAAMAgwhgAAIBBhDEAAACDCGMAAAAGEcYAAAAM6m+6AAAwISUlJSrvwyYnAI6GMAYgKRGSAMQKLlMCQB80NTXpwgsv1FlnnaVx48bp6aefNl0SgDjDRuEAEIKamho9+uijOnjwoKZPn66ysjJJ0q5du9TW1qb8/Hy1trZq/Pjxevvtt3X88ccbrhhAvOAyJQAcxW9/+1utWLFCv/71r5WRkaFbb71Vra2tuu2223Tqqafq1FNPlSQNGzZMQ4cO1QcffEAYAxAyLlMCSBovvPCCRowYcczPW7x4sYqKilRVVaXHH39cAwcO1AsvvKB9+/b5nbd+/Xp1dHTI4XBIkpYtW6a33norHKUDSGCEMQDowZ49e5Sbm6v//M//1KZNm/TII4/o/vvvV2Fhod5++23feR988IFmzJihxYsX+8ZeeeUVPfLIIybKBhBHCGMA0IMTTjhBu3bt0lNPPSVJOvfcc/XLX/5SW7Zs8c2AHThwQKWlpZo7d64mTZrke+60adO0cuVKI3UDiB+EMQDoQWpqqiZNmqTnnntOK1eu1G9/+1vt2bNHxx9/vE455RTZti23262LLrpIP/jBD/yee/HFF6utrU1/+9vfDFUPIB4QxgAkvRdffFEFBQXKyMjQ0KFD9eCDD/odr6io0Mknn6yvfvWrmjBhgnbv3q1FixZJkurr67VixQrV1NQoPz9f+fn52rx5syQpPT1dl1xyCbNjAHpEawsASeOFF16Q2+3Wjh07fGNr1qyRy+XSfffdp0mTJqm6ulqVlZX6xz/+0avF/keqrq7WQw89pJdffrnPrwUgMTEzBiBpffrpp7r++uv14IMPyu1268wzz9Sdd96p448/Xn/961/D8h6XXXaZGhsbtXv37rC8HoDEQxgDkLT+8pe/6JNPPtH06dN9Y/369VNKSorS09PD8h7bt2/XkCFDNGTIkLC8HoDEQxgDkLTWrVun/Px89evXzze2bds27du3T+eee66krsD2i1/8otfvsXLlSl122WXq358e2wAC498OAJLWa6+9poMHD/qN/frXv9b48eN15plnSpIuuugiXXTRRb1+j5UrV+qOO+7oU50AEhszYwCS1muvvaa//e1v+s1vfqMtW7bo7rvv1kMPPaQlS5b4zrniiit8d0ceq+3bt2vLli269NJLw1UygATEzBiApLRz50598MEHWr16tebOnau3335b48aN09q1a32XKCXprbfe0tixY3v1HitXrtSFF16owYMHh6tsAAmIMAYgKW3cuFEnnniipkyZoilTpgQ8Z9++fcrIyNCAAQN69R4rV67UlVde2ZcyASQBwhiApPTaa6/pnHPO6fGcN954Q1/5yld6/R61tbW9fi6A5MGaMQBJY8SIEZozZ46krjA2bty4Hs/fvHnzUQMbAPQVHfgBIIibbrpJJSUluuKKK0yXAiCBEcYAAAAM4jIlAACAQYQxAAAAgwhjAAAABhHGAAAADCKMAQAAGEQYAwAAMIgwBgAAYBBhDAAAwCDCGAAAgEGEMQAAAIMIYwAAAAb9f2Zhhf3Du/MnAAAAAElFTkSuQmCC",
      "text/plain": [
       "<Figure size 600x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1,1,figsize=(6,6), constrained_layout=True)\n",
    "ax.plot(ref_manual[:,22,0], ref_manual[:,22,1], 'g.')\n",
    "ax.plot(ref_manual[:,33,0], ref_manual[:,33,1], 'r.')\n",
    "ax.plot(ref_manual[:,44,0], ref_manual[:,44,1], 'b.')\n",
    "ax.set_xlabel(r\"$|\\overline{\\rho_i^{\\otimes 2}}\\rangle$\")\n",
    "ax.set_ylabel(\"ACSF\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id=\"variations\"> </a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Equivariance: a primer"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "_Equivariance_ indicates the property of a function for which the inputs and outputs are subject to the action of the same symmetries, and which commutes with the application of the symmetries, that is: $f(\\hat{S}A) = \\hat{S} f(A)$. _Invariance_ can be seen as a special case, in which  $f(\\hat{S}A) = f(A)$.\n",
    "This section focuses in particular on the case of 3D rotations and inversion - in technical terms the $O(3)$ group symmetries - and their combination with translations - the three-dimensional Euclidean group $E(3)$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Rotations and vectors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's now start looking into how a vector such as the dipole moment rotates. The key concept is that a vectorial property such as the atomic positions or the dipole moment transform, under the action of a rotation operation $\\hat{R}$ in a way consistent with the application of a rotation matrix $\\mathbf{R}$, i.e. if a structure $A$ has atomic positions $\\mathbf{r}_i$ and dipole moment $\\mathbf{y}_i$ (each of these being a 3-vector corresponding to the Cartesian coordinates $(x,y,z)$) then the rotated structure $\\hat{R}A$ has atomic coordinates $\\mathbf{R}\\mathbf{r}_i$ and dipole moment $\\mathbf{R}\\mathbf{y}$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We begin generating a dataset of rotated molecules, rotating also the dipoles"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "rotated_structures = []\n",
    "rotated_wdipole = []\n",
    "# this is the reference molecule we use \n",
    "selected_frame = h2o_frames[81].copy()\n",
    "selected_dipole = h2o_dipole[81].copy()\n",
    "selected_frame.info['alpha'] = 0\n",
    "selected_frame.info['beta'] = 0\n",
    "selected_frame.info['gamma'] = 0\n",
    "for alpha in np.linspace(0, 2*np.pi, 8):\n",
    "    for beta in np.linspace(0, np.pi, 4):\n",
    "        for gamma in np.linspace(0, 2*np.pi, 8):\n",
    "            rot_frame = selected_frame.copy() \n",
    "            rot_frame.info['alpha'] = alpha\n",
    "            rot_frame.info['beta'] = beta\n",
    "            rot_frame.info['gamma'] = gamma\n",
    "            # rotates the frame\n",
    "            R = rotation_matrix(alpha, beta, gamma)\n",
    "            rot_frame.positions = rot_frame.positions@R.T\n",
    "            rot_frame.cell = rot_frame.cell@R.T\n",
    "            rot_frame.info['dipole'] = rot_frame.info['dipole']@R.T\n",
    "            rotated_structures.append(rot_frame)            "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "56bb39b4164045269af2fa54c012445b",
       "version_major": 2,
       "version_minor": 0
      },
      "text/html": [
       "<ChemiscopeWidget(meta={'name': ' '}, settings={'map': {'x': {'property': 'alpha'}, 'y': {'property': 'beta'}, ...}, structures=[{'size': 3, 'names': ['O', 'H', 'H'], 'x': [0.0, -0.80434667, 0...}, shapes={'dipole': {'kind': 'arrow', 'parameters': {'global': {'color': ...}, properties={'OH2': {'target': 'structure', 'values': [0.8499999999999999, 0...}, )>"
      ],
      "text/plain": [
       "<ChemiscopeWidget(meta={'name': ' '}, settings={'map': {'x': {'property': 'alpha'}, 'y': {'property': 'beta'},…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dipole_arrows = chemiscope.ase_vectors_to_arrows(rotated_structures, \"dipole\", scale=4, head_length_scale=3);\n",
    "dipole_arrows[\"parameters\"][\"global\"].update({ \"color\": 0x60A0FF })\n",
