{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "ad6b13cb01007316fa509551e4c8b998",
     "grade": false,
     "grade_id": "cell-98f724ece1aacb67",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "# CDS: Numerical Methods -- Final Assignment\n",
    "\n",
    "- See lecture notes and documentation on Brightspace for Python and Jupyter basics. If you are stuck, try to google or get in touch via Discord.\n",
    "\n",
    "- Solutions must be submitted <font color=red>**individually**</font> via the Jupyter Hub until <font color=red>**Monday, April 4th, 23:59**</font>.\n",
    "\n",
    "- Make sure you fill in any place that says `YOUR CODE HERE` or \"YOUR ANSWER HERE\".\n",
    "\n",
    "- Remember to document your source codes (docstrings, comments where necessary) and to write it as clear as possible.\n",
    "\n",
    "- Do not forget to fully annotate all of your plots.\n",
    "\n",
    "## Submission\n",
    "\n",
    "1. make sure everything runs as expected\n",
    "2. **restart the kernel** (in the menubar, select Kernel$\\rightarrow$Restart)\n",
    "3. **run all cells** (in the menubar, select Cell$\\rightarrow$Run All)\n",
    "4. Check all outputs (Out[\\*]) for errors and **resolve them if necessary**\n",
    "5. submit your solutions  **in time (before the deadline)**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "23a115c4a147aab2185c76637a509f7f",
     "grade": false,
     "grade_id": "cell-fd297f265de59887",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "## Tight-Binding Propagation Method Module\n",
    "\n",
    "### Tight-Binding Theory\n",
    "\n",
    "Solid state theory aims to describe crystalline structures defined by periodic arrangements of atomic positions $\\vec{R}_i$ with $i= 1 \\dots n$. To model the electronic properties of such a structure, we can use the so-called tight-binding method. Here one assumes that the problem for a single atom described by the Hamiltonian $H_{at}(\\vec{r})$ has already been solved, so that the atomic wave functions $\\phi_m(\\vec{r})$ are known. The Hamiltonian of the crystalline structure is then constructed from these atomic Hamiltonians as follows \n",
    "\n",
    "\\begin{align*}\n",
    "    H(\\vec{r}) = \\sum_{i} H_{at}(\\vec{r} - \\vec{R}_i) + \\Delta V(\\vec{r}),\n",
    "\\end{align*}\n",
    "\n",
    "where $\\Delta V(\\vec{r})$ describes the changes to the atomic potentials due to the periodic arrangement. Solutions to the time-dependent Schrödinger equation $\\psi_n(\\vec{r})$ can then be approximated by linear combinations of the atomic orbitals, i.e. \n",
    "\n",
    "\\begin{align*}\n",
    "    \\psi_m(\\vec{r}) = \\sum_{i} \\, c_{i,m} \\, \\phi_m(\\vec{r}-\\vec{R}_i). \n",
    "\\end{align*}\n",
    "\n",
    "Thus, our task is to find the coefficients $c_{i,m}$, which are the eigenfunctions of the tight-binding Hamiltonian $H_{tb}$. In the basis of the atomic orbitals $H_{tb}$ is an $n \\times n$ matrix which describes the \"hopping\" of an electron from one atomic position to the other. In this description the electrons are assumed to be tightly bound to the atomic positions, hence the name of the approach. In summary, we have reduced our original problem $H(\\vec{r})$, described in a continuous space $\\vec{r}$, to a strongly discretized problem $H_{tb}$ in the space of lattice coordinates $\\vec{R}_i$.\n",
    "\n",
    "### Propagation Method\n",
    "\t\n",
    "While this reduction already helps a lot, full diagonalizations of the tight-binding matrix is still not feasible if we need to describe realistic structures with thousands of atoms. For this case we like to have a method which allows us to study the electronic properties, without the need of fully diagonalizing the tight-binding matrix. The tight-biding propagation method allows for exactly this. By analyzing the propagation of an initial electronic state through the crystalline structure we also have access to the full eigenspectrum of $H_{tb}$, without explicit diagonalization.\n",
    "\t\n",
    "### Your Goal\n",
    "    \n",
    "In the following you will setup the tight-binding Hamiltonian for a one-dimensional chain of atoms and numerically study its properties using exact diagonalization. Then you will compare it to the results obtained using the tight-binding propagation method. You will need some of the algorithms which you have implemented in the weekly assignments before. Additionally, you will need to implement a few new algorithms, which we have discussed in the last lecture. In principle there will be no need to use Numpy or Scipy (except for Numpy's array handling and a few other exceptions). However, if you encounter any problems with your own implementations of specific functionalities, you can use the Numpy and Scipy alternatives. Therefore you should be able to perform all of the following tasks in any case.\n",
    "\n",
    "Let us start by importing the necessary packages."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "5d73a1e28cac71eb63db02e72960f030",
     "grade": true,
     "grade_id": "cell-9a7b93b917f8bfed",
     "locked": false,
     "points": 0,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from matplotlib import pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "46edf5bfda2392bd3743329097a4e7ae",
     "grade": false,
