tutorial/402_excitedstate.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Calculating excited states with the Variational Quantum Eigensolver(VQE)\n",
    "\n",
    "VQE optimizes a wave function to minimize the expectation value of Hamiltonian. Therefore, VQE can only calculate the ground state. In this time, we will explain how to calculate the excited state using VQE, and we will show the example of the calculation for the excited state of the hydrogen molecule. \n",
    "\n",
    "## Theoretical Outline\n",
    "This time we will use an [Overlap-based method](https://quantum-journal.org/papers/q-2019-07-01-156/) to calculate the $1^{st}$ excited state. This method uses a penalty term for the overlap between the wavefunctions from VQE and the ground state. The excited state can be obtained by adding the penalty term to Hamiltonian. An equation is as follows\n",
    "$$\n",
    "H^\\prime=H+\\alpha \\mid \\psi_{gs}\\rangle\\langle\\psi_{gs} \\vert\n",
    "$$\n",
    "The $H$ on the right side is the original Hamiltonian. The second term on the right side evaluates the value of the overlap with the ground-state wave function $ \\mid \\psi_{gs}\\rangle$ (In this tutorial, $ \\mid \\psi_{gs}\\rangle$ is obtained by a classical calculation, but it is also possible to use VQE). $\\alpha$ in the second term is a parameter that represents a penalty, and we take a large enough value about it. By adding the penalty term to the original $H$ term, we define $H^\\prime$ on the left side.\n",
    "\n",
    "We evaluate the expected value by using the wave function $ \\mid \\psi_{VQE}\\rangle$ obtained from the VQE.\n",
    "$$\n",
    "\\langle\\psi_{VQE} \\mid H^\\prime \\mid \\psi_{VQE}\\rangle=\\langle\\psi_{VQE} \\mid H \\mid \\psi_{VQE}\\rangle+\\alpha\\langle\\psi_{VQE} \\mid \\psi_{gs}\\rangle\\langle\\psi_{gs} \\mid \\psi_{VQE}\\rangle \\\\\n",
    "=\\langle\\psi_{VQE} \\mid H \\mid \\psi_{VQE}\\rangle+\\alpha \\mid \\langle\\psi_{gs} \\mid \\psi_{VQE}\\rangle \\mid ^2\n",
    "$$\n",
    "Since the excited state is orthogonal to the ground state, the second term in the above equation is zero if $ \\mid \\psi_{VQE}\\rangle$ becomes the excited state. In this time, we will use this expectation value to implement the excited-state calculation of hydrogen molecule in VQE.\n",
    "\n",
    "Incidentally, If the wanted wavefunction is not the $1^{st}$ excited state but the $N^{th}$ excited state, penalty terms are created by using the wavefunction from the ground state to the ${N-1}^{th}$ excited state.\n",
    "$$\n",
    "H^\\prime=H+\\sum_{i=gs}^{N-1}\\alpha_i \\mid \\psi_{i}\\rangle\\langle\\psi_{i} \\mid \n",
    "$$\n",
    "\n",
    "## Preparation\n",
    "Install the necessary libraries. The Hamiltonian is obtained with OpenFermion."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "!pip3 install blueqat openfermion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Import the necessary libraries. The optimization of VQE uses SciPy minimize."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "from blueqat import Circuit\n",
    "from openfermion.hamiltonians import MolecularData\n",
    "from openfermion.transforms import get_fermion_operator, jordan_wigner, get_sparse_operator\n",
    "import numpy as np\n",
    "from scipy.optimize import minimize"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ansatz definition and molecular information acquisition\n",
    "Define the functions for calculating VQE and obtaining information about the molecule. The calculations are performed using Hardware Efficient Ansatz. The initial wavefunction is set to $\\mid1010\\rangle$.\n",
    "The molecular information is obtained using OpenFermion. The basis set is STO-3G."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "def HEA(params,n_qubits,n_depth):\n",
    "    #Wave function initialization |1010>\n",
    "    circ=Circuit().x[1, 3]\n",
    "\n",
    "    #Circuit creation\n",
    "    params_devided=np.array_split(params,n_depth)\n",
    "    for params_one_depth in params_devided:\n",
    "        for i,param in enumerate(params_one_depth):\n",
    "            if i < n_qubits:\n",
    "                circ.ry(param)[i]\n",
    "            else:\n",
    "                circ.rz(param)[i%n_qubits]\n",
    "        for qbit in range(n_qubits):\n",
    "            if qbit < n_qubits-1:\n",
    "                circ.cz[qbit,qbit+1]\n",
    "    \n",
    "    #Running the circuit\n",
    "    wf = circ.run(backend=\"numpy\")\n",
    "    return wf\n",
    "\n",
    "def get_molecule(length):\n",
    "    geometry = [('H',(0.,0.,0.)),('H',(0.,0.,length))]\n",
    "    try:\n",
    "        description = f'{length:.2f}'\n",
    "        molecule = MolecularData(geometry, \"sto-3g\",1,description=description)\n",
    "        molecule.load()\n",
    "    except:\n",
    "        description = f'{length:.1f}'\n",
    "        molecule = MolecularData(geometry, \"sto-3g\",1,description=description)\n",
    "        molecule.load()\n",
    "    return molecule"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Expected value and cost function\n",
    "The expected value is calculated using the wave function obtained by VQE. In the expected value calculation, the term concerning the overlap between the ground state and VQE wave functions is added to the original expected value of the Hamiltonian."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "def expect_excited_state(wf,hamiltonian,penalty,wf_gs):\n",
    "    #Calculating Hamiltonian+penalty \n",