    "cs = chemiscope.show(rotated_structures, properties = chemiscope.extract_properties(rotated_structures),\n",
    "        shapes={ \"dipole\" : dipole_arrows},\n",
    "        settings={\n",
    "            'map' : { 'x':{'property' : \"alpha\"},  'y':{'property' : \"beta\"}, 'z' : {'property' : 'gamma'},\n",
    "                      'color' : {'property': \"dipole[1]\"} },\n",
    "            'structure': [{'axes': 'off','keepOrientation': True, 'shape': ['dipole']}]\n",
    "        })\n",
    "\n",
    "if chemiscope.jupyter._is_running_in_notebook():\n",
    "    from IPython.display import display\n",
    "    display(cs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id=\"spherical-tensors\"> </a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Rotations and arbitrary tensors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vectorial quantities such as the dipole $\\mathbf{y}$ are just one of the possible types of equivariant properties associated with a molecular structure. We consider as an example the tensor\n",
    "\n",
    "$$\n",
    "\\mathbf{Y} = \\mathbf{y}\\mathbf{y}^T + |\\mathbf{y}|^2 \\mathbf{1}/2,  \n",
    "$$\n",
    "\n",
    "built starting from the dipole moment. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "def y_to_Y(y):\n",
    "    return y.reshape(-1,1)@y.reshape(1,-1) + 0.5* (y@y)*np.eye(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "selected_frame.info['Y'] = y_to_Y(selected_frame.info['dipole'])\n",
    "for rot_frame in rotated_structures:\n",
    "    rot_frame.info['Y'] = y_to_Y(rot_frame.info['dipole'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This quantity transforms under rotation as $\\mathbf{Y}(\\hat{R}A) = \\mathbf{R}\\mathbf{Y}(A)\\mathbf{R}^T$. \n",
    "While this is a perfectly fine way of expressing the transformation, it is more complicated than necessary: all the $9$ components of $\\mathbf{Y}$ are formally mixed in the process, and even though it is linear in $\\mathbf{Y}$, to explicitly cast it as such requires combining the two multiplications by $\\mathbf{R}$ into the multiplication by a large, $9\\times 9$ matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "# picks one of the rotated structures\n",
    "test_frame = rotated_structures[21]\n",
    "selected_Y = selected_frame.info['Y']\n",
    "test_Y = test_frame.info['Y']\n",
    "# builds the rotation matrix\n",
    "R = rotation_matrix(test_frame.info['alpha'], test_frame.info['beta'], test_frame.info['gamma'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1000x400 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1,3,figsize=(10,4), sharey=True, constrained_layout=True)\n",
    "ax[0].matshow(selected_Y)\n",
    "# rotated Y matches the Y of the rotated structure\n",
    "ax[1].matshow(test_Y)\n",
    "ax[2].matshow(R@selected_Y@R.T)\n",
    "\n",
    "ax[0].set_title(\"Reference\")\n",
    "ax[1].set_title(\"Rotated molecule\")\n",
    "ax[2].set_title(\"Reference\")\n",
    "\n",
    "for axi in ax:\n",
    "    axi.set_xticks(np.arange(3))\n",
    "    axi.set_xticklabels(['x', 'y', 'z'])\n",
    "    axi.set_yticks(np.arange(3))\n",
    "    axi.set_yticklabels(['x', 'y', 'z'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "# builds the 9x9 rotation matrix - can you follow the index juggling?\n",
    "flat_idx = np.array([(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2)], dtype=int)\n",
    "RR = R[flat_idx[:,0]][:,flat_idx[:,0]]*R[flat_idx[:,1]][:,flat_idx[:,1]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x7f171c6ba8b0>"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 480x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# the 9x9 matrix is almost entirely full\n",
    "plt.matshow(RR, cmap=plt.cm.RdBu, vmin=-max(np.abs(RR.flatten())), vmax=max(np.abs(RR.flatten())))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1000x400 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# the 9x9 matrix rotates the tensor with a single multiplication\n",
    "rotated_9x9=(RR@(selected_Y[flat_idx[:,0],flat_idx[:,1]])).reshape(3,3)\n",
    "\n",
    "fig, ax = plt.subplots(1,3,figsize=(10,4), sharey=True, constrained_layout=True)\n",
    "ax[0].matshow(selected_Y)\n",
    "# rotated Y matches the Y of the rotated structure\n",
    "ax[1].matshow(test_Y)\n",
    "ax[2].matshow(rotated_9x9)\n",
    "\n",
    "ax[0].set_title(\"Reference\")\n",
    "ax[1].set_title(\"Rotated molecule\")\n",
    "ax[2].set_title(\"Manual rotation (9x9)\")\n",
    "\n",
    "for axi in ax:\n",
    "    axi.set_xticks(np.arange(3))\n",
    "    axi.set_xticklabels(['x', 'y', 'z'])\n",
    "    axi.set_yticks(np.arange(3))\n",
    "    axi.set_yticklabels(['x', 'y', 'z'])\n",
    "    \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This brings us to the concept of _irreducible_ representations of the rotation group. In very hand-wavy terms, we look for ways to simplify the transformation by recombining entries in $\\mathbf{Y}$ so that they are less mixed up when acted upon by a rotation $\\hat{R}$. \n",
    "That this might be possible can be seen by noticing that the _trace_ of $\\mathbf{Y}$, $Y_{xx}+Y_{yy}+ Y_{zz}$ is left unchanged by the rotation: it is _invariant_!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.40527335997463276 0.4052733599746325\n"
     ]
    }
   ],
   "source": [
    "print(np.trace(test_Y), np.trace(selected_Y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It turns out that products of Cartesian coordinates can be rearranged into blocks that transform under rotation as _spherical harmonics_, $Y^m_l$. The theory is not entirely trivial, and gets even messier if (as we do here) one wants to express everything in terms of _real_ quantities. \n",
