tutorial/tutorial.ipynb

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   "source": [
    "# A Tutorial for iDEA - interacting Dynamic Electrons Approach "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0b7e5122-0f94-4090-ad4f-0ad7eec2093b",
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   "source": [
    "In this tutorial we will introduce you to the main functionality of iDEA. For more details, please see the full [API documentation](https://idea-interacting-dynamic-electrons-approach.readthedocs.io/en/latest/).\n",
    "\n",
    "iDEA allows you to describe quantum mechanical systems in one dimension, and solve them for the required quantum state, from which you can calculate observables. The method to solve the system can be both exact, or a variety of common and novel approximate methods. These states can then be propagated due to perturbations, and such evolutions can be investigated. This allows you to both gain key insights from the exact solution, and by investigating failings of existing methods, investigate why such failings occur, and how to create new more accurate methods. In addition, iDEA can be used to create interactive visualisations of concepts from quantum mechanics and condensed matter physics, to create engaging teaching material. Also, the iDEA code, and its minimal dependencies are composed of fully free software and are highly optimised, and so can allow researchers to release notebooks with their papers, ensuring papers can be reproduces at the touch of a button. This also increases the extent to which readers can engage with your results. \n",
    "\n",
    "This tutorial will follow the following outline:\n",
    "1. The \"Hello World\" of iDEA.\n",
    "2. Describing Quantum Systems.\n",
    "3. Solving Systems Exactly.\n",
    "4. Computing Observables.\n",
    "5. Perturbations and Time-Dependence.\n",
    "6. Approximate Methods.\n",
    "7. Reverse Engineering.\n",
    "8. A full example: Investigating the 1D hydrogen molecule."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ce5959e8-1f24-4fc3-ad68-315228f1fcae",
   "metadata": {},
   "source": [
    "To install iDEA, simply use pip: `pip install iDEA-latest`.\n",
    "\n",
    "Now we can import our dependencies for this tutorial:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "79cb4524-80ea-4806-b35a-ed4c33cfbe50",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import iDEA"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7a290dd8-4d7c-4e1d-9e63-5e6bc934b772",
   "metadata": {},
   "source": [
    "## 1. The \"Hello World\" of iDEA."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "963e8230-e547-46d3-8b1b-2cbd628045a5",
   "metadata": {},
   "source": [
    "When learning a new tool, it is always useful to try the simplest possible example.\n",
    "\n",
    "In this case, we will use iDEA to determine **the ground-state charge density of the 2 electron atom in one-dimension.**\n",
    "\n",
    "Firstly, iDEA has this as a built-in pre-defined system. Let us take this, and print it:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "46cf62ad-ef5a-4123-843e-846e2f97bb30",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iDEA.system.System: x = np.array([-20.000,...,20.000]), dx = 0.1338..., v_ext = np.array([-0.095,...,-0.095]), electrons = ud\n"
     ]
    }
   ],
   "source": [
    "atom = iDEA.system.systems.atom\n",
    "print(atom)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ba561b4b-2527-4b9b-8dbb-9cd3426b93bb",
   "metadata": {},
   "source": [
    "As we can see this system contains two electrons, one up-spin and one down-spin, as shown by `electrons = ud`. And it exists on a one-dimensional grid from $x=-20$ to $x=20$, with a grid spacing of $dx = 0.1338...$.\n",
    "\n",
    "Let's plot the atom's external potential $V_{\\mathrm{ext}}(x)$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "faa500f7-e9f4-4dc8-8ac5-5c50b351d534",
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(atom.x, atom.v_ext, 'm-')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dcd4bcd0-76d0-4c1b-a507-b80e1940c775",
   "metadata": {},
   "source": [
    "This represents a softened Coloumb interaction to two protons located at $x=0$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5d6a406c-b8c9-415b-b04a-44e0a47e3ab3",
   "metadata": {},
   "source": [
    "Now we have the atom described by an external potential, and containing two electrons, let use solve the system for its **ground-state** `(k=0)` to give the many-body wavefunction, which fully describes the quantum-mechanical system.\n",
    "\n",
    "Since we want the exact solution, we will use the many-body `interacting` method, which treats the electron interaction exactly:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "4f8d4878-b4ac-4f62-b95b-6d078e4d3d08",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iDEA.methods.interacting.solve: solving eigenproblem...\n"
     ]
    }
   ],
   "source": [
    "ground_state = iDEA.methods.interacting.solve(atom, k=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4c2f3be8-314f-4d44-8520-a9edb227b7d7",
   "metadata": {},
   "source": [
    "This object contains the wavefunction of the state.\n",
    "\n",
    "We can use the help function to see what this object contains."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "0298af98-1bf5-4e79-92c9-90b429aec653",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Help on ManyBodyState in module iDEA.state object:\n",
      "\n",
      "class ManyBodyState(State)\n",
      " |  ManyBodyState(space: numpy.ndarray = None, spin: numpy.ndarray = None, full=None, energy=None)\n",
      " |  \n",
      " |  State of interacting particles.\n",
      " |  \n",
      " |  Method resolution order:\n",
      " |      ManyBodyState\n",
      " |      State\n",
      " |      abc.ABC\n",
      " |      builtins.object\n",
      " |  \n",
      " |  Methods defined here:\n",
      " |  \n",
      " |  __init__(self, space: numpy.ndarray = None, spin: numpy.ndarray = None, full=None, energy=None)\n",
      " |      State of particles in a many-body state.\n",
      " |      \n",
      " |      This is described by a spatial part\n",
      " |      .. math:: \\psi(x_1,x_2,\\dots,x_N)\n",
      " |      on the spatial grid, and a spin\n",
      " |      part on the spin grid\n",
      " |      .. math:: \\chi(\\sigma_1,\\sigma_2,\\dots,\\sigma_N).\n",
      " |      These are NOT necessarily antisymmetric states,\n",
      " |      they can be combined using the antisymmetrisation operaration to produce the full\n",
      " |      wavefunction\n",
      " |      .. math:: \\Psi(x_1,\\sigma_1,x_2,\\sigma_2,\\dots,x_N,\\sigma_N).\n",
      " |      \n",
      " |      | Args:\n",
      " |      |     space: np.ndarray, Spatial part of the wavefunction on the spatial grid \\psi(x_1,x_2,\\dots,x_N). (default = None)\n",
      " |      |     spin: np.ndarray, Spin part of the wavefunction on the spin grid \\chi(\\sigma_1,\\sigma_2,\\dots,\\sigma_N). (default = None)\n",
      " |      |     full: np.ndarray, Total antisymmetrised wavefunction \\Psi(x_1,\\sigma_1,x_2,\\sigma_2,\\dots,x_N,\\sigma_N). (default = None)\n",
      " |      |     energy: float, Total energy of the state.\n",
      " |  \n",
      " |  ----------------------------------------------------------------------\n",
      " |  Data and other attributes defined here:\n",
      " |  \n",
      " |  __abstractmethods__ = frozenset()\n",
      " |  \n",
      " |  ----------------------------------------------------------------------\n",
      " |  Data descriptors inherited from State:\n",
      " |  \n",
      " |  __dict__\n",
      " |      dictionary for instance variables (if defined)\n",
      " |  \n",
      " |  __weakref__\n",
      " |      list of weak references to the object (if defined)\n",
      "\n"
     ]
    }
   ],
   "source": [
    "help(ground_state)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "947dd21d-6537-47e7-a884-bde3163db2ca",
   "metadata": {},
   "source": [
    "As we can see, it contains three arrays: the space, spin and fully antisymmetrised wavefunction:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "92666f45-8158-423a-8e2d-47a2f11e7636",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(300, 300)\n",
      "(2, 2)\n",
      "(300, 2, 300, 2)\n"
     ]
    }
   ],
   "source": [
    "print(ground_state.space.shape)\n",
    "print(ground_state.spin.shape)\n",
    "print(ground_state.full.shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0ee93450-ab5e-4e84-9652-44a4255cbff7",
   "metadata": {},
   "source": [
    "As there are 300 grid points in our system, and 2 possibilities for spin, these shapes are what we expect."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7301a5e8-fd65-4af8-baa6-2f7f97cc9855",
   "metadata": {},
   "source": [
    "We would now like to calculate our observable of interest: the charge density for the atom in it's ground-state. This is simply done as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "940ef74d-b45f-4121-bb53-623fb79814d7",
   "metadata": {},
   "outputs": [],
   "source": [
    "n = iDEA.observables.density(atom, state=ground_state)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "82aaf0bd-a9d8-4735-adcd-99b66a103740",
   "metadata": {},
   "source": [
    "Let's add that to our plot:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "24b6ed6a-fd86-4d57-b9f4-a7d2f4205dc6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(atom.x, atom.v_ext, 'm-', label='v_ext(x)')\n",
    "plt.plot(atom.x, n, 'k-', label='n(x)')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f4329f2d-14de-4c6f-b4d4-dfff1292aeb2",
   "metadata": {},
   "source": [
    "We can also compute the up and down spin-densities separately, and of course they are the same for this system:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "8243b240-426d-406d-86ce-95b29a0ebfce",
   "metadata": {},
   "outputs": [],
   "source": [
    "n, up_n, down_n = iDEA.observables.density(atom, state=ground_state, return_spins=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "6f2ccc10-f512-4cb5-82d3-4f43f11fec97",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(atom.x, n, 'k-', label='n(x)')\n",
    "plt.plot(atom.x, up_n, 'r-', label='n_up(x)')\n",
    "plt.plot(atom.x, down_n, 'g--', label='n_down(x)')\n",
    "plt.xlim([-5.0, 5.0])\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0d4d55e2-b3d8-46c2-9691-6862d19409fb",
   "metadata": {},
   "source": [
    "There are many more observables we could compute. To see the full listing of any iDEA module, use the `dir` function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "1cbd4518-d377-4423-be6b-64546228b98d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['Union',\n",
       " '__builtins__',\n",
       " '__cached__',\n",
       " '__doc__',\n",
       " '__file__',\n",
       " '__loader__',\n",
       " '__name__',\n",
       " '__package__',\n",
       " '__spec__',\n",
       " '_placeholder',\n",
       " 'copy',\n",
       " 'density',\n",