    "    hamiltonian_overlap = np.vdot(wf, hamiltonian.dot(wf)).real+penalty*np.abs(np.vdot(wf_gs,wf))**2\n",
    "    return hamiltonian_overlap\n",
    "\n",
    "def cost(params,hamiltonian,n_qubits,n_depth,penalty,wf_gs):\n",
    "    wf=HEA(params,n_qubits,n_depth)\n",
    "    return expect_excited_state(wf,hamiltonian,penalty,wf_gs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculation execution and plotting\n",
    "Run VQE on each bond length (this will take a few minutes). This time, the wave function of the ground state is classically obtained, but it can be calculated by VQE. The penalty value is 10 Hartree.\n",
    "After the calculation, compare the calculated results of VQE, ground state, and first excited state for energy and bond length."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "<matplotlib.legend.Legend at 0x7f26a3366760>"
     },
     "metadata": {},
     "execution_count": 14
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "<Figure size 432x288 with 1 Axes>",
      "image/svg+xml": "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n  \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n<!-- Created with matplotlib (https://matplotlib.org/) -->\n<svg height=\"262.19625pt\" version=\"1.1\" viewBox=\"0 0 400.729029 262.19625\" width=\"400.729029pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n <defs>\n  <style type=\"text/css\">\n*{stroke-linecap:butt;stroke-linejoin:round;}\n  </style>\n </defs>\n <g id=\"figure_1\">\n  <g id=\"patch_1\">\n   <path d=\"M 0 262.19625 \nL 400.729029 262.19625 \nL 400.729029 0 \nL 0 0 \nz\n\" style=\"fill:none;\"/>\n  </g>\n  <g id=\"axes_1\">\n   <g id=\"patch_2\">\n    <path d=\"M 52.160938 224.64 \nL 386.960938 224.64 \nL 386.960938 7.2 \nL 52.160938 7.2 \nz\n\" style=\"fill:#ffffff;\"/>\n   </g>\n   <g id=\"matplotlib.axis_1\">\n    <g id=\"xtick_1\">\n     <g id=\"line2d_1\">\n      <defs>\n       <path d=\"M 0 0 \nL 0 3.5 \n\" id=\"m07b821a253\" style=\"stroke:#000000;stroke-width:0.8;\"/>\n      </defs>\n      <g>\n       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"108.883252\" xlink:href=\"#m07b821a253\" y=\"224.64\"/>\n      </g>\n     </g>\n     <g id=\"text_1\">\n      <!-- 0.5 -->\n      <defs>\n       <path d=\"M 31.78125 66.40625 \nQ 24.171875 66.40625 20.328125 58.90625 \nQ 16.5 51.421875 16.5 36.375 \nQ 16.5 21.390625 20.328125 13.890625 \nQ 24.171875 6.390625 31.78125 6.390625 \nQ 39.453125 6.390625 43.28125 13.890625 \nQ 47.125 21.390625 47.125 36.375 \nQ 47.125 51.421875 43.28125 58.90625 \nQ 39.453125 66.40625 31.78125 66.40625 \nz\nM 31.78125 74.21875 \nQ 44.046875 74.21875 50.515625 64.515625 \nQ 56.984375 54.828125 56.984375 36.375 \nQ 56.984375 17.96875 50.515625 8.265625 \nQ 44.046875 -1.421875 31.78125 -1.421875 \nQ 19.53125 -1.421875 13.0625 8.265625 \nQ 6.59375 17.96875 6.59375 36.375 \nQ 6.59375 54.828125 13.0625 64.515625 \nQ 19.53125 74.21875 31.78125 74.21875 \nz\n\" id=\"DejaVuSans-48\"/>\n       <path d=\"M 10.6875 12.40625 \nL 21 12.40625 \nL 21 0 \nL 10.6875 0 \nz\n\" id=\"DejaVuSans-46\"/>\n       <path d=\"M 10.796875 72.90625 \nL 49.515625 72.90625 \nL 49.515625 64.59375 \nL 19.828125 64.59375 \nL 19.828125 46.734375 \nQ 21.96875 47.46875 24.109375 47.828125 \nQ 26.265625 48.1875 28.421875 48.1875 \nQ 40.625 48.1875 47.75 41.5 \nQ 54.890625 34.8125 54.890625 23.390625 \nQ 54.890625 11.625 47.5625 5.09375 \nQ 40.234375 -1.421875 26.90625 -1.421875 \nQ 22.3125 -1.421875 17.546875 -0.640625 \nQ 12.796875 0.140625 7.71875 1.703125 \nL 7.71875 11.625 \nQ 12.109375 9.234375 16.796875 8.0625 \nQ 21.484375 6.890625 26.703125 6.890625 \nQ 35.15625 6.890625 40.078125 11.328125 \nQ 45.015625 15.765625 45.015625 23.390625 \nQ 45.015625 31 40.078125 35.4375 \nQ 35.15625 39.890625 26.703125 39.890625 \nQ 22.75 39.890625 18.8125 39.015625 \nQ 14.890625 38.140625 10.796875 36.28125 \nz\n\" id=\"DejaVuSans-53\"/>\n      </defs>\n      <g transform=\"translate(100.931689 239.238437)scale(0.1 -0.1)\">\n       <use xlink:href=\"#DejaVuSans-48\"/>\n       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-46\"/>\n       <use x=\"95.410156\" xlink:href=\"#DejaVuSans-53\"/>\n      </g>\n     </g>\n    </g>\n    <g id=\"xtick_2\">\n     <g id=\"line2d_2\">\n      <g>\n       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"178.056805\" xlink:href=\"#m07b821a253\" y=\"224.64\"/>\n      </g>\n     </g>\n     <g id=\"text_2\">\n      <!-- 1.0 -->\n      <defs>\n       <path d=\"M 12.40625 8.296875 \nL 28.515625 8.296875 \nL 28.515625 63.921875 \nL 10.984375 60.40625 \nL 10.984375 69.390625 \nL 28.421875 72.90625 \nL 38.28125 72.90625 \nL 38.28125 8.296875 \nL 54.390625 8.296875 \nL 54.390625 0 \nL 12.40625 0 \nz\n\" id=\"DejaVuSans-49\"/>\n      </defs>\n      <g transform=\"translate(170.105243 239.238437)scale(0.1 -0.1)\">\n       <use xlink:href=\"#DejaVuSans-49\"/>\n       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-46\"/>\n       <use x=\"95.410156\" xlink:href=\"#DejaVuSans-48\"/>\n      </g>\n     </g>\n    </g>\n    <g id=\"xtick_3\">\n     <g id=\"line2d_3\">\n      <g>\n       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"247.230359\" xlink:href=\"#m07b821a253\" y=\"224.64\"/>\n      </g>\n     </g>\n     <g id=\"text_3\">\n      <!-- 1.5 -->\n      <g transform=\"translate(239.278796 239.238437)scale(0.1 -0.1)\">\n       <use xlink:href=\"#DejaVuSans-49\"/>\n       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-46\"/>\n       <use x=\"95.410156\" xlink:href=\"#DejaVuSans-53\"/>\n      </g>\n     </g>\n    </g>\n    <g id=\"xtick_4\">\n     <g id=\"line2d_4\">\n      <g>\n       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"316.403913\" xlink:href=\"#m07b821a253\" y=\"224.64\"/>\n      </g>\n     </g>\n     <g id=\"text_4\">\n      <!