    "The manipulations here are somewhat opaque (and rely heavily on utility functions) but allow to follow the construction of the _spherical tensor_ form of $\\mathbf{Y}$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{(1, 1): {0: array([-0.23398468]),\n",
       "  1: array([0., 0., 0.]),\n",
       "  2: array([-0.01584898,  0.        , -0.06618086,  0.        , -0.11352766])}}"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# we need Clebsch-Gordan coefficients to transform the Cartesian tensor into an irreducible form\n",
    "cg = ClebschGordanReal(l_max=2)\n",
    "# xyz_to_spherical transforms Cartesian coordinates into a form consistent \n",
    "# with the definition of l=1 real spherical harmonics we use \n",
    "cg.couple(xyz_to_spherical(selected_Y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The output indicates that the tensor was built as the product of two vector-like ($l=1$) spherical Harmonics, and it has been decomposed into a scalar ($l=0$), pseudo-vector ($l=1$, zero because the tensor is symmetric), and higher-order ($l=2$) term. Each block contains $2m+1$ components. We can therefore stack these in a a single array with 9 components, that could (in principle) be written as a unitary transformation of the initial Cartesian form."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.23398468  0.          0.          0.         -0.01584898  0.\n",
      " -0.06618086  0.         -0.11352766]\n",
      "[-0.23398468  0.          0.          0.          0.01748918  0.03029216\n",
      "  0.07917508  0.09690396  0.02524021]\n"
     ]
    }
   ],
   "source": [
    "def to_coupled(cartesian):\n",
    "    coupled = cg.couple(xyz_to_spherical(cartesian))\n",
    "    return np.hstack( [coupled[(1,1)][l] for l in [0,1,2]] )\n",
    "selected_coupled_Y = to_coupled(selected_Y)\n",
    "test_coupled_Y = to_coupled(test_Y)\n",
    "# note that elements [1:4] are zero: this is a consequence of the symmetry of Y\n",
    "print(selected_coupled_Y)\n",
    "print(test_coupled_Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compare the $9\\times 9$ matrix above with that associated with rotating the tensor in this new basis. Each of the irreducible blocks transform separately, so the rotation matrix is block-diagonal. _Any_ Cartesian tensor can be expressed in terms of a collection of $Y^m_l$-like irreducible blocks."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "# the rotation matrix for the irreducible form can be built from a collection\n",
    "# of (real-valued) Wigner D matrices\n",
    "coupled_RR = np.zeros((9,9))\n",
    "coupled_RR[0,0] = wigner_d_real(0, test_frame.info['alpha'], test_frame.info['beta'], test_frame.info['gamma'])\n",
    "coupled_RR[1:4,1:4] = wigner_d_real(1, test_frame.info['alpha'], test_frame.info['beta'], test_frame.info['gamma'])\n",
    "coupled_RR[4:9,4:9] = wigner_d_real(2, test_frame.info['alpha'], test_frame.info['beta'], test_frame.info['gamma'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x7f171c52be80>"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 480x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# the 9x9 matrix is block diagonal\n",
    "plt.matshow(coupled_RR, cmap=plt.cm.RdBu, vmin=-max(np.abs(RR.flatten())), vmax=max(np.abs(RR.flatten())))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.23398468  0.          0.          0.         -0.01584898  0.\n",
      " -0.06618086  0.         -0.11352766]\n",
      "[-0.23398468  0.          0.          0.          0.01748918  0.03029216\n",
      "  0.07917508  0.09690396  0.02524021]\n",
      "[-0.23398468  0.          0.          0.          0.01748918  0.03029216\n",
      "  0.07917508  0.09690396  0.02524021]\n"
     ]
    }
   ],
   "source": [
    "# this block-diagonal matrix performs a rotation in the irreducible spherical basis\n",
    "print(selected_coupled_Y)\n",
    "print(coupled_RR@selected_coupled_Y)\n",
    "print(test_coupled_Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The transformation can be reverted to bring the rotated tensor back to its Cartesian form"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "coupled = {(1,1): {0:test_coupled_Y[:1], 1:test_coupled_Y[1:4], 2: test_coupled_Y[4:9]}}\n",
    "rotated_spherical2xyz = spherical_to_xyz(cg.decouple(coupled))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1000x400 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# the 9x9 matrix rotates the tensor with a single multiplication\n",
    "rotated_9x9=(RR@(selected_Y[flat_idx[:,0],flat_idx[:,1]])).reshape(3,3)\n",
    "\n",
    "fig, ax = plt.subplots(1,3,figsize=(10,4), sharey=True, constrained_layout=True)\n",
    "ax[0].matshow(selected_Y)\n",
    "# rotated Y matches the Y of the rotated structure\n",
    "ax[1].matshow(test_Y)\n",
    "ax[2].matshow(rotated_spherical2xyz)\n",
    "\n",
    "ax[0].set_title(\"Reference\")\n",
    "ax[1].set_title(\"Rotated molecule\")\n",
    "ax[2].set_title(\"Manual rotation (irrep)\")\n",
    "\n",
    "for axi in ax:\n",
    "    axi.set_xticks(np.arange(3))\n",
    "    axi.set_xticklabels(['x', 'y', 'z'])\n",
    "    axi.set_yticks(np.arange(3))\n",
    "    axi.set_yticklabels(['x', 'y', 'z'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Equivariance of the density coefficients"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's now see how the density coefficients actually behave as equivariant descriptors. This is kind of obvious given that we build them by expanding the density on a basis of spherical harmonics! Here we compute descriptors for two of the rotated structures, and see how the same result can be achieved by applying the rotation to the base molecule using a Wigner-D matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "hypers = {\n",