     "grade_id": "cell-0f4a00fe587d193a",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "## Step 1: Crystal Lattice\n",
    "\n",
    "### Task 1.1 [3 points]\n",
    "\n",
    "In the following exercises the atomic positions of the 1D crystal lattice will be fixed to $\\vec{R}_i = x_i = i a$, with $i = 0 \\dots n-1$ and $a$ being the lattice constant.\n",
    "\n",
    "Write a simple Python function that takes the chain length $n$ as an argument and returns the atomic positions $x_i$. Set $a = 1$ for all the following exercises."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "bad6e1d563be71de711926b41649c875",
     "grade": true,
     "grade_id": "cell-65a97e8f9f981da1",
     "locked": false,
     "points": 3,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "def atomic_positions(n, a=1):\n",
    "    \"\"\"\n",
    "    Creates an array of atomic position in a 1D crystal lattice\n",
    "    for lattice constant a having default value a = 1.\n",
    "    \n",
    "    Args:\n",
    "        n: number of atoms in the 1D lattice string\n",
    "        a: numerical value for the lattice constant\n",
    "\n",
    "    Returns:\n",
    "        A 1D array of atomic positions.\n",
    "    \"\"\"\n",
    "    \n",
    "    return np.arange(n)*a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "684ad7f7808a1d5b4360a0acb4e52921",
     "grade": false,
     "grade_id": "cell-a61043ba1148856d",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "## Step 2: Atomic Basis Functions\n",
    "\n",
    "Our atomic basis functions will be Gaussians of the form\n",
    "$$\n",
    "\\large\n",
    "\\phi(x, \\mu, \\sigma) = \\frac{1}{\\pi^{1/4} \\sigma^{1/2}} e^{-\\frac{1}{2} \\left(\\frac{x-\\mu}{\\sigma}\\right)^2},\n",
    "$$\n",
    "\twhere $\\mu$ is their localization position and $\\sigma$ their broadenings. We also choose to have just one orbital per atom so that we can drop the index $m$ from now on. \n",
    "\t\n",
    "### Task 2.1 [4 points]\n",
    "Implement a Python function which calculates $\\phi(x, \\mu, \\sigma)$ for a whole array of arbitrary $x$, centered at given $\\mu$ with a given broadening $\\sigma$.\n",
    "\n",
    "Plot all the atomic basis functions for a chain with $n = 10$ atoms, using $\\sigma = 0.25$. I.e. plot $\\phi(x, x_i, \\sigma)$ vs $x$, for all atomic positions $x_i$ in the chain."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "2ad9cc4c03612d5b9bba4824cff364cb",
     "grade": true,
     "grade_id": "cell-4689e172e70a4762",
     "locked": false,
     "points": 4,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "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": [
    "def atomic_basis(x, mu, sigma):\n",
    "    \"\"\"\n",
    "    Calculates the atomic basis functions for the 1D chain of atoms.\n",
    "    \n",
    "    Args:\n",
    "        x:     array of positions to calculate the wavefunction at\n",
    "        mu:    atomic position to center Gaussian wavefunction at\n",
    "        sigma: broadening constant for Gaussian function\n",
    "\n",
    "    Returns:\n",
    "        A 1D array of values for the wavefunction over the positions\n",
    "        as given by x.\n",
    "    \"\"\"\n",
    "    \n",
    "    return np.pi**(-1/4)*sigma**(-1/2)*np.exp(-1/2*((x - mu)/sigma)**2)\n",
    "\n",
    "n = 10\n",
    "sigma = .25\n",
    "x = np.linspace(0, 10, 1000)\n",
    "\n",
    "plt.figure()\n",
    "plt.xlabel(\"$x$\")\n",
    "plt.ylabel(\"$\\phi$\")\n",
    "\n",
    "for mu in atomic_positions(n):\n",
    "    plt.plot(x, atomic_basis(x, mu, sigma), label=\"n = \" + str(mu))\n",
    "\n",
    "plt.legend()\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "e1c7774260f02916e34521c6236638f4",
     "grade": false,
     "grade_id": "cell-e5c9315357a401f9",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "### Task 2.2 [6 points]\n",
    "Implement a Python function to calculate numerical integrals (using for example the composite trapezoid or Simpson rule). This one should be general enough to calculate integrals $\\int_a^b f(x) dx$ for arbitrary functions $f(x)$, as you will need it for other tasks as well.\n",
    "\n",
    "Implement a simple unit test for your integration function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "aecc6d50a1ffd4e4bfbfe3573847edf8",
     "grade": true,
     "grade_id": "cell-d851197b213e5d2d",
     "locked": false,
     "points": 3,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "def integrate(yk, x):\n",
    "    \"\"\"\n",
    "    Numerically integrates function yk over [x[0], x[-1]] using Simpson's 3/8\n",
    "    rule over the grid provided by x.\n",
    "    \n",
    "    Args:\n",
    "        yk: function of one numerical argument that returns a numeric\n",
    "            or an array of function values such that x[i] corresponds to yk[i]\n",
    "        x:  array of numerics as argument to yk\n",
    "\n",
    "    Returns:\n",
    "        A numeric value for the quadrature of yk over x with error\n",
    "        of order \n",
    "    \"\"\"\n",
    "    \n",
    "    # If yk is callable, we use it to determine the function values\n",
    "    # over array x.\n",
    "    if callable(yk):\n",
    "        yk = yk(x)\n",
    "    \n",
    "    # The distance h_i = x[i + 1] - x[i] is not necessarily constant. The choice of\n",
    "    # partitioning of the interval is subject to mathematical considerations I will\n",