-- 2.0 -->\n      <defs>\n       <path d=\"M 19.1875 8.296875 \nL 53.609375 8.296875 \nL 53.609375 0 \nL 7.328125 0 \nL 7.328125 8.296875 \nQ 12.9375 14.109375 22.625 23.890625 \nQ 32.328125 33.6875 34.8125 36.53125 \nQ 39.546875 41.84375 41.421875 45.53125 \nQ 43.3125 49.21875 43.3125 52.78125 \nQ 43.3125 58.59375 39.234375 62.25 \nQ 35.15625 65.921875 28.609375 65.921875 \nQ 23.96875 65.921875 18.8125 64.3125 \nQ 13.671875 62.703125 7.8125 59.421875 \nL 7.8125 69.390625 \nQ 13.765625 71.78125 18.9375 73 \nQ 24.125 74.21875 28.421875 74.21875 \nQ 39.75 74.21875 46.484375 68.546875 \nQ 53.21875 62.890625 53.21875 53.421875 \nQ 53.21875 48.921875 51.53125 44.890625 \nQ 49.859375 40.875 45.40625 35.40625 \nQ 44.1875 33.984375 37.640625 27.21875 \nQ 31.109375 20.453125 19.1875 8.296875 \nz\n\" id=\"DejaVuSans-50\"/>\n      </defs>\n      <g transform=\"translate(308.45235 239.238437)scale(0.1 -0.1)\">\n       <use xlink:href=\"#DejaVuSans-50\"/>\n       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-46\"/>\n       <use x=\"95.410156\" xlink:href=\"#DejaVuSans-48\"/>\n      </g>\n     </g>\n    </g>\n    <g id=\"xtick_5\">\n     <g id=\"line2d_5\">\n      <g>\n       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"385.577466\" xlink:href=\"#m07b821a253\" y=\"224.64\"/>\n      </g>\n     </g>\n     <g id=\"text_5\">\n      <!-- 2.5 -->\n      <g transform=\"translate(377.625904 239.238437)scale(0.1 -0.1)\">\n       <use xlink:href=\"#DejaVuSans-50\"/>\n       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-46\"/>\n       <use x=\"95.410156\" xlink:href=\"#DejaVuSans-53\"/>\n      </g>\n     </g>\n    </g>\n    <g id=\"text_6\">\n     <!-- Bond length (Angstrom) -->\n     <defs>\n      <path d=\"M 19.671875 34.8125 \nL 19.671875 8.109375 \nL 35.5 8.109375 \nQ 43.453125 8.109375 47.28125 11.40625 \nQ 51.125 14.703125 51.125 21.484375 \nQ 51.125 28.328125 47.28125 31.5625 \nQ 43.453125 34.8125 35.5 34.8125 \nz\nM 19.671875 64.796875 \nL 19.671875 42.828125 \nL 34.28125 42.828125 \nQ 41.5 42.828125 45.03125 45.53125 \nQ 48.578125 48.25 48.578125 53.8125 \nQ 48.578125 59.328125 45.03125 62.0625 \nQ 41.5 64.796875 34.28125 64.796875 \nz\nM 9.8125 72.90625 \nL 35.015625 72.90625 \nQ 46.296875 72.90625 52.390625 68.21875 \nQ 58.5 63.53125 58.5 54.890625 \nQ 58.5 48.1875 55.375 44.234375 \nQ 52.25 40.28125 46.1875 39.3125 \nQ 53.46875 37.75 57.5 32.78125 \nQ 61.53125 27.828125 61.53125 20.40625 \nQ 61.53125 10.640625 54.890625 5.3125 \nQ 48.25 0 35.984375 0 \nL 9.8125 0 \nz\n\" id=\"DejaVuSans-66\"/>\n      <path d=\"M 30.609375 48.390625 \nQ 23.390625 48.390625 19.1875 42.75 \nQ 14.984375 37.109375 14.984375 27.296875 \nQ 14.984375 17.484375 19.15625 11.84375 \nQ 23.34375 6.203125 30.609375 6.203125 \nQ 37.796875 6.203125 41.984375 11.859375 \nQ 46.1875 17.53125 46.1875 27.296875 \nQ 46.1875 37.015625 41.984375 42.703125 \nQ 37.796875 48.390625 30.609375 48.390625 \nz\nM 30.609375 56 \nQ 42.328125 56 49.015625 48.375 \nQ 55.71875 40.765625 55.71875 27.296875 \nQ 55.71875 13.875 49.015625 6.21875 \nQ 42.328125 -1.421875 30.609375 -1.421875 \nQ 18.84375 -1.421875 12.171875 6.21875 \nQ 5.515625 13.875 5.515625 27.296875 \nQ 5.515625 40.765625 12.171875 48.375 \nQ 18.84375 56 30.609375 56 \nz\n\" id=\"DejaVuSans-111\"/>\n      <path d=\"M 54.890625 33.015625 \nL 54.890625 0 \nL 45.90625 0 \nL 45.90625 32.71875 \nQ 45.90625 40.484375 42.875 44.328125 \nQ 39.84375 48.1875 33.796875 48.1875 \nQ 26.515625 48.1875 22.3125 43.546875 \nQ 18.109375 38.921875 18.109375 30.90625 \nL 18.109375 0 \nL 9.078125 0 \nL 9.078125 54.6875 \nL 18.109375 54.6875 \nL 18.109375 46.1875 \nQ 21.34375 51.125 25.703125 53.5625 \nQ 30.078125 56 35.796875 56 \nQ 45.21875 56 50.046875 50.171875 \nQ 54.890625 44.34375 54.890625 33.015625 \nz\n\" id=\"DejaVuSans-110\"/>\n      <path d=\"M 45.40625 46.390625 \nL 45.40625 75.984375 \nL 54.390625 75.984375 \nL 54.390625 0 \nL 45.40625 0 \nL 45.40625 8.203125 \nQ 42.578125 3.328125 38.25 0.953125 \nQ 33.9375 -1.421875 27.875 -1.421875 \nQ 17.96875 -1.421875 11.734375 6.484375 \nQ 5.515625 14.40625 5.515625 27.296875 \nQ 5.515625 40.1875 11.734375 48.09375 \nQ 17.96875 56 27.875 56 \nQ 33.9375 56 38.25 53.625 \nQ 42.578125 51.265625 45.40625 46.390625 \nz\nM 14.796875 27.296875 \nQ 14.796875 17.390625 18.875 11.75 \nQ 22.953125 6.109375 30.078125 6.109375 \nQ 37.203125 6.109375 41.296875 11.75 \nQ 45.40625 17.390625 45.40625 27.296875 \nQ 45.40625 37.203125 41.296875 42.84375 \nQ 37.203125 48.484375 30.078125 48.484375 \nQ 22.953125 48.484375 18.875 42.84375 \nQ 14.796875 37.203125 14.796875 27.296875 \nz\n\" id=\"DejaVuSans-100\"/>\n      <path id=\"DejaVuSans-32\"/>\n      <path d=\"M 9.421875 75.984375 \nL 18.40625 75.984375 \nL 18.40625 0 \nL 9.421875 0 \nz\n\" id=\"DejaVuSans-108\"/>\n      <path d=\"M 56.203125 29.59375 \nL 56.203125 25.203125 \nL 14.890625 25.203125 \nQ 15.484375 15.921875 20.484375 11.0625 \nQ 25.484375 6.203125 34.421875 6.203125 \nQ 39.59375 6.203125 44.453125 7.46875 \nQ 49.3125 8.734375 54.109375 11.28125 \nL 54.109375 2.78125 \nQ 49.265625 0.734375 44.1875 -0.34375 \nQ 39.109375 -1.421875 33.890625 -1.421875 \nQ 20.796875 -1.421875 13.15625 6.1875 \nQ 5.515625 13.8125 5.515625 26.8125 \nQ 5.515625 40.234375 12.765625 48.109375 \nQ 20.015625 56 32.328125 56 \nQ 43.359375 56 49.78125 