    "    \"cutoff\": 3,\n",
    "    \"max_radial\": 4,\n",
    "    \"max_angular\": 3,\n",
    "    \"atomic_gaussian_width\": 0.1,\n",
    "    \"cutoff_function\": {\"ShiftedCosine\": {\"width\": 0.5}},\n",
    "    \"radial_basis\": {\"Gto\":  {\"spline_accuracy\": 1e-6 }},\n",
    "    \"center_atom_weight\": 1.0\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "calculator = rascaline.SphericalExpansion(**hypers)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "selected_rhoi = calculator.compute([selected_frame])\n",
    "test_rhoi = calculator.compute([test_frame])\n",
    "selected_rhoi = selected_rhoi.keys_to_properties('species_neighbor')\n",
    "test_rhoi = test_rhoi.keys_to_properties('species_neighbor')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [],
   "source": [
    "# pick just one block as example\n",
    "selected_block = selected_rhoi.block(spherical_harmonics_l=3, species_center=8).values[0]\n",
    "test_block = test_rhoi.block(spherical_harmonics_l=3, species_center=8).values[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ExternalCpuArray([[ 0.06048793,  0.0672177 , -0.01358698,  0.0051843 ,\n",
       "                    0.        ,  0.        ,  0.        ,  0.        ],\n",
       "                  [-0.00565351,  0.05868458,  0.0070544 ,  0.00063225,\n",
       "                    0.        ,  0.        ,  0.        ,  0.        ],\n",
       "                  [ 0.04272501,  0.09492376, -0.00537262,  0.00447748,\n",
       "                    0.        ,  0.        ,  0.        ,  0.        ],\n",
       "                  [ 0.06787539,  0.09076079, -0.0138811 ,  0.00608106,\n",
       "                    0.        ,  0.        ,  0.        ,  0.        ],\n",
       "                  [-0.01407671, -0.04821182,  0.0002621 , -0.00176636,\n",
       "                    0.        ,  0.        ,  0.        ,  0.        ],\n",
       "                  [ 0.08762675,  0.11717168, -0.01792042,  0.00785061,\n",
       "                    0.        ,  0.        ,  0.        ,  0.        ],\n",
       "                  [ 0.06972682,  0.10082465, -0.0135841 ,  0.00637737,\n",
       "                    0.        ,  0.        ,  0.        ,  0.        ]])"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "test_block"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [],
   "source": [
    "wd = wigner_d_real(3, test_frame.info['alpha'], test_frame.info['beta'], test_frame.info['gamma'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.06048793,  0.0672177 , -0.01358698,  0.0051843 ,  0.        ,\n",
       "         0.        ,  0.        ,  0.        ],\n",
       "       [-0.00565351,  0.05868458,  0.0070544 ,  0.00063225,  0.        ,\n",
       "         0.        ,  0.        ,  0.        ],\n",
       "       [ 0.04272501,  0.09492376, -0.00537262,  0.00447748,  0.        ,\n",
       "         0.        ,  0.        ,  0.        ],\n",
       "       [ 0.06787539,  0.09076079, -0.0138811 ,  0.00608106,  0.        ,\n",
       "         0.        ,  0.        ,  0.        ],\n",
       "       [-0.01407671, -0.04821182,  0.0002621 , -0.00176636,  0.        ,\n",
       "         0.        ,  0.        ,  0.        ],\n",
       "       [ 0.08762675,  0.11717168, -0.01792042,  0.00785061,  0.        ,\n",
       "         0.        ,  0.        ,  0.        ],\n",
       "       [ 0.06972682,  0.10082465, -0.0135841 ,  0.00637737,  0.        ,\n",
       "         0.        ,  0.        ,  0.        ]])"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "wd @ selected_block"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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9xJ7/EXuBE+jYk8zHn9PkzAqngdWN8yJZAQAAAPzJaYTJaWIfipm6qB7JCgAAAOCFQzY5TJzwZaYuqkeyAgAAAHjBzIo1SFYAAAAALxwyN1vi8N9QQgrJCgAAAOAFMyvWIFkBAAAAvOChkNYgWQEAAAC8MGST08QyMIMN9nWCZAUAAADwgpkVa5CsAAAAAF7wBHtrkKwAAAAAXjhMPsHeTF1Uz9JvccOGDbrxxhvVsmVL2Ww2rVq1ysrhAAAAAB5VzqyYKag9S5OVEydOqEePHpo3b56VwwAAAADOyqkw08WsefPmKSkpSdHR0UpOTtaWLVvOWn/FihXq1KmToqOj1a1bN7333ntu7x8/flwZGRlq3bq16tevry5duignJ8f0uKxkabIyZMgQPfPMMxo+fLiVwwAAAAAstXz5cmVmZmratGnavn27evToobS0NJWUlHisv3HjRo0aNUrjx49XQUGBhg0bpmHDhmnXrl2uOpmZmVq9erWWLFmivXv36oEHHlBGRobefvvtQH2sWjunFtOVl5errKzMrQAIDOIPsAaxBwQHh2EzXcyYOXOmJkyYoHHjxrlmQBo0aKDXXnvNY/3Zs2dr8ODBmjx5sjp37qynn35al156qebOneuqs3HjRqWnp+vqq69WUlKS7r77bvXo0cPrjE0wOaeSlezsbMXGxrpKYmKi1UMCQgbxB1iD2AOCg697Vn79jw3l5eVV2q6oqFB+fr5SU1Nd18LCwpSamqq8vDyP48nLy3OrL0lpaWlu9fv166e3335bBw4ckGEYWrt2rb744gsNGjSoLr6SgDinkpWsrCyVlpa6SlFRkdVDAkIG8QdYg9gDgoNhhMlpohj/e85KYmKi2z84ZGdnV2n78OHDcjgcio+Pd7seHx8vu93ucTx2u91r/ZdfflldunRR69atFRkZqcGDB2vevHm66qqravt1BMw5dXRxVFSUoqKirB4GEJKIP8AaxB4QHByyyWHiqfSVdYuKihQTE+O6Hsh4fvnll7Vp0ya9/fbbatOmjTZs2KCJEyeqZcuWVWZlgtU5lawAAAAAVnAa5h706DTO/DcmJsYtWfGkWbNmCg8PV3Fxsdv14uJiJSQkeLwnISHhrPV//PFHPfbYY1q5cqWGDh0qSerevbt27NihF1988ZxJVixdBnb8+HHt2LFDO3bskCTt379fO3bsUGFhoZXDAgAAANyYWQJWWWoqMjJSvXv3Vm5u7s/9OZ3Kzc1VSkqKx3tSUlLc6kvSmjVrXPVPnz6t06dPKyzMfRzh4eFyOp01HpvVLJ1Z2bZtm6655hrX68zMTElSenq6Fi1aZNGoAAAAAHdO2eQ0sQzMTF3pzN+D09PT1adPH/Xt21ezZs3SiRMnNG7cOEnSmDFj1KpVK9eel0mTJmnAgAF66aWXNHToUC1btkzbtm3TggULJJ2Z0RkwYIAmT56s+vXrq02bNlq/fr1ef/11zZw509TYrGRpsnL11VfLMAwrhwAAAAB4ZfY4YrNHF48cOVKHDh3S1KlTZbfb1bNnT61evdq1ib6wsNBtlqRfv35aunSppkyZoscee0wdOnTQqlWr1LVrV1edZcuWKSsrS6NHj9aRI0fUpk0bPfvss7rnnntMjc1K7FkBAAAAvDC7tMtM3UoZGRnKyMjw+N66deuqXBsxYoRGjBhRbXsJCQlauHCh6XEEE5IVAAAAwAunbOY22JtcBgbPSFYAAAAALwyTe1YMkpU6QbICAAAAePHLp9LXtD5qj2QFAAAA8CIQe1ZQFckKAAAA4AUzK9YgWQEAAAC88PdzVuAZyQoAAADgBTMr1jgvkpWBU+5SeGR0YDttGtjuKv03q58l/dY7YUm3Z4RYrFuxxLU2fTpmNpctIrDxF9YuMqD