    "    # not go into.\n",
    "    h = x[1:] - x[:-1]\n",
    "    \n",
    "    integral = 0\n",
    "    integral += 3/8*(x[1] - x[0])*yk[0]\n",
    "    integral += 9/8*h[1::3]@yk[1:-1:3]\n",
    "    integral += 9/8*h[2::3]@yk[2:-1:3]\n",
    "    integral += 6/8*h[ ::3]@yk[ :-1:3]\n",
    "    integral += 3/8*(x[-1] - x[-2])*yk[-1]\n",
    "    return integral"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "6ab06c87cf65c73463ed243e46d63b3d",
     "grade": true,
     "grade_id": "cell-59912b2862fbce5a",
     "locked": false,
     "points": 3,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "def test_integrate():\n",
    "    # Test integral 1 of f with F its primitive with integration constant 0\n",
    "    f = lambda x: x**2\n",
    "    F = lambda x: x**3/3\n",
    "    x = np.logspace(0, 3, 1000000)\n",
    "    assert np.isclose(integrate(f, x), F(x[-1]) - F(x[0]))\n",
    "    \n",
    "    # Test integral 2 of f with F its primitive with integration constant 0\n",
    "    f = lambda x: np.sin(2*x)/(2 + np.cos(2*x))\n",
    "    F = lambda x: -.5*np.log(np.cos(2*x) + 2)\n",
    "    x = np.linspace(0, 10, 1000)\n",
    "    assert np.isclose(integrate(f, x), F(x[-1]) - F(x[0]))\n",
    "    \n",
    "test_integrate()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "8c1413a8a11006398e962e8c803ae001",
     "grade": false,
     "grade_id": "cell-86005829da536b5b",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "### Task 2.3 [2 points]\n",
    "Use your Python integration function to check the orthogonality of the Gaussian basis functions by verifying the following condition $$\\delta_{ij} = \\int_{-\\infty}^{+\\infty} \\phi(x, x_i, \\sigma) \\, \\phi(x, x_j, \\sigma) \\, dx,$$ where $\\delta_{ii} \\approx 1$ and $\\delta_{ij} \\approx 0$ for $\\sigma = 0.25$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "4751becb5d3cb7663536a0624b3d9c54",
     "grade": true,
     "grade_id": "cell-8a6a8db84dcef484",
     "locked": false,
     "points": 2,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "delta_00 = 0.999999999998739\n",
      "delta_01 = 0.018315638888710697\n",
      "delta_02 = 1.125351747191182e-07\n",
      "delta_03 = 2.319522830240641e-16\n",
      "delta_04 = 1.6038108905482657e-28\n",
      "delta_05 = 3.720075976030123e-44\n",
      "delta_06 = 2.894640311655602e-63\n",
      "delta_07 = 7.555819019718062e-86\n",
      "delta_08 = 6.616261056701202e-112\n",
      "delta_09 = 1.9435148500468318e-141\n",
      "delta_10 = 0.018315638888710697\n",
      "delta_11 = 0.9999999999987479\n",
      "delta_12 = 0.018315638888711055\n",
      "delta_13 = 1.1253517471923307e-07\n",
      "delta_14 = 2.3195228302493623e-16\n",
      "delta_15 = 1.6038108905526817e-28\n",
      "delta_16 = 3.7200759760238336e-44\n",
      "delta_17 = 2.89464031164467e-63\n",
      "delta_18 = 7.555819019702406e-86\n",
      "delta_19 = 6.616261056701183e-112\n",
      "delta_20 = 1.125351747191182e-07\n",
      "delta_21 = 0.018315638888711055\n",
      "delta_22 = 0.9999999999997685\n",
      "delta_23 = 0.018315638888779923\n",
      "delta_24 = 1.1253517471954291e-07\n",
      "delta_25 = 2.3195228302454397e-16\n",
      "delta_26 = 1.6038108905466284e-28\n",
      "delta_27 = 3.720075976016127e-44\n",
      "delta_28 = 2.8946403116446785e-63\n",
      "delta_29 = 7.55581901970241e-86\n",
      "delta_30 = 2.319522830240641e-16\n",
      "delta_31 = 1.1253517471923307e-07\n",
      "delta_32 = 0.018315638888779923\n",
      "delta_33 = 1.000000000002522\n",
      "delta_34 = 0.018315638888748948\n",
      "delta_35 = 1.1253517471911809e-07\n",
      "delta_36 = 2.319522830240635e-16\n",
      "delta_37 = 1.6038108905466259e-28\n",
      "delta_38 = 3.720075976016144e-44\n",
      "delta_39 = 2.8946403116461607e-63\n",
      "delta_40 = 1.6038108905482657e-28\n",
      "delta_41 = 2.3195228302493623e-16\n",
      "delta_42 = 1.1253517471954291e-07\n",
      "delta_43 = 0.018315638888748948\n",
      "delta_44 = 0.9999999999987468\n",
      "delta_45 = 0.018315638888711006\n",
      "delta_46 = 1.1253517471911791e-07\n",
      "delta_47 = 2.31952283024064e-16\n",
      "delta_48 = 1.6038108905474523e-28\n",
      "delta_49 = 3.7200759760298655e-44\n",
      "delta_50 = 3.720075976030123e-44\n",
      "delta_51 = 1.6038108905526817e-28\n",
      "delta_52 = 2.3195228302454397e-16\n",
      "delta_53 = 1.1253517471911809e-07\n",
      "delta_54 = 0.018315638888711006\n",
      "delta_55 = 0.9999999999987453\n",
      "delta_56 = 0.01831563888871106\n",
      "delta_57 = 1.1253517471917587e-07\n",
      "delta_58 = 2.3195228302491977e-16\n",
      "delta_59 = 1.6038108905526882e-28\n",
      "delta_60 = 2.894640311655602e-63\n",
      "delta_61 = 3.7200759760238336e-44\n",
      "delta_62 = 1.6038108905466284e-28\n",
      "delta_63 = 2.319522830240635e-16\n",
      "delta_64 = 1.1253517471911791e-07\n",
      "delta_65 = 0.01831563888871106\n",
      "delta_66 = 0.9999999999992604\n",
      "delta_67 = 0.01831563888877863\n",
      "delta_68 = 1.1253517471954331e-07\n",
      "delta_69 = 2.319522830246971e-16\n",
      "delta_70 = 7.555819019718062e-86\n",
      "delta_71 = 2.89464031164467e-63\n",
      "delta_72 = 3.720075976016127e-44\n",
      "delta_73 = 1.6038108905466259e-28\n",
      "delta_74 = 2.31952283024064e-16\n",