48.890625 \nQ 56.203125 41.796875 56.203125 29.59375 \nz\nM 47.21875 32.234375 \nQ 47.125 39.59375 43.09375 43.984375 \nQ 39.0625 48.390625 32.421875 48.390625 \nQ 24.90625 48.390625 20.390625 44.140625 \nQ 15.875 39.890625 15.1875 32.171875 \nz\n\" id=\"DejaVuSans-101\"/>\n      <path d=\"M 45.40625 27.984375 \nQ 45.40625 37.75 41.375 43.109375 \nQ 37.359375 48.484375 30.078125 48.484375 \nQ 22.859375 48.484375 18.828125 43.109375 \nQ 14.796875 37.75 14.796875 27.984375 \nQ 14.796875 18.265625 18.828125 12.890625 \nQ 22.859375 7.515625 30.078125 7.515625 \nQ 37.359375 7.515625 41.375 12.890625 \nQ 45.40625 18.265625 45.40625 27.984375 \nz\nM 54.390625 6.78125 \nQ 54.390625 -7.171875 48.1875 -13.984375 \nQ 42 -20.796875 29.203125 -20.796875 \nQ 24.46875 -20.796875 20.265625 -20.09375 \nQ 16.0625 -19.390625 12.109375 -17.921875 \nL 12.109375 -9.1875 \nQ 16.0625 -11.328125 19.921875 -12.34375 \nQ 23.78125 -13.375 27.78125 -13.375 \nQ 36.625 -13.375 41.015625 -8.765625 \nQ 45.40625 -4.15625 45.40625 5.171875 \nL 45.40625 9.625 \nQ 42.625 4.78125 38.28125 2.390625 \nQ 33.9375 0 27.875 0 \nQ 17.828125 0 11.671875 7.65625 \nQ 5.515625 15.328125 5.515625 27.984375 \nQ 5.515625 40.671875 11.671875 48.328125 \nQ 17.828125 56 27.875 56 \nQ 33.9375 56 38.28125 53.609375 \nQ 42.625 51.21875 45.40625 46.390625 \nL 45.40625 54.6875 \nL 54.390625 54.6875 \nz\n\" id=\"DejaVuSans-103\"/>\n      <path d=\"M 18.3125 70.21875 \nL 18.3125 54.6875 \nL 36.8125 54.6875 \nL 36.8125 47.703125 \nL 18.3125 47.703125 \nL 18.3125 18.015625 \nQ 18.3125 11.328125 20.140625 9.421875 \nQ 21.96875 7.515625 27.59375 7.515625 \nL 36.8125 7.515625 \nL 36.8125 0 \nL 27.59375 0 \nQ 17.1875 0 13.234375 3.875 \nQ 9.28125 7.765625 9.28125 18.015625 \nL 9.28125 47.703125 \nL 2.6875 47.703125 \nL 2.6875 54.6875 \nL 9.28125 54.6875 \nL 9.28125 70.21875 \nz\n\" id=\"DejaVuSans-116\"/>\n      <path d=\"M 54.890625 33.015625 \nL 54.890625 0 \nL 45.90625 0 \nL 45.90625 32.71875 \nQ 45.90625 40.484375 42.875 44.328125 \nQ 39.84375 48.1875 33.796875 48.1875 \nQ 26.515625 48.1875 22.3125 43.546875 \nQ 18.109375 38.921875 18.109375 30.90625 \nL 18.109375 0 \nL 9.078125 0 \nL 9.078125 75.984375 \nL 18.109375 75.984375 \nL 18.109375 46.1875 \nQ 21.34375 51.125 25.703125 53.5625 \nQ 30.078125 56 35.796875 56 \nQ 45.21875 56 50.046875 50.171875 \nQ 54.890625 44.34375 54.890625 33.015625 \nz\n\" id=\"DejaVuSans-104\"/>\n      <path d=\"M 31 75.875 \nQ 24.46875 64.65625 21.28125 53.65625 \nQ 18.109375 42.671875 18.109375 31.390625 \nQ 18.109375 20.125 21.3125 9.0625 \nQ 24.515625 -2 31 -13.1875 \nL 23.1875 -13.1875 \nQ 15.875 -1.703125 12.234375 9.375 \nQ 8.59375 20.453125 8.59375 31.390625 \nQ 8.59375 42.28125 12.203125 53.3125 \nQ 15.828125 64.359375 23.1875 75.875 \nz\n\" id=\"DejaVuSans-40\"/>\n      <path d=\"M 34.1875 63.1875 \nL 20.796875 26.90625 \nL 47.609375 26.90625 \nz\nM 28.609375 72.90625 \nL 39.796875 72.90625 \nL 67.578125 0 \nL 57.328125 0 \nL 50.6875 18.703125 \nL 17.828125 18.703125 \nL 11.1875 0 \nL 0.78125 0 \nz\n\" id=\"DejaVuSans-65\"/>\n      <path d=\"M 44.28125 53.078125 \nL 44.28125 44.578125 \nQ 40.484375 46.53125 36.375 47.5 \nQ 32.28125 48.484375 27.875 48.484375 \nQ 21.1875 48.484375 17.84375 46.4375 \nQ 14.5 44.390625 14.5 40.28125 \nQ 14.5 37.15625 16.890625 35.375 \nQ 19.28125 33.59375 26.515625 31.984375 \nL 29.59375 31.296875 \nQ 39.15625 29.25 43.1875 25.515625 \nQ 47.21875 21.78125 47.21875 15.09375 \nQ 47.21875 7.46875 41.1875 3.015625 \nQ 35.15625 -1.421875 24.609375 -1.421875 \nQ 20.21875 -1.421875 15.453125 -0.5625 \nQ 10.6875 0.296875 5.421875 2 \nL 5.421875 11.28125 \nQ 10.40625 8.6875 15.234375 7.390625 \nQ 20.0625 6.109375 24.8125 6.109375 \nQ 31.15625 6.109375 34.5625 8.28125 \nQ 37.984375 10.453125 37.984375 14.40625 \nQ 37.984375 18.0625 35.515625 20.015625 \nQ 33.0625 21.96875 24.703125 23.78125 \nL 21.578125 24.515625 \nQ 13.234375 26.265625 9.515625 29.90625 \nQ 5.8125 33.546875 5.8125 39.890625 \nQ 5.8125 47.609375 11.28125 51.796875 \nQ 16.75 56 26.8125 56 \nQ 31.78125 56 36.171875 55.265625 \nQ 40.578125 54.546875 44.28125 53.078125 \nz\n\" id=\"DejaVuSans-115\"/>\n      <path d=\"M 41.109375 46.296875 \nQ 39.59375 47.171875 37.8125 47.578125 \nQ 36.03125 48 33.890625 48 \nQ 26.265625 48 22.1875 43.046875 \nQ 18.109375 38.09375 18.109375 28.8125 \nL 18.109375 0 \nL 9.078125 0 \nL 9.078125 54.6875 \nL 18.109375 54.6875 \nL 18.109375 46.1875 \nQ 20.953125 51.171875 25.484375 53.578125 \nQ 30.03125 56 36.53125 56 \nQ 37.453125 56 38.578125 55.875 \nQ 39.703125 55.765625 41.0625 55.515625 \nz\n\" id=\"DejaVuSans-114\"/>\n      <path d=\"M 52 44.1875 \nQ 55.375 50.25 60.0625 53.125 \nQ 64.75 56 71.09375 56 \nQ 79.640625 56 84.28125 50.015625 \nQ 88.921875 44.046875 88.921875 33.015625 \nL 88.921875 0 \nL 79.890625 0 \nL 79.890625 32.71875 \nQ 79.890625 40.578125 77.09375 44.375 \nQ 74.3125 48.1875 68.609375 48.1875 \nQ 61.625 48.1875 57.5625 43.546875 \nQ 53.515625 38.921875 53.515625 30.90625 \nL 53.515625 0 \nL 44.484375 0 \nL 44.484375 32.71875 \nQ 44.484375 40.625 41.703125 44.40625 \nQ 38.921875 48.1875 33.109375 48.1875 \nQ 26.21875 48.1875 22.15625 43.53125 \nQ 18.109375 38.875 18.109375 30.90625 \nL 18.109375 0 \nL 9.078125 0 \nL 9.078125 54.6875 \nL 18.109375 54.6875 \nL 18.109375 46.1875 \nQ 21.1875 51.21875 25.484375 53.609375 \nQ 29.78125 56 35.6875 56 \nQ 41.65625 56 45.828125 52.96875 \nQ 50 49.953125 52 44.1875 \nz\n\" id=\"DejaVuSans-109\"/>\n      <path d=\"M 8.015625 75.875 \nL 15.828125 75.875 \nQ 23.140625 64.359375 26.78125 53.3125 \nQ 30.421875 