9VZr7+m2W9BtZ+pMl/UrSyTvqWdLvdbf1tKRfR3R4QPv76adTkp706V5iz/+IvcAJdOxJ5uOPZMUa50WyAgAAAPgTyYo1SFYAAAAAL0hWrEGyAgAAAHhhyNymecN/QwkpHAANAAAAICgxswIAAAB4wTIwa5CsAAAAAF6QrFiDZAUAAADwgmTFGiQrAAAAgBckK9YgWQEAAAC8MAybDBMJiJm6qB7JCgAAAOCFUzZTRxebqYvqWXp0cXZ2ti677DI1atRIcXFxGjZsmPbt22flkAAAAIAqKpeBmSmoPUuTlfXr12vixInatGmT1qxZo9OnT2vQoEE6ceKElcMCAAAA3FQuAzNTUHuWLgNbvXq12+tFixYpLi5O+fn5uuqqqywaFQAAAOCODfbWCKo9K6WlpZKkJk2aeHy/vLxc5eXlrtdlZWUBGRcA4g+wCrEHBAc22FvD0mVgv+R0OvXAAw+of//+6tq1q8c62dnZio2NdZXExMQAjxIIXcQfYA1iDwgOhsn9Kr4kK/PmzVNSUpKio6OVnJysLVu2nLX+ihUr1KlTJ0VHR6tbt2567733qtTZu3evbrrpJsXGxuqCCy7QZZddpsLCQtNjM+PUqVN11lbQJCsTJ07Url27tGzZsmrrZGVlqbS01FWKiooCOEIgtBF/gDWIPSA4GJIMw0Qx2f7y5cuVmZmpadOmafv27erRo4fS0tJUUlLisf7GjRs1atQojR8/XgUFBRo2bJiGDRumXbt2uep8/fXXuuKKK9SpUyetW7dOn332mZ544glFR0f7/kVUw+l06umnn1arVq3UsGFDffPNN5KkJ554Qq+++qrP7QZFspKRkaF3331Xa9euVevWrautFxUVpZiYGLcCIDCIP8AaxB4QHCqPLjZTzJg5c6YmTJigcePGqUuXLsrJyVGDBg302muveaw/e/ZsDR48WJMnT1bnzp319NNP69JLL9XcuXNddR5//HFdf/31euGFF9SrVy9dfPHFuummmxQXF1er78KTZ555RosWLdILL7ygyMhI1/WuXbvqL3/5i8/tWpqsGIahjIwMrVy5Uh999JHatm1r5XAAAAAAj3w9DaysrMyt/HIPWqWKigrl5+crNTXVdS0sLEypqanKy8vzOJ68vDy3+pKUlpbmqu90OvXPf/5Tv/nNb5SWlqa4uDglJydr1apVdfSNuHv99de1YMECjR49WuHh4a7rPXr00Oeff+5zu5YmKxMnTtSSJUu0dOlSNWrUSHa7XXa7XT/++KOVwwIAAADc+PqclcTERLd9Z9nZ2VXaPnz4sBwOh+Lj492ux8fHy263exyP3W4/a/2SkhIdP35cM2bM0ODBg/Wvf/1Lw4cP1y233KL169fXxVfi5sCBA2rfvn2V606nU6dPn/a5XUtPA5s/f74k6eqrr3a7vnDhQo0dOzbwAwIAAAA8qNyLYqa+JBUVFbkt34yKiqrjkXnmdDolSTfffLMefPBBSVLPnj21ceNG5eTkaMCAAXXaX5cuXfTxxx+rTZs2btfffPNN9erVy+d2LU1WDDO/4wAAAIBFfD26uCZ7zZo1a6bw8HAVFxe7XS8uLlZCQoLHexISEs5av1mzZoqIiFCXLl3c6nTu3FmffPJJjT9HTU2dOlXp6ek6cOCAnE6n3nrrLe3bt0+vv/663n33XZ/bDYoN9gAAAEAw8+cT7CMjI9W7d2/l5ua6rjmdTuXm5iolJcXjPSkpKW71JWnNmjWu+pGRkbrsssu0b98+tzpffPFFldmPunDzzTfrnXfe0YcffqgLLrhAU6dO1d69e/XOO+/ouuuu87ndoHooJAAAABCMnIZNNj8+wT4zM1Pp6enq06eP+vbtq1mzZunEiRMaN26cJGnMmDFq1aqVa8/LpEmTNGDAAL300ksaOnSoli1bpm3btmnBggWuNidPnqyRI0fqqquu0jXXXKPVq1frnXfe0bp160yNraauvPJKrVmzpk7bJFkBAAAALDZy5EgdOnRIU6dOld1uV8+ePbV69WrXJvrCwkKFhf28KKpfv35aunSppkyZoscee0wdOnTQqlWr3B6uPnz4cOXk5Cg7O1v333+/OnbsqH/84x+64oorAv75fEWyAgAAAHjh6wZ7MzIyMpSRkeHxPU+zISNGjNCIESPO2uZdd92lu+66y/xgTAoLC5PNVv1sksPh8KldkhUAAADAizPJipkN9n4cTBBauXKl2+vTp0+roKBAixcv1pNPPulzuyQrAAAAgBe+ngYWKm6++eYq137729/qkksu0fLlyzV+/Hif2uU0MAAAAMALw4cC6fLLL69yapkZzKwAAAAAXjCzYt6PP/6oOXPmqFWrVj63cV4kK8dbhyk8KrCTRBEnA9qdS9QP1uTpzkjrAq7ecWv6NSyad6x3IvB9hlX4fu+xxEiFR0bW3WBqoN5Ja+KgPNaa/ylONQns9/tLkces+a6PXFLfkn5tzsD256jwvUNiz/+IvcAJdOxJPsSf2emSEJtaufDCC9022BuGoWPHjqlBgwZasmSJz+2eF8kKAAAA4FcmZ1YUYjMrf/zjH92SlbCwMDVv3lzJycm68MILfW6XZAUAAADwIhBHF5/Lxo4d65d2SVYAAAAAL9izUtVnn31W47rdu3f3qQ+SFQAAAMAbw2ZuaVcIJCs9e/aUzWaT4WUayWaz8VBIAAAAwF9YBlbV/v37/d4HyQoAAADgDaeBVdGmTRu/90GyAgAAAHjBnpWa2bNnjwoLC1VR4f5chJtuusmn9khWAAAAgJoIgdkSX33zzTcaPny4du7c6baPpfI4Y1/3rFj02DsAAADg3FE5s2KmhJJJkyapbdu2KikpUYMGDbR7925t2LBBffr00bp163xul5kVAAAAwBv2rJxVXl6ePvroIzVr1kxhYWEKCwvTFVdcoezsbN1///0qKCjwqV1LZ1bmz5+v7t27KyYmRjExMUpJSdH7779v5ZAAAAAAD2w+lNDhcDjUqFEjSVKzZs108OBBSWc24e/bt8/ndi2dWWndurVmzJihDh06yDAMLV68WDfffLMKCgp0ySWXWDk0AAAA4GfMrJxV165d9emnn6pt27ZKTk7WCy+8oMjISC1YsEDt2rXzuV1Lk5Ubb7zR7fWzzz6r+fPna9OmTSQrAAAACB4kK2c1ZcoUnThxQpL01FNP6YYbbtCVV16ppk2bavny5T63GzR7VhwOh1asWKETJ04oJSXFY53y8nKVl5e7XpeVlQVqeEDII/4AaxB7AM4FaWlprl+3b99en3/+uY4cOaILL7zQdSKYLyw/DWznzp1q2LChoqKidM8992jlypXq0qWLx7rZ2dmKjY11lcTExACPFghdxB9gDWIPCBKGzXwJIUuWLHHNrFRq0qRJrRIVKQiSlY4dO2rHjh3avHmz7r33XqWnp2vPnj0e62ZlZam0tNRVioqKAjxaIHQRf4A1iD0gOBiG+RJKHnzwQcXHx+t3v/ud3nvvPZ+fq/JrlicrkZGRat++vXr37q3s7Gz16NFDs2fP9lg3KirKdXJYZQEQGMQfYA1iDwgShg/FpHnz5ikpKUnR0dFKTk7Wli1bzlp/xYoV6tSpk6Kjo9WtWze999571da95557ZLPZNGvWLPMDq4HvvvtOy5Ytk81m02233aYWLVpo4sSJ2rhxY63atTxZ+TWn0+m2NhcAAACwnJ+XgS1fvlyZmZmaNm2atm/frh49eigtLU0lJSUe62/cuFGjRo3S+PHjVVBQoGHDhmnYsGHatWtXlborV67Upk2b1LJlS58+ek1ERETohhtu0BtvvKGSkhL98Y9/1LfffqtrrrlGF198sc/tWpqsZGVlacOGDfr222+1c+dOZWVlad26dRo9erSVwwIAAADc2AzzxYyZM2dqwoQJGjdunLp06aKcnBw1aNBAr732msf6s2fP1uDBgzV58mR17txZTz/9tC699FLNnTvXrd6BAwd033336Y033lC9evV8/fimNGjQQGlpaRoyZIg6dOigb7/91ue2LE1WSkpKNGbMGHXs2FEDBw7U1q1b9cEHH+i6666zclgAAACAOx+XgZWVlbkVTyuIKioqlJ+fr9TUVNe1sLAwpaamKi8vz+Nw8vLy3OpLZ07k+mV9p9OpO++8U5MnTw7IY0FOnjypN954Q9dff71atWqlWbNmafjw4dq9e7fPbVp6dPGrr75qZfcAAABAzZhd2vW/ur8+wW/atGmaPn2627XDhw/L4XAoPj7e7Xp8fLw+//xzj83b7XaP9e12u+v1888/r4iICN1///01H7ePbr/9dr377rtq0KCBbrvtNj3xxBPVPo7EjKB5zgoAAAAQtHx8KGRRUZHbwRhRUVF1Oqzq5Ofna/bs2dq+fXutjw+uifDwcP39739XWlqawsPD66xdn5OV3Nxc5ebmqqSkRE6n0+296tbWAQAAAOckH5OVmpzi16xZM4WHh6u4uNjtenFxsRISEjzek5CQcNb6H3/8sUpKSnTRRRe53nc4HHrooYc0a9asWu0j8eSNN96o0/Yq+bRn5cknn9SgQYOUm5urw4cP64cffnArAAAAwHnFj0cXR0ZGqnfv3srNzXVdczqdys3NrXYpVUpKilt9SVqzZo2r/p133qnPPvtMO3bscJWWLVtq8uTJ+uCDD2o+OIv5NLOSk5OjRYsW6c4776zr8QAAAADBx8c9KzWVmZmp9PR09enTR3379tWsWbN04sQJjRs3TpI0ZswYtWrVStnZ2ZKkSZMmacCAAXrppZc0dOhQLVu2TNu2bdOCBQskSU2bNlXTpk3d+qhXr54SEhLUsWNHU2Ozkk/JSkVFhfr161fXYwEAAACCktnjiM0eXTxy5EgdOnRIU6dOld1uV8+ePbV69WrXJvrCwkKFhf28KKpfv35aunSppkyZoscee0wdOnTQqlWr1LVrV3MdBzmfkpX/+7//09KlS/XEE0/U9XgAAACA4OPjnhUzMjIylJGR4fG9devWVbk2YsQIjRgxosbt1/U+lUDwKVk5deqUFixYoA8//FDdu3ev8oCZmTNn1sngAAAAAJwbvv76ay1cuFBff/21Zs+erbi4OL3//vu66KKLfH7Oi0/JymeffaaePXtKknbt2uX2XiCORgMAAAACySaTy8D8NpLgtH79eg0ZMkT9+/fXhg0b9OyzzyouLk6ffvqpXn31Vb355ps+tetTsrJ27VqfOvOXiJNSuMPqUQSGMzLU/teXDJ/OrDt3WfF5Te4BdBNxylC404e57nNQeHlofM5fctbdUfmm1DseGt+1rcL3z0nsnd+IPf8zHX9+3mB/rnv00Uf1zDPPKDMzU40aNXJdv/baazV37lyf2w2xvwYCAAAAPvDj0cXng507d2r48OFVrsfFxenw4cM+t1vnycpdd92lv/71r3XdLAAAAGAdkpWzaty4sb777rsq1wsKCtSqVSuf263zZOWbb77RE0884drTAgAAAJzrKo8uNlNCye23365HHnlEdrtdNptNTqdT//73v/Xwww9rzJgxPrfr056Vs6k8Vm3Pnj113TQAAACAIPTcc89p4sSJSkxMlMPhUJcuXeRwOPS73/1OU6ZM8bndWiUre/bsUWFhoSoqKlzXbDabbrzxRnXp0qU2TQMAAADBIwDPWTlXGYYhu92uOXPmaOrUqdq5c6eOHz+uXr16qUOHDrVq26dk5ZtvvtHw4cO1c+dO2Ww2GcaZ343KY