{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Correcting for spin contamination in zero-field splitting calculations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The accuracy of the computed zero-field splitting (ZFS) tensor can be affected by spin contamination in spin-polarized density functional theory (DFT) calculations, where the Kohn–Sham states in the two spin channels have different energies and spatial distributions. [Biktagirov et al.](https://doi.org/10.1103/PhysRevResearch.2.022024) proposed a correction scheme that effectively cancels spin contamination errors, providing more accurate computation of ZFS tensors while reducing the dependence on the exchange-correlation functional employed in DFT calculations.\n",
    "\n",
    "In this tutorial, we demonstrate how to apply this correction to the negatively charged nitrogen-vacancy (NV$^-$) center in diamond."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "/Users/vwzyu/Soft/pyzfs/doc/tutorials/diamond_nv_qe_hdf5_correction_data\n"
     ]
    }
   ],
   "source": [
    "%cd diamond_nv_qe_hdf5_correction_data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1 Compute magnetic dipolar interaction for triplet spin state"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, we perform a regular PyZFS calculation for the triplet spin state ($S=1$) of the NV$^-$ center in diamond, obtaining the uncorrected spin-spin ZFS tensor $\\mathbf{D}^{\\mathrm{SS}} = \\frac{\\mathbf{d}}{S(S-1/2)}$. The magnetic dipolar interaction $\\mathbf{d}$ is a 3 $\\times$ 3 tensor defined as\n",
    "\n",
    "$d_{ab} = \\frac{\\alpha^2}{8} \\sum_{ij} \\chi_{ij} \\int \\frac{|\\mathbf{r} - \\mathbf{r}'|^2 \\delta_{ab} - 3(\\mathbf{r} - \\mathbf{r}')_a(\\mathbf{r} - \\mathbf{r}')_b}{|\\mathbf{r} - \\mathbf{r}'|^5} \\times [n_{ii}(\\mathbf{r}) n_{jj}^*(\\mathbf{r}') - n_{ij}(\\mathbf{r}) n_{ij}^*(\\mathbf{r}')] d\\mathbf{r}d\\mathbf{r}'$\n",
    "\n",
    "where $a, b = x, y, z$, $\\alpha$ is the fine-structure constant, $i, j$ are indices of occupied Kohn-Sham states $\\psi(\\mathbf{r})$, $n_{ij}(\\mathbf{r}) = \\psi_i(\\mathbf{r}) \\psi_j(\\mathbf{r})$, and $\\chi_{ij} = 1$ when $i$ and $j$ belong to the same spin channel, and $\\chi_{ij} = -1$ otherwise. The $\\mathbf{d}$ tensor will be used to carry out the correction for spin contamination errors.\n",
    "\n",
    "We run the `pw.x` code of Quantum ESPRESSO for the DFT calculation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "&CONTROL\n",
      "calculation = 'scf'\n",
      "pseudo_dir = './'\n",
      "prefix = 'triplet'\n",
      "/\n",
      "&SYSTEM\n",
      "ibrav = 0\n",
      "ntyp = 2\n",
      "nat = 215\n",
      "tot_charge = -1.0\n",
      "tot_magnetization = 2.0\n",
      "nspin = 2\n",
      "nbnd = 628\n",
      "ecutwfc = 60.0\n",
      "/\n",
      "&ELECTRONS\n",
      "diago_full_acc = .true.\n",
      "/\n",
      "ATOMIC_SPECIES\n",
      "C  12.01099968  C_ONCV_PBE-1.2.upf\n",
      "N  14.00699997  N_ONCV_PBE-1.2.upf\n",
      "CELL_PARAMETERS angstrom\n",
      "10.704  0.000  0.000\n",