42.28125 30.421875 31.390625 \nQ 30.421875 20.453125 26.78125 9.375 \nQ 23.140625 -1.703125 15.828125 -13.1875 \nL 8.015625 -13.1875 \nQ 14.5 -2 17.703125 9.0625 \nQ 20.90625 20.125 20.90625 31.390625 \nQ 20.90625 42.671875 17.703125 53.65625 \nQ 14.5 64.65625 8.015625 75.875 \nz\n\" id=\"DejaVuSans-41\"/>\n     </defs>\n     <g transform=\"translate(159.510156 252.916562)scale(0.1 -0.1)\">\n      <use xlink:href=\"#DejaVuSans-66\"/>\n      <use x=\"68.603516\" xlink:href=\"#DejaVuSans-111\"/>\n      <use x=\"129.785156\" xlink:href=\"#DejaVuSans-110\"/>\n      <use x=\"193.164062\" xlink:href=\"#DejaVuSans-100\"/>\n      <use x=\"256.640625\" xlink:href=\"#DejaVuSans-32\"/>\n      <use x=\"288.427734\" xlink:href=\"#DejaVuSans-108\"/>\n      <use x=\"316.210938\" xlink:href=\"#DejaVuSans-101\"/>\n      <use x=\"377.734375\" xlink:href=\"#DejaVuSans-110\"/>\n      <use x=\"441.113281\" xlink:href=\"#DejaVuSans-103\"/>\n      <use x=\"504.589844\" xlink:href=\"#DejaVuSans-116\"/>\n      <use x=\"543.798828\" xlink:href=\"#DejaVuSans-104\"/>\n      <use x=\"607.177734\" xlink:href=\"#DejaVuSans-32\"/>\n      <use x=\"638.964844\" xlink:href=\"#DejaVuSans-40\"/>\n      <use x=\"677.978516\" xlink:href=\"#DejaVuSans-65\"/>\n      <use x=\"746.386719\" xlink:href=\"#DejaVuSans-110\"/>\n      <use x=\"809.765625\" xlink:href=\"#DejaVuSans-103\"/>\n      <use x=\"873.242188\" xlink:href=\"#DejaVuSans-115\"/>\n      <use x=\"925.341797\" xlink:href=\"#DejaVuSans-116\"/>\n      <use x=\"964.550781\" xlink:href=\"#DejaVuSans-114\"/>\n      <use x=\"1003.414062\" xlink:href=\"#DejaVuSans-111\"/>\n      <use x=\"1064.595703\" xlink:href=\"#DejaVuSans-109\"/>\n      <use x=\"1162.007812\" xlink:href=\"#DejaVuSans-41\"/>\n     </g>\n    </g>\n   </g>\n   <g id=\"matplotlib.axis_2\">\n    <g id=\"ytick_1\">\n     <g id=\"line2d_6\">\n      <defs>\n       <path d=\"M 0 0 \nL -3.5 0 \n\" id=\"m65582b6be1\" style=\"stroke:#000000;stroke-width:0.8;\"/>\n      </defs>\n      <g>\n       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"52.160938\" xlink:href=\"#m65582b6be1\" y=\"202.288285\"/>\n      </g>\n     </g>\n     <g id=\"text_7\">\n      <!-- −1.0 -->\n      <defs>\n       <path d=\"M 10.59375 35.5 \nL 73.1875 35.5 \nL 73.1875 27.203125 \nL 10.59375 27.203125 \nz\n\" id=\"DejaVuSans-8722\"/>\n      </defs>\n      <g transform=\"translate(20.878125 206.087503)scale(0.1 -0.1)\">\n       <use xlink:href=\"#DejaVuSans-8722\"/>\n       <use x=\"83.789062\" xlink:href=\"#DejaVuSans-49\"/>\n       <use x=\"147.412109\" xlink:href=\"#DejaVuSans-46\"/>\n       <use x=\"179.199219\" xlink:href=\"#DejaVuSans-48\"/>\n      </g>\n     </g>\n    </g>\n    <g id=\"ytick_2\">\n     <g id=\"line2d_7\">\n      <g>\n       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"52.160938\" xlink:href=\"#m65582b6be1\" y=\"156.513526\"/>\n      </g>\n     </g>\n     <g id=\"text_8\">\n      <!-- −0.5 -->\n      <g transform=\"translate(20.878125 160.312744)scale(0.1 -0.1)\">\n       <use xlink:href=\"#DejaVuSans-8722\"/>\n       <use x=\"83.789062\" xlink:href=\"#DejaVuSans-48\"/>\n       <use x=\"147.412109\" xlink:href=\"#DejaVuSans-46\"/>\n       <use x=\"179.199219\" xlink:href=\"#DejaVuSans-53\"/>\n      </g>\n     </g>\n    </g>\n    <g id=\"ytick_3\">\n     <g id=\"line2d_8\">\n      <g>\n       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"52.160938\" xlink:href=\"#m65582b6be1\" y=\"110.738767\"/>\n      </g>\n     </g>\n     <g id=\"text_9\">\n      <!-- 0.0 -->\n      <g transform=\"translate(29.257812 114.537985)scale(0.1 -0.1)\">\n       <use xlink:href=\"#DejaVuSans-48\"/>\n       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-46\"/>\n       <use x=\"95.410156\" xlink:href=\"#DejaVuSans-48\"/>\n      </g>\n     </g>\n    </g>\n    <g id=\"ytick_4\">\n     <g id=\"line2d_9\">\n      <g>\n       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"52.160938\" xlink:href=\"#m65582b6be1\" y=\"64.964008\"/>\n      </g>\n     </g>\n     <g id=\"text_10\">\n      <!-- 0.5 -->\n      <g transform=\"translate(29.257812 68.763226)scale(0.1 -0.1)\">\n       <use xlink:href=\"#DejaVuSans-48\"/>\n       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-46\"/>\n       <use x=\"95.410156\" xlink:href=\"#DejaVuSans-53\"/>\n      </g>\n     </g>\n    </g>\n    <g id=\"ytick_5\">\n     <g id=\"line2d_10\">\n      <g>\n       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"52.160938\" xlink:href=\"#m65582b6be1\" y=\"19.189249\"/>\n      </g>\n     </g>\n     <g id=\"text_11\">\n      <!-- 1.0 -->\n      <g transform=\"translate(29.257812 22.988467)scale(0.1 -0.1)\">\n       <use xlink:href=\"#DejaVuSans-49\"/>\n       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-46\"/>\n       <use x=\"95.410156\" xlink:href=\"#DejaVuSans-48\"/>\n      </g>\n     </g>\n    </g>\n    <g id=\"text_12\">\n     <!