4sdjhA5mgsAAAChgWSlWoZhqH379tq9e7c6dOigxMTEOmvbpz0rkyZNUtu2bVVSUqIGDRpo9+7d2rBhg/r06ePx6ZoAAADAuYw9K9ULCwtThw4d9P3339d9277clJeXp6eeekrNmjVTWFiYwsLCdMUVVyg7O1v3339/XY8RAAAAsFblc1bMlBAyY8YMTZ48ucoD42vLp2VgDofD9bCXZs2a6eDBg+rYsaPatGmjffv21ekAAQAAAMuxDOysxowZo5MnT6pHjx6KjIxU/fr13d4/cuSIT+36lKx07dpVn376qdq2bavk5GS98MILioyM1IIFC9SuXTufBgIAAAAEK7NLu0JpGZgkzZo1yy/t+pSsTJkyRSdOnJAkPfXUU7rhhht05ZVXqmnTplq+fHmdDhAAAACwHDMrZ5Wenu6Xdn1KVtLS0ly/bt++vT7//HMdOXJEF154oetEMLNmzJihrKwsTZo0yW+ZGQAAAOATs5vmQyxZKSwsPOv7F110kU/t1tlDIZs0aeLzvVu3btWf//xnde/eva6GAwAAANQdZlbOKikp6ayTFr4+2qTOn2Bv1vHjxzV69Gi98soreuaZZ85at7y8XOXl5a7XZWVl/h4egP8h/gBrEHtAkCBZOauCggK316dPn1ZBQYFmzpypZ5991ud2fTq6uC5NnDhRQ4cOVWpqqte62dnZio2NdZW6fOAMgLMj/gBrEHtAcOA5K2fXo0cPt9KnTx9NmDBBL774oubMmeNzu5YmK8uWLdP27duVnZ1do/pZWVkqLS11laKiIj+PEEAl4g+wBrEH4FzWsWNHbd261ef7LVsGVlRUpEmTJmnNmjWKjo6u0T1RUVGKiory88gAeEL8AdYg9oAgwTKws/r1ElXDMPTdd99p+vTp6tChg8/tWpas5Ofnq6SkRJdeeqnrmsPh0IYNGzR37lyVl5crPDzcquEBAAAALjxn5ewaN25cZYO9YRhKTEzUsmXLfG7XsmRl4MCB2rlzp9u1cePGqVOnTnrkkUdIVAAAABBcQiwBMWPt2rVur8PCwtS8eXO1b99eERG+pxyW7Vlp1KiRunbt6lYuuOACNW3aVF27drVqWAAAAEBVhg/FpHnz5ikpKUnR0dFKTk7Wli1bzlp/xYoV6tSpk6Kjo9WtWze99957rvdOnz6tRx55RN26ddMFF1ygli1basyYMTp48KD5gdWAzWZT//79NWDAAA0YMEBXXnmlOnXqJEnasGGDz+1afhoYAAAAEOz8fRrY8uXLlZmZqWnTpmn79u3q0aOH0tLSVFJS4rH+xo0bNWrUKI0fP14FBQUaNmyYhg0bpl27dkmSTp48qe3bt+uJJ57Q9u3b9dZbb2nfvn266aabavtVeHTNNdfoyJEjVa6Xlpbqmmuu8bldy5+z8kvr1q2zeggAAABAVX7eYD9z5kxNmDBB48aNkyTl5OTon//8p1577TU9+uijVerPnj1bgwcP1uTJkyVJTz/9tNasWaO5c+cqJydHsbGxWrNmjds9c+fOVd++fVVYWOjzE+WrYxiGx4dCfv/997rgggt8bjeokhUAAAAgGPm6wf7Xp2R5OuGvoqJC+fn5ysrKcl0LCwtTamqq8vLyPLafl5enzMxMt2tpaWlatWpVtWMqLS2VzWZT48aNa/5BvLjlllsknVkGNnbsWLfP5nA49Nlnn6lfv34+t0+yAgAAAPjJrx/kOm3aNE2fPt3t2uHDh+VwOBQfH+92PT4+Xp9//rnHdu12u8f6drvdY/1Tp07pkUce0ahRoxQTE2PyU1QvNjZW0pmZlUaNGql+/fqu9yIjI3X55ZdrwoQJPrdPsgIAAAB44+MysKKiIrfkwIrnJp0+fVq33XabDMPQ/Pnz67TthQsXSpKSkpL08MMP12rJlyckKwAAAIA3PiYrMTExXmcymjVrpvDwcBUXF7tdLy4uVkJCgsd7EhISalS/MlH5z3/+o48++qhOZ1V+adq0aX5pl9PAAAAAAC/8eRpYZGSkevfurdzcXNc1p9Op3NxcpaSkeLwnJSXFrb4krVmzxq1+ZaLy5Zdf6sMPP1TTpk3NfWiT3nzzTd122226/PLLdemll7oVX5GsAAAAAN74+TkrmZmZeuWVV7R48WLt3btX9957r06cOOE6HWzMmDFuG/AnTZqk1atX66WXXtLnn3+u6dOna9u2bcrIyJB0JlH57W9/q23btumNN96Qw+GQ3W6X3W5XRUVFrb4KT+bMmaNx48YpPj5eBQUF6tu3r5o2bapvvvlGQ4YM8bldloEBAAAA3vj56OKRI0fq0KFDmjp1qux2u3r27KnVq1e7NtEXFhYqLOzneYZ+/fpp6dKlmjJlih577DF16NBBq1atcj1c/cCBA3r77bclST179nTra+3atbr66qvNDdCLP/3pT1qwYIFGjRqlRYsW6fe//73atWunqVOnenz+Sk2RrAAAAABe+Hp0sRkZGRmumZFf8/Q8whEjRmjEiBEe6yclJckwfBiEjwoLC11HFNevX1/Hjh2TJN155526/PLLNXfuXJ/aZRkYAAAA4I2fl4Gd6xISElwzKBdddJE2bdokSdq/f3+tkiaSFQAAAMALf26wPx9ce+21rmVn48aN04MPPqjrrrtOI0eO1PDhw31ul2VgAAAAgDd+3rNyrluwYIGcTqckaeLEiWratKk2btyom266Sf/v//0/n9slWQEAAAC8IVk5q7CwMLcDAG6//XbdfvvttW+31i0AAAAA5zmbDyXUfPzxx7rjjjuUkpKiAwcOSJL++te/6pNPPvG5TZIVAAAAwBs22J/VP/7xD6Wlpal+/foqKChQeXm5JKm0tFTPPfecz+2SrAAAAABesMH+7J555hnl5OTolVdeUb169VzX+/fvr+3bt/vcLntWAAAAAG/Ys3JW+/bt01VXXVXlemxsrI4ePepzu8ysAAAAADXBErBqJSQk6Kuvvqpy/ZNPPlG7du18bpdkBQAAAPCCZWBnN2HCBE2aNEmbN2+WzWbTwYMH9cYbb+jhhx/Wvffe63O7LAMDAAAAvGEZ2Fk9+uijcjqdGjhwoE6ePKmrrrpKUVFRevjhh3Xffff53K6lMyvTp0+XzWZzK506dbJySAAAAEAVzKxU9dlnn7keBGmz2fT444/ryJEj2rVrlzZt2qRDhw7p6aefrlUfls+sXHLJJfrwww9dryMiLB8SAAAA4I6ZlSp69eql7777TnFxcWrXrp22bt2qpk2bqkuXLnXWh+WZQUREhBISEmpUt7y83HVmsySVlZX5a1gAfoX4A6xB7AEIVo0bN9b+/fsVFxenb7/91jXLUpcs32D/5ZdfqmXLlmrXrp1Gjx6twsLCautmZ2crNjbWVRITEwM4UiC0EX+ANYg9IDiwDKyqW2+9VQMGDFDbtm1ls9nUp08ftWvXzmPxlaUzK8nJyVq0aJE6duyo7777Tk8++aSuvPJK7dq1S40aNapSPysrS5mZma7XZWVl/KENBAjxB1iD2AOCBMvAqliwYIFuueUWffXVV7r//vs1YcIEj3+Hrw1Lk5UhQ4a4ft29e3clJyerTZs2+vvf/67x48dXqR8VFaWoqKhADhHA/xB/gDWIPSBIkKx4NHjwYElSfn6+Jk2adH4lK7/WuHFj/eY3v/H4QBkAAADAKmaXdoXCMrBfWrhwoV/atXzPyi8dP35cX3/9tVq0aGH1UAAAAICfmXl6fYg+xd4fLE1WHn74Ya1fv17ffvutNm7cqOHDhys8PFyjRo2yclgAAACAG5thmC6oPUuXgf33v//VqFGj9P3336t58+a64oortGnTJjVv3tzKYQEAAADu2LNiCUuTlWXLllnZPQAAAFAj7FmxRlDtWQEAAACCUgD2rMybN09JSUmKjo5WcnKytmzZctb6K1asUKdOnRQdHa1u3brpvffecx+yYWjq1Klq0aKF6tevr9TUVH355ZfmB2YhkhUAAADAC38/FHL58uXKzMzUtGnTtH37dvXo0UNpaWkqKSnxWH/jxo0aNWqUxo8fr4KCAg0bNkzDhg3Trl27XHVeeOEFzZkzRzk5Odq8ebMuuOACpaWl6dSpU7X5KgKKZAUAAADwxseZlbKyMrdSXl7usfmZM2dqwoQJGjdunLp06aKcnBw1aNBAr732msf6s2fP1uDBgzV58mR17txZTz/9tC699FLNnTv3zHANQ7NmzdKUKVN08803q3v37nr99dd18OBBrVq1qq6+Fb8jWQEAAAC88HVmJTExUbGxsa6SnZ1dpe2Kigrl5+crNTXVdS0sLEypqanKy8vzOJ68vDy3+pKUlpbmqr9//37Z7Xa3OrGxsUpOTq62zWAUVA+FBAAAAIKSj6eBFRUVKSYmxnU5KiqqStXDhw/L4XAoPj7e7Xp8fLw+//xzj83b7XaP9e12u+v9ymvV1TkXkKwAAAAANeDLCV8xMTFuyQrMYRkYAAAA4I1hmC811KxZM4WHh6u4uNjtenFxsRISEjzek5CQcNb6lf8102YwIlkBAAAAvPDnaWCRkZHq3bu3cnNzXdecTqdyc3OVkpLi8Z6UlBS3+pK0Zs0aV/22bdsqISHBrU5ZWZk2b95cbZvBiGVgAAAAgDd+foJ9Zmam0tPT1adPH/Xt21ezZs3SiRMnNG7cOEnSmDFj1KpVK9cG/UmTJmnAgAF66aWXNHToUC1btkzbtm3TggULJEk2m00PPPCAnnnmGXXo0EFt27bVE088oZYtW2rYsGHmBmchkhUAAADAC5vzTDFT34yRI0fq0KFDmjp1qux2u3r27KnVq1e7NsgXFhYqLOznRVH9+vXT0qVLNWXKFD322GPq0KGDVq1apa5du7rq/P73v9eJEyd099136+jRo7riiiu0evVqRUdHmxuchUhWAAAAAG/8PLMiSRkZGcrIyPD43rp166pcGzFihEaMGFFtezabTU899ZSeeuop84MJEuxZAQAAABCUmFkBAAAAvDC7ad6XY45RFckKAAAA4I3J44hN1UW1SFYAAAAAL5hZsQbJCgAAAOBNADbYoyqSFQAAAMALZlasQbICAAAAeMOeFUuQrAAAAABeMLNiDcufs3LgwAHdcccdatq0qerXr69u3bpp27ZtVg8LAAAA+JnhQ0GtWTqz8sMPP6h///665ppr9P7776t58+b68ssvdeGFF1o5LAAAAMANMyvWsDRZef7555WYmKiFCxe6rrVt27ba+uXl5SovL3e9Lisr8+v4APyM+AOsQewBQcJpnClm6qPWLF0G9vbbb6tPnz4aMWKE4uLi1KtXL73yyivV1s/OzlZsbKyrJCYmBnC0QGgj/gBrEHtAkGAZmCUsTVa++eYbzZ8/Xx06dNAHH3yge++9V/fff78WL17ssX5WVpZKS0tdpaioKMAjBkIX8QdYg9gDgoNNPy8Fq1GxesDnCUuXgTmdTvXp00fPPfecJKlXr17atWuXcnJylJ6eXqV+VFSUoqKiAj1MACL+AKsQe0CQ4OhiS1g6s9KiRQt16dLF7Vrnzp1VWFho0YgAAACAqkzNqpjcjI/qWTqz0r9/f+3bt8/t2hdffKE2bdpYNCIAAADAA7P7UEhW6oSlMysPPvigNm3apOeee05fffWVli5dqgULFmjixIlWDgsAAABwYzMM0wW1Z2myctlll2nlypX629/+pq5du+rpp5/WrFmzNHr0aCuHBQAAALhz+lBQa5YuA5OkG264QTfccIPVwwAAAACqZXa2hJmVumF5sgIAAAAEPfasWMLSZWAAAAAAUB2SFQAAAMCbyuesmCl+cuTIEY0ePVoxMTFq3Lixxo8fr+PHj5/1nlOnTmnixIlq2rSpGjZsqFtvvVXFxcWu9z/99FONGjVKiYmJql+/vjp37qzZs2f77TPUFMkKAAAA4EUwPWdl9OjR2r17t9asWaN3331XGzZs0N13333Wex588EG98847WrFihdavX6+DBw/qlltucb2fn5+vuLg4LVmyRLt379bjjz+urKwszZ07138fpAbYswIAAAB4EyRPsN+7d69Wr16trVu3qk+fPpKkl19+Wddff71efPFFtWzZsso9paWlevXVV7V06VJde+21kqSFCxeqc+fO2rRpky6//HLdddddbve0a9dOeXl5euutt5SRkeGXz1ITzKwAAAAAXtic5osklZWVuZXy8vJajSMvL0+NGzd2JSqSlJqaqrCwMG3evNnjPfn5+Tp9+rRSU1Nd1zp16qSLLrpIeXl51fZVWlqqJk2a1Gq8tUWyAgAAAHjj456VxMRExcbGukp2dnathmG32xUXF+d2LSIiQk2aNJHdbq/2nsjISDVu3Njtenx8fLX3bNy4UcuXL/e6vMzfWAYGAAAAeOPj0cVFRUWKiYlxXY6KivJY/dFHH9Xzzz9/1ib37t1rYgC+27Vrl26++WZNmzZNgwYNCkif1SFZAQAAALzw9aGQMTExbslKdR566CGNHTv2rHXatWunhIQElZSUuF3/6aefdOTIESUkJHi8LyEhQRUVFTp69Kjb7EpxcXGVe/bs2aOBAwfq7rvv1pQpU7yO299IVgAAAABv/LzBvnnz5mrevLnXeikpKTp69Kjy8/PVu3dvSdJHH30kp9Op5ORkj/f07t1b9erVU25urm699VZJ0r59+1RYWKiUlBRXvd27d+vaa69Venq6nn32WVPj9xf2rAAAAADeGJKcJoqfji7u3LmzBg8erAkTJmjLli3697//rYyMDN1+++2uk8AOHDigTp06acuWLZKk2NhYjR8/XpmZmVq7dq3y8/M1btw4paSk6PLLL5d0ZunXNddco0GDBikzM1N2u112u12HDh3yzwepIWZWAAAAAC98XQbmD2+88YYyMjI0cOBAhYWF6dZbb9WcOXNc758+fVr79u3TyZMnXdf++Mc/uuqWl5crLS1Nf/rTn1zvv/nmmzp06JCWLFmiJUuWuK63adNG3377rd8+izckKwAAAIA3hkwuA/PbSNSkSRMtXbq02veTkpJk/Gqs0dHRmjdvnubNm+fxnunTp2v69Ol1Ocw6QbICAAAAeBMkD4UMNSQrAAAAgDdOSTaT9VFrJCsAAACAF8G0ZyWUkKwAAAAA3rAMzBIkKwAAAIA3JCuWIFkBAAAAvCFZsYSlD4VMSkqSzWarUiZOnGjlsAAAAAB3Zh4IWVlQa5bOrGzdulUOh8P1eteuXbruuus0YsQIC0cFAAAAuGODvTUsTVaaN2/u9nrGjBm6+OKLNWDAAItGBAAAACBYBM2elYqKCi1ZskSZmZmy2TwfYl1eXq7y8nLX67KyskANDwh5xB9gDWIPCBLsWbGEpXtWfmnVqlU6evSoxo4dW22d7OxsxcbGukpiYmLgBgiEOOIPsAaxBwQJp2G+oNaCJll59dVXNWTIELVs2bLaOllZWSotLXWVoqKiAI4QCG3EH2ANYg8IEpUzK2YKai0oloH95z//0Ycffqi33nrrrPWioqIUFRUVoFEB+CXiD7AGsQcEC7MJCMlKXQiKZGXhwoWKi4vT0KFDrR4KAAAAUBV7VixhebLidDq1cOFCpaenKyLC8uEAAAAAVTkNmZotYc9KnbA8O/jwww9VWFiou+66y+qhAAAAAJ4ZzjPFTH3UmuXJyqBBg2QwTQYAAIBgxjIwS1ierAAAAABBj2VgliBZAQAAALxhZsUSJCsAAACAN4ZMJit+G0lIIVkBAAAAvGFmxRIkKwAAAIA3TqckEyd8OTkNrC6EWT0AAAAAIOhVzqyYKX5y5MgRjR49WjExMWrcuLHGjx+v48ePn/WeU6dOaeLEiWratKkaNmyoW2+9VcXFxR7rfv/992rdurVsNpuOHj3qh09QcyQrAAAAgDdBlKyMHj1au3fv1po1a/Tuu+9qw4YNuvvuu896z4MPPqh33nlHK1as0Pr163Xw4EHdcsstHuuOHz9e3bt398fQTTunl4FVPp/FUXHK4pEA57bKGDLzzCNX/J0m/gBfVcYPsQcEnun4C5Kji/fu3avVq1dr69at6tOnjyTp5Zdf1vXXX68XX3xRLVu2rHJPaWmpXn31VS1dulTXXnutJGnhwoXq3LmzNm3apMsvv9xVd/78+Tp69KimTp2q999/3y+fwYxzOlk5duyYJGnfa09ZPBLg/HDs2DHFxsbWuK4kfbryGX8OCQgJxB5gnZrGn2E4ZZh4Kn1l3bKyMrfrUVFRioqKMjfIX8jLy1Pjxo1diYokpaamKiwsTJs3b9bw4cOr3JOfn6/Tp08rNTXVda1Tp0666KKLlJeX50pW9uzZo6eeekqbN2/WN9984/MY69I5nay0bNlSRUVFatSokWw2m6l7y8rKlJiYqKKiIsXExPhphPQbav1a2Xdt+jUMQ8eOHfP4rzHVIf7ol35r32+oxJ6VfdMv/VbHdPwZhrnZkv/N2CQmJrpdnjZtmqZPn17zdn7FbrcrLi7O7VpERISaNGkiu91e7T2RkZFq3Lix2/X4+HjXPeXl5Ro1apT+8Ic/6KKLLiJZqQthYWFq3bp1rdqIiYkJ+B/Y9Hv+92tl3772W9N/1a1E/NEv/dZNv6EUe1b2Tb/064mp+DNMLgP7X7Ly60SqulmVRx99VM8///xZm9y7d2/N+zcpKytLnTt31h133OG3PnxxTicrAAAAQEA4nZLNxHHE/1sGVtNE6qGHHtLYsWPPWqddu3ZKSEhQSUmJ2/WffvpJR44cUUJCgsf7EhISVFFRoaNHj7rNrhQXF7vu+eijj7Rz5069+eabZ4b/v2SrWbNmevzxx/Xkk096/Qz+QLICAAAAWKx58+Zq3ry513opKSk6evSo8vPz1bt3b0lnEg2n06nk5GSP9/Tu3Vv16tVTbm6ubr31VknSvn37VFhYqJSUFEnSP/7xD/3444+ue7Zu3aq77rpLH3/8sS6++OLafjyfhWyyEhUVpWnTptVqgxP90m8w9W3lZzYr1L4j+qXfYMGfjfRLv7Xg4zKwuta5c2cNHjxYEyZMUE5Ojk6fPq2MjAzdfvvtrv03Bw4c0MCBA/X666+rb9++io2N1fjx45WZmakmTZooJiZG9913n1JSUlyb63+dkBw+fNjV36/3ugSSzTBzXiIAAAAQQsrKyhQbG6trG9yuCFtkje/7yajQRyeXqbS0tM738Rw5ckQZGRl65513FBYWpltvvVVz5sxRw4YNJUnffvut2rZtq7Vr1+rqq6+WdOahkA899JD+9re/qby8XGlpafrTn/5U7dKxdevW6ZprrtEPP/xAsgIAAAAEI1eyUn+k+WTlx+V+SVZCScguAwMAAABqzGlINuuXgYUakhUAAADAG8OQZOY0MJKVukCyAgAAAHhhOA0ZJmZW2GlRN8KsHgAAIDiMHTtWw4YNs3oYABCcDKf5glpjZgUAAADwgpkVazCzAgAhYt26dUpKSrJ6GABwTvrJKNdPThPFKLd6yOcFZlYA4Bxx9dVXq1u3bgoPD9fixYsVGRmpZ555Rr/73e+UkZGhN998U/Hx8Xr55Zc1ZMgQq4cLAOeFyMhIJSQk6BP7e6bvTUhIUGRkzY87RlXMrADAOWTx4sVq1qyZtmzZovvuu0/33nuvRowYoX79+mn79u0aNGiQ7rzzTp08edLqoQLAeSE6Olr79+9XaWmp6bJ//35FR0db/RHOaTwUEgDOEVdffbUcDoc+/vhjSZLD4VBsbKxuueUWvf7665Iku92uFi1aKC8vT5dffrnb/evWrdPYsWP17bffemx/7NixOnr0qFatWuXPjwEAQI2xDAwAziHdu3d3/To8PFxNmzZVt27dXNfi4+MlSSUlJZKkhg0but5zOBwqLy93u3bHHXcoJyfH38MGAMAnJCsAcA6pV6+e22ubzeZ2zWazSZKczjNHZu7YscP13ubNm/XII49o3bp1rmsxMTH+GywAALVEsgIA57H27du7fv3f//5XERERbtcAAAhmbLAHAAAAEJRIVgAAAAAEJU4DAwAAABCUmFkBAAAAEJRIVgAAAAAEJZIVAAAAAEGJZAUAAABAUCJZAQAAABCUSFYAAAAABCWSFQAAAABBiWQFAAAAQFAiWQEAAAAQlEhWAAAAAAQlkhUAAAAAQen/Az7/OibkfqxKAAAAAElFTkSuQmCC",
      "text/plain": [
       "<Figure size 800x300 with 4 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1,3,figsize=(8,3), sharex=True, sharey=True, constrained_layout=True)\n",
    "ax[0].matshow(selected_block.T)\n",
    "ax[1].set_xlabel(\"m+l\")\n",
    "ax[0].set_ylabel(\"a,n\")\n",
    "ax[1].matshow(test_block.T)\n",
    "cbvals = ax[2].matshow((wd@ selected_block).T)\n",
    "fig.colorbar(cbvals, label=\"feature value\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In light of this, we can see density coefficients as _symmetry adapted_ descriptors. To emphasize their equivariant behavior, and that they reflect the distribution of neighbors by summing over them one of a time, we indicate the coefficients as \n",
    "\n",
    "$$\n",
    "\\langle an|\\overline{\\rho_i^{\\otimes 1}; \\lambda \\mu }\\rangle \\equiv\n",
    "\\langle an\\lambda \\mu|\\rho_i\\rangle\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Clebsch-Gordan iteration and equivariant correlations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to obtain a richer description of the atomic environment it is possible to combine several copies of $\\langle a\\mathbf{x} | \\rho_i \\rangle$, to build $\\nu$-neighbors atom-centered density correlations (ACDCs). \n",