      "delta_75 = 1.1253517471917587e-07\n",
      "delta_76 = 0.01831563888877863\n",
      "delta_77 = 1.0000000000025255\n",
      "delta_78 = 0.01831563888876104\n",
      "delta_79 = 1.1253517471912122e-07\n",
      "delta_80 = 6.616261056701202e-112\n",
      "delta_81 = 7.555819019702406e-86\n",
      "delta_82 = 2.8946403116446785e-63\n",
      "delta_83 = 3.720075976016144e-44\n",
      "delta_84 = 1.6038108905474523e-28\n",
      "delta_85 = 2.3195228302491977e-16\n",
      "delta_86 = 1.1253517471954331e-07\n",
      "delta_87 = 0.01831563888876104\n",
      "delta_88 = 0.9999999999987745\n",
      "delta_89 = 0.018315638888711034\n",
      "delta_90 = 1.9435148500468318e-141\n",
      "delta_91 = 6.616261056701183e-112\n",
      "delta_92 = 7.55581901970241e-86\n",
      "delta_93 = 2.8946403116461607e-63\n",
      "delta_94 = 3.7200759760298655e-44\n",
      "delta_95 = 1.6038108905526882e-28\n",
      "delta_96 = 2.319522830246971e-16\n",
      "delta_97 = 1.1253517471912122e-07\n",
      "delta_98 = 0.018315638888711034\n",
      "delta_99 = 0.9999999999987306\n"
     ]
    }
   ],
   "source": [
    "n = 10\n",
    "sigma = .25\n",
    "\n",
    "positions = atomic_positions(n)\n",
    "infty = 10000\n",
    "x = np.linspace(-infty, infty, 1000000)\n",
    "\n",
    "for i in range(n):\n",
    "    for j in range(n):\n",
    "        integrand = lambda x: atomic_basis(x, positions[i], sigma)*atomic_basis(x, positions[j], sigma)\n",
    "        print(\"delta_{}{} = {}\".format(i, j, integrate(integrand, x)))\n",
    "\n",
    "# Yann had output:\n",
    "#delta_00 = 1.00000\n",
    "#delta_01 = 0.01832\n",
    "#delta_02 = 0.00000\n",
    "#delta_34 = 0.01832\n",
    "# Explanation: next neighbours migth have some overlap. Further away, no overlap at all."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "e9ccbed5ba3e6b844bcc6e326053d8da",
     "grade": false,
     "grade_id": "cell-3cba7034f4eac62f",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "## Step 3: Tight-Binding Hamiltonian\n",
    "\n",
    "The tight-binding Hamiltonian for our 1D chain should describe the hopping of an electron from all atomic positions to their nearest left and right neighbours (i.e. no long-range hopping). The resulting matrix representation in the basis of the discrete $x_i$ positions is therefore given as a tri-diagonal $n \\times n$ matrix of the form\n",
    "\n",
    "\\begin{align}\n",
    "    \\mathbf{H}_{tb} =\n",
    "    \\left( \\begin{array}{cccc}\n",
    "    0 & t & & 0\\\\\n",
    "    t & \\ddots & \\ddots &  \\\\\n",
    "      & \\ddots & \\ddots &  t \\\\\n",
    "    0 &  & t & 0\n",
    "    \\end{array}  \\right),\n",
    "\\end{align}\n",
    "\n",
    "where $t = t_{i,i\\pm1}$ is the nearest-neighbour hopping matrix element. A hopping matrix element $t_{i,j}$ is a measure for the probability of an electron to hop from site $i$ to site $j$. They are defined as\n",
    "\n",
    "\\begin{align}\n",
    "    t_{i,j} = \\int_{-\\infty}^{+\\infty} \\phi(x, x_i, \\sigma) \\, \\Delta V(x) \\, \\phi(x, x_j, \\sigma) \\, dx,\n",
    "\\end{align}\n",
    "\n",
    "with the potential fixed to\n",
    "\n",
    "\\begin{align}\n",
    "    \\Delta V(x) = \\sum_i \\frac{-1}{|x - x_i| + 0.001}.\n",
    "\\end{align}\n",
    "\n",
    "### Task 3.1 [3 points]\n",
    "Write a Python function to calculate $t_{i,j}$, using $\\sigma = 0.25$. The function should have as input the indices $i$ and $j$, and the chain length $n$. Verify that the long-range hoppings $t_{i,i\\pm2}$ and $t_{i,i\\pm3}$ are negligible compared to $t_{i,i\\pm1}$.\n",
    "\n",
    "Hint: use your integration function from task 2.2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "260ae3c806429aee5900599c01cb65c6",
     "grade": true,
     "grade_id": "cell-0abfcd1aa9fad2fa",
     "locked": false,
     "points": 3,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "def hopping(i, j, n):\n",
    "    \"\"\"\n",
    "    \n",
    "    \"\"\"\n",
    "    \n",
    "    x = np.linspace(-1e12, 1e12, 1e12)\n",
    "    V = lambda x: -1 \n",
    "    integrand = lambda x: atomic_basis(x, positions[i], sigma)*V(x)*atomic_basis(x, positions[j], sigma)\n",
    "    return integrate(integrand, x)\n",
    "\n",
    "# Yann had\n",
    "# 1) a plot of the peaks as before\n",
    "# 2) a weird line under it with peaks downward at these x\n",
    "#    and slowly rising energy values outside the chain\n",
    "# 3)\n",
    "# For i = 1 ...\n",
    "#     t_{i,i-1} = -0.1203\n",
    "#     t_{i,i+1} = ..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "b1a56ecde33e723ff450defcf5dc2e74",
     "grade": true,
     "grade_id": "cell-ea36ee5a2b35154c",
     "locked": false,
     "points": 0,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "# Verify that long-range hoppings are negligible here ...\n",
    "\n",
    "# YOUR CODE HERE\n",
    "raise NotImplementedError()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "7d2c8f74993fe38c2c979376961f869a",
     "grade": false,
     "grade_id": "cell-8a0f18c44306ae00",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "### Task 3.2 [3 points]\n",
    "Implement a diagonalization routine for tri-diagonal matrices which returns all eigenvalues, for example using the $QR$ decomposition (it is fine to use Numpy's $\\text{qr()}$). \n",
    "\n",