-- Energy (Hartree) -->\n     <defs>\n      <path d=\"M 9.8125 72.90625 \nL 55.90625 72.90625 \nL 55.90625 64.59375 \nL 19.671875 64.59375 \nL 19.671875 43.015625 \nL 54.390625 43.015625 \nL 54.390625 34.71875 \nL 19.671875 34.71875 \nL 19.671875 8.296875 \nL 56.78125 8.296875 \nL 56.78125 0 \nL 9.8125 0 \nz\n\" id=\"DejaVuSans-69\"/>\n      <path d=\"M 32.171875 -5.078125 \nQ 28.375 -14.84375 24.75 -17.8125 \nQ 21.140625 -20.796875 15.09375 -20.796875 \nL 7.90625 -20.796875 \nL 7.90625 -13.28125 \nL 13.1875 -13.28125 \nQ 16.890625 -13.28125 18.9375 -11.515625 \nQ 21 -9.765625 23.484375 -3.21875 \nL 25.09375 0.875 \nL 2.984375 54.6875 \nL 12.5 54.6875 \nL 29.59375 11.921875 \nL 46.6875 54.6875 \nL 56.203125 54.6875 \nz\n\" id=\"DejaVuSans-121\"/>\n      <path d=\"M 9.8125 72.90625 \nL 19.671875 72.90625 \nL 19.671875 43.015625 \nL 55.515625 43.015625 \nL 55.515625 72.90625 \nL 65.375 72.90625 \nL 65.375 0 \nL 55.515625 0 \nL 55.515625 34.71875 \nL 19.671875 34.71875 \nL 19.671875 0 \nL 9.8125 0 \nz\n\" id=\"DejaVuSans-72\"/>\n      <path d=\"M 34.28125 27.484375 \nQ 23.390625 27.484375 19.1875 25 \nQ 14.984375 22.515625 14.984375 16.5 \nQ 14.984375 11.71875 18.140625 8.90625 \nQ 21.296875 6.109375 26.703125 6.109375 \nQ 34.1875 6.109375 38.703125 11.40625 \nQ 43.21875 16.703125 43.21875 25.484375 \nL 43.21875 27.484375 \nz\nM 52.203125 31.203125 \nL 52.203125 0 \nL 43.21875 0 \nL 43.21875 8.296875 \nQ 40.140625 3.328125 35.546875 0.953125 \nQ 30.953125 -1.421875 24.3125 -1.421875 \nQ 15.921875 -1.421875 10.953125 3.296875 \nQ 6 8.015625 6 15.921875 \nQ 6 25.140625 12.171875 29.828125 \nQ 18.359375 34.515625 30.609375 34.515625 \nL 43.21875 34.515625 \nL 43.21875 35.40625 \nQ 43.21875 41.609375 39.140625 45 \nQ 35.0625 48.390625 27.6875 48.390625 \nQ 23 48.390625 18.546875 47.265625 \nQ 14.109375 46.140625 10.015625 43.890625 \nL 10.015625 52.203125 \nQ 14.9375 54.109375 19.578125 55.046875 \nQ 24.21875 56 28.609375 56 \nQ 40.484375 56 46.34375 49.84375 \nQ 52.203125 43.703125 52.203125 31.203125 \nz\n\" id=\"DejaVuSans-97\"/>\n     </defs>\n     <g transform=\"translate(14.798437 157.852812)rotate(-90)scale(0.1 -0.1)\">\n      <use xlink:href=\"#DejaVuSans-69\"/>\n      <use x=\"63.183594\" xlink:href=\"#DejaVuSans-110\"/>\n      <use x=\"126.5625\" xlink:href=\"#DejaVuSans-101\"/>\n      <use x=\"188.085938\" xlink:href=\"#DejaVuSans-114\"/>\n      <use x=\"227.449219\" xlink:href=\"#DejaVuSans-103\"/>\n      <use x=\"290.925781\" xlink:href=\"#DejaVuSans-121\"/>\n      <use x=\"350.105469\" xlink:href=\"#DejaVuSans-32\"/>\n      <use x=\"381.892578\" xlink:href=\"#DejaVuSans-40\"/>\n      <use x=\"420.90625\" xlink:href=\"#DejaVuSans-72\"/>\n      <use x=\"496.101562\" xlink:href=\"#DejaVuSans-97\"/>\n      <use x=\"557.380859\" xlink:href=\"#DejaVuSans-114\"/>\n      <use x=\"598.494141\" xlink:href=\"#DejaVuSans-116\"/>\n      <use x=\"637.703125\" xlink:href=\"#DejaVuSans-114\"/>\n      <use x=\"676.566406\" xlink:href=\"#DejaVuSans-101\"/>\n      <use x=\"738.089844\" xlink:href=\"#DejaVuSans-101\"/>\n      <use x=\"799.613281\" xlink:href=\"#DejaVuSans-41\"/>\n     </g>\n    </g>\n   </g>\n   <g id=\"line2d_11\">\n    <path clip-path=\"url(#pf0f2eccdba)\" d=\"M 67.379119 96.321353 \nL 81.21383 165.833607 \nL 95.048541 194.428732 \nL 108.883252 207.338138 \nL 122.717962 212.934213 \nL 136.552673 214.756364 \nL 150.387384 214.569439 \nL 164.222095 213.32552 \nL 178.056805 211.548549 \nL 191.891516 209.538361 \nL 205.726227 207.482873 \nL 219.560937 205.50957 \nL 233.395648 203.704395 \nL 247.230359 202.118859 \nL 261.06507 200.775221 \nL 274.89978 199.672412 \nL 288.734491 198.792645 \nL 302.569202 198.108029 \nL 316.403913 197.586403 \nL 330.238623 197.195814 \nL 344.073334 196.907373 \nL 357.908045 196.696659 \nL 371.742756 196.544006 \n\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-width:1.5;\"/>\n   </g>\n   <g id=\"line2d_12\">\n    <path clip-path=\"url(#pf0f2eccdba)\" d=\"M 67.379119 17.083652 \nL 81.21383 91.60136 \nL 95.048541 125.308769 \nL 108.883252 142.980122 \nL 122.717962 152.874614 \nL 136.552673 158.517138 \nL 150.387384 165.410107 \nL 164.222095 173.2533 \nL 178.056805 179.02297 \nL 191.891516 183.333845 \nL 205.726227 186.582356 \nL 219.560937 189.035297 \nL 233.395648 190.883773 \nL 247.230359 192.271374 \nL 261.06507 193.308359 \nL 274.89978 194.07972 \nL 288.734491 194.650758 \nL 302.569202 195.071412 \nL 316.403913 195.379713 \nL 330.238623 195.604501 \nL 344.073334 195.767545 \nL 357.908045 195.885187 \nL 371.742756 195.969617 \n\" style=\"fill:none;stroke:#0000ff;stroke-linecap:square;stroke-width:1.5;\"/>\n   </g>\n   <g id=\"line2d_13\">\n    <defs>\n     <path d=\"M 0 3 \nC 0.795609 3 1.55874 2.683901 2.12132 2.12132 \nC 2.683901 1.55874 3 0.795609 3 0 \nC 3 -0.795609 2.683901 -1.55874 2.12132 -2.12132 \nC 1.55874 -2.683901 0.795609 -3 0 -3 \nC -0.795609 -3 -1.55874 -2.683901 -2.12132 -2.12132 \nC -2.683901 -1.55874 -3 -0.795609 -3 0 \nC -3 0.795609 -2.683901 1.55874 -2.12132 2.12132 \nC -1.55874 2.683901 -0.795609 3 0 3 \nz\n\" id=\"m10ea729c3a\" style=\"stroke:#ff0000;\"/>\n    </defs>\n    <g clip-path=\"url(#pf0f2eccdba)\">\n     <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"67.379119\" xlink:href=\"#m10ea729c3a\" y=\"17.083636\"/>\n     <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"81.21383\" xlink:href=\"#m10ea729c3a\" y=\"91.601359\"/>\n     <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"95.048541\" xlink:href=\"#m10ea729c3a\" y=\"125.308768\"/>\n     <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"108.883252\" xlink:href=\"#m10ea729c3a\" y=\"142.980122\"/>\n     <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"122.717962\" xlink:href=\"#m10ea729c3a\" y=\"152.874611\"/>\n     <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"136.552673\" xlink:href=\"#m10ea729c3a\" y=\"158.517123\"/>\n     <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"150.387384\" xlink:href=\"#m10ea729c3a\" y=\"165.410101\"/>\n     <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"164.222095\" xlink:href=\"#m10ea729c3a\" y=\"173.253288\"/>\n     <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"178.056805\" xlink:href=\"#m10ea729c3a\" y=\"179.022961\"/>\n     <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"191.891516\" xlink:href=\"#m10ea729c3a\" y=\"183.333835\"/>\n     <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"205.726227\" xlink:href=\"#m10ea729c3a\" y=\"186.582354\"/>\n     <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"219.560937\" xlink:href=\"#m10ea729c3a\" y=\"189.035293\"/>\n     <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"233.395648\" xlink:href=\"#m10ea729c3a\" y=\"190.883758\"/>\n     <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"247.230359\" xlink:href=\"#m10ea729c3a\" y=\"192.271373\"/>\n     <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"261.06507\" xlink:href=\"#m10ea729c3a\" y=\"193.308354\"/>\n     <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"274.89978\" xlink:href=\"#m10ea729c3a\" y=\"194.079714\"/>\n     <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"288.734491\" xlink:href=\"#m10ea729c3a\" y=\"194.65075\"/>\n     <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"302.569202\" xlink:href=\"#m10ea729c3a\" y=\"195.071406\"/>\n     <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"316.403913\" xlink:href=\"#m10ea729c3a\" y=\"195.379705\"/>\n     <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"330.238623\" xlink:href=\"#m10ea729c3a\" y=\"195.604492\"/>\n     <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"344.073334\" xlink:href=\"#m10ea729c3a\" y=\"195.767536\"/>\n     <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"357.908045\" xlink:href=\"#m10ea729c3a\" y=\"195.88518\"/>\n     <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"371.742756\" xlink:href=\"#m10ea729c3a\" y=\"195.969615\"/>\n    </g>\n   </g>\n   <g id=\"patch_3\">\n    <path d=\"M 52.160938 224.64 \nL 52.160938 7.2 \n\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;stroke-width:0.8;\"/>\n   </g>\n   <g id=\"patch_4\">\n    <path d=\"M 386.960938 224.64 \nL 386.960938 7.2 \n\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;stroke-width:0.8;\"/>\n   </g>\n   <g id=\"patch_5\">\n    <path d=\"M 52.160938 224.64 \nL 386.960938 224.64 \n\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;stroke-width:0.8;\"/>\n   </g>\n   <g id=\"patch_6\">\n    <path d=\"M 52.160938 7.2 \nL 386.960938 7.2 \n\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;stroke-width:0.8;\"/>\n   </g>\n   <g id=\"legend_1\">\n    <g id=\"patch_7\">\n     <path d=\"M 282.675 59.234375 \nL 379.960938 59.234375 \nQ 381.960938 59.234375 381.960938 57.234375 \nL 381.960938 14.2 \nQ 381.960938 12.2 379.960938 12.2 \nL 282.675 12.2 \nQ 280.675 12.2 280.675 14.2 \nL 280.675 57.234375 \nQ 280.675 59.234375 282.675 59.234375 \nz\n\" style=\"fill:#ffffff;opacity:0.8;stroke:#cccccc;stroke-linejoin:miter;\"/>\n    </g>\n    <g id=\"line2d_14\">\n     <path d=\"M 284.675 20.298437 \nL 304.675 20.298437 \n\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-width:1.5;\"/>\n    </g>\n    <g id=\"line2d_15\"/>\n    <g id=\"text_13\">\n     <!-- Ground state -->\n     <defs>\n      <path d=\"M 59.515625 10.40625 \nL 59.515625 29.984375 \nL 43.40625 29.984375 \nL 43.40625 38.09375 \nL 69.28125 38.09375 \nL 69.28125 6.78125 \nQ 63.578125 2.734375 56.6875 0.65625 \nQ 49.8125 -1.421875 42 -1.421875 \nQ 24.90625 -1.421875 15.25 8.5625 \nQ 5.609375 18.5625 5.609375 36.375 \nQ 5.609375 54.25 15.25 64.234375 \nQ 24.90625 74.21875 42 74.21875 \nQ 49.125 74.21875 55.546875 72.453125 \nQ 61.96875 70.703125 67.390625 67.28125 \nL 67.390625 56.78125 \nQ 61.921875 61.421875 55.765625 63.765625 \nQ 49.609375 66.109375 42.828125 66.109375 \nQ 29.4375 66.109375 22.71875 58.640625 \nQ 16.015625 51.171875 16.015625 36.375 \nQ 16.015625 21.625 22.71875 14.15625 \nQ 29.4375 6.6875 42.828125 6.6875 \nQ 48.046875 6.6875 52.140625 7.59375 \nQ 56.25 8.5 59.515625 10.40625 \nz\n\" id=\"DejaVuSans-71\"/>\n      <path d=\"M 8.5 21.578125 \nL 8.5 54.6875 \nL 17.484375 54.6875 \nL 17.484375 21.921875 \nQ 17.484375 14.15625 20.5 10.265625 \nQ 23.53125 6.390625 29.59375 6.390625 \nQ 36.859375 6.390625 41.078125 11.03125 \nQ 45.3125 15.671875 45.3125 23.6875 \nL 45.3125 54.6875 \nL 54.296875 54.6875 \nL 54.296875 0 \nL 45.3125 0 \nL 45.3125 8.40625 \nQ 42.046875 3.421875 37.71875 1 \nQ 33.40625 -1.421875 27.6875 -1.421875 \nQ 18.265625 -1.421875 13.375 4.4375 \nQ 8.5 10.296875 8.5 21.578125 \nz\nM 31.109375 56 \nz\n\" id=\"DejaVuSans-117\"/>\n     </defs>\n     <g transform=\"translate(312.675 23.798437)scale(0.1 -0.1)\">\n      <use xlink:href=\"#DejaVuSans-71\"/>\n      <use x=\"77.490234\" xlink:href=\"#DejaVuSans-114\"/>\n      <use x=\"116.353516\" xlink:href=\"#DejaVuSans-111\"/>\n      <use x=\"177.535156\" xlink:href=\"#DejaVuSans-117\"/>\n      <use x=\"240.914062\" xlink:href=\"#DejaVuSans-110\"/>\n      <use x=\"304.292969\" xlink:href=\"#DejaVuSans-100\"/>\n      <use x=\"367.769531\" xlink:href=\"#DejaVuSans-32\"/>\n      <use x=\"399.556641\" xlink:href=\"#DejaVuSans-115\"/>\n      <use x=\"451.65625\" xlink:href=\"#DejaVuSans-116\"/>\n      <use x=\"490.865234\" xlink:href=\"#DejaVuSans-97\"/>\n      <use x=\"552.144531\" xlink:href=\"#DejaVuSans-116\"/>\n      <use x=\"591.353516\" xlink:href=\"#DejaVuSans-101\"/>\n     </g>\n    </g>\n    <g id=\"line2d_16\">\n     <path d=\"M 284.675 34.976562 \nL 304.675 34.976562 \n\" style=\"fill:none;stroke:#0000ff;stroke-linecap:square;stroke-width:1.5;\"/>\n    </g>\n    <g id=\"line2d_17\"/>\n    <g id=\"text_14\">\n     <!