      " 0.000 10.704  0.000\n",
      " 0.000  0.000 10.704\n",
      "K_POINTS gamma\n",
      "ATOMIC_POSITIONS crystal\n",
      "N 0.09152860 0.09152860 0.09152860\n",
      "C 0.00406095 0.16656435 0.16656435\n",
      "C 0.08375956 0.25132100 0.25132100\n",
      "C 0.16656435 0.00406095 0.16656435\n",
      "C 0.25132100 0.08375956 0.25132100\n",
      "C 0.16656435 0.16656435 0.00406095\n",
      "C 0.25132100 0.25132100 0.08375956\n",
      "C 0.00008479 0.00008479 0.33283360\n",
      "C 0.08339786 0.08339786 0.41672965\n",
      "C 0.00001566 0.16656881 0.50004871\n",
      "C 0.08326671 0.24989560 0.58351323\n",
      "C 0.16656881 0.00001566 0.50004871\n",
      "C 0.24989560 0.08326671 0.58351323\n",
      "C 0.16713153 0.16713153 0.33408428\n",
      "C 0.25020487 0.25020487 0.41712458\n",
      "C 0.99984338 0.99984338 0.66702816\n",
      "C 0.08281135 0.08281135 0.75072909\n",
      "C 0.99895986 0.16485431 0.83461141\n",
      "C 0.08251591 0.24836734 0.91825974\n",
      "C 0.16485431 0.99895986 0.83461141\n",
      "C 0.24836734 0.08251591 0.91825974\n",
      "C 0.16622186 0.16622186 0.66720992\n",
      "C 0.24953177 0.24953177 0.75057394\n",
      "C 0.00008479 0.33283360 0.00008479\n",
      "C 0.08339786 0.41672965 0.08339786\n",
      "C 0.00001566 0.50004871 0.16656881\n",
      "C 0.08326671 0.58351323 0.24989560\n",
      "C 0.16713153 0.33408428 0.16713153\n",
      "C 0.25020487 0.41712458 0.25020487\n",
      "C 0.16656881 0.50004871 0.00001566\n",
      "C 0.24989560 0.58351323 0.08326671\n",
      "C 0.00027688 0.33386603 0.33386603\n",
      "C 0.08336456 0.41697495 0.41697495\n",
      "C 0.00001683 0.49998105 0.49998105\n",
      "C 0.08321204 0.58306345 0.58306345\n",
      "C 0.16670944 0.33337114 0.50028918\n",
      "C 0.24993513 0.41657415 0.58357284\n",
      "C 0.16670944 0.50028918 0.33337114\n",
      "C 0.24993513 0.58357284 0.41657415\n",
      "C 0.99985602 0.33340447 0.66654379\n",
      "C 0.08316950 0.41622885 0.75039993\n",
      "C 0.99998161 0.49983489 0.83346972\n",
      "C 0.08322304 0.58335107 0.91669677\n",
      "C 0.16638542 0.33231108 0.83427654\n",
      "C 0.24992639 0.41652052 0.91689915\n",
      "C 0.16665894 0.49975729 0.66687415\n",
      "C 0.24993176 0.58319109 0.75005301\n",
      "C 0.99984338 0.66702816 0.99984338\n",
      "C 0.08281135 0.75072909 0.08281135\n",
      "C 0.99895986 0.83461141 0.16485431\n",
      "C 0.08251591 0.91825974 0.24836734\n",
      "C 0.16622186 0.66720992 0.16622186\n",
      "C 0.24953177 0.75057394 0.24953177\n",
      "C 0.16485431 0.83461141 0.99895986\n",
      "C 0.24836734 0.91825974 0.08251591\n",
      "C 0.99985602 0.66654379 0.33340447\n",
      "C 0.08316950 0.75039993 0.41622885\n",
      "C 0.99998161 0.83346972 0.49983489\n",
      "C 0.08322304 0.91669677 0.58335107\n",
      "C 0.16665894 0.66687415 0.49975729\n",
      "C 0.24993176 0.75005301 0.58319109\n",
      "C 0.16638542 0.83427654 0.33231108\n",
      "C 0.24992639 0.91689915 0.41652052\n",
      "C 0.00008183 0.66614375 0.66614375\n",