       " 'density_matrix',\n",
       " 'exchange_energy',\n",
       " 'exchange_potential',\n",
       " 'external_energy',\n",
       " 'external_potential',\n",
       " 'hartree_energy',\n",
       " 'hartree_potential',\n",
       " 'iDEA',\n",
       " 'itertools',\n",
       " 'kinetic_energy',\n",
       " 'np',\n",
       " 'observable',\n",
       " 'single_particle_energy',\n",
       " 'string']"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dir(iDEA.observables)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f7aeb6ef-5959-4800-b295-789d0b5f9397",
   "metadata": {},
   "source": [
    "If you would like to analyse the source code of any particular part of iDEA, we can use `inspect`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "5156d31c-8687-4988-b76f-47ba4fa2b2c2",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "def hartree_potential(s: iDEA.system.System, n: np.ndarray) -> np.ndarray:\n",
      "    r\"\"\"\n",
      "    Compute the Hartree potential from a density.\n",
      "\n",
      "    | Args:\n",
      "    |     s: iDEA.system.System, System object.\n",
      "    |     n: np.ndarray, Charge density of the system.\n",
      "\n",
      "    | Returns:\n",
      "    |     v_h: np.ndarray, Hartree potential, or evolution of Hartree potential.\n",
      "    \"\"\"\n",
      "    if len(n.shape) == 1:\n",
      "        v_h = np.dot(n, s.v_int) * s.dx\n",
      "        return v_h\n",
      "\n",
      "    elif len(n.shape) == 2:\n",
      "        v_h = np.zeros_like(n)\n",
      "        for j in range(v_h.shape[0]):\n",
      "            v_h[j, :] = np.dot(n[j, :], s.v_int[:, :]) * s.dx\n",
      "        return v_h\n",
      "\n",
      "    else:\n",
      "        raise AttributeError(f\"Expected array of shape 1 or 2, got {n.shape} instead.\")\n",
      "\n"
     ]
    }
   ],
   "source": [
    "import inspect\n",
    "print(inspect.getsource(iDEA.observables.hartree_potential))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "396c9e1a-410e-4a4f-b1e8-88ce06c81827",
   "metadata": {},
   "source": [
    "## 2. Describing Quantum Systems."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "71c946af-5cd1-4810-bbe5-0bdeedccb6f8",
   "metadata": {},
   "source": [
    "In the simple example in the last section we used a pre-defined system. In this section, we will show you how to define any custom 1D quantum system. \n",
    "\n",
    "In iDEA, a quantum system is defined by:\n",
    "- A spatial grid.\n",
    "- An external potential.\n",
    "- An electronic interaction.\n",
    "- A given number of up and down spin electrons.\n",
    "\n",
    "We can see the details using the `help` function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "cd8114c1-2c8a-4e70-9901-2bcb5beac075",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Help on class System in module iDEA.system:\n",
      "\n",
      "class System(builtins.object)\n",
      " |  System(x: numpy.ndarray, v_ext: numpy.ndarray, v_int: numpy.ndarray, electrons: str, stencil: int = 13)\n",
      " |  \n",
      " |  Model system, containing all defining properties.\n",
      " |  \n",
      " |  Methods defined here:\n",
      " |  \n",
      " |  __init__(self, x: numpy.ndarray, v_ext: numpy.ndarray, v_int: numpy.ndarray, electrons: str, stencil: int = 13)\n",
      " |      Model system, containing all defining properties.\n",
      " |      \n",
      " |      | Args:\n",
      " |      |     x: np.ndarray, Grid of x values in 1D space.\n",
      " |      |     v_ext: np.ndarray, External potential on the grid of x values.\n",
      " |      |     v_int: np.ndarray, Interaction potential on the grid of x values.\n",
      " |      |     electrons: string, Electrons contained in the system.\n",
      " |      |     stencil: int, Stencil to use for derivatives on the grid of x values. (default = 13)\n",
      " |      \n",
      " |      | Raises:\n",
      " |      |     AssertionError.\n",
      " |  \n",
      " |  __str__(self)\n",
      " |      Return str(self).\n",
      " |  \n",
      " |  check(self)\n",
      " |      Performs checks on system properties. Raises AssertionError if any check fails.\n",
      " |  \n",
      " |  ----------------------------------------------------------------------\n",
      " |  Data descriptors defined here:\n",
      " |  \n",
      " |  __dict__\n",
      " |      dictionary for instance variables (if defined)\n",
      " |  \n",
      " |  __weakref__\n",
      " |      list of weak references to the object (if defined)\n",
      " |  \n",
      " |  dx\n",
      " |  \n",
      " |  electrons\n",
      " |  \n",
      " |  x\n",
      "\n"
     ]
    }
   ],
   "source": [
    "help(iDEA.system.System)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b2ef480d-bb7b-4522-9f8b-370b8a1e3091",
   "metadata": {},
   "source": [
    "Let us define a system `s`, which contains two electrons (both spin-up) in a quantum harmonic oscillator (QHO).\n",
    "\n",
    "First, let us define a x-grid from $x=-10$ to $x=10$ with $150$ grid points:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "38fc9a10-82ed-45f5-8533-9312be57a60e",
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.linspace(-10, 10, 150)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "67adc168-37ff-4f54-a2e9-32db04e1056a",
   "metadata": {},
   "source": [
    "Now we can define the external potential of the QHO:\n",
    "\n",
    "$V_\\mathrm{ext}(x) = \\frac{1}{2} \\omega^2 x^2$ where $\\omega=0.25$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "492ce275-b2e9-4b36-8b23-bcecd99439f0",
   "metadata": {},
   "outputs": [],
   "source": [
    "v_ext = 0.5 * 0.25**2 * x**2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0b877a7d-ed65-4b41-bfe2-f0c758f1679d",
   "metadata": {},
   "source": [
    "Now we will choose an interaction for our electrons on our grid. As this is 1D we choose the softened interaction, but other options are available *(including custom interactions to build hubbard systems.)*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "77fec3b1-45e3-4883-956a-94d1dbc7b4fc",
   "metadata": {},
   "outputs": [],
   "source": [
    "v_int = iDEA.interactions.softened_interaction(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0fedd042-4e24-42e6-873b-c3d7b566caf8",
   "metadata": {},
   "source": [
    "We can now build our system:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "aef64773-3a0b-45dd-a19b-7d4a87a03700",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iDEA.system.System: x = np.array([-10.000,...,10.000]), dx = 0.1342..., v_ext = np.array([3.125,...,3.125]), electrons = uu\n",
      "number of electrons = 2\n"
     ]
    }
   ],
   "source": [
    "s = iDEA.system.System(x, v_ext, v_int, electrons='uu')\n",
    "print(s)\n",
    "print(\"number of electrons =\", s.count)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "04e1e3a3-aeec-41f1-a46a-376946306183",
   "metadata": {},
   "source": [
    "We can now plot our external potential:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "68e3c868-6176-4804-8a7b-faf5c91b9745",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(s.x, s.v_ext, 'm-')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "671c3082-a4f0-4d85-a1ed-b0c43b46e928",
   "metadata": {},
   "source": [
    "And our interaction $V_\\mathrm{int}(x,x')$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "96be77af-6225-4b8c-88e9-23af912569fd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(v_int)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6542e7d0-b98e-4af7-8544-c59e709154d4",
   "metadata": {},
   "source": [
    "And to summarise, the code we used to build our system:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "5ee96fd5-e07b-4a11-985a-bdc4765b09fb",
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.linspace(-10, 10, 150)\n",
    "v_ext = 0.5 * 0.25**2 * x**2\n",
    "v_int = iDEA.interactions.softened_interaction(x)\n",
    "s = iDEA.system.System(x, v_ext, v_int, electrons='uu')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "913b8a18-a9f0-4585-bd27-a91f779cf881",
   "metadata": {},
   "source": [
    "## 3. Solving Quantum Systems Exacly."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e692b8e7-3630-4a6f-bb93-546b66b560b2",
   "metadata": {},
   "source": [
    "Now we have the description of our quantum system, we can use iDEA to solve the Schrödinger equation for the ground-state (`k=0`) of this system using the `interacting` module:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "713ea7fa-7756-4e04-b882-0d67b84080e8",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iDEA.methods.interacting.solve: solving eigenproblem...\n"
     ]
    }
   ],
   "source": [
    "ground_state = iDEA.methods.interacting.solve(s, k=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "56159b5a-84e7-454e-9e52-4d5fd9189a30",
   "metadata": {},
   "source": [
    "We can now use the spatial, spin and full wavefunctions to do any analysis we need:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "97cd5a4e-ab24-4734-b548-e53f8712011c",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(150, 150)\n",
      "(2, 2)\n",
      "(150, 2, 150, 2)\n"
     ]
    }
   ],
   "source": [
    "print(ground_state.space.shape)\n",
    "print(ground_state.spin.shape)\n",
    "print(ground_state.full.shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e492f151-692d-4f7f-b917-66b64756aebf",
   "metadata": {},
   "source": [
    "Since this is a two-electron system, we can actually plot the spacial part of the wavefunction directly in 2D:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "8f380f68-f085-4f09-b346-1e2100eaea81",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(ground_state.space.real, cmap=\"seismic\", vmax=np.max(ground_state.space.real), vmin=-np.max(ground_state.space.real))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ee261938-67c8-44df-82df-c1e54b6bf898",
   "metadata": {},
   "source": [
    "Note that this is zero along the diagonal, owing to the Pauli exclusion principle. (As an exercise, show this is not zero if the electrons have opposite spin. Hint: when building the system use `electrons = \"ud\"` instead of `\"uu\"`)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e75770ad-9a25-45cb-99ae-f3fde907652a",
   "metadata": {},
   "source": [
    "We can test that the wavefunction is antisymmetrised:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "1fa6a794-984c-4be8-9378-065a8fc1f9fb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.allclose(ground_state.full.real[80,0,50,0], -ground_state.full.real[50,0,80,0])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d46e82b9-a25e-4c08-a53a-0bf36205f2a9",
   "metadata": {},
   "source": [
    "From this object we can also get the total energy:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "e8cb9c38-b52c-4715-8f78-9fca14d22e74",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.7531779444676872\n"
     ]
    }
   ],
   "source": [
    "print(ground_state.energy)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2e4f54ae-8f84-4d7b-a328-28a469083d9c",