    "Hint: For tri-diagonal matrices with vanishing diagonal elements, the $QR$-decomposition-based diagonalization algorithm gets trapped. To get around this you could, for example, add a diagonal $1$ to your matrix, and later subtract $1$ from each eigenvalue."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "c20cbcce0a7df50b6ae7b90c7aa35721",
     "grade": true,
     "grade_id": "cell-9d4942b717eadeb2",
     "locked": false,
     "points": 3,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "# YOUR CODE HERE\n",
    "raise NotImplementedError()\n",
    "\n",
    "# Yann said we just have to be able to solve tri-diagonal matrix diagonalization, but more general is ok."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "19976946c5746804cb08c34f0bda50fc",
     "grade": false,
     "grade_id": "cell-2d8fb5c080951dd5",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "### Task 3.3 [3 points]\n",
    "Implement a unit test for your diagonalization routine."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "d95777361c07514a97ff1458f26f4f44",
     "grade": true,
     "grade_id": "cell-001cb3c043c4e371",
     "locked": false,
     "points": 3,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "# YOUR CODE HERE\n",
    "raise NotImplementedError()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "83df149b46d779a846f9de925342b681",
     "grade": false,
     "grade_id": "cell-85c89b0eb0930f2b",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "### Task 3.4 [4 points]\n",
    "First, write a function that generates your tight-binding Hamiltonian $\\mathbf{H}_{tb}$, for a given chain length $n$. Use $t = t_{i,i\\pm1}$, as calculated in task 3.1. You can choose any $i$ near the center of the chain for the calculation of $t$, as the chain is (approximately) periodic.\n",
    "\n",
    "Second, use your diagonalization routine to calculate all the eigenvalues $E_m$, for a variety of $n=10,20,40,80,100$. Sort the resulting $E_m$ and plot them vs. $m$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "a0779b496fd41a4664bb0cdd857c70fc",
     "grade": true,
     "grade_id": "cell-764cb41c37700042",
     "locked": false,
     "points": 3,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "def TBHamiltonian(n):\n",
    "    # YOUR CODE HERE\n",
    "    raise NotImplementedError()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "634e139137eead8808d1d8ccb793d5a5",
     "grade": true,
     "grade_id": "cell-39ada0528e69d2e5",
     "locked": false,
     "points": 1,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "# Do your plotting here ...\n",
    "\n",
    "# YOUR CODE HERE\n",
    "raise NotImplementedError()\n",
    "\n",
    "# Yann had a plot with a couple of logisticly looking rising lines,\n",
    "# starting at the same x point, but some higher starting y value,\n",
    "# ending nonintersection some higher some lower."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "c2b46d2fef4b0c243103a5a6f1111e2d",
     "grade": false,
     "grade_id": "cell-b7c84b8c4ed4c1be",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "### Task 3.5 [3 points]\n",
    "Implement a function to calculate the so-called density-of-states \n",
    "\n",
    "\\begin{align*}\n",
    "    \\rho(\\omega) = \\frac{1}{N} \\sum_i \\delta(\\omega - E_i),\n",
    "\\end{align*}\n",
    "\n",
    "for a variable energy grid $\\omega$. Do this by approximating the $\\delta$-distribution with a Gaussian. In detail, you can use your atomic orbital function $\\delta(\\omega - E_i) \\approx \\phi(\\omega, E_i, \\sigma_\\rho)$. Calculate the normalization factor $N$ such that $\\int \\rho(\\omega) dw = 1$ is fulfilled.\n",
    "\n",
    "Your function should take as input the energy grid $\\omega$, the eigenenergies $E_i$ and the broadening $\\sigma_\\rho$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "ca46cf0b09305fafb522fc0395d1e495",
     "grade": true,
     "grade_id": "cell-d7c225b7687b5a9c",
     "locked": false,
     "points": 3,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "def getDOS_ED(w, Ei, sigma):\n",
    "    # YOUR CODE HERE\n",
    "    raise NotImplementedError()"
   ]
  },
  {
   "attachments": {
    "dosN010.png": {
     "image/png": "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"
    },
    "dosN100.png": {
     "image/png": "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"
    }
   },
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "b2049bb4d5fb56c59e7d0e742a91d3c8",
     "grade": false,
     "grade_id": "cell-7560c4658b1da5d3",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "### Task 3.6 [3 points]\n",
    "Use your density-of-states routine to calculate $\\rho(\\omega)$ for $n=10,20,40,80,100$ for $\\sigma_\\rho \\approx 0.005$. See below for two examples with $t \\approx -0.195$ and $n=10$ and $n=100$.\n",
    "\n",
    "Hint: if your plots look like they are smoothed out, try decreasing $\\sigma_\\rho$. If they look like there is a lot of noise, try increasing $\\sigma_\\rho$.\n",
    "\n",
    "$n = 10$ | $n = 100$\n",
    ":-: | :-:\n",
    "![dosN010.png](attachment:dosN010.png) | ![dosN100.png](attachment:dosN100.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "10cb847540f9e1998c9c1b40c5e43a7b",
     "grade": true,
     "grade_id": "cell-c3083a03553a2aa9",
     "locked": false,
     "points": 3,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "# YOUR CODE HERE\n",