      "C 0.08289904 0.74902978 0.74902978\n",
      "C 0.00053850 0.83348289 0.83348289\n",
      "C 0.08713894 0.91048546 0.91048546\n",
      "C 0.16657469 0.66657430 0.83322405\n",
      "C 0.25035752 0.74957641 0.91684864\n",
      "C 0.16657469 0.83322405 0.66657430\n",
      "C 0.25035752 0.91684864 0.74957641\n",
      "C 0.33283360 0.00008479 0.00008479\n",
      "C 0.41672965 0.08339786 0.08339786\n",
      "C 0.33408428 0.16713153 0.16713153\n",
      "C 0.41712458 0.25020487 0.25020487\n",
      "C 0.50004871 0.00001566 0.16656881\n",
      "C 0.58351323 0.08326671 0.24989560\n",
      "C 0.50004871 0.16656881 0.00001566\n",
      "C 0.58351323 0.24989560 0.08326671\n",
      "C 0.33386603 0.00027688 0.33386603\n",
      "C 0.41697495 0.08336456 0.41697495\n",
      "C 0.33337114 0.16670944 0.50028918\n",
      "C 0.41657415 0.24993513 0.58357284\n",
      "C 0.49998105 0.00001683 0.49998105\n",
      "C 0.58306345 0.08321204 0.58306345\n",
      "C 0.50028918 0.16670944 0.33337114\n",
      "C 0.58357284 0.24993513 0.41657415\n",
      "C 0.33340447 0.99985602 0.66654379\n",
      "C 0.41622885 0.08316950 0.75039993\n",
      "C 0.33231108 0.16638542 0.83427654\n",
      "C 0.41652052 0.24992639 0.91689915\n",
      "C 0.49983489 0.99998161 0.83346972\n",
      "C 0.58335107 0.08322304 0.91669677\n",
      "C 0.49975729 0.16665894 0.66687415\n",
      "C 0.58319109 0.24993176 0.75005301\n",
      "C 0.33386603 0.33386603 0.00027688\n",
      "C 0.41697495 0.41697495 0.08336456\n",
      "C 0.33337114 0.50028918 0.16670944\n",
      "C 0.41657415 0.58357284 0.24993513\n",
      "C 0.50028918 0.33337114 0.16670944\n",
      "C 0.58357284 0.41657415 0.24993513\n",
      "C 0.49998105 0.49998105 0.00001683\n",
      "C 0.58306345 0.58306345 0.08321204\n",
      "C 0.33347050 0.33347050 0.33347050\n",
      "C 0.41673247 0.41673247 0.41673247\n",
      "C 0.33329711 0.50005503 0.50005503\n",
      "C 0.41664684 0.58334948 0.58334948\n",
      "C 0.50005503 0.33329711 0.50005503\n",
      "C 0.58334948 0.41664684 0.58334948\n",
      "C 0.50005503 0.50005503 0.33329711\n",
      "C 0.58334948 0.58334948 0.41664684\n",
      "C 0.33309935 0.33309935 0.66702679\n",
      "C 0.41654604 0.41654604 0.75010768\n",
      "C 0.33320910 0.49986185 0.83336463\n",
      "C 0.41672856 0.58316778 0.91673106\n",
      "C 0.49986185 0.33320910 0.83336463\n",
      "C 0.58316778 0.41672856 0.91673106\n",
      "C 0.49997743 0.49997743 0.66672237\n",
      "C 0.58336242 0.58336242 0.75004203\n",
      "C 0.33340447 0.66654379 0.99985602\n",
      "C 0.41622885 0.75039993 0.08316950\n",
      "C 0.33231108 0.83427654 0.16638542\n",
      "C 0.41652052 0.91689915 0.24992639\n",
      "C 0.49975729 0.66687415 0.16665894\n",
      "C 0.58319109 0.75005301 0.24993176\n",
      "C 0.49983489 0.83346972 0.99998161\n",
      "C 0.58335107 0.91669677 0.08322304\n",
      "C 0.33309935 0.66702679 0.33309935\n",
      "C 0.41654604 0.75010768 0.41654604\n",
      "C 0.33320910 0.83336463 0.49986185\n",
      "C 0.41672856 0.91673106 0.58316778\n",