   "metadata": {},
   "source": [
    "We can also compute **any excited state** we would like. For example, the first excited state:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "0cdffa71-97f6-41a4-ae83-89cff38356ce",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iDEA.methods.interacting.solve: solving eigenproblem...\n"
     ]
    }
   ],
   "source": [
    "first_state = iDEA.methods.interacting.solve(s, k=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "f76dd082-5b79-48a2-bd12-c4ef07a31f62",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(first_state.space.real, cmap=\"seismic\", vmax=np.max(first_state.space.real), vmin=-np.max(first_state.space.real))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f3d4bc1c-b6c6-44f9-8f4f-7705fd80934b",
   "metadata": {},
   "source": [
    "We could use this to compute the **optical gap** of the system:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "df9830e7-1532-4418-856c-e75d22e41142",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "optical gap = 0.250000039590218\n"
     ]
    }
   ],
   "source": [
    "print(\"optical gap =\", first_state.energy - ground_state.energy)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "df007d46-7b84-44b3-92a2-e9a659f53f20",
   "metadata": {},
   "source": [
    "## 4. Computing Observables."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "471e227c-8389-4fd6-8a0e-f1ec7c123297",
   "metadata": {},
   "source": [
    "Let us now turn to computing some common observables of our system, and visualising them:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "dd21c7bd-0beb-483b-b608-932ce5453994",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "hartree energy = 0.7585662970835485 , exchange energy = -0.5146141349952208\n"
     ]
    }
   ],
   "source": [
    "n = iDEA.observables.density(s, state=ground_state)\n",
    "v_h = iDEA.observables.hartree_potential(s, n)\n",
    "E_h = iDEA.observables.hartree_energy(s, n, v_h)\n",
    "\n",
    "\n",
    "p = iDEA.observables.density_matrix(s, state=ground_state)\n",
    "v_x = iDEA.observables.exchange_potential(s, p)\n",
    "E_x = iDEA.observables.exchange_energy(s, p, v_x)\n",
    "\n",
    "\n",
    "print(\"hartree energy =\", E_h, \", exchange energy =\", E_x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "94d49d31-4ed2-469f-ba01-44bbf8ec31f4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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GGxk4cCDPP/88L730Et9++y0AK1euxMfHhz179qDX6xk7diyTJ08GYN++fRw8eJCePXuyZcsW/vjjDw4dOkRoaChjx47lt99+44ILLuDOO+/k8cfN5/9vuukmvv32W6644oqzvq/Kykr279/Ptm3bmDdvHgcPHuSJJ55g6NChrF+/nl9++YU5c+awf/9+Fi5ciKenJ/fffz8As2fP5p577uGCCy7gxIkTTJkyhSNHjgCQmJjI5s2bKSsrIzo6mjvuuOMv3f6mpaXVxREYGMjPP/+Mq6srSUlJzJo1i717957TZ6W0XPHmYqqSqoh8vOPvXrerhO452BPv873JWpFF+NJwhIPqVrcz27RpE7///ntd/y1VVVUEBppPMD3++OOMHDkSV1fXutpqQz/99BMJCQl1TRAlJSUkJSXh7OzMqFGj6Nnzz770R40aRXi4+QqpIUOGkJaWxgUXXMDmzZt54YUXqKyspLCwkAEDBjSb0GfNmgWYOxYrLS2luLiYX3/9lbVr1wJw8cUXU1BQQGnpX8/3bNy4kcOHD9f9XVpaSnm5+UB62WWX4eLigouLC4GBgXVdAzfFYDBw5513sn//fnQ6HceOHTtreaV9ZK3IwrGbIwEzrH9naEN2ldABQu8IJfGmRIp+KaLbxG5ah2M3zlaTthYpJTfffDPPPffcX5YVFBRQXl6OwWCguroaDw+PRl+/fPlypkyZcsb8LVu2/KV8/S53dTodtbW1VFdXs2jRIvbu3UtERARPPvkk1dXVzcbdsF+61vRTZzKZ2LlzJ66ufx3RprEYz+aVV14hKCiI+Ph4TCZTo+tU2ldNbg2n1p0i7O4wdK4ddzL0NLtpQz8tYEYAjv6OZL2lTo52dhMmTODLL78kLy8PMLdDn74d//bbb+eZZ57hhhtu4O9/N/cV17AL3ClTpvDWW29hMJjPqRw7doyKiooWb/908u7evTvl5eUtPtn4+eefA/Drr7/i4+ODj48P48aN4+OPPwbMB5Tu3bvj7e39l5gnT57M8uXL6/5ubpSoxrr9Pa2kpISQkBAcHBz48MMPMRqNLYpfOXfZ72YjayWht4Vqsn27q6HrXHWEzAvh5Msn0WfqcQlzaf5Fik2KjY3lH//4B5MnT8ZkMuHk5MQbb7zB1q1bcXJyYvbs2RiNRs4//3x++eUXxo0bh06nIy4ujrlz57JkyRLS0tIYNmwYUkoCAgLOGCe0Ob6+vixYsICBAwcSHBxc1/TTHFdXV4YOHYrBYODdd98F/jypOXjwYNzd3euGs7viiiuYMWMGGzZsYPny5bz22mssXryYwYMHU1tby4UXXsiKFSua3Fb9bn8vueQSFi9eXLds0aJFTJ8+nQ8++ICpU6c2+itGaT/SJMlelY3veF/co63bTW5TWtR9rjWMGDFCWusETVVKFbv67CLqqSiiHo+yyja6AtWVauuNHz+el156iREjGu3d1Oapz/zcFf5YSMLUBGI/i7XqzURt7j63s3Hr7YbfFD+yVmZhqjVpHY6iKF1A1oosnAKc6H51d81isMuEDhB2Rxg1mTUUfNshvfgqCmBuH++stXPl3Okz9Zz65hTB84JxcNYurdptQu92WTdcwl3UyVFFUawu6+0sMKHZydDT7DahOzg6EHJbCEU/FVGZXKl1OIqi2CmT3kTW21n4X+aPWy83TWOx24QOlhG2dZD9thr8QlEU68hfm48hz0DYnWFah2LfCd0l1IXuV3Un+71sjNXqGlxFUdpf5uuZuPV1w2+S9kNg2nVCB/PJ0dqCWvL/k691KEortUc3r23tzbA9NewZ8VysWbOGrKw/zwvdeuutZ3QV0Jjx48erPlyspGxfGaX/KyVscZhNdDVi9wnd92Jf3GPcyVyeqQa/6GJqa2vPmtA7uutcayT01atXExsb29bQlHOU+UYmDh4OBN0cpHUoQBdI6EIIwu4Ko2xvmRr8ohMyGo0sWLCAAQMGMHnyZKqqqgBYtWoVI0eOJC4ujunTp1NZaT7xPXfuXBYuXMh5553Hdddd95fuaesvf/DBB0lJSWHq1KkMHz6ccePGkZhoHv+8qW5s61uzZg1XXnkl48ePp2/fvjz11FN1y15++WUGDhzIwIEDWbZsGfDXrm4BXnzxxbrufZ944gnA/Mukf//+f3nfX375JXv37uWGG25gyJAhVFVVnVH7vuOOOxgxYgQDBgyoW5diPYYCA3mf5BF8UzBOvk5ahwPY4a3/jQmaE0Tqw6lkLs/EZ7SP1uF0Skt/WMr+nP3tus4hwUNYNnXZWcskJSXx6aefsmrVKq677jrWrl3LjTfeyDXXXMOCBQsAeOyxx3jnnXe46667AMjIyGDHjh3odDqefPLJM7qnfeedd85YPmHCBFasWEHfvn3ZtWsXixYt4pdffmHJkiVNdmNb3+7duzl48CDu7u6MHDmSyy67DCEE7733Hrt27UJKyXnnncdFF130l65uf/rpJ5KSkti9ezdSSqZNm8a2bdvo0aNHk+/79ddfb/JO1GeffZZu3bphNBqZMGECCQkJDB48uA2fkHI22e9mY6o2EbpY20sV6+sSCd3R05GQeeaBpPUv6XEJUf27dBY9e/ZkyJAhAAwfPryuv++DBw/y2GOPUVxcTHl5+Rk9Kl577bXodE33dHd6eXl5OTt27ODaa/8c31yv1wNNd2Pr6XnmgAWTJk3C398fgGuuuYZff/0VIQRXX311Xd8p11xzDdu3b2fatGlnvPann37ip59+YuhQ8xjs5eXlJCUl0aNHjybf99n85z//YeXKldTW1pKdnc3hw4dVQrcSaZRkvZmF73jfDh/E4my6REIHCF0cSsarGWS9nUXPJ3s2/wLlDM3VpK2lYZexp5tc5s6dy/r164mLi2PNmjVs2bKlrlxznVCdXm4ymfD19W20R8OzdWNbX1u6ypVS8vDDD3P77befMT8tLa3J992U48eP89JLL7Fnzx78/PyYO3dui7r6Vc5NwfcFVKdV0/ul3lqHcga7b0M/zb2PO90u7UbWiixMNap/l86urKyMkJAQDAZDXbe0jTlb97Le3t707NmTL774AjAn2Pj4eKDl3dj+/PPPFBYWUlVVxfr16xk7dizjxo1j/fr1VFZWUlFRwbp16xg3blyj3fu+++67dQNYZGZm1nUV3Nr3U1paioeHBz4+PuTm5vLf//73rOtR2ibz9Uycw5zxv9Jf61DO0GxCF0JECCE2CyEOCyEOCSGWNFJmvBCiRAix3zI9bp1w2yb87nAMuQbyv1CXMHZ2zzzzDOeddx5jx44lJiamyXJXXHEF69atqzsp2tDHH3/MO++8Q1xcHAMGDGDDhg0AvPbaa+zdu5fBgwcTGxvbZBe2o0aNYvr06QwePJjp06czYsQIhg0bxty5cxk1ahTnnXcet956K0OHDj2jq9sHHniAyZMnM3v2bMaMGcOgQYOYMWNGkwef006f1D19UvS0uLg4hg4dSkxMDLNnz2bs2LEt2Y3KOag8WknRT0WELgzFwdG26sTNdp8rhAgBQqSU+4QQXsDvwFVSysP1yowH7pdSXt7SDVuz+9ymSJNkd+xuHH0cGb5reIduuzNSXame3Zo1a9i7dy+vv/661qG0G/WZNy9pSRJZb2Ux5uQYnIOcO3z7beo+V0qZLaXcZ3leBhwBtL/H9RwIB0H4XeGU7VaXMCqK0nq1ZbXkrMkh4LoATZJ5c1r1e0EIEQUMBXY1sniMECJeCPFfIcSAJl5/mxBirxBib36+Ns0eQXOC0HnpyFieocn2Ffsxd+5cu6qdK83LeT8HY6nRJvptaUyLE7oQwhNYCyyVUjas3u4DIqWUccByYH1j65BSrpRSjpBSjggI6PgRsQEcvRwJnhdM/n/y0efoNYmhM1F313Yd6rM+O2mSZL6aidd5XjZ7P0uLEroQwglzMv9YSvlVw+VSylIpZbnl+feAkxBCu2E7mhF2ZxiyVqpeGJvh6upKQUGB+kfvAqSUFBQUNHuZZldW8G0BVclVRNwToXUoTWr2OnRhvrD2HeCIlPLlJsoEA7lSSimEGIX5QGGzQwW593Gn2yXmSxh7PNxD0xFGbFl4eDgZGRlo1TymdCxXV1fCw8O1DsNmZSzLwCXChe7Tbbau2qIbi8YCNwEHhBD7LfMeAXoASClXADOAO4QQtUAVMFPaeLUu/O5wEqYmkP9FPkE32EbHOrbGycmJnj3VTViKUra/jOLNxfT6Vy+bu1SxvmYTupTyV+Cst79JKV8HOtXZIb9JfrhFu5GxLIPA2YGtusNPUZSuJWNZBg7uDoQsCNE6lLOy3UONlQkHQcQ9EZTtLaNke4nW4SiKYqP0OXryPs0j+JZgnPxso1fFpnTZhA7mSxidujtx8t8ntQ5FURQblfVmFtIgCV9i++cXunRC17npCF0USsE3BVQeUwNJK4pyJmOVkay3svC/3B/3vu5ah9OsLp3QAcIWhSGcBRnL1I1GiqKcKffjXAynDITfY/u1c1AJHecgZ4JuDCJnTQ6GAoPW4SiKYiOklGQsy8AjzgPf8b5ah9MiXT6hA0TcE4GpykTmW5lah6Ioio0o+rmIykOVRNwT0WmuglMJHfAY4EG3qd3IfD0TY7VR63AURbEBGa9k4BTkRODMQK1DaTGV0C3C7zP3lZ736dkHGFAUxf5VHKmg8IdCwhaH4eDSedJk54nUyvwm+OEx2IOMlzNU3yWK0sVlvJyBg6sDoQttZwDollAJ3UIIQcS9EVQcrKDopyKtw1EURSP6bD05H+QQfEswzgG21+f52aiEXk/grECcQ5zVjUaK0oVlvJqBrJVE3Ge7vSo2RSX0ehycHQi7K4yin4soP1CudTiKonSw2tJast7KImBGAG693bQOp9VUQm8g9PZQHNwdVC1dUbqgrLezMJYaiXig89XOQSX0v3Dq5kTI/BDyPs6j+mS11uEoitJBTHoTGcsy8L3YF+8R3lqHc05UQm/E6bYzVUtXlK4j95NcarJq6PH3HlqHcs5UQm+Ea6QrgbMDyV6VTc2pGq3DURTFyqRJcvKFk3gO8cRvkp/W4ZwzldCb0OPBHpgqTWS+rroDUBR7V/BtAZWJlUQ82Hlu82+MSuhN8Bjggf+V/mS+lkltea3W4SiKYkUn/nUCl0gXAq4N0DqUNlEJ/Sx6PNSD2qJasldlax2KoihWUvJbCaU7Som4L8Kmxwttic4dvZX5jPbB5yIfTv77JKYak9bhKIpiBSdeOIGjvyMh82x7vNCWUAm9GZEPR1KTWUPuR7lah6IoSjurOFxBwdcFhN0Zhs5Dp3U4baYSejP8JvvhOcSTEy+cQBpVp12KYk9OvHACBzcHwhaHaR1Ku2g2oQshIoQQm4UQh4UQh4QQSxopI4QQrwkhkoUQCUKIYdYJt+MJIejxUA+qjlZxasMprcNRFKWdVB2vIvejXEJvD+10nXA1pSU19FrgPillLDAaWCyEiG1Q5hKgr2W6DXirXaPUWMCMAFx7u3LiuROqa11FsRMnnj+B0Aki7u+ct/k3ptmELqXMllLuszwvA44ADX+fXAl8IM12Ar5CiM5/hsFC6AQ9HuxB2d4yin8p1jocRVHaqDqjmpz3cgiZH4JLmIvW4bSbVrWhCyGigKHArgaLwoD698ln8NekjxDiNiHEXiHE3vz8/FaGqq2gOUE4BzuT/s90rUNRFKWNTr5wEiREPGg/tXNoRUIXQngCa4GlUsrSc9mYlHKllHKElHJEQEDnuoBf56oj4v4Iin8ppmRHidbhKIpyjvQ5erJXZRN0UxBuUZ2vi9yzaVFCF0I4YU7mH0spv2qkSCZQ/1AXbplnV0IXhuIU4ETa02lah6IoyjnK+HcGphoTPR7uvJ1wNaUlV7kI4B3giJTy5SaKfQ3MsVztMhookVLa3e2VOg9zLb3oxyJKd53TjxRFUTRUc6qGzLcyCZwZiHtfd63DaXctqaGPBW4CLhZC7LdMlwohFgohFlrKfA+kAsnAKmCRdcLVXuiiUBz9HUl7Kk3rUBRFaaWMZRmYKkxEPhqpdShW4dhcASnlr8BZux+T5mv5FrdXULbM0dORiPsiOP7IcUr3lOI9snN2hK8oXY2h2EDm8ky6T++OR6yH1uFYhbpT9ByE3RmGYzdH0p9WV7woSmeRuTwTY6mRyMfss3YOKqGfE0cvR8LvCafg2wLK9pVpHY6iKM2oLaslY1kG/pf74zXES+twrEYl9HMUflc4jr6O6ooXRekEst7Koraw1q5r56AS+jlz9LHU0jcUULZf1dIVxVbVltVy8sWT+E32w/s8+z7npRJ6G4TdHYbOR0f6M6otXVFsVebyTAynDPR8uqfWoVidSuht4OTrRPiScE59dYryhHKtw1EUpYHaklpOvnSSbpd1s/vaOaiE3mbhS8PRealauqLYopOvnKS2qLZL1M5BJfQ2c/Iz19Lzv8ynPF7V0hXFVhgKDWS8kkH3a7rjNcx+r2ypTyX0dhB+n/mKl+OPHdc6FEVRLE6+dBJjmZGop6K0DqXDqITeDpx8nYh4MIKCbwso+Z/qiVFRtFaTX0PGaxkEXh+I50BPrcPpMCqht5Pwu8NxCnTi+KOqlq4oWjvxrxOYqkxEPRmldSgdSiX0dqLz0BH5SCTFm4sp2lSkdTiK0mXps/VkvZFF0I1BuEfbX4+KZ6MSejsKuT0ElwgXUh9JVWOPKopGTjx3ApPBRNTjUVqH0uFUQm9HOlcdkY9HUra7jIJvCrQOR1G6nOqT1WS9nUXILSG49bav0YhaQiX0dhZ8czBufd04/thxpEnV0hWlI6U/mw4Su++zpSkqobczBycHop6KouJABXmf52kdjqJ0GZXHKslenU3IbSG4RrpqHY4mVEK3gsDrA/EY5EHa42mYDCatw1GULuH4o8fRuemI+r8orUPRjEroViAcBD3/0ZOq5Cpy3s/ROhxFsXulu0rJ/zKfiPsjcA5y1joczaiEbiX+V/jjNcqL9KfSMVYbtQ5HUeyWlJKUB1NwCnQi/L5wrcPRlEroViKEoNdzvdBn6Ml8LVPrcBTFbhV+X0jJthKinojC0bPZYZLtmkroVuR3sR/dLu1G+j/TMRQYtA5HUeyONEpSH0rFrY8bIQtCtA5HcyqhW1nvF3pjLDOS9kya1qEoit3J+TCHioMV9PxnTxycVDpTe8DKPAZ4EDI/hKw3s6hKqdI6HEWxG8YqI2n/l4bXKC8CZgRoHY5NaDahCyHeFULkCSEONrF8vBCiRAix3zI93v5hdm5RT0UhnAWpD6dqHYqi2I3M1zPRZ+jp9a9eCCG0DscmtKSGvgaY2kyZ7VLKIZbp6baHZV9cQlyIuD+C/C/yVfe6itIODEUGTvzzBN0u7YbfeD+tw7EZzSZ0KeU2oLADYrFrEfdH4BzsTMr9KarjLkVpoxPPnaC2pJZez/XSOhSb0l5t6GOEEPFCiP8KIQY0VUgIcZsQYq8QYm9+fn47bbpzcPR0JOrpKEp3lHJq3Smtw1GUTqvqeBUZr2UQNCcIz8FdZ/CKlmiPhL4PiJRSxgHLgfVNFZRSrpRSjpBSjggI6HonMYJvCcY91p3Uh1JVlwCKco5SH0xF6AS9nlW184banNCllKVSynLL8+8BJyFE9zZHZoccHB3o/UJvqpKqyHo7S+twFKXTKd5aTP6X+fR4qAcuYS5ah2Nz2pzQhRDBwnKKWQgxyrJO1Rl4E7pd2g3fv/mS/lQ6tSW1WoejKJ2GNEqSlybjEmG+yED5q5Zctvgp8D8gWgiRIYSYL4RYKIRYaCkyAzgohIgHXgNmSnXWr0lCCHq/1BtDgYG0p9O0DkdROo2cNTmU7y+n1wu90LnptA7HJjXb8YGUclYzy18HXm+3iLoAr2FehNwaQuZrmYTcGoJHfw+tQ1IUm1ZbWkvqI6l4j/Um8PpArcOxWepOUY30fLYnDh4OJC9NVpcxKkoz0p9Nx5BnoM+yPuomorNQCV0jzgHO9Hy6J0U/FanxRxXlLKpSqshYlkHw3GC8R3hrHY5NUwldQ6F3hOI+wJ3ke5JVn+mK0oSU+1NwcHag5z97ah2KzVMJXUMOTg70fbUv1anVZLycoXU4imJzin4p4tT6U/R4pAcuIeoyxeaohK4xvwl+dL+mO+nPplOdUa11OIpiM0y1JpKXJuMa5Ur4PV17JKKWUgndBvT+d28wme+AUxTFLOutLCoOVND7pd7oXNVlii2hEroNcItyI+LBCPI+zaN4e7HW4SiK5mpyazj+2HH8Jpt/wSotoxK6jejx9x64RLiQdFcS0qguY1S6tpQHUjBVm+i7vK+6TLEVVEK3ETp3Hb3/3ZuK+AqyVqh+XpSuq3hbMbkf5hLxQATu/dy1DqdTUQndhgTMCMBvoh+pj6Siz9FrHY6idDiTwUTS4iRcIl2IfCRS63A6HZXQbYgQgr5v9sWkN5Fyb4rW4ShKh8t8PZOKgxX0fbUvOnd1IrS1VEK3Me593Yl8OJK8T/Mo/FkNFKV0HfosPWlPpNHt0m74T/PXOpxOSSV0GxTx9wjc+riRtChJ3UGqdBlJdyUhDZK+r6kToedKJXQbpHPV0ffNvlQlV3Hi+RNah6MoVnfq61Oc+uoUkU9E4tbbTetwOi2V0G1Ut0ndCJwVyInnTlB5rFLrcBTFamrLaklanITHIA8i7lMDV7SFSug2rPfLvXFwc+DobUeRJnVtumKfjj92HH2mnuhV0Tg4qZTUFmrv2TCXYBd6v9Sbkq0lZK/O1jocRWl3pbtLyVyeSeiiULzPU13jtpVK6DYuZH4Ivn/zJeWBFPSZ6tp0xX6Yak0cve0oziHO9PpnL63DsQsqods4IQT9VvZD1kiOLT6mRjdS7MbJF09SEV9B3+V9cfRudjRMpQVUQu8E3Pu4E/V0FAUbCsj/Ml/rcBSlzSoOV5D2ZBoBMwIIuCZA63DshkronUT4PeF4Dvck6c4kDIUGrcNRlHMmjZLEeYnovHT0faOv1uHYFZXQOwkHRweiV0djKDCQvDRZ63AU5ZxlLMugbFcZfZf3xTnQWetw7IpK6J2I1xAvIh+JJPfDXE59fUrrcBSl1SqPVXL8seP4T/MncGag1uHYnWYTuhDiXSFEnhDiYBPLhRDiNSFEshAiQQgxrP3DVE6LfCwSzyGeHL3tKIYC1fSidB7SJDk6/ygOrg70W9FP3d5vBS2poa8Bpp5l+SVAX8t0G/BW28NSmuLg7EDM+zHUFtZybPExrcNRlBbLeDWDkl9L6LOsjxrw2UqaTehSym3A2br9uxL4QJrtBHyFECHtFaDyV56DPYl6Ior8z/PJ+yJP63AUpVkVhytIfTgV/yv8CZoTpHU4dqs92tDDgJP1/s6wzPsLIcRtQoi9Qoi9+fnq8ru2iPh7BF4jvEhalERNXo3W4ShKk0w1Jo7cdARHb0eiV0WrphYr6tCTolLKlVLKEVLKEQEB6trTtnBwtDS9lNVydMFRdcORYrPSn0mnfF85/Vb2wzlIXdViTe2R0DOB+l2khVvmKVbmEetBr+d6UfB1AdmrVF8viu0p2VlC+j/TCZ4bTMBVqhJnbe2R0L8G5liudhkNlEgpVXbpIOFLwvGb6EfyPclUHlXd7Cq2w1hhJPGmRFwiXOjzah+tw+kSWnLZ4qfA/4BoIUSGEGK+EGKhEGKhpcj3QCqQDKwCFlktWuUvhIMg5v0YHFwdOHzDYUw1Jq1DUhQAku9Npiqlipg1Maqvlg7S7F6WUs5qZrkEFrdbREqruYS6EL0qmkPTD5H2ZJrquU7RXN6XeWSvzKbHQz3wG++ndThdhrpT1E4EXBNA8PxgTjx/guKtxVqHo3Rh1enVHFtwDK9RXkQ9HaV1OF2KSuh2pM+yPrj1duPIjUfUXaSKJky1Jg7fcBhplMR+EqtGIOpgam/bEUdPR2I/j6Umr4YjNx9Rw9YpHS79mXRKfyul34p+arBnDagzFXbGa5gXvV/qTfLdyZx8+SQ97u+hdUhWU1FTQUZpBiX6Ekr1pZRUl1BpqMRBOKBz0OEgHHB0cMTV0RU3Rzfzo5PbGc9PL3PSOTW7PSkleqOeSkMllYZKKmoqzI+GijP+NkkTzjpnnHROODk44ePqQ5hXGKFeobg52W+SK95aTPo/0gm6OYig2epuUC2ohG6Hwu4Mo3hLMccfPo7PBT74jPbROqQ2kVKSXJjM9hPb+SP7D44WHCXxVCInS082/+IW0gndGQnfxdEFg9FAdW113aQ3tn0IQD9XP3r69WRw0GCGBQ/jgh4XMDhoMDoHXTu8C+3oc/QcnnkYt95u9F2u+jjXitDqDsMRI0bIvXv3arLtrsBQbOD3ob8jpWTEHyNw8mu+BmpLjCYjm45vYt2RdXyX9F1d8vZy9iK6ezQx3WOI9o8myjcKX1dffFx88HH1wd3JHZM0YZImjCYjtaZaqmurqaqtospQVfe8uraaKkNV488tj846Z1x1rrg6njm5O7nj7uSOh7OH+dHJo+65u5M7DsIBg9GAwWSgxlhDcXUxmaWZZJVlkVmWSVJhEvE58eRW5ALmJD+1z1SujL6SadHTOl0tXhol8ZPiKd1ZyrBdw/Ac5Kl1SHZNCPG7lHJEY8tUDd1OOfk6Eft5LH+M/YPEuYkMXDcQ4WD7fWikFqWyZv8a1uxfw8nSk3g4eTC592QeHfcoF0ZeSEz3GLvpC+REyQm2p29n4/GNfHfsOz49+Cm+rr7MHjibeUPnMSxkWKd4r8efOE7x5mKi34tWyVxjqoZu5zJezSB5aTI9n+1J5CORWofTpG3p2/jn9n/yY8qPCASTe09m3tB5TIuehqujq9bhWZ1JmtiStoV3/niHtYfXojfqGRI8hIcveJjp/afbbJNMwXcFHLj8AMHzg4lZHaN1OF3C2WroKqHbOSklR248Qt6neQz+YTDdJnfTOqQ6Uko2Hd/EM9ueYVv6NgI9Alk8cjG3DLmFCJ+I5ldgp4qqivjs4Ge8tvs1Ek8lEtM9hkfHPcrMgTNxdLCdH9XV6dXsHboXlx4uDPvfMHRutnnQsTcqoXdxxgoj+8bsQ5+pZ/jvw3GL0r6Ndnfmbu798V5+O/kboV6hPHj+gywYvgB3J3etQ7MZRpORtUfW8sy2ZziYd5A+3frwwsQXuCrmKs2bYoyVRv4Y9wdVyVUM/3047n3U59ZRzpbQ1XXoXYDOQ8eArwYgjZJD0w9hrDJqFktmaSZz1s3hvNXnkVyYzBuXvkHK3SksGb1EJfMGdA46rhtwHfEL4/nquq9w1jlzzX+uYcIHE0jITdAsLiklR287Svkf5fT/qL9K5jZEJfQuwr2PO/0/6k/5vnKO3XGsw/tPrzHW8Oy2Z+n3ej8+P/Q5D419iKS7klg0clGXaCNvCwfhwNX9ryZ+YTyvX/I68bnxDH17KAu/XUhh1dkGE7OOjJczyPs4j6ino+h+RfcO377SNJXQu5Dul3cn8olIct/PJeOVjA7b7q6MXQxfOZzHNj/G1D5TObL4CM9NfA4vF68Oi8EeODo4snjUYpLvSuauUXexet9q+r/Rn88OftZhB+jCnwpJeTCFgBkBRD5quyfZuyqV0LuYqMej6D69OykPpFDwfYFVt1VRU8G9P97LmHfGUFRVxDezvmHtdWvp5ad6g2wLPzc/lk1dxu+3/U4Pnx7MWjuLKz69ghMlJ6y63aqUKg7PPIzHAA+i31NDydkildC7GOEg6P9+fzzjPDk88zAVhyqssp1NqZsY9NYgXtn5CgtHLOTw4sNc3u9yq2yrq4oLjmPn/J28PPllNqdtJvaNWF7d+SpGU/ufIzEUGzhwxQEQMHDDQBw9bedqG+VPKqF3QToPHQO/HojOQ8eBaQeoOdV+g0wXVRUxf8N8Jn44EUcHR7bO3cqbl72Jt4t3u21D+ZPOQcc9Y+7h0KJDjIscx9Ifl3L+u+dzMO9gu23DZDBxaMYhqpKrGLB2AG49tb9KSmmcSuhdlGu4KwPXD0SfqefQ9EOY9G0f6eirI18R+2Ys78e/z0NjHyJ+YTwXRl7YDtEqzYnyjeL72d/z8TUfk1qUyrC3h/HUlqeoMbbtYC2l5NgdxyjeVEz06mg1WIWNUwm9C/M+z5uY92Io2VZC4tzEc+5uN6c8hxn/mcH0/0wn2DOY3Qt289zE5zpdnySdnRCC2YNmc3jRYWbEzuDJrU8yYuUI9mTuOed1nnzhJDnv5BD5WCTBc4LbMVrFGlRC7+KCZgXR6/le5H2WR+rDqa16rZSSNfvXEPtGLN8e+5bnJjzH7lt3MyxkmJWiVVoiwCOAT6Z/wtczv6agqoDR74zmwZ8fpMpQ1ar15H2ZR+pDqQTODFQjD3USKqErRDwYQeiiUE6+cJKM11t2OWNqUSpTPprCLRtuYUDgAOIXxvPQBQ+1qF9xpWNcEX0FhxYdYt6Qeby440XiVsSxPX17i15bvLWYIzcewft8b3VFSyeiErqCEIK+r/XF/0p/ku9OJn9dfpNla4w1/HP7Pxnw5gB2ZuzkzUvfZOvcrUR3j+7AiJWW8nX1ZdW0VWy8aSO1plouXHMhi79bTJm+rMnXlO0v48C0A7j1cmPQ14PQuao+WjoLldAVAIROEPtJLF6jvDg86zBFW4r+UmZ7+naGvj2UR395lMv7Xc6RxUe4Y+QdOAj1NbJ1E3pN4MAdB1h63lLe2vsWMW/ENHpDUlVqFQlTE3D0dmTwj4Nx8le/uDqTFv0nCiGmCiGOCiGShRAPNbJ8rhAiXwix3zLd2v6hKtamc9cx6NtBuPVy4+AVByndUwpAQWUB8zfM58I1F1JRU8G3s77li2u/IMw7TOOIldbwcPbglamvsPPWnYR4hjBr7SwmfDCBw/mHAajJrSF+cjzSIBn802BcI1SXDJ1Ns70tCiF0wDFgEpAB7AFmSSkP1yszFxghpbyzpRtWvS3aLn2mnj/G/YG+VE/Cewk8nfg0xdXF3Dv6Xh6/6HE8nD20DlFpI6PJyKp9q3hk0yOU1ZRx95C7mfb0NMQhQdwvcZ1+2EJ71tbeFkcByVLKVCllDfAZcGV7BqjYFpcwFwo/LOTW2bdy97676efRj3237eNfk/6lkrmd0DnoWDhiIcfuOsac2Dm8vO9lZlw4g8MrD+N1nupjp7NqSUIPA+qPxpthmdfQdCFEghDiSyFEo6MTCCFuE0LsFULszc9v+sSbop0j+UeY9uk0Ltt4GYYIA09//zQvvfgSffVq4F975GvyZfHLi3njvTcICwrjjpQ7GL5yOD+l/KR1aMo5aK+zWd8AUVLKwcDPwPuNFZJSrpRSjpBSjggICGinTSvtIbkwmZvW3cSANwewJW0Lz094nsR7ElnyxhJqi2rZP34/1enVWoeptCNjpdF8rmRXKTNemsG+B/bx8TUfU1xdzJSPpjDpw0ltuilJ6XgtSeiZQP0ad7hlXh0pZYGUUm/5czUwvH3CU6zteNFxbv36VmJej2Ht4bXcf/79pC5J5e8X/B1XR1e8R3gT93MchkID+8fvpyqtdTenKLbJWGnk4JUHKd5aTP8P+hM4IxAH4cDsQbNJXJzIsinL+CP7D0atHsUlH1/CjpM7tA5ZaYGWJPQ9QF8hRE8hhDMwE/i6fgEhREi9P6cBR9ovRMUa9mXvY9baWfRZ3ocPEz5k8cjFpC5J5YVJL9Dd/cxBC7xHehO3MY7a4lqV1O1AbVktCZckUPRLETHvxRA0O+iM5S6OLiwZvYTUJak8N+E59mbtZey7Y5nwwQQ2pW7q8MFRlJZr0ZiiQohLgWWADnhXSvmsEOJpYK+U8mshxHOYE3ktUAjcIaVMPNs61VUuHc8kTfyY/CMv73yZjakb8XL24vbht7Nk9BLCvcObfX3ZvjLiJ8aj89IRtylODT3WCRmKDCRMTaDs9zL6f9SfoJlBzb6moqaCt39/mxd3vEhOeQ6DAgdx93l3c8OgG1R/PRpQg0R3cXkVebz7x7u8/fvbpBWnEeIZwtLRS7l9+O34uLbu8rSyP8qInxSPcBTE/RiHZ5ynlaJW2ltNfg0JkxOoOFzBgP8MoPuVrRs+rrq2mk8PfMqru14lPjcefzd/bh12K/OHzqevvzpp3lFUQu+CDEYDPyT/wIcJH7I+cT0Gk4G/Rf2NhSMWclXMVTjrnM953RWJFSRMSqC2rJZB3wzCd5xv+wWuWEX1yWoSpiRQfbyagesH0m1Kt3Nel5SSbenbeHXXq2w4ugGTNDGuxzjmDZ3HjNgZeDqrg7w1qYTeRZikiZ0ZO/nkwCd8dvAzCqoK6O7enRsG3cDCEQuJ6R7TbtuqPlFN/OR49Ol6Bnw5AP/L/Ntt3Ur7Kk8oJ+GSBIzlRgZ9PQjfi3zbbd1ZZVl8GP8h7+5/l2MFx/Bw8uCK6Cu4fsD1TO0zVQ0AbgUqoduxKkMVm45vYkPiBr4+9jV5FXm4OrpyZfSV3DT4Jib3nmy1HhBr8mtIuCSB8v3lRL8dTcj8kOZfpHSoos1FHLzqIDovHYP/OxjPQdapPUsp+fXEr3yU8BFrj6yloKoAbxdvpkVP44p+VzC592R8XX2tsu2uRiV0O5NRmsHG1I18c+wbfkz+kQpDBV7OXlza91KujL6Sy/pd1mFDvtWW1nLo2kMU/VRExN8j6PXPXggH1dWqLcj9LJfEOYm49XNj8H87rm8Wg9HA5rTNfH7wc9YfXU9hVSE6oWNsj7Fc1vcyLu17KQMCBqguec+RSuidXH5FPlvStvDL8V/YdHwTSYVJAIR6hXJl9JVcGX0l46PG4+Lookl8JoOJpLuSyH47m4AZAcR8EIPOrW1drhoMBv744w8OHTrE8ePHKSwspKamBicnJwICAoiIiGDQoEEMHDgQd3fbv9rGYDCQmJjIgQMHSE1NJT8/n8rKSpycnPD29iYqKoro6GhGjRqFh0fbuleQJkn6M+mkPZmGz4U+DFw/ECc/bXpNNJqM7MrcxXfHvuO7pO+Iz40HoIdPD8ZHjeeiyIu4KPIievn1Ugm+hVRC70SklBwrOMbOjJ3szNjJjowdJOQmAODl7MVFURdxcdTFXNzzYgYFDbKZrmullGS8kkHK/Sl4jfRi4PqBuIS07gBTVFTEunXr+OKLL9i6dStVVebr3R0cHPD19cXZ2Rm9Xk9R0Z9d+zo5OTFmzBimTJnCzJkz6dWrV7u+r7bIzs7mq6++4ocffmDLli2Ul5fXLfPy8sLT0xODwUBJSQkGgwEAnU7HqFGjmD59Otdeey09evRo1TaNFUaO3HyEU2tPEXRzEP1W9LOp/swzSzP5Pul7fkj5gW3p2zhVeQqAMK8wLooyJ/fR4aOJDYjF0cFR42htk0roNkpKycnSk8TnxPNHzh91Sbyo2pywfFx8GBU2ivFR47m458WMCB1h81/yUxtOcXj2YRy9HYn9IhbfC3ybfU1CQgKvvvoqn3zyCdXV1fTs2ZPLL7+ccePGMXToUHr06IGz859X5dTW1pKWlsaBAwfYuXMnGzduZN++fQCMHj2aG264geuuu47AwEBrvc0mVVdX89VXX7F69Wq2bNmClJI+ffowceJELrjgAuLi4ujTpw+urn82fxiNRrKysjhw4AC//fYbP/zwA/v27UMIwWWXXcbdd9/NxIkTm63BVqdXc+DKA1QcqKD3i70Jvyfcpmu9UkoO5x9ma/pWtqVvY2v6VnLKcwBwd3JnWMgwRoaOZFTYKEaGjlS1eAuV0G2AvlbP4fzDxOfGsz9nP/G58cTnxNclb4FgQOAARoeNZkzEGEaHjyame4zN1MBbo/xgOYeuOUT18Wp6/7s3YXeFNfqPmJyczP/93//x2Wef4e7uzk033cT8+fMZMWJEq/9x09PT+eyzz/j44485cOAAjo6OTJs2jQULFjBp0iR0OuvWUg8ePMiqVav48MMPKSoqolevXsyZM4drr72W2NjYVq8vJSWFNWvWsHLlSvLy8jjvvPN46qmnmDx5cqP7pvDnQo7MPoLJYGLA5wPadFmiVqSUJBcmsztzN3uy9rA7czd/5PxBda25DyEvZy8GBQ1iUKB5Ghw0mEFBg7rcyVaV0DtQYVUhiacSOXrqqPmxwPyYUpRCrakWMNc+BgUOYkjwEOKC4ogLjmNQ4CC8XOyn29LaklqOzDlCwdcFBM4OJHplNDoPc1LNysrimWeeYfXq1Tg7O3Pvvfdy77334ufn1y7bPnDgAO+//z7vv/8+p06dokePHsyfP5958+YRHt78HbEtVVxczJdffsnq1avZtWsXzs7OXH311SxYsIC//e1vODi0/WCs1+v54IMP+Mc//sGJEyeYMmUKy5YtIybGfAmqNErSnkoj/R/puPd3Z+BXA3GPtv1zCi1lMBo4mHeQPVl7SMhN4EDeARJyEyiuLq4rE+4dzqDAQcQGxNLPv1/dFOIZYpc1epXQ25FJmsguy+Z48XGOFx0ntSiV48XHSSlKIfFUYl2bIICzzpm+3foS3T2aGP8Y4oLjGBI8hN5+vdE52E67prVIk+TEcyc4/n/HcevrRsxHMazZsYZHH30UvV7P7bffzmOPPUZwcLBVtq/X69mwYQOrVq1i48aNODg4MHXqVGbNmsW0adPw9m79lUB6vZ7vv/+ejz76iO+++w69Xk///v1ZsGABN910E927t+7uy5aqqanhjTfe4Mknn6SyspIlS5bw8G0Pc/L2kxRvKSb4lmD6Lu9bd9C0Z1JKMssyOZB7oC7BH8g7wLGCY3W1eQAPJ48zEnxvv95E+UYR5RtFmHeYzTdfNkUl9FaoMdaQVZZFZmkmmWWZpBenc7z4z8SdXpyO3qg/4zWhXqH08utFjH8MMd3NU3T3aKJ8ozrtl6Y9Ff1SRMLMBAynDKyWqymdWsry15fTu3fvDovh+PHjvPPOO7z//vtkZGTg7OzM2LFjmThxIiNHjiQuLo6AgIC/1OhKS0s5cOAAv//+O5s2bWLz5s2UlZURGBjIzJkzufHGG8+piehc5eXl8cgjj3DknSM84PAA3k7e9H+7PyE3q3sATNLEyZKTJBUmcazg2BnT8eLjmKSprqxO6IjwiahL8JE+kWc8D/UK1eyqseaohI75qF5cXWxO1mWZZJRm1CXtzLJMMkvN8/Ir/zrwRje3bvT07UlPv57mx3rPI30j1d1wZ1FVVcVTTz3FyhdX8pDTQ4zSj8J3gi8xa2JwDe/4/WYymdi5cyfr1q3j559/Jj4+vm6Zi4sLQUFBODk5YTQayc/Pp6Kiom55z549mTRpEldffTUTJ07E0bHjD9a1JbUk35NMzns5ZLll8XDVw8ROjeWtt94iKiqqw+PpLPS1ek6UnCCtOO3PqSSN9OJ00orTyCrLQnJmLvR38yfUK5Qw7zBCPUP/fO4VSpiX+THQI7DDf23bbUKvMdaQX5FPbkUueRV55FXkkVtueV7Z4O+KPAwmw1/W0d29O2FeYYR5hxHuFU6YdxhhXmGEe5ufR3hHtLoDK8Vs48aNLFy4kJSUFObNm8cLL7yA/is9yUuTEY6C3i/0JmRBiKY3IhUWFhIfH09CQgKZmZnk5uZiNBoRQtC9e3dCQ0OJjY1lyJAhhIVpOyh20aYiEm9JRJ+pp8fDPYh4NIK3Vr3FI488gpSSp59+miVLlmhyoOns9LV6TpaeJK3YnOSzy7PJLM0kqzyr7hd7bkXuGbV8MNf0gzyDCPIIItAj8IwpwD3gL/Pao3dKu0ro/036L0t/XEpeRd4ZJ0bqc9G5EOT55w4+vbMD3APqEvbpI62qXbe/U6dOcf/99/P+++/Tp08fVq5cyd/+9re65VUpVRxdcJTizcX4XORD9Mpo3PvZz4m89lZzqobUh1LJeScHt35u9P+gP97n/dn+f+LECRYvXsy3337LsGHDWLlyJcOHqzFm2lutqZa8ijxzoi/LqpsyyzLJr8w/o1JZVdv4mAGezp4EegSyaMQi7jv/vnOK42wJvdMdyru5dWNI8JBGj4in53k6e9rl2W1bJ6Xkww8/5L777qO4uJhHHnmExx57DDe3M2slbr3diNsUR867OSTfl8yewXuIfDSSiPsj2nyHqT2RJkn2u9mk/j0VY6mRiAciiHoyCp37mfuoR48efP3116xdu5a77rqLUaNGsXTpUp5++uk233Wq/MnRwZFQL3PTS3MqairqEnzDKb8ynxAvK53zkFJqMg0fPlwq9uPw4cPyoosukoAcPXq0jI+Pb9HrqrOq5cFrD8rNbJY7InfI3M9zpclksnK0tq9kT4n8fczvcjOb5b5x+2TZgbIWva6oqEjefvvtEpCRkZHy+++/t3KkSkfDPLBQo3m18921otiU8vJyHn30UeLi4khISGDlypX89ttvDB48uEWvdwlxYcB/BhC3OQ5HX0cOX3+Y/Rftp3RvqZUjt02VyZUcuv4Q+0buoyq5ipj3YxiydQieA1vWS6Kvry8rVqxg+/btuLu7c+mllzJr1iyysrKsHLliE5rK9NaeVA29czMYDHLFihUyKChIAnLOnDkyNze3Tes01Zpk5tuZ8teAX+VmNsuEKxJk6b7SdorYtlVnVcuji4/KLY5b5Fb3rTL1/1KlocTQtnVWV8unnnpKOjs7S3d3d/nYY4/JkpKSdopY0QpnqaGrhK60islkkuvWrZPR0dESkBdccIHcsWNHu27DUGKQx585Lrf7bpeb2SwPXHXAbhN7ZXKlTLw9UW5x2SI36zbLo3ccldXZ1e26jZSUFDlz5kwJyICAALl8+XKp1+vbdRtKx1EJXWkzvV4v16xZIwcNGiQBGRMTIzds2GDV9u6aohp5/Mnjcpv3NnNb8oX7ZN5XedJU27nb2E0mkyzZXSIPzTwkNztslluct8jE2xNlZXKlVbe7e/duOX78eAnIsLAw+a9//UsWFRVZdZtK+ztbQu90ly0qHSs3N5f33nuP1157jezsbAYOHMj999/PDTfc0GHXOxuKDWSvzibz9Uz06Xpco1wJvSOUoBuDcAm1zbv5GlNbVkveJ3lkvZ1F+R/l6Lx0hN4RSvjS8FZ3NXyupJT89NNPvPjii2zatAlPT0/mz5/PokWL6NevX4fEoLSNXV2HrlhfaWkp69at45NPPmHjxo2YTCYmTZrEfffd12Rvfx3BVGuiYEMBGa9mULK9BBzAb6IfwXOC6X5Vd5vsx8RUY6Lo5yLyvsgj/8t8TBUmPAZ7EHp7KEE3BOHoo92Vw/v37+ff//43n332GbW1tYwYMYLZs2czc+ZMQkJUVwK2SiV05ayMRiPx8fFs2rSJX375hS1bttT1Sz579mxmz559Tl3AWlPl0UpyP8ol58Mc9Ol6HNwc8Jvoh//l/vhf5o9LmHY1d0OxgeItxZxad4pTG05hLDGi89ERMD2A0NtC8RrlZVP3SWRlZfHpp5/yySef1PXDPmbMGCZMmMCECRMYPXo0Li6d55eQvWtzQhdCTAVeBXTAainl8w2WuwAfAMOBAuB6KWXa2dapEro2ampqSEpKIiEhgfj4eOLj49m9ezeFhYUA9O/fn0mTJjFz5kxGjx5tU4mnMdIkKdleQv6X+RR8W0B1mrm3PY/BHvhc4IPP+T54n++Na5Sr1d6LPkdP+R/lFG8tpnhTMWX7ysAEjr6O+F/pT+C1gfhN9MPBxfavEk5MTOTTTz/lhx9+YO/evZhMJtzc3Oo6MDs9xcbGdoqh/+xRmxK6EEIHHAMmARnAHmCWlPJwvTKLgMFSyoVCiJnA1VLK68+2XpXQz53JZEKv158xVVdXU1ZWRmFhYd1UVFREYWEhWVlZpKenk56eTlZWFqc/cycnJ/r378/w4cO5+OKLufjiiwkNbf4uOFslpaTycCUF3xZQ+HMhZbvKMJYbAXAKdMIj1gP3GHfc+7vj1s8NlxAXnAKdcOruhINT08lWmiTGciP6DD3V6dXmKa2aigMVlP9RTk12DQDCUeA92hvfCb74TfDD+zxvHJxtP4k3pbi4mK1bt7Jp0yb27NnDgQMHzuisLCAggMjISCIjIwkPD6dbt251k5+fH926dcPHxwdXV1dcXFxwcXGpe27tAUfsWVsT+hjgSSnlFMvfDwNIKZ+rV+ZHS5n/CSEcgRwgQJ5l5eea0H/88UfuueceLDGc8djU87bM66jXtGQ9NTU16PV6amtraSl3d3eCgoLq/vEiIyPp06cPcXFxxMTEnDG0m72RRknFwQpKdpRQtqeMysRKKo9UUlv81/3n6OeIg4sDwkkgHAVCJzBWGjGWGc0HhQbfZOEocI9xx3OoZ93kNcILR89O15tGi5lMJlJSUkhISODo0aN1lYT09HQyMzMpKytr8bp0Oh0uLi44Ozuj0+kQQuDg4PCXx8bm1X+0Jmuu/9Zbb+Xee+89p9e2tS+XMOBkvb8zgPOaKiOlrBVClAD+wKn6hYQQtwG3Aa0e/PY0b29vBg4cWH+dZzw29bwt8zrqNc2tx9nZua6mU7+2c3ry9vY+o3bk5+fXpds+hU7gGeeJZ5wn3GGeJ6XEkGegMqkSQ66BmrwaDHkGavJrkDUSWSuRBvOjg4cDjl6O6Lx06Lx1uIS64BrpikukCy4hLgidbTdHtTcHBwf69u1L3759G11uMBgoLi6u+2VYWFhISUkJ1dXVjf6iPP1cSonJZPrLY2Pz6i+zJmufWwwKCrLKeju0OiGlXAmsBHMN/VzWMWbMGMaMGdOucSldhxAC5yBnnIPs95eJVpycnAgICCAgIEDrULqsljTwZQIR9f4Ot8xrtIylycUH88lRRVEUpYO0JKHvAfoKIXoKIZyBmcDXDcp8DdxseT4D+OVs7eeKoihK+2u2ycXSJn4n8CPmyxbflVIeEkI8jfkW1K+Bd4APhRDJQCHmpK8oiqJ0oBa1oUspvwe+bzDv8XrPq4Fr2zc0RVEUpTU670WyiqIoyhlUQlcURbETKqEriqLYCZXQFUVR7IRmvS0KIfKB9HN8eXca3IVqI2w1LrDd2FRcraPiah17jCtSStno3VuaJfS2EELsbaovAy3Zalxgu7GpuFpHxdU6XS0u1eSiKIpiJ1RCVxRFsROdNaGv1DqAJthqXGC7sam4WkfF1TpdKq5O2YauKIqi/FVnraEriqIoDaiEriiKYidsNqELIa4VQhwSQpiEECMaLHtYCJEshDgqhJjSxOt7CiF2Wcp9bun6t71j/FwIsd8ypQkh9jdRLk0IccBSzuoDqQohnhRCZNaL7dImyk217MNkIcRDHRDXi0KIRCFEghBinRDCt4lyHbK/mnv/QggXy2ecbPkuRVkrlnrbjBBCbBZCHLZ8/5c0Uma8EKKk3uf7eGPrslJ8Z/1shNlrln2WIIQY1gExRdfbF/uFEKVCiKUNynTIPhNCvCuEyBNCHKw3r5sQ4mchRJLl0a+J195sKZMkhLi5sTLNklLa5AT0B6KBLcCIevNjgXjABegJpAC6Rl7/H2Cm5fkK4A4rx/tv4PEmlqUB3Ttw3z0J3N9MGZ1l3/UCnC37NNbKcU0GHC3P/wX8S6v91ZL3DywCVliezwQ+74DPLgQYZnnuhXmA9oZxjQe+7ajvU2s+G+BS4L+AAEYDuzo4Ph3mMY0jtdhnwIXAMOBgvXkvAA9Znj/U2Pce6AakWh79LM/9Wrt9m62hSymPSCmPNrLoSuAzKaVeSnkcSAZG1S8gzINwXgx8aZn1PnCVtWK1bO864FNrbcMKRgHJUspUKWUN8BnmfWs1UsqfpJSnR2jeiXn0K6205P1fifm7A+bv0gQhrDsysZQyW0q5z/K8DDiCeczezuJK4ANpthPwFUKEdOD2JwApUspzvQu9TaSU2zCPCVFf/e9RU7loCvCzlLJQSlkE/AxMbe32bTahn0Vjg1Y3/ML7A8X1kkdjZdrTOCBXSpnUxHIJ/CSE+F2YB8ruCHdafvK+28RPvJbsR2uah7km15iO2F8tef9nDH4OnB78vENYmniGArsaWTxGCBEvhPivEGJAR8VE85+N1t+rmTRdsdJqnwVJKbMtz3OAxkaIbpf91qGDRDckhNgIBDey6FEp5YaOjqcxLYxxFmevnV8gpcwUQgQCPwshEi1HcqvEBbwFPIP5n+8ZzM1B89qyvfaI6/T+EkI8CtQCHzexmnbfX52NEMITWAsslVKWNli8D3OTQrnl/Mh6oG8HhWazn43lPNk04OFGFmu5z+pIKaUQwmrXimua0KWUE8/hZS0ZtLoA8089R0vNqrEy7RKjMA+KfQ0w/CzryLQ85gkh1mH+ud+mf4KW7jshxCrg20YWtWQ/tntcQoi5wOXABGlpPGxkHe2+vxrRmsHPM0QHDn4uhHDCnMw/llJ+1XB5/QQvpfxeCPGmEKK7lNLqnVC14LOxyveqhS4B9kkpcxsu0HKfAblCiBApZbal+SmvkTKZmNv5TwvHfP6wVTpjk8vXwEzLFQg9MR9ld9cvYEkUmzEPWA3mAaytVeOfCCRKKTMaWyiE8BBCeJ1+jvnE4MHGyraXBm2WVzexvZYM/t3ecU0FHgSmSSkrmyjTUfvLJgc/t7TRvwMckVK+3ESZ4NNt+UKIUZj/jzviQNOSz+ZrYI7lapfRQEm95gZra/KXslb7zKL+96ipXPQjMFkI4WdpIp1smdc61j7re64T5kSUAeiBXODHessexXyFwlHgknrzvwdCLc97YU70ycAXgIuV4lwDLGwwLxT4vl4c8ZbpEOamB2vvuw+BA0CC5csU0jAuy9+XYr6KIqWD4krG3E643zKtaBhXR+6vxt4/8DTmAw6Aq+W7k2z5LvXqgH10AeamsoR6++lSYOHp7xlwp2XfxGM+uXy+teM622fTIDYBvGHZpweod4WalWPzwJygferN6/B9hvmAkg0YLPlrPubzLpuAJGAj0M1SdgSwut5r51m+a8nALeeyfXXrv6Ioip3ojE0uiqIoSiNUQlcURbETKqEriqLYCZXQFUVR7IRK6IqiKHZCJXRFURQ7oRK6oiiKnVAJXVEshBAjLR2auVruijwkhBiodVyK0lLqxiJFqUcI8Q/Md4i6ARlSyuc0DklRWkwldEWpx9Kvyx6gGvPt4UaNQ1KUFlNNLopyJn/AE/NoQa4ax6IoraJq6IpSjxDia8yjF/XE3KnZnRqHpCgtpml/6IpiS4QQcwCDlPITIYQO2CGEuFhK+YvWsSlKS6gauqIoip1QbeiKoih2QiV0RVEUO6ESuqIoip1QCV1RFMVOqISuKIpiJ1RCVxRFsRMqoSuKotiJ/wdW9aXdUbU0swAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(s.x, n, 'k-', label=\"density\")\n",
    "plt.plot(s.x, s.v_ext, 'm-', label=\"external potential\")\n",
    "plt.plot(s.x, v_h, 'g-', label=\"hartree potential\")\n",
    "plt.legend()\n",
    "plt.xlabel(\"x\")\n",
    "plt.title(\"Ground state QHO\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "a248f5f7-13fc-4c9a-96b8-53133ab96402",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(p, cmap=\"seismic\", vmax=np.max(p), vmin=-np.max(p))\n",
    "plt.title(\"Density matrix of ground-state QHO\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "4e92698d-f6b9-4837-9934-b81619a64189",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(v_x, cmap=\"seismic\", vmax=np.max(v_x), vmin=-np.max(v_x))\n",
    "plt.title(\"Exchange potential of ground-state QHO\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d36e5edf-28a2-49dc-9fe4-45b5c7fe8b03",
   "metadata": {},
   "source": [
    "And we can of course repeat the above for the first excited-state:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "e17f1102-aad8-4d02-a12a-facf34288347",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "hartree energy = 0.6990259617571019 , exchange energy = -0.43974894134460124\n"
     ]
    }
   ],
   "source": [
    "n = iDEA.observables.density(s, state=first_state)\n",
    "v_h = iDEA.observables.hartree_potential(s, n)\n",
    "E_h = iDEA.observables.hartree_energy(s, n, v_h)\n",
    "\n",
    "\n",
    "p = iDEA.observables.density_matrix(s, state=first_state)\n",
    "v_x = iDEA.observables.exchange_potential(s, p)\n",
    "E_x = iDEA.observables.exchange_energy(s, p, v_x)\n",
    "\n",
    "\n",
    "print(\"hartree energy =\", E_h, \", exchange energy =\", E_x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "8eb775b7-d6b9-4a29-8c00-dd6b91a59b7e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(s.x, n, 'k-', label=\"density\")\n",
    "plt.plot(s.x, s.v_ext, 'm-', label=\"external potential\")\n",
    "plt.plot(s.x, v_h, 'g-', label=\"hartree potential\")\n",
    "plt.legend()\n",
    "plt.xlabel(\"x\")\n",
    "plt.title(\"First excited-state QHO\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "3971b9cb-9eee-4d20-8fea-0417dc4332be",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(p, cmap=\"seismic\", vmax=np.max(p), vmin=-np.max(p))\n",
    "plt.title(\"Density matrix of first excited-state QHO\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "538870c3-98e0-44f9-b9f2-b663e81b8c20",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.imshow(v_x, cmap=\"seismic\", vmax=np.max(v_x), vmin=-np.max(v_x))\n",
    "plt.title(\"Exchange potential of first excited-state QHO\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8a8a3138-ed3f-45ba-9640-e3f03871f7f1",
   "metadata": {},
   "source": [
    "## 5. Perturbations and Time-Dependence."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "96f0e288-b0ba-49b8-97ab-955064767d87",
   "metadata": {},
   "source": [
    "Now we know how to; describe systems, solve them for given states, and compute observables, we can now look at driving these systems out of thier stationary states using a external perturbation."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "611a1645-7c79-4182-b67b-87883b0265a9",
   "metadata": {},
   "source": [
    "First we define a time grid $t=0 \\rightarrow t=100$: wih $1000$ timesteps:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "21f9048d-47a2-40a2-b8c7-c272c6863eff",
   "metadata": {},
   "outputs": [],
   "source": [
    "t = np.linspace(0, 50, 500)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4905e03f-c525-4463-9fae-0b1a4a6b63d7",
   "metadata": {},
   "source": [
    "For our perturbation, we choose a oscillating linear electric field:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "7992dbb8-ae1e-4121-a68e-d329996b1fb0",
   "metadata": {},
   "outputs": [],
   "source": [
    "v_ptrb = np.zeros(shape=t.shape+x.shape)\n",
    "for j, ti in enumerate(t):\n",
    "    v_ptrb[j,:] = -0.1 * x * np.sin(0.5*ti)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0f680b90-bf8a-41b4-a0e0-416c2d945f1a",
   "metadata": {},
   "source": [
    "We can animate our field:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "06716ac8-ef81-4221-a1bf-f9cf2e243fc9",
   "metadata": {},
   "outputs": [
    {
     "data": {
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      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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JyisXE7mlhciEkeqXFykDBb8MWLo9SfuuOLHWGO0vHGbigzcR6X6bqj7tUlbL8FlJhq2aSHRhlOj8HxMa//OeD2lFpLwU/NIvl0wRf2QbyfXP9Uwi4uJJXuLOnjaNJEgTJF47je7J87KTiNywjPAVs5iowcpEBi0Fv+Cco+uPXbS1tpFa9wjRJ35C9am9VBPv1S+fIUB0ulGz5ELqmutIjXuS4IpLqa3WubzIUKLg96GeSUSe3kRgzxaOxFs4FLsagJEc5iK2AvRMIpKZ00Tl1UuIrF7GgjH1OWu6qPTFi0jeFPznuXRnmti2GJkfryWw6WlC7+wknDrUa7CyOCGOjLyWuuY66mZ/irbwxYRXLyc8c7wGKxM5Dyn4zyN9JxF5PfjXxPamIQ2XcT/DaAU4YxKRYas/xrJll2Jm3prmlO+PEJGiU/APUc45EjuOkrzjQdwLrVS+tu2MSUSM5RC4jNq5tcRHf4GTY1YT/HQL4evmaxIRER9T8A8RyT0Hid/zDIlX2jlyajFtrW1UHj/IYr7cq13PJCJzm5h82xXUfHIqgeoA0FSewkVk0FHwD0Kpd08SX/c8qQ0vULFjK6HDOwin3yUIBJjEce4AwEaN52TVNbhp06hcsZTwLS2aREREzknBX2aZjuwkIm2H6ji1L0xbaxtj9vwDDfy6V7s0ETqHzSA1rYkZ35hO3aJhhCaEMNOgqCLy4Sj4S8il0iSe3E3yN8+R6TOJyCH+E+9yPQBtFbPoCE+ju3EuLGrWJCIiUlBKkiLq2vMOsf1B2ja3EdvUxtTfX0skc/iM8WniVQ1E59cz+c8mE10YpXbucgLh28tSs4ic/xT8BZL64zHi9z6X7Zf3JhGpSp9kD4/gvNFs4oyloiJJfPRsMpc1ZScRuWU5kUZNIiIipaPg/wgyXRnad7QT2xyj++E/cOHvbyecPEAU16tdimpGL+4g9PHZRJujVM/aQHDKCHQhpYiUk4L/HHomEVn/PGzMTiJyIj6d/ZmvAhAhTSNvkaGKzpop2UlEFjcTvKGFyNVzmK5JRERkkFHw53DO0XWoi1hrDLd2LZHW9URO7KGajj6DlaWonl5NtDlKXdMltEd/T/WNi6mpU4eNiAx+vg7+vpOIvJ76K069nx147BK2M9ob4qArMPpPk4hctYTILS00jx+Rs6YJZaheROSj8U3wpxNp2je+h/vxz+ClLYQO7ThjEpEIC+mobyDaHMUavkDb6FWEVrcQmtuoSURE5LxxXgZ/zyQi//Y83S8f5kDyFjp2dkAqRQvfJ0AC4IxJRCZ+9mqmLpvkDVamgcpE5Pw05IPfOUfXtrfpuvtxMs+1Uvnqtl6TiKQJsZdP4ioC1Myq41joKwSnjqbqU8uI3LBQk4iIiO8MueDvfvUw8XXPEHt3GMcOXESsNcbI9x5iGj/s1a4rMIbE2Dlk5ixgzn+cTu3S0VTWVgILy1O4iMggMaiDP/1+jM57niP12ItUbN/SM4lIFdDGTRwne0llx7DZtFUtIz1jPoFPLCG8ejmhmePVLy8i0o9BE/yZRJLE+q2cPDGB2NYO2lrbmLrj89Sxr1e705OIhBbMZPqa6dQ11xG++GOY/XmZKhcRGVryCn4zWw18B5gONDvntgz0dxNP7yXxwLPw/KZek4js4hfEaQCgzWZQGUqTbJwLzc29JhGpzadwEREfy/eMfzfwGeBnH+aX3NbthD8x84xx4xOV47jwciNw3SXUNddRO3cdgZrKXl+eEhGR/OQV/M65l4GcuVoHxsiQtHoSo2aTnrmAwIrsYGXhS8fSmE9BIiJyTiXr4zezNcAagIvHjKfq0AGCFRWl2ryIiHjOGfxm9gQwpp9FtzvnHhrohpxza4G1AE1NTc4U+iIiZXHO4HfOXVmKQkREpDR02i0i4jN5Bb+Z3WhmB4ElwCNm9lhhyhIRkWLJ96qeB4EHC1SLiIiUgLp6RER8RsEvIuIzCn4REZ9R8IuI+IyCX0TEZxT8IiI+o+AXEfEZBb+IiM8o+EVEfEbBLyLiMwp+ERGfUfCLiPiMgl9ExGcU/CIiPqPgFxHxGQW/iIjPKPhFRHxGwS8i4jMKfhERn1Hwi4j4jIJfRMRnFPwiIj6j4BcR8RkFv4iIzyj4RUR8RsEvIuIzCn4REZ/JK/jN7Idmts/MdprZg2ZWX6C6RESkSPI9498AzHLOzQZeBb6df0kiIlJMeQW/c+5x51zKu7sRGJ9/SSIiUkyF7OP/IvDbAq5PRESKoPJcDczsCWBMP4tud8495LW5HUgBd3/AetYAawAaGho+UrEiIpK/cwa/c+7KD1puZl8ArgOucM65D1jPWmAtQFNT01nbiYhIcZ0z+D+Ima0Evgl8zDnXWZiSRESkmPLt4/8nIApsMLPtZvbTAtQkIiJFlNcZv3NucqEKERGR0tA3d0VEfEbBLyLiMwp+ERGfUfCLiPiMgl9ExGcU/CIiPqPgFxHxGQW/iIjPKPhFRHxGwS8i4jMKfhERn1Hwi4j4jIJfRMRnFPwiIj6j4BcR8RkFv4iIzyj4RUR8RsEvIuIzCn4REZ9R8IuI+IyCX0TEZxT8IiI+o+AXEfEZBb+IiM8o+EVEfEbBLyLiMwp+ERGfUfCLiPhMXsFvZt81s51mtt3MHjeziwpVmIiIFEe+Z/w/dM7Nds7NBR4G/ib/kkREpJjyCn7nXFvO3RrA5VeOiIgUW2W+KzCz7wGfB04Bn/iAdmuANd7dLjPbne+2S+AC4P1yFzEAqrNwhkKNoDoLbajUObUQKzHnPvgk3cyeAMb0s+h259xDOe2+DYSdc397zo2abXHONX3YYktNdRbWUKhzKNQIqrPQ/FbnOc/4nXNXDnBddwOPAucMfhERKZ98r+qZknN3FbAvv3JERKTY8u3j/76ZTQUywAHgSwP8vbV5brdUVGdhDYU6h0KNoDoLzVd1nrOPX0REzi/65q6IiM8o+EVEfKZowW9mq81sj5llzKypz7Jvm9l+M3vFzD55lt+/2Mw2ee3WmVmwWLXmbHOdN/zEdjN7y8y2n6XdW2a2y2u3pdh19bP975jZoZxarz1Lu5XePt5vZt8qcY0/NLN93pAeD5pZ/VnalWVfnmvfmFnIOx72e8dhY6lqy6lhgpk9ZWZ7vf9LX+unzcfN7FTOsVCWb8+f63m0rB97+3Onmc0vQ41Tc/bTdjNrM7Ov92lTlv1pZnea2dHc7zeZ2Qgz22Bmr3n