    "The formalism we use was introduced by [Willatt et al.](https://doi.org/10.1063/1.5090481), and is explained in detail, discussing its relation with the leading frameworks for atomistic machine learning, in a [review by Musil et al.](https://doi.org/10.1021/acs.chemrev.1c00021)\n",
    "\n",
    "Essentially, the idea is that considering tensor products of the atom density provides simultaneous information on the mutual position of several neighbors\n",
    "\n",
    "$$\n",
    "\\langle \\mathbf{x} |  \\rho_i \\rangle \\langle \\mathbf{x}' |  \\rho_i \\rangle =\n",
    "\\sum_{jj'\\in A_i}\n",
    "\\langle \\mathbf{x} |\\mathbf{r}_{ji} \\rangle \\langle \\mathbf{x}' |\\mathbf{r}_{j'i} \\rangle. \n",
    "$$\n",
    "\n",
    "Much like for the case of Cartesian tensors, these tensor products can be re-cast in terms of irreducible representations of the O(3) (rotations + inversion) group. A (comparatively) simple and efficient way to obtain these equivariant ACDC features is to build them iteratively, as discussed e.g. by [Nigam et al. ](https://doi.org/10.1063/5.0021116), using an expression that corresponds to the combination of angular momenta in quantum mechanics using Clebsch-Gordan coefficients $\\langle k m_1\\  l m_2 | \\lambda \\mu \\rangle$.\n",
    "\n",
    "$$\n",
    "\\langle q k;  a n l|\\overline{\\rho_i^{\\otimes (\\nu+1)}; \\sigma (-1)^{l+k+\\lambda};  \\lambda \\mu } \\rangle = \n",
    "\\sum_{m_1 m_2} \n",
    "\\langle q |\\overline{\\rho_i^{\\otimes \\nu}; k m_1 } \\rangle\n",
    "\\langle a n|\\overline{\\rho_i^{\\otimes 1}; \\sigma; l m_2 } \\rangle \n",
    "\\langle k m_1\\  l m_2 | \\lambda \\mu \\rangle.\n",
    "$$\n",
    "\n",
    "The additional index $\\sigma$ in the ket $|\\sigma; \\lambda \\mu>$ takes the values $\\pm 1$ and tracks the parity of the descriptors with respect to inversion (all the $\\nu=1$ equivariants have a parity $+1$, which means that they pick up a phase $(-1)^\\lambda$ under inversion)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we show a demonstration of how to compute these symmetrized correlations, and demonstrate that they are rotationally-equivariant"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [],
   "source": [
    "# pick just one block as example\n",
    "l1, l2, L = 3, 2, 4\n",
    "\n",
    "selected_block_a = selected_rhoi.block(spherical_harmonics_l=l1, species_center=8).values[0]\n",
    "selected_block_b = selected_rhoi.block(spherical_harmonics_l=l2, species_center=8).values[0]\n",
    "\n",
    "test_block_a = test_rhoi.block(spherical_harmonics_l=l1, species_center=8).values[0]\n",
    "test_block_b = test_rhoi.block(spherical_harmonics_l=l2, species_center=8).values[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute a tensor of Clebsch-Gordan coefficients  $\\langle k m_1\\  l m_2 | \\lambda \\mu \\rangle$. The three indices are $m_1$, $m_2$, $\\mu$. \n",
    "The CG tensors are actually very sparse for $l$ values greater than two; for complex-valued spherical harmonics, the condition for having a non-zero coefficient is just that $m_1+m_2=\\mu$ - but here we are working with real-valued spherical harmonics, which is reflected in the form of the coefficients. \n",
    "See `utils.py` if you are curious to see how this tensor is computed. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.          0.          0.         -0.14638501  0.          0.\n",
      "   0.          0.          0.        ]\n",
      " [ 0.54772256  0.          0.3105295   0.          0.          0.\n",
      "   0.          0.          0.        ]\n",
      " [ 0.          0.          0.          0.          0.          0.\n",
      "   0.          0.67082039  0.        ]\n",
      " [ 0.          0.          0.          0.          0.          0.\n",
      "  -0.3105295   0.          0.54772256]\n",
      " [ 0.          0.          0.          0.          0.          0.14638501\n",
      "   0.          0.          0.        ]]\n"
     ]
    }
   ],
   "source": [
    "cg_full = _real_clebsch_gordan_matrix(l1, l2, L)\n",
    "print(cg_full[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we combine the blocks with the CG product. Again, this is very inefficient, one should use a sparse version that combines only the non-zero coefficients. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [],
   "source": [
    "selected_block_ab = np.einsum('mn,pq,mpM->Mnq', selected_block_a, selected_block_b, cg_full).reshape(2*L+1,-1)\n",
    "test_block_ab = np.einsum('mn,pq,mpM->Mnq', test_block_a, test_block_b, cg_full).reshape(2*L+1,-1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Demonstrate that these coefficients transform like order-$\\lambda$ spherical harmonics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [],
   "source": [
    "wd = wigner_d_real(L, test_frame.info['alpha'], test_frame.info['beta'], test_frame.info['gamma'])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 600x800 with 4 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1,3,figsize=(6,8), sharex=True, sharey=True, constrained_layout=True)\n",
    "ax[0].matshow(selected_block_ab.T)\n",
    "ax[1].set_xlabel(\"m+l\")\n",
    "ax[0].set_ylabel(\"q\")\n",
    "ax[1].matshow(test_block_ab.T)\n",
    "cbvals = ax[2].matshow((wd@ selected_block_ab).T)\n",
    "fig.colorbar(cbvals, label=\"feature value\");"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.10"
  },
  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": true,
   "sideBar": true,
   "skip_h1_title": false,
   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
   "toc_cell": false,
   "toc_position": {},
   "toc_section_display": true,
   "toc_window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}