      "C 0.49997743 0.66672237 0.49997743\n",
      "C 0.58336242 0.75004203 0.58336242\n",
      "C 0.49986185 0.83336463 0.33320910\n",
      "C 0.58316778 0.91673106 0.41672856\n",
      "C 0.33323057 0.66669044 0.66669044\n",
      "C 0.41664480 0.74999600 0.74999600\n",
      "C 0.33322293 0.83331221 0.83331221\n",
      "C 0.41644973 0.91672376 0.91672376\n",
      "C 0.49995303 0.66666581 0.83331752\n",
      "C 0.58318248 0.74995219 0.91666536\n",
      "C 0.49995303 0.83331752 0.66666581\n",
      "C 0.58318248 0.91666536 0.74995219\n",
      "C 0.66702816 0.99984338 0.99984338\n",
      "C 0.75072909 0.08281135 0.08281135\n",
      "C 0.66720992 0.16622186 0.16622186\n",
      "C 0.75057394 0.24953177 0.24953177\n",
      "C 0.83461141 0.99895986 0.16485431\n",
      "C 0.91825974 0.08251591 0.24836734\n",
      "C 0.83461141 0.16485431 0.99895986\n",
      "C 0.91825974 0.24836734 0.08251591\n",
      "C 0.66654379 0.99985602 0.33340447\n",
      "C 0.75039993 0.08316950 0.41622885\n",
      "C 0.66687415 0.16665894 0.49975729\n",
      "C 0.75005301 0.24993176 0.58319109\n",
      "C 0.83346972 0.99998161 0.49983489\n",
      "C 0.91669677 0.08322304 0.58335107\n",
      "C 0.83427654 0.16638542 0.33231108\n",
      "C 0.91689915 0.24992639 0.41652052\n",
      "C 0.66614375 0.00008183 0.66614375\n",
      "C 0.74902978 0.08289904 0.74902978\n",
      "C 0.66657430 0.16657469 0.83322405\n",
      "C 0.74957641 0.25035752 0.91684864\n",
      "C 0.83348289 0.00053850 0.83348289\n",
      "C 0.91048546 0.08713894 0.91048546\n",
      "C 0.83322405 0.16657469 0.66657430\n",
      "C 0.91684864 0.25035752 0.74957641\n",
      "C 0.66654379 0.33340447 0.99985602\n",
      "C 0.75039993 0.41622885 0.08316950\n",
      "C 0.66687415 0.49975729 0.16665894\n",
      "C 0.75005301 0.58319109 0.24993176\n",
      "C 0.83427654 0.33231108 0.16638542\n",
      "C 0.91689915 0.41652052 0.24992639\n",
      "C 0.83346972 0.49983489 0.99998161\n",
      "C 0.91669677 0.58335107 0.08322304\n",
      "C 0.66702679 0.33309935 0.33309935\n",
      "C 0.75010768 0.41654604 0.41654604\n",
      "C 0.66672237 0.49997743 0.49997743\n",
      "C 0.75004203 0.58336242 0.58336242\n",
      "C 0.83336463 0.33320910 0.49986185\n",
      "C 0.91673106 0.41672856 0.58316778\n",
      "C 0.83336463 0.49986185 0.33320910\n",
      "C 0.91673106 0.58316778 0.41672856\n",
      "C 0.66669044 0.33323057 0.66669044\n",
      "C 0.74999600 0.41664480 0.74999600\n",
      "C 0.66666581 0.49995303 0.83331752\n",
      "C 0.74995219 0.58318248 0.91666536\n",
      "C 0.83331221 0.33322293 0.83331221\n",
      "C 0.91672376 0.41644973 0.91672376\n",
      "C 0.83331752 0.49995303 0.66666581\n",
      "C 0.91666536 0.58318248 0.74995219\n",
      "C 0.66614375 0.66614375 0.00008183\n",
      "C 0.74902978 0.74902978 0.08289904\n",
      "C 0.66657430 0.83322405 0.16657469\n",
      "C 0.74957641 0.91684864 0.25035752\n",
      "C 0.83322405 0.66657430 0.16657469\n",
      "C 0.91684864 0.74957641 0.25035752\n",
      "C 0.83348289 0.83348289 0.00053850\n",
      "C 0.91048546 0.91048546 0.08713894\n",