    "raise NotImplementedError()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "69c457b9ef8fbd13ad935fe12c37c81c",
     "grade": false,
     "grade_id": "cell-362439917c95705f",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "## Step 4: Tight-Binding Propagation Method\n",
    "\n",
    "Now we turn to the time-dependent Schrödinger equation\n",
    "\n",
    "\\begin{align}\n",
    "    i\\hbar\\frac{\\partial}{\\partial t} \\psi(x,t) = H \\psi(x,t),\n",
    "\\end{align}\n",
    "\n",
    "which has the formal solution\n",
    "\n",
    "\\begin{align}\n",
    "    \\psi(x,t) = U(t) \\psi(x,t=0),\n",
    "\\end{align}\n",
    "\n",
    "with \n",
    "\n",
    "\\begin{align}\n",
    "    U(t) = e^{-i \\hbar H t}\n",
    "\\end{align}\n",
    "\n",
    "being the time-propagation operator. Within the propagation method we can calculate the so-called local density-of-states\n",
    "\n",
    "\\begin{align}\n",
    "    \\rho_{loc}(\\omega) = \\frac{1}{2\\pi} \\int_{-\\infty}^{+\\infty} \\, e^{i\\omega t} \\, f(t) \\ dt,\n",
    "\\end{align}\n",
    "\n",
    "with respect to an (arbitrary) initial state $\\psi(x,t=0)$, where\n",
    "\n",
    "\\begin{align}\n",
    "    f(t) &= \\int_{-\\infty}^{+\\infty} \\, \\psi^*(x,t) \\, \\psi(x,t=0) \\, dx \\\\\n",
    "         &\\approx \\int_{-\\infty}^{+\\infty} \\sum_i c_i^*(t) \\phi(x,x_i,\\sigma) \\, \\sum_j c_j(0) \\phi(x,x_j,\\sigma) \\, dx \\notag \\\\\n",
    "         &\\approx \\sum_i c_i^*(t) c_i(0). \\notag\n",
    "\\end{align}\n",
    "\n",
    "Thus, the time propagation of an initial state towards positive *and* negative times followed by a Fourier transform of $f(t)$ yields the local density-of-states. To obtain the full density-of-states we need to average $\\rho_{loc}(\\omega)$ as follows\n",
    "\n",
    "\\begin{align}\n",
    "    \\rho(\\omega) = \\lim_{S \\to \\infty} \\frac{1}{S} \\sum_p^S \\rho^{(p)}_{loc}(\\omega)\n",
    "\\end{align}\n",
    "\n",
    "over a variety of *random* initial states $p$.\n",
    "\n",
    "### Task 4.1 [3 points]\n",
    "Implement a function which calculates the exact time-propagation matrix $U(\\tau)$ for a small time-step $\\tau$ given the Hamiltonian $H$. For simplicity, set $\\hbar = 1$ in the following. \n",
    "\n",
    "Hint: Use Scipy's $\\text{expm()}$ function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "6cb01e4b3c6c192a0df3c4111b91c8fa",
     "grade": true,
     "grade_id": "cell-42a7aac3f0fa4d1b",
     "locked": false,
     "points": 3,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "def getU_exact(tau, H):\n",
    "    # YOUR CODE HERE\n",
    "    raise NotImplementedError()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "79f8101b73fbb28ff9138437e9767178",
     "grade": false,
     "grade_id": "cell-9b02ad5515424242",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "### Task 4.2 [3 points]\n",
    "Implement a function which performs the step-by-step time propagation given an initial state $\\vec{c}(0)$, the matrix $U(\\tau)$ and the discretized time grid $t_j$. In other words, your function should calculate \n",
    "\n",
    "$$\\vec{c}(j+1) = U(\\tau) \\cdot \\vec{c}(j)$$ \n",
    "\n",
    "for all $j$ of a given discretized time grid $t_j = j \\tau$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "dc040cc32e832b097bfb8c367f4203a1",
     "grade": true,
     "grade_id": "cell-4e444f44bf3bc9c1",
     "locked": false,
     "points": 3,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "def timePropagate(U, c0, t):\n",
    "    # YOUR CODE HERE\n",
    "    raise NotImplementedError()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "61362905e7a2d19219ae21f10a417823",
     "grade": false,
     "grade_id": "cell-62bfe608c358ff6d",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "### Task 4.3 [4 points]\n",
    "Use both of the above functions to calculate and animate the time propagation of an initial state\n",
    "\n",
    "$$\\psi(x,t=0) = \\phi(x, x_{i=n/2}, \\sigma) \\leftrightarrow \\vec{c}(0) = [c_{i=n/2}(0) = 1, c_{i\\neq n/2}(0) = 0]$$\n",
    "\n",
    "for a $n=100$ chain. Discretize your time grid as $t_j=j\\tau$ with $j=0 \\dots 200$, and $\\tau=1.5$. Use again $a = 1$ and $\\sigma=0.25$. \n",
    "\n",
    "To plot / animate the time propagation you should plot the real-space wave function $\\psi(x,t) \\approx \\sum_i c_i(t) \\phi(x, x_i, \\sigma)$.\n",
    "\n",
    "Hint: use your function from task 3.4 to get the Hamiltonian $H$.\n",
    "\n",
    "For the animation you can use the following draft:\n",
    "```python\n",
    "# use matplotlib's animation package\n",
    "import matplotlib.pylab as plt\n",
    "import matplotlib\n",
    "import matplotlib.animation as animation\n",
    "# set the animation style to \"jshtml\" (for the use in Jupyter)\n",
    "matplotlib.rcParams['animation.html'] = 'jshtml'\n",
    "\n",
    "# create a figure for the animation\n",
    "fig = plt.figure()\n",
    "plt.grid(True)\n",
    "plt.xlim( ... )     # fix x limits\n",
    "plt.ylim( ... )     # fix y limits\n",
    "\n",
    "# Create an empty plot object and prevent its showing (we will fill it each frame)\n",
    "myPlot, = plt.plot([0], [0])\n",
    "plt.close()\n",
    "\n",
    "# This function is called each frame to generate the animation (f is the frame number)\n",