/Dj/L797mtXnNzG4b0Aadc0X5AaaT/bLB00BTzuMzgB1ACLgYeB0I9PP79wC3erd/Cny5WLWepf7/CfzNWZa9BVxQynr6bP87wH85R5uAt28nAUFvn88oYY1XA5Xe7R8APxgs+3Ig+wb4CvBT7/atwLoyPM9jgfne7Sjwaj91fhx4uNS1fdjnEbgW+C1gwGJgU5nrDQDvAhMHw/4ELgfmA7tzHvsfwLe829/q7/8QMAJ4w/t3uHd7+Lm2V7Qzfufcy865V/pZtAr4tXOuyzn3JrAfaM5tYGYGrADu8x76BXBDsWrty9v+LcCvSrXNImgG9jvn3nDOJYFfk933JeGce9w5l/LubgTGl2rbAzCQfbOK7HEH2ePwCu+4KBnn3GHn3Dbvdgx4GRhXyhoKaBXwry5rI1BvZmPLWM8VwOvOuQNlrKGHc+4Z4Hifh3OPwbNl4CeBDc654865E8AGYOW5tleOPv5xwNs59w9y5sE8EjiZExz9tSmm5cAR59xrZ1nugMfNbKs3FEU5fNV7y3znWd4CDmQ/l8oXyZ7t9acc+3Ig+6anjXccniJ7XJaF19U0D9jUz+IlZrbDzH5rZjNLW1mPcz2Pg+l4hOy7uLOd2A2G/QlwoXPusHf7XeDCftp8pP2a13X8NsDhHAaTAdb8WT74bL/FOXfIzEYDG8xsn/eKXZI6gX8Bvkv2P9t3yXZLfbGQ2x+IgexLM7sdSJH9Znd/ir4vhzozqwXuB77ueg+MCLCNbHdFu/dZz2+AKZTekHkevc8Lrwe+3c/iwbI/e3HOOTMr2LX3eQW/G/hwDrkOARNy7o/3Hst1jOxbwUrvbKu/Nh/JuWo2s0rgM8CCD1jHIe/fo2b2INmug4Ie5APdt2b2f8gOid3XQPZzXgawL78AXAdc4bwOyX7WUfR92Y+B7JvTbQ56x8QwssdlSZlZFdnQv9s590Df5bkvBM65R83sn83sAudcSQccG8DzWPTj8UO4BtjmnDvSd8Fg2Z+eI2Y21jl32OsWO9pPm0NkP5c4bTzZz1U/UDm6etYDt3pXTVxM9tW0NbeBFxJPATd7D90GlOodxJXAPufcwf4WmlmNmUVP3yb7IWZJRxrt0zd641m2vxmYYtmro4Jk39quL0V9kL1qBvgmcL1zrvMsbcq1Lweyb9aTPe4gexz+/mwvXsXifaZwB/Cyc+5HZ2kz5vRnD2bWTPb/dElfoAb4PK4HPu9d3bMYOJXTjVFqZ31HPxj2Z47cY/BsGfgYcLWZDfe6fK/2HvtgRfyU+kay/U1dwBHgsZxlt5O9quIV4Jqcxx8FLvJuTyL7grAfuBcIFavWPnXfBXypz2MXAY/m1LXD+9lDtluj1FcA/BLYBez0Do6xfev07l9L9kqQ10tdp/e8vQ1s935+2rfGcu7L/vYN8HdkX6gAwt5xt987DieV4XluIdudtzNnP15LdmiUL3ltvurtux1kP0RfWoY6+30e+9RpwE+8/b2LnCv9SlxrDdkgH5bzWNn3J9kXosNAt5ebf0n2M6UngdeAJ4ARXtsm4Oc5v/tF7zjdD/z7gWxPQzaIiPiMvrkrIuIzCn4REZ9R8IuI+IyCX0TEZxT8IiI+o+AXEfEZBb+IiM/8f/C47uTyaU5nAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from matplotlib.animation import FuncAnimation\n",
    "from IPython.display import HTML\n",
    "fig = plt.figure()\n",
    "ax = plt.axes(xlim=(s.x[0], s.x[-1]), ylim=(-3.0, 3.0))\n",
    "line1, = ax.plot([], [], lw=2, c='m')\n",
    "line2, = ax.plot([], [], lw=2, c='r', ls='--')\n",
    "def init():\n",
    "    line1.set_data([], [])\n",
    "    line2.set_data([], [])\n",
    "    return line1, line2\n",
    "def animate(i):\n",
    "    line1.set_data(s.x, v_ptrb[i,:])\n",
    "    line2.set_data(s.x, v_ptrb[i,:])\n",
    "    return line1, line2\n",
    "anim = FuncAnimation(fig, animate, init_func=init, frames=100, interval=10, blit=True)\n",
    "HTML(anim.to_html5_video())"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1e83a614-e91f-4c25-acae-101bda5881ea",
   "metadata": {},
   "source": [
    "To propagate our ground-state, due to this perturbation: *(this can take some time on slower machine, to speed up simply reduce the number of timesteps)*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "1ac06da7-8f0c-4d1f-b575-a63042c6048b",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "iDEA.methods.interacting.propagate: propagating state: 100%|██████████████████████████████████████████████████████████████████████████| 500/500 [02:54<00:00,  2.87it/s]\n"
     ]
    }
   ],
   "source": [
    "evolution = iDEA.methods.interacting.propagate(s, ground_state, v_ptrb, t)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9a924c77-140e-4c1f-aee5-bf0a2f4f66fd",
   "metadata": {},
   "source": [
    "From this evolution, we can compute dynamic observables in much the same way as previously."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "5f4f5d5f-081a-4d84-9ea9-2ea774cb265d",
   "metadata": {},
   "outputs": [],
   "source": [
    "n = iDEA.observables.density(s, evolution=evolution)\n",
    "v_h = iDEA.observables.hartree_potential(s, n)\n",
    "E_h = iDEA.observables.hartree_energy(s, n, v_h)\n",
    "\n",
    "\n",
    "p = iDEA.observables.density_matrix(s, evolution=evolution)\n",
    "v_x = iDEA.observables.exchange_potential(s, p)\n",
    "E_x = iDEA.observables.exchange_energy(s, p, v_x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "dc14e513-c81a-4e5a-ae7b-9ead9da098aa",
   "metadata": {},
   "outputs": [
    {
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      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure()\n",
    "ax = plt.axes(xlim=(s.x[0], s.x[-1]), ylim=(-3.0, 3.0))\n",
    "line1, = ax.plot([], [], lw=2, c='m', label=\"driving potential\")\n",
    "line2, = ax.plot([], [], lw=2, c='k', label=\"density\")\n",
    "line3, = ax.plot([], [], lw=2, c='g', label=\"hartree potential\")\n",
    "def init():\n",
    "    line1.set_data([], [])\n",
    "    line2.set_data([], [])\n",
    "    line3.set_data([], [])\n",
    "    return line1, line2, line3\n",
    "def animate(i):\n",
    "    line1.set_data(s.x, s.v_ext[:] + v_ptrb[i,:])\n",
    "    line2.set_data(s.x, n[i,:])\n",
    "    line3.set_data(s.x, v_h[i,:])\n",
    "    return line1, line2, line3\n",
    "plt.legend()\n",
    "anim = FuncAnimation(fig, animate, init_func=init, frames=100, interval=10, blit=True)\n",
    "HTML(anim.to_html5_video())"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c59430f7-b6d8-4f7b-a8f7-06825e3725d3",
   "metadata": {},
   "source": [
    "## 6. Approximate Methods."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e2eee005-c9cd-40c6-b8a7-7e100de19c93",
   "metadata": {},
   "source": [
    "As well as the exact solution, we can also compute everything we have in the previous sections, but with common approximations. To see the current list of approximations (more are added each release!) we can use the `dir` function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "23f80fdd-aaea-4830-8f18-841a576924c7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['Enum',\n",
       " 'Methods',\n",
       " '__builtins__',\n",
       " '__cached__',\n",
       " '__doc__',\n",
       " '__file__',\n",
       " '__loader__',\n",
       " '__name__',\n",
       " '__package__',\n",
       " '__path__',\n",
       " '__spec__',\n",
       " 'hartree',\n",
       " 'hartree_fock',\n",
       " 'hybrid',\n",
       " 'interacting',\n",
       " 'lda',\n",
       " 'non_interacting']"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dir(iDEA.methods)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ea833c6b-b8f5-4e75-9266-033e10d1ca0c",
   "metadata": {},
   "source": [
    "We can investigate a given method in a little more detail:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "ff7dcf47-ccd0-4b37-bcaf-b79e679c66fd",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Help on module iDEA.methods.hartree_fock in iDEA.methods:\n",
      "\n",
      "NAME\n",
      "    iDEA.methods.hartree_fock - Contains all Hartree Fock functionality and solvers.\n",
      "\n",
      "FUNCTIONS\n",
      "    exchange_potential_operator(s: iDEA.system.System, p: numpy.ndarray) -> numpy.ndarray\n",
      "        Compute the exchange potential operator.\n",
      "        \n",
      "        | Args:\n",
      "        |     s: iDEA.system.System, System object.\n",
      "        |     p: np.ndarray, Charge density matrix.\n",
      "        \n",
      "        | Returns:\n",
      "        |     Vx: np.ndarray, Exchange potential energy operator.\n",
      "    \n",
      "    hamiltonian(s: iDEA.system.System, up_n: numpy.ndarray, down_n: numpy.ndarray, up_p: numpy.ndarray, down_p: numpy.ndarray, K: numpy.ndarray = None, Vext: numpy.ndarray = None) -> numpy.ndarray\n",
      "        Compute the Hamiltonian from the kinetic and potential terms.\n",
      "        \n",
      "        | Args:\n",
      "        |     s: iDEA.system.System, System object.\n",
      "        |     up_n: np.ndarray, Charge density of up electrons.\n",
      "        |     down_n: np.ndarray, Charge density of down electrons.\n",
      "        |     up_p: np.ndarray, Charge density matrix of up electrons.\n",
      "        |     down_p: np.ndarray, Charge density matrix of down electrons.\n",
      "        |     K: np.ndarray, Single-particle kinetic energy operator [If None this will be computed from s]. (default = None)\n",
      "        |     Vext: np.ndarray, Potential energy operator [If None this will be computed from s]. (default = None)\n",
      "        \n",
      "        | Returns:\n",
      "        |     H: np.ndarray, Hamiltonian, up Hamiltonian, down Hamiltonian.\n",
      "    \n",
      "    propagate(s: iDEA.system.System, state: iDEA.state.SingleBodyState, v_ptrb: numpy.ndarray, t: numpy.ndarray, hamiltonian_function: collections.abc.Callable = None, restricted: bool = False) -> iDEA.state.SingleBodyEvolution\n",
      "        Propagate a set of orbitals forward in time due to a dynamic local pertubation.\n",
      "        \n",
      "        | Args:\n",
      "        |     s: iDEA.system.System, System object.\n",
      "        |     state: iDEA.state.SingleBodyState, State to be propigated.\n",
      "        |     v_ptrb: np.ndarray, Local perturbing potential on the grid of t and x values, indexed as v_ptrb[time,space].\n",
      "        |     t: np.ndarray, Grid of time values.\n",
      "        |     hamiltonian_function: Callable, Hamiltonian function [If None this will be the non_interacting function]. (default = None)\n",
      "        |     restricted: bool, Is the calculation restricted (r) on unrestricted (u). (default=False)\n",
      "        \n",
      "        | Returns:\n",
      "        |     evolution: iDEA.state.SingleBodyEvolution, Solved time-dependent evolution.\n",
      "    \n",
      "    solve(s: iDEA.system.System, k: int = 0, restricted: bool = False, mixing: float = 0.5, tol: float = 1e-10, initial: tuple = None, silent: bool = False) -> iDEA.state.SingleBodyState\n",
      "        Solves the Schrodinger equation for the given system.\n",
      "        \n",
      "        | Args:\n",
      "        |     s: iDEA.system.System, System object.\n",
      "        |     k: int, Energy state to solve for. (default = 0, the ground-state)\n",
      "        |     restricted: bool, Is the calculation restricted (r) on unrestricted (u). (default=False)\n",
      "        |     mixing: float, Mixing parameter. (default = 0.5)\n",
      "        |     tol: float, Tollerance of convergence. (default = 1e-10)\n",
      "        |     initial: tuple. Tuple of initial values used to begin the self-consistency (n, up_n, down_n, p, up_p, down_p). (default = None)\n",
      "        |     silent: bool, Set to true to prevent printing. (default = False)\n",
      "        \n",
      "        | Returns:\n",
      "        |     state: iDEA.state.SingleBodyState, Solved state.\n",
      "    \n",
      "    total_energy(s: iDEA.system.System, state: iDEA.state.SingleBodyState) -> float\n",
      "        Compute the total energy.\n",
      "        \n",
      "        | Args:\n",
      "        |     s: iDEA.system.System, System object.\n",
      "        |     state: iDEA.state.SingleBodyState, State. (default = None)\n",
      "        \n",
      "        | Returns:\n",
      "        |     E: float, Total energy.\n",
      "\n",
      "DATA\n",
      "    name = 'hartree_fock'\n",
      "\n",
      "FILE\n",
      "    /home/mjph501/.local/lib/python3.8/site-packages/iDEA/methods/hartree_fock.py\n",