      "C 0.66669044 0.66669044 0.33323057\n",
      "C 0.74999600 0.74999600 0.41664480\n",
      "C 0.66666581 0.83331752 0.49995303\n",
      "C 0.74995219 0.91666536 0.58318248\n",
      "C 0.83331752 0.66666581 0.49995303\n",
      "C 0.91666536 0.74995219 0.58318248\n",
      "C 0.83331221 0.83331221 0.33322293\n",
      "C 0.91672376 0.91672376 0.41644973\n",
      "C 0.66676564 0.66676564 0.66676564\n",
      "C 0.75009290 0.75009290 0.75009290\n",
      "C 0.66679851 0.83348831 0.83348831\n",
      "C 0.75027490 0.91667191 0.91667191\n",
      "C 0.83348831 0.66679851 0.83348831\n",
      "C 0.91667191 0.75027490 0.91667191\n",
      "C 0.83348831 0.83348831 0.66679851\n",
      "C 0.91667191 0.91667191 0.75027490\n"
     ]
    }
   ],
   "source": [
    "%%bash\n",
    "cat pw_triplet.in\n",
    "\n",
    "mpirun -n 4 pw.x -i pw_triplet.in > pw_triplet.out"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, we run PyZFS to compute the ZFS tensor."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%%bash\n",
    "mpirun -n 4 pyzfs --wfcfmt qeh5 --prefix triplet --memory low > zfs_triplet.out\n",
    "mv zfs.xml zfs_triplet.xml"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From the XML output of PyZFS, we extract the uncorrected ZFS parameters D and E, as well as the magnetic dipolar interaction $\\mathbf{d}$ tensor."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "D = 3033.97 MHz\n",
      "E = -0.68 MHz\n",
      "d tensor:\n",
      "[[-1.37407286e-01  5.05645425e+02  5.05897945e+02]\n",
      " [ 5.05645425e+02  1.36650451e-01  5.05442520e+02]\n",
      " [ 5.05897945e+02  5.05442520e+02  7.56833945e-04]]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import xml.etree.ElementTree as ET\n",
    "\n",
    "tree = ET.parse(\"zfs_triplet.xml\")\n",
    "root = tree.getroot()\n",
    "\n",
    "ZFS_D = float(root.find(\"D\").text)\n",
    "ZFS_E = float(root.find(\"E\").text)\n",
    "\n",
    "print(f\"D = {ZFS_D} MHz\")\n",
    "print(f\"E = {ZFS_E} MHz\")\n",
    "\n",
    "D_triplet = root.find(\"DTensor\").text\n",
    "D_triplet = D_triplet.replace(\"[\", \"\").replace(\"]\", \"\").replace(\"\\n\", \" \")\n",
    "D_triplet = np.fromstring(D_triplet, sep=\" \").reshape(3, 3)\n",
    "\n",
    "# PyZFS internally assumes S = 1, so d = D / 2\n",
    "d_triplet = D_triplet / 2\n",
    "\n",
    "print(\"d tensor:\")\n",
    "print(d_triplet)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2 Compute magnetic dipolar interaction for three singlet spin states"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To correct for spin contamination in the spin-spin ZFS tensor, we compute\n",
    "\n",
    "$\\tilde{\\mathbf{D}}^{\\mathrm{SS}} = \\frac{\\mathbf{d}_{(m_S=S)} - \\bar{\\mathbf{d}}_{(m_S=S-1)}}{2S-1}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where $\\bar{\\mathbf{d}}_{(m_S=S-1)}$ is the average $\\mathbf{d}_{(m_S=S−1)}$ among all the possible $m_S=S−1$ configurations. For our target triplet state ($S=1$), we compute and average $\\mathbf{d}$ for the singlet states ($S=0$).\n",