-- Exited state -->\n     <defs>\n      <path d=\"M 54.890625 54.6875 \nL 35.109375 28.078125 \nL 55.90625 0 \nL 45.3125 0 \nL 29.390625 21.484375 \nL 13.484375 0 \nL 2.875 0 \nL 24.125 28.609375 \nL 4.6875 54.6875 \nL 15.28125 54.6875 \nL 29.78125 35.203125 \nL 44.28125 54.6875 \nz\n\" id=\"DejaVuSans-120\"/>\n      <path d=\"M 9.421875 54.6875 \nL 18.40625 54.6875 \nL 18.40625 0 \nL 9.421875 0 \nz\nM 9.421875 75.984375 \nL 18.40625 75.984375 \nL 18.40625 64.59375 \nL 9.421875 64.59375 \nz\n\" id=\"DejaVuSans-105\"/>\n     </defs>\n     <g transform=\"translate(312.675 38.476562)scale(0.1 -0.1)\">\n      <use xlink:href=\"#DejaVuSans-69\"/>\n      <use x=\"63.183594\" xlink:href=\"#DejaVuSans-120\"/>\n      <use x=\"122.363281\" xlink:href=\"#DejaVuSans-105\"/>\n      <use x=\"150.146484\" xlink:href=\"#DejaVuSans-116\"/>\n      <use x=\"189.355469\" xlink:href=\"#DejaVuSans-101\"/>\n      <use x=\"250.878906\" xlink:href=\"#DejaVuSans-100\"/>\n      <use x=\"314.355469\" xlink:href=\"#DejaVuSans-32\"/>\n      <use x=\"346.142578\" xlink:href=\"#DejaVuSans-115\"/>\n      <use x=\"398.242188\" xlink:href=\"#DejaVuSans-116\"/>\n      <use x=\"437.451172\" xlink:href=\"#DejaVuSans-97\"/>\n      <use x=\"498.730469\" xlink:href=\"#DejaVuSans-116\"/>\n      <use x=\"537.939453\" xlink:href=\"#DejaVuSans-101\"/>\n     </g>\n    </g>\n    <g id=\"line2d_18\"/>\n    <g id=\"line2d_19\">\n     <g>\n      <use style=\"fill:#ff0000;stroke:#ff0000;\" x=\"294.675\" xlink:href=\"#m10ea729c3a\" y=\"49.654687\"/>\n     </g>\n    </g>\n    <g id=\"text_15\">\n     <!-- VQE -->\n     <defs>\n      <path d=\"M 28.609375 0 \nL 0.78125 72.90625 \nL 11.078125 72.90625 \nL 34.1875 11.53125 \nL 57.328125 72.90625 \nL 67.578125 72.90625 \nL 39.796875 0 \nz\n\" id=\"DejaVuSans-86\"/>\n      <path d=\"M 39.40625 66.21875 \nQ 28.65625 66.21875 22.328125 58.203125 \nQ 16.015625 50.203125 16.015625 36.375 \nQ 16.015625 22.609375 22.328125 14.59375 \nQ 28.65625 6.59375 39.40625 6.59375 \nQ 50.140625 6.59375 56.421875 14.59375 \nQ 62.703125 22.609375 62.703125 36.375 \nQ 62.703125 50.203125 56.421875 58.203125 \nQ 50.140625 66.21875 39.40625 66.21875 \nz\nM 53.21875 1.3125 \nL 66.21875 -12.890625 \nL 54.296875 -12.890625 \nL 43.5 -1.21875 \nQ 41.890625 -1.3125 41.03125 -1.359375 \nQ 40.1875 -1.421875 39.40625 -1.421875 \nQ 24.03125 -1.421875 14.8125 8.859375 \nQ 5.609375 19.140625 5.609375 36.375 \nQ 5.609375 53.65625 14.8125 63.9375 \nQ 24.03125 74.21875 39.40625 74.21875 \nQ 54.734375 74.21875 63.90625 63.9375 \nQ 73.09375 53.65625 73.09375 36.375 \nQ 73.09375 23.6875 67.984375 14.640625 \nQ 62.890625 5.609375 53.21875 1.3125 \nz\n\" id=\"DejaVuSans-81\"/>\n     </defs>\n     <g transform=\"translate(312.675 53.154687)scale(0.1 -0.1)\">\n      <use xlink:href=\"#DejaVuSans-86\"/>\n      <use x=\"68.408203\" xlink:href=\"#DejaVuSans-81\"/>\n      <use x=\"147.119141\" xlink:href=\"#DejaVuSans-69\"/>\n     </g>\n    </g>\n   </g>\n  </g>\n </g>\n <defs>\n  <clipPath id=\"pf0f2eccdba\">\n   <rect height=\"217.44\" width=\"334.8\" x=\"52.160938\" y=\"7.2\"/>\n  </clipPath>\n </defs>\n</svg>\n",
      "image/png": "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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    }
   ],
   "source": [
    "#For recording bond length, VQE, ground state and 1st excited state results\n",
    "bond_len_list = [];VQE_energy_list=[];gs_list=[];es_list=[]\n",
    "#Execute the calculation for each bond length\n",
    "for bond_len in np.arange(0.2,2.5,0.1):\n",
    "    molecule = get_molecule(bond_len)\n",
    "    #Determination of the number of bits, depth, initial parameter, and penalty\n",
    "    n_qubits=molecule.n_qubits\n",
    "    n_depth=4\n",
    "    init_params=np.random.rand(2*n_qubits*n_depth)*0.1\n",
    "    penalty=10\n",
    "    #Hamiltonian Definition\n",
    "    hamiltonian_gs = get_sparse_operator(jordan_wigner(get_fermion_operator(molecule.get_molecular_hamiltonian())))\n",
    "\n",
    "    #Obtaining the eigenvalues and eigenvectors using classical calculation\n",
    "    eigen_energy, eigen_vec =np.linalg.eigh((hamiltonian_gs).toarray())\n",
    "    #Obtaining the wave function of the ground state\n",
    "    wf_gs=eigen_vec.T[np.argmin(eigen_energy)]\n",
    "\n",
    "    #Optimization run\n",
    "    result=minimize(cost,x0=init_params,args=(hamiltonian_gs,n_qubits,n_depth,penalty,wf_gs))\n",
    "\n",
    "    #Recording bond length, VQE, ground state and 1st excited state results\n",
    "    bond_len_list.append(bond_len)\n",
    "    VQE_energy_list.append(result.fun)\n",
    "    gs_list.append(sorted(eigen_energy)[0])\n",
    "    es_list.append(sorted(eigen_energy)[1])\n",
    "\n",
    "#Plotting\n",
    "import matplotlib.pyplot as plt\n",
    "plt.plot(bond_len_list,gs_list,label=\"Ground state\",color=\"black\")\n",
    "plt.plot(bond_len_list,es_list,label=\"Excited state\",color=\"blue\")\n",
    "plt.plot(bond_len_list,VQE_energy_list, marker=\"o\",label=\"VQE\",color=\"red\",linestyle='None')\n",
    "plt.xlabel(\"Bond length (Angstrom)\")\n",
    "plt.ylabel(\"Energy (Hartree)\")\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The results are in good agreement with the exact value of the excited states. While the method used here is the Overlap-based method,  there are various methods to calculate the excited states. Hence, you can try a method that suits your purpose by referencing the papers, for example, [this review](https://arxiv.org/abs/1808.10402)."
   ]
  }
 ],
 "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": "3.7.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}