      "\n",
      "\n"
     ]
    }
   ],
   "source": [
    "help(iDEA.methods.hartree_fock)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a6196509-990e-45e9-9afb-43ff1df2ad74",
   "metadata": {},
   "source": [
    "We can solve our system using **LDA** and compare this to our exact result:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "822036cb-3652-4114-b5d7-908faead067c",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iDEA.methods.lda.solve: convergence = 6.9313e-11, tolerance = 1e-10\n"
     ]
    }
   ],
   "source": [
    "ground_state_lda = iDEA.methods.lda.solve(s, k=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "id": "5a5856da-d1c7-4a59-8ce4-a9c7649439cb",
   "metadata": {},
   "outputs": [],
   "source": [
    "n_exact = iDEA.observables.density(s, state=ground_state)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "id": "831de8bb-cfcc-49e9-8a3f-11d3849d1aa8",
   "metadata": {},
   "outputs": [],
   "source": [
    "n_lda = iDEA.observables.density(s, state=ground_state_lda)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "id": "7b5043c4-7c22-4100-b2f7-0a250336dd62",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(s.x, n_exact, \"k-\", label=\"density (exact)\")\n",
    "plt.plot(s.x, n_lda, \"r--\", label=\"density (LDA)\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7fc503e8-10c0-4f81-a107-a2eb49ae86f0",
   "metadata": {},
   "source": [
    "`iDEA` makes it easy to compare many different approximations. For example, **we can compare the first-excited state charge density, for the two electron (ud) QHO for all of the methods using the following simple code:**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "279f6731-5ea0-49a3-a03c-72bf21e2b378",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iDEA.methods.interacting.solve: solving eigenproblem...\n",
      "iDEA.methods.non_interacting.solve: convergence = 9.0251e-12, tolerance = 1e-06\n",
      "iDEA.methods.hartree.solve: convergence = 7.6079e-07, tolerance = 1e-06\n",
      "iDEA.methods.hartree_fock.solve: convergence = 7.9506e-07, tolerance = 1e-06\n",
      "iDEA.methods.lda.solve: convergence = 9.5944e-07, tolerance = 1e-06\n",
      "iDEA.methods.hybrid.solve: convergence = 8.8926e-07, tolerance = 1e-06\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-15, 15, 200)\n",
    "v_ext = 0.5 * 0.25**2 * x**2\n",
    "v_int = iDEA.interactions.softened_interaction(x)\n",
    "s = iDEA.system.System(x, v_ext, v_int, electrons='ud')\n",
    "\n",
    "\n",
    "state_exact = iDEA.methods.interacting.solve(s, k=1)\n",
    "n_exact = iDEA.observables.density(s, state_exact)\n",
    "plt.plot(s.x, n_exact, label=\"exact\")\n",
    "\n",
    "for method in iDEA.iterate_sb_methods:\n",
    "    state = method.solve(s, k=1, tol=1e-6)\n",
    "    n = iDEA.observables.density(s, state)\n",
    "    plt.plot(s.x, n, label=method.name)\n",
    "plt.title(\"Comparing different approximations\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ceaccf84-033b-460d-ac08-210e9a58d32a",
   "metadata": {},
   "source": [
    "## 7. Reverse Engineering."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1ccfddcd-d694-4f30-8b57-11bf5efca6eb",
   "metadata": {},
   "source": [
    "`iDEA` also gives you the ability to reverse-engineer charge densities. **This tells you what fictitious system would need to be solved using a given method, to reproduce a target density.**\n",
    "\n",
    "The reverse function provides this functionality:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "id": "8c5a818c-faa8-4771-8cf4-72a0f4d2da2b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Help on function reverse in module iDEA.reverse_engineering:\n",
      "\n",
      "reverse(s: iDEA.system.System, target_n: numpy.ndarray, method: collections.abc.Container, v_guess: numpy.ndarray = None, mu: float = 1.0, pe: float = 0.1, tol: float = 1e-12, silent: bool = False, **kwargs) -> iDEA.state.State\n",
      "    Determines what ficticious system is needed for a given method, when solving the system, to produce a given target density.\n",
      "    If the given target density is from solving the interacting electron problem (iDEA.methods.interacting), and the method is the non-interacting electron solver (iDEA.methods.non_interacting)\n",
      "    the output is the Kohn-Sham system.\n",
      "    \n",
      "    The iterative method used is defined by the following formula:\n",
      "    .. math:: \\mathrm{V}_\\mathrm{ext} \\rightarrow \\mu * (\\mathrm{n}^p - \\mathrm{target_n}^p)\n",
      "    \n",
      "    | Args:\n",
      "    |     s: iDEA.system.System, System object.\n",
      "    |     target_n: np.ndarray, Target density to reverse engineer.\n",
      "    |     method: Container, The method used to solve the system.\n",
      "    |     v_guess: np.ndarray, The initial guess of the fictitious potential. (default = None)\n",
      "    |     mu: float = 1.0, Reverse engineering parameter mu. (default = 1.0)\n",
      "    |     pe: float = 0.1, Reverse engineering parameter p. (default = 0.1)\n",
      "    |     tol: float, Tollerance of convergence. (default = 1e-12)\n",
      "    |     silent: bool, Set to true to prevent printing. (default = False)\n",
      "    |     kwargs: Other arguments that will be given to the method's solve function.\n",
      "    \n",
      "    | Returns:\n",
      "    |     s_fictitious: iDEA.system.System, fictitious system object.\n",
      "\n"
     ]
    }
   ],
   "source": [
    "help(iDEA.reverse_engineering.reverse)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d6eade5e-4063-4f32-8df0-1cd527930602",
   "metadata": {},
   "source": [
    "This is a general method, but can be used to perform many common tasks in condensed matter physics."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dc77c517-c358-465f-b125-e8e926482f3d",
   "metadata": {},
   "source": [
    "For example, we can determine the exact Kohn-Sham (KS) and exchange-correlation (xc) potential of a system, by asking: **What fictitious system, when solved with the non-interacting method, gives us the same density as that solved by the interacting method.** The external potential of this fictitious system is none other than the exact KS potential, from which the exact xc potential can be found:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "id": "f2b25329-3650-440e-b7b6-628058ff978a",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iDEA.methods.interacting.solve: solving eigenproblem...\n",
      "iDEA.reverse_engineering.reverse: convergence = 1.0135e-12, tolerance = 1e-12\n",
      "iDEA.methods.non_interacting.solve: convergence = 6.8035e-12, tolerance = 1e-10\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-15, 15, 200)\n",
    "v_ext = 0.5 * 0.25**2 * x**2\n",
    "v_int = iDEA.interactions.softened_interaction(x)\n",
    "s = iDEA.system.System(x, v_ext, v_int, electrons='uu')\n",
    "\n",
    "\n",
    "state = iDEA.methods.interacting.solve(s, k=0)\n",
    "n = iDEA.observables.density(s, state)\n",
    "\n",
    "\n",
    "s_fictitious = iDEA.reverse_engineering.reverse(s, target_n=n, method=iDEA.methods.non_interacting)\n",
    "state_fictitious = iDEA.methods.non_interacting.solve(s_fictitious, k=0)\n",
    "\n",
    "\n",
    "v_ks = s_fictitious.v_ext\n",
    "n_fictitious = iDEA.observables.density(s, state_fictitious)\n",
    "v_h = iDEA.observables.hartree_potential(s_fictitious, n_fictitious)\n",
    "v_xc = v_ks - s.v_ext - v_h\n",
    "\n",
    "\n",
    "plt.plot(s_fictitious.x, s.v_ext, 'm-', label=\"V_ext\")\n",
    "plt.plot(s_fictitious.x, v_ks, 'r-', label=\"V_ks\")\n",
    "plt.plot(s_fictitious.x, v_h, 'g-', label=\"V_h\")\n",
    "plt.plot(s_fictitious.x, v_xc, 'c-', label=\"V_xc\")\n",
    "plt.legend()\n",
    "plt.ylim([-1.5, 2.0])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b14f21a8-83e2-48fe-8e76-7a20552ce305",
   "metadata": {},
   "source": [
    "Reverse-engineering can also be used to determine what time-dependent potential is needed in a fictitious evolution, in order to recover a given target density. A full example of this is shown below:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "b62d3572-305c-4e7c-ba97-bdd33b649490",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iDEA.methods.interacting.solve: solving eigenproblem...\n",
      "iDEA.reverse_engineering.reverse: convergence = 1.0093e-12, tolerance = 1e-12\n",
      "iDEA.methods.non_interacting.solve: convergence = 2.7436e-11, tolerance = 1e-10\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "iDEA.methods.interacting.propagate: propagating state: 100%|██████████████████████████████████████████████████████████████████████████| 100/100 [00:01<00:00, 58.52it/s]\n"
     ]
    },
    {
     "ename": "AttributeError",
     "evalue": "module 'numpy' has no attribute 'complex'.\n`np.complex` was a deprecated alias for the builtin `complex`. To avoid this error in existing code, use `complex` by itself. Doing this will not modify any behavior and is safe. If you specifically wanted the numpy scalar type, use `np.complex128` here.\nThe aliases was originally deprecated in NumPy 1.20; for more details and guidance see the original release note at:\n    https://numpy.org/devdocs/release/1.20.0-notes.html#deprecations",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mAttributeError\u001b[0m                            Traceback (most recent call last)",
      "Input \u001b[0;32mIn [4]\u001b[0m, in \u001b[0;36m<cell line: 24>\u001b[0;34m()\u001b[0m\n\u001b[1;32m     22\u001b[0m evolution \u001b[38;5;241m=\u001b[39m iDEA\u001b[38;5;241m.\u001b[39mmethods\u001b[38;5;241m.\u001b[39minteracting\u001b[38;5;241m.\u001b[39mpropagate(s, state, v_ptrb, t)\n\u001b[1;32m     23\u001b[0m target_n \u001b[38;5;241m=\u001b[39m iDEA\u001b[38;5;241m.\u001b[39mobservables\u001b[38;5;241m.\u001b[39mdensity(s, evolution\u001b[38;5;241m=\u001b[39mevolution)\n\u001b[0;32m---> 24\u001b[0m evolution_fictitious, error \u001b[38;5;241m=\u001b[39m \u001b[43miDEA\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mreverse_engineering\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mreverse_propagation\u001b[49m\u001b[43m(\u001b[49m\u001b[43ms_fictitious\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mstate_fictitious\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mtarget_n\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43miDEA\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mmethods\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mnon_interacting\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mv_ptrb\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mt\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m     25\u001b[0m evolution_fictitious \u001b[38;5;241m=\u001b[39m iDEA\u001b[38;5;241m.\u001b[39mmethods\u001b[38;5;241m.\u001b[39mnon_interacting\u001b[38;5;241m.\u001b[39mpropagate(s_fictitious, state_fictitious, evolution_fictitious\u001b[38;5;241m.\u001b[39mv_ptrb, t)\n\u001b[1;32m     28\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m ti \u001b[38;5;129;01min\u001b[39;00m \u001b[38;5;28mrange\u001b[39m( \u001b[38;5;28mlen\u001b[39m(t) ):\n",