    "def animate(f):                   \n",
    "    myPlot.set_data( ... )  # update plot\n",
    "\n",
    "# Show the animation\n",
    "frames = np.arange(1, np.size(t))  # t is the time grid here\n",
    "myAnimation = animation.FuncAnimation(fig, animate, frames, interval = 20)\n",
    "myAnimation\n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "dac0e917be4cfe57c7d30715f3f61912",
     "grade": true,
     "grade_id": "cell-dd676b90f6a61df6",
     "locked": false,
     "points": 4,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "# YOUR CODE HERE\n",
    "raise NotImplementedError()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "6786036a70e4fffbda4c92e340ff90de",
     "grade": true,
     "grade_id": "cell-70e223783d806888",
     "locked": false,
     "points": 0,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "# Animate here ...\n",
    "\n",
    "# YOUR CODE HERE\n",
    "raise NotImplementedError()\n",
    "\n",
    "# Yann has an animation about an atomic orbital that starts\n",
    "# moving to left and right and then bounce back."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "74ab18f8b5e98bc5456ef221449f9299",
     "grade": false,
     "grade_id": "cell-0395602360fd9e4c",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "### Task 4.4 [3 points]\n",
    "Implement a function which calculates the Crank-Nicolson time-propagation matrix \n",
    "\n",
    "\\begin{align*}\n",
    "    U_{CN}(\\tau) = (I - i \\tau H / 2)\\cdot(I + i \\tau H / 2)^{-1}.\n",
    "\\end{align*}\n",
    "\n",
    "Here, $I$ is the diagonal identity matrix. Use Numpy's $\\text{inv()}$ function to invert the needed expression."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "1b2677753953d9a528f0dbb71d4077bb",
     "grade": true,
     "grade_id": "cell-d74914e5d0a13365",
     "locked": false,
     "points": 3,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "def getU_CN(tau, H):\n",
    "    # YOUR CODE HERE\n",
    "    raise NotImplementedError()\n",
    "\n",
    "# Yann notes that the definition of $U_{CN}(\\tau)$ here is a little\n",
    "# different from what Malte used on the slides. He recommends using\n",
    "# what is stated here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "3746f3298575d0e0c37d35c01039e60e",
     "grade": false,
     "grade_id": "cell-1daec83575502040",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "### Task 4.5 [5 points]\n",
    "Implement a function which calculates the time-propagation matrix using the Trotter-Suzuki decomposition \n",
    "\n",
    "\\begin{align*}\n",
    "    U_{TZ}(\\tau) = e^{-i\\tau H_1} \\cdot e^{-i \\tau H_2}.\n",
    "\\end{align*}\n",
    "\n",
    "In this approach you choose a decomposition of the tight-binding Hamiltonian $H = H_1 + H_2$, which allows you to analytically diagonalize $H_1$ and $H_2$ (see last lecture). From this analytic diagonalization you will be able to calculate the matrix exponentials $e^{-i\\tau H_1}$ and $e^{-i \\tau H_2}$.\n",
    "\n",
    "Write your definition of the 2x2 blocks in $e^{-i\\tau H_1}$ and $e^{-i \\tau H_2}$ in the Markdown cell below. (Double click on \"YOUR ANSWER HERE\" to open the cell, and ctrl+enter to compile.) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "566fe9a7f8031baea9812438b155671c",
     "grade": true,
     "grade_id": "cell-bef909a443eb2a68",
     "locked": false,
     "points": 2,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "source": [
    "YOUR ANSWER HERE"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "f9f25759b1a81bbac8c1834c2f4565b8",
     "grade": true,
     "grade_id": "cell-1425de6027596dea",
     "locked": false,
     "points": 3,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "def getU_TZ(tau, H):\n",
    "    # YOUR CODE HERE\n",
    "    raise NotImplementedError()\n",
    "\n",
    "# Yann mentions again that this is slightly different wrong what\n",
    "# is in the slides/lecture."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "1747285f36e24921cb5c2811632f33c3",
     "grade": false,
     "grade_id": "cell-f53dc443bd1858b1",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "### Task 4.6 [3 points]\n",
    "In your implementation of $U_{TZ}(\\tau)$ you analytically evaluate the matrix exponentials $e^{-i\\tau H_1}$ and $e^{-i \\tau H_2}$. Test your implementation by comparing your results for these matrix exponentials to those obtained using Scipy's $\\text{expm()}$ function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "684e4173792cb10809386ef097c561e4",
     "grade": true,
     "grade_id": "cell-5aa3ffce9359fa7e",
     "locked": false,
     "points": 3,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "# YOUR CODE HERE\n",
    "raise NotImplementedError()\n",
    "\n",
    "# Yann printed\n",
    "#Biggest differences of U1 with Scipy:\n",
    "#Real: 1e-16 \n",
    "#Imag: 2.77e-17\n",
    "#    \n",
    "# and difference with U_exact in the order of 1e-1 or 1e-2."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "94ca5bdd479043f3c73214a3c4916923",
     "grade": false,
     "grade_id": "cell-c255a2bf5eac4e2b",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "### Task 4.7 [6 points]\n",
    "In the next task you will need a Fourier transform to calculate the local density-of-states. Therefore you will need to implement a function that returns the Fourier transform $f(\\omega)$ of a given function $f(t)$ defined on a time grid $t$, for a given energy grid $\\omega$. I.e. it should calculate:\n",