      "File \u001b[0;32m~/.local/lib/python3.8/site-packages/iDEA/reverse_engineering.py:198\u001b[0m, in \u001b[0;36mreverse_propagation\u001b[0;34m(s_fictitious, state_fictitious, target_n, method, v_ptrb, t, restricted, tol, **kwargs)\u001b[0m\n\u001b[1;32m    190\u001b[0m \u001b[38;5;66;03m# Initilise the single-body time-dependent evolution.\u001b[39;00m\n\u001b[1;32m    191\u001b[0m evolution_fictitious \u001b[38;5;241m=\u001b[39m iDEA\u001b[38;5;241m.\u001b[39mstate\u001b[38;5;241m.\u001b[39mSingleBodyEvolution(state_fictitious)\n\u001b[1;32m    192\u001b[0m evolution_fictitious\u001b[38;5;241m.\u001b[39mup\u001b[38;5;241m.\u001b[39mtd_orbitals \u001b[38;5;241m=\u001b[39m np\u001b[38;5;241m.\u001b[39mzeros(\n\u001b[1;32m    193\u001b[0m     shape\u001b[38;5;241m=\u001b[39m(\n\u001b[1;32m    194\u001b[0m         t\u001b[38;5;241m.\u001b[39mshape[\u001b[38;5;241m0\u001b[39m],\n\u001b[1;32m    195\u001b[0m         s_fictitious\u001b[38;5;241m.\u001b[39mx\u001b[38;5;241m.\u001b[39mshape[\u001b[38;5;241m0\u001b[39m],\n\u001b[1;32m    196\u001b[0m         state_fictitious\u001b[38;5;241m.\u001b[39mup\u001b[38;5;241m.\u001b[39moccupied\u001b[38;5;241m.\u001b[39mshape[\u001b[38;5;241m0\u001b[39m],\n\u001b[1;32m    197\u001b[0m     ),\n\u001b[0;32m--> 198\u001b[0m     dtype\u001b[38;5;241m=\u001b[39m\u001b[43mnp\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mcomplex\u001b[49m,\n\u001b[1;32m    199\u001b[0m )\n\u001b[1;32m    200\u001b[0m evolution_fictitious\u001b[38;5;241m.\u001b[39mdown\u001b[38;5;241m.\u001b[39mtd_orbitals \u001b[38;5;241m=\u001b[39m np\u001b[38;5;241m.\u001b[39mzeros(\n\u001b[1;32m    201\u001b[0m     shape\u001b[38;5;241m=\u001b[39m(\n\u001b[1;32m    202\u001b[0m         t\u001b[38;5;241m.\u001b[39mshape[\u001b[38;5;241m0\u001b[39m],\n\u001b[0;32m   (...)\u001b[0m\n\u001b[1;32m    206\u001b[0m     dtype\u001b[38;5;241m=\u001b[39mnp\u001b[38;5;241m.\u001b[39mcomplex,\n\u001b[1;32m    207\u001b[0m )\n\u001b[1;32m    208\u001b[0m evolution_fictitious\u001b[38;5;241m.\u001b[39mup\u001b[38;5;241m.\u001b[39mtd_orbitals[\u001b[38;5;241m0\u001b[39m, :, :] \u001b[38;5;241m=\u001b[39m state_fictitious\u001b[38;5;241m.\u001b[39mup\u001b[38;5;241m.\u001b[39morbitals[\n\u001b[1;32m    209\u001b[0m     :, state_fictitious\u001b[38;5;241m.\u001b[39mup\u001b[38;5;241m.\u001b[39moccupied\n\u001b[1;32m    210\u001b[0m ]\n",
      "File \u001b[0;32m~/.local/lib/python3.8/site-packages/numpy/__init__.py:305\u001b[0m, in \u001b[0;36m__getattr__\u001b[0;34m(attr)\u001b[0m\n\u001b[1;32m    300\u001b[0m     warnings\u001b[38;5;241m.\u001b[39mwarn(\n\u001b[1;32m    301\u001b[0m         \u001b[38;5;124mf\u001b[39m\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mIn the future `np.\u001b[39m\u001b[38;5;132;01m{\u001b[39;00mattr\u001b[38;5;132;01m}\u001b[39;00m\u001b[38;5;124m` will be defined as the \u001b[39m\u001b[38;5;124m\"\u001b[39m\n\u001b[1;32m    302\u001b[0m         \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mcorresponding NumPy scalar.\u001b[39m\u001b[38;5;124m\"\u001b[39m, \u001b[38;5;167;01mFutureWarning\u001b[39;00m, stacklevel\u001b[38;5;241m=\u001b[39m\u001b[38;5;241m2\u001b[39m)\n\u001b[1;32m    304\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m attr \u001b[38;5;129;01min\u001b[39;00m __former_attrs__:\n\u001b[0;32m--> 305\u001b[0m     \u001b[38;5;28;01mraise\u001b[39;00m \u001b[38;5;167;01mAttributeError\u001b[39;00m(__former_attrs__[attr])\n\u001b[1;32m    307\u001b[0m \u001b[38;5;66;03m# Importing Tester requires importing all of UnitTest which is not a\u001b[39;00m\n\u001b[1;32m    308\u001b[0m \u001b[38;5;66;03m# cheap import Since it is mainly used in test suits, we lazy import it\u001b[39;00m\n\u001b[1;32m    309\u001b[0m \u001b[38;5;66;03m# here to save on the order of 10 ms of import time for most users\u001b[39;00m\n\u001b[1;32m    310\u001b[0m \u001b[38;5;66;03m#\u001b[39;00m\n\u001b[1;32m    311\u001b[0m \u001b[38;5;66;03m# The previous way Tester was imported also had a side effect of adding\u001b[39;00m\n\u001b[1;32m    312\u001b[0m \u001b[38;5;66;03m# the full `numpy.testing` namespace\u001b[39;00m\n\u001b[1;32m    313\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m attr \u001b[38;5;241m==\u001b[39m \u001b[38;5;124m'\u001b[39m\u001b[38;5;124mtesting\u001b[39m\u001b[38;5;124m'\u001b[39m:\n",
      "\u001b[0;31mAttributeError\u001b[0m: module 'numpy' has no attribute 'complex'.\n`np.complex` was a deprecated alias for the builtin `complex`. To avoid this error in existing code, use `complex` by itself. Doing this will not modify any behavior and is safe. If you specifically wanted the numpy scalar type, use `np.complex128` here.\nThe aliases was originally deprecated in NumPy 1.20; for more details and guidance see the original release note at:\n    https://numpy.org/devdocs/release/1.20.0-notes.html#deprecations"
     ]
    }
   ],
   "source": [
    "import scipy.special as spspec\n",
    "\n",
    "x = np.linspace(-10, 10, 100)\n",
    "v_ext = -2.0 / (abs(x) + 1.0)\n",
    "v_int = iDEA.interactions.softened_interaction(x)\n",
    "electrons = 'uu'\n",
    "s = iDEA.system.System(x, v_ext, v_int, electrons)\n",
    "\n",
    "\n",
    "t = np.linspace(0, 10, 100)\n",
    "v_ptrb = np.zeros(shape=t.shape+x.shape)\n",
    "for j, ti in enumerate(t):\n",
    "    v_ptrb[j,:] = -0.1 * x * 0.5*(spspec.erf(ti - 2.0) + 1.0)\n",
    "\n",
    "\n",
    "state = iDEA.methods.interacting.solve(s, k=0)\n",
    "target_n = iDEA.observables.density(s, state=state)\n",
    "s_fictitious = iDEA.reverse_engineering.reverse(s, target_n, method=iDEA.methods.non_interacting)\n",
    "state_fictitious = iDEA.methods.non_interacting.solve(s_fictitious)\n",
    "\n",
    "\n",
    "evolution = iDEA.methods.interacting.propagate(s, state, v_ptrb, t)\n",
    "target_n = iDEA.observables.density(s, evolution=evolution)\n",
    "evolution_fictitious, error = iDEA.reverse_engineering.reverse_propagation(s_fictitious, state_fictitious, target_n, iDEA.methods.non_interacting, v_ptrb, t)\n",
    "evolution_fictitious = iDEA.methods.non_interacting.propagate(s_fictitious, state_fictitious, evolution_fictitious.v_ptrb, t)\n",
    "\n",
    "\n",
    "for ti in range( len(t) ):\n",
    "    evolution_fictitious.v_ptrb[ti,:] -= evolution_fictitious.v_ptrb[ti,int(0.5*len(x))]\n",
    "\n",
    "\n",
    "n = iDEA.observables.density(s, evolution=evolution_fictitious)\n",
    "v_ext = iDEA.observables.external_potential(s)\n",
    "v_h = iDEA.observables.hartree_potential(s, n)\n",
    "\n",
    "\n",
    "fig = plt.figure()\n",
    "ax = plt.axes(xlim=(s.x[0], s.x[-1]), ylim=(-3.0, 3.0))\n",
    "line1, = ax.plot([], [], lw=2, c=\"k\", label=\"target density\")\n",
    "line2, = ax.plot([], [], lw=2, c=\"r\", ls=\"--\", label=\"fictitious density\")\n",
    "line3, = ax.plot([], [], lw=2, c=\"c\", label=\"exact KS potential\")\n",
    "line4, = ax.plot([], [], lw=2, c=\"g\", ls=\"--\", label=\"driving potential\")\n",
    "def init():\n",
    "    line1.set_data([], [])\n",
    "    line2.set_data([], [])\n",
    "    line3.set_data([], [])\n",
    "    line4.set_data([], [])\n",
    "    return line1, line2, line3, line4\n",
    "def animate(i):\n",
    "    line1.set_data(s.x, n[i,:])\n",
    "    line2.set_data(s.x, target_n[i,:])\n",
    "    line3.set_data(s.x, evolution_fictitious.v_ptrb[i,:])\n",
    "    line4.set_data(s.x, v_ptrb[i,:])\n",
    "    return line1, line2, line3, line4\n",
    "plt.legend()\n",
    "anim = FuncAnimation(fig, animate, init_func=init, frames=100, interval=10, blit=True)\n",
    "HTML(anim.to_html5_video())"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c20e85eb-b5b6-4729-be53-120d23884b8e",
   "metadata": {},
   "source": [
    "## 8. A full example: Investigating 1D hydrogen molecule."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fe903879-7d35-4bf7-ad91-f2f9eff13f08",
   "metadata": {},
   "source": [
    "Now you know all the basics of using iDEA to investigate and simulate quantum systems, here we will show you a \"real-world\" example.\n",
    "\n",
    "**In the following section of code, we will compute the exact bonding curve of the 1D hydrogen molecule, and compare it to a conventional figure.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "eb87a7b5-9ab8-4418-bba5-84e61bdbb51a",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Build the base system.\n",
    "x = np.linspace(-10, 10, 200)\n",
    "v_ext = np.zeros_like(x) \n",
    "v_int = iDEA.interactions.softened_interaction(x)\n",
    "electrons = 'ud'\n",
    "s = iDEA.system.System(x, v_ext, v_int, electrons)\n",
    "\n",
    "\n",
    "# Function to build the hydrogen molecule at each given distance d.\n",
    "d = 8.0\n",
    "def v(x):\n",
    "    return -1.0/(np.abs(x + 0.5*d) + 1.0) -1.0/(np.abs(x - 0.5*d) + 1.0)\n",
    "\n",
    "\n",
    "# Define range of bond distances.\n",
    "distances = np.linspace(0.0, 8.0, 50)\n",
    "energies = np.zeros_like(distances)\n",
    "\n",
    "\n",
    "# Compute exact electron energy at each distance.\n",
    "for i in range(distances.shape[0]):\n",
    "    d = distances[i]\n",
    "    print('distance = {0}'.format(d))\n",
    "    s.v_ext = np.copy(v(x))                     \n",
    "    state = iDEA.methods.interacting.solve(s)\n",
    "    energies[i] = state.energy\n",
    "\n",
    "\n",
    "# Add the proton energy.\n",
    "E_proton = 1.0 / (np.abs(distances) + 1.0)\n",
    "energies += E_proton\n",
    "\n",
    "\n",
    "# Plot energy curve.\n",
    "plt.axhline(y=-1.0, xmin=0.0, xmax=8.2, linewidth=1, color='k')\n",
    "plt.plot(distances, energies, linestyle='-', color='k', linewidth=2)\n",
    "plt.xlabel('distance (a.u.)', size=25)\n",
    "plt.ylabel('E (a.u.)', size=25)\n",
    "plt.tick_params(top='on', right='on')\n",
    "plt.tick_params(direction='in')\n",
    "plt.gcf().subplots_adjust(left=0.18)\n",
    "plt.gcf().subplots_adjust(bottom=0.18)\n",
    "plt.plot()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "96048e21-a1e9-4554-a02f-d34fcb036204",
   "metadata": {},
   "source": [
    "We can see this follows the expected curve:\n",
    "\n",
    "![Curve](diagram.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4ff78ef0-fc02-4ea9-b8f2-8b87235bd12a",
   "metadata": {},
   "source": [
    "Now we recommend, using the tools we have walked through in this tutorial, you gain more experience with the iDEA code by trying to solve some of these more advanced open-ended problems:\n",
    "\n",
    "1. What does the bonding curve from part 8 look like if the electrons have the same spin?\n",
    "2. What is the exact xc potential for the three electron 1D lithium atom.\n",
    "3. Which approximation yields the most accurate density matrix for the QHO.\n",
    "4. What kind of field is needed to ionise an electron out of an atom in real time?\n",
    "5. Can you modify the occupations of a Hartree-Fock calculation, to yield the exact density?"
   ]
  }
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