    "\n",
    "\\begin{align}\n",
    "    f(\\omega) = \\frac{1}{2\\pi} \\int_{-\\infty}^{+\\infty} \\, e^{i\\omega t} \\, f(t) \\ dt.\n",
    "\\end{align}\n",
    "\n",
    "Hint: use your integration function from task 2.2.\n",
    "\n",
    "Then implement a unit test for your function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "37055009cb70e69bc9b1dbc761859c51",
     "grade": true,
     "grade_id": "cell-87ece8e50b1f8de5",
     "locked": false,
     "points": 3,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "# YOUR CODE HERE\n",
    "raise NotImplementedError()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "49aa23055a91a51494bcb9d64924cc75",
     "grade": true,
     "grade_id": "cell-46e1530333341bc6",
     "locked": false,
     "points": 3,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "# Implement your unit test here ...\n",
    "\n",
    "# YOUR CODE HERE\n",
    "raise NotImplementedError()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "028bf13b6bf982c70fd1057c9d6f23f6",
     "grade": false,
     "grade_id": "cell-dc5656a6bdea875a",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "### Task 4.8 [3 points]\n",
    "Calculate the local density-of-states $\\rho_{loc}(\\omega)$ from the Fourier transform of $f(t)$ using all three time propagation methods: $U(\\tau)$, $U_{CN}(\\tau)$ and $U_{TZ}(\\tau)$.\n",
    "\n",
    "Start from $\\psi(x,t=0) = \\phi(x, x_{i=0}, \\sigma)$ and $\\psi(x,t=0) = \\phi(x, x_{i=n/2}, \\sigma)$, using a $n=100$ chain. Discretize your integration time grid as $t_j=j\\tau$, with $j=-150 \\dots 150$ and $\\tau=1.5$. Use again $a = 1$ and $\\sigma=0.25$.\n",
    "\n",
    "Be careful: for the Fourier transform you will need positive *and* negative time steps! Thus you will need to do two time propagations: one using $U(\\tau)$ towards positive times and one using $U(-\\tau)$ towards negative times, both starting from $\\psi(x,t=0)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "29ff620823bca3839839fbc35ba9b236",
     "grade": true,
     "grade_id": "cell-316f9c26031f89df",
     "locked": false,
     "points": 3,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "# YOUR CODE HERE\n",
    "raise NotImplementedError()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "1c27de03eb5f84673d52e1e621c316ee",
     "grade": true,
     "grade_id": "cell-d7a678fdeef64ea2",
     "locked": false,
     "points": 0,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "# Do your own testing here ...\n",
    "\n",
    "# YOUR CODE HERE\n",
    "raise NotImplementedError()\n",
    "\n",
    "# Yann had a plot for Tau = 1.5\n",
    "# DOS: looking like a hill (\"like a dome with a peak around zero energy 0\")\n",
    "# for CN, TS and the exact one\n",
    "# a plot of f(t)\n",
    "# a plot of local DOS\n",
    "# in the title he mentiones the inital values."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": false,
    "editable": false,
    "nbgrader": {
     "cell_type": "markdown",
     "checksum": "c3e0ecb1b67f93590abf1a796bd507b8",
     "grade": false,
     "grade_id": "cell-ffbf1e8460ac69d8",
     "locked": true,
     "schema_version": 3,
     "solution": false,
     "task": false
    }
   },
   "source": [
    "### Task 4.9 [6 points]\n",
    "Use the Trotter-Suzuki decomposition to calculate the full density-of-states by averaging over about $100$ local density-of-states you obtained from the time propagation of $100$ random initial states $\\vec{c}(0)$. To this end, you will need to make sure that each $\\vec{c}(0)$ is (a) normalized and (b) can have positive *and* negative elements. \n",
    "\n",
    "Compare this approximation to the total density-of-states to the exact one from task 3.6, which you obtained directly from the eigenvalues.\n",
    "\n",
    "Hint: don't expect the results to be the exact same. Check for the location of the peaks, and whether they have a similar order of magnitude.\n",
    "\n",
    "Hint: if you did not get the Trotter-Suzuki decomposition to work, you can instead use the exact or the Crank-Nicolson time-propagation matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "81edbb8d07068d29021696fd87a961ba",
     "grade": true,
     "grade_id": "cell-2493a46a63277eda",
     "locked": false,
     "points": 3,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "# YOUR CODE HERE\n",
    "raise NotImplementedError()\n",
    "\n",
    "# Yann says the initial states do need to be negative, too."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "deletable": false,
    "nbgrader": {
     "cell_type": "code",
     "checksum": "5438067dfec55e69ee224e67178d9e36",
     "grade": true,
     "grade_id": "cell-a40dfcd993da467c",
     "locked": false,
     "points": 3,
     "schema_version": 3,
     "solution": true,
     "task": false
    }
   },
   "outputs": [],
   "source": [
    "# Do your plotting here ...\n",
    "\n",
    "# YOUR CODE HERE\n",
    "raise NotImplementedError()\n",
    "\n",
    "# Yann plotted the exact diagonalisation and the TS propagation results\n",
    "# he had two plots, one peaky, one with peaks on the edges (looking a little\n",
    "# like my 1f/2f results in my bachelor internship hmmpfff)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}