    "\n",
    "The unpaired electrons of the NV$^-$ center in diamond are\n",
    "equally distributed among the three carbon dangling bonds\n",
    "surrounding the vacancy. In an $m_S=0$ configuration, the spin up electron localizes at one of the three dangling bonds, while the spin down electron is shared by the other two adjacent carbon atoms, resulting in three configurations of singlet states.\n",
    "\n",
    "To simulate these singlet states, we change the `tot_magnetization` input keyword of `pw.x` from 2.0 to 0.0, and we initialize the magnetization by setting the `starting_magnetization` such that the spin up electron localizes at one of the three carbon atoms around the vacancy (for example, `C1` in the input file below), while the spin down electron is shared by the other two carbon atoms around the vacancy (`C2` and `C3`)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "&CONTROL\n",
      "calculation = 'scf'\n",
      "pseudo_dir = './'\n",
      "prefix = 'singlet_1'\n",
      "/\n",
      "&SYSTEM\n",
      "ibrav = 0\n",
      "ntyp = 5\n",
      "nat = 215\n",
      "tot_charge = -1.0\n",
      "tot_magnetization = 0.0\n",
      "starting_magnetization(1) =  0.5  ! C1\n",
      "starting_magnetization(2) = -0.25 ! C2\n",
      "starting_magnetization(3) = -0.25 ! C3\n",
      "starting_magnetization(4) =  0.0  ! C\n",
      "starting_magnetization(5) =  0.0  ! N\n",
      "nspin = 2\n",
      "nbnd = 628\n",
      "ecutwfc = 60.0\n",
      "/\n",
      "&ELECTRONS\n",
      "diago_full_acc = .true.\n",
      "mixing_beta = 0.2\n",
      "/\n",
      "ATOMIC_SPECIES\n",
      "C1  12.01099968  C_ONCV_PBE-1.2.upf\n",
      "C2  12.01099968  C_ONCV_PBE-1.2.upf\n",
      "C3  12.01099968  C_ONCV_PBE-1.2.upf\n",
      "C   12.01099968  C_ONCV_PBE-1.2.upf\n",
      "N   14.00699997  N_ONCV_PBE-1.2.upf\n",
      "CELL_PARAMETERS angstrom\n",
      "10.704  0.000  0.000\n",
      " 0.000 10.704  0.000\n",
      " 0.000  0.000 10.704\n",
      "K_POINTS gamma\n",
      "ATOMIC_POSITIONS crystal\n",
      "N 0.09152860 0.09152860 0.09152860\n",
      "C 0.00406095 0.16656435 0.16656435\n",
      "C 0.08375956 0.25132100 0.25132100\n",
      "C 0.16656435 0.00406095 0.16656435\n",
      "C 0.25132100 0.08375956 0.25132100\n",
      "C 0.16656435 0.16656435 0.00406095\n",
      "C 0.25132100 0.25132100 0.08375956\n",
      "C 0.00008479 0.00008479 0.33283360\n",
      "C 0.08339786 0.08339786 0.41672965\n",
      "C 0.00001566 0.16656881 0.50004871\n",
      "C 0.08326671 0.24989560 0.58351323\n",
      "C 0.16656881 0.00001566 0.50004871\n",
      "C 0.24989560 0.08326671 0.58351323\n",
      "C 0.16713153 0.16713153 0.33408428\n",
      "C 0.25020487 0.25020487 0.41712458\n",
      "C 0.99984338 0.99984338 0.66702816\n",
      "C 0.08281135 0.08281135 0.75072909\n",
      "C 0.99895986 0.16485431 0.83461141\n",
      "C 0.08251591 0.24836734 0.91825974\n",
      "C 0.16485431 0.99895986 0.83461141\n",
      "C 0.24836734 0.08251591 0.91825974\n",
      "C 0.16622186 0.16622186 0.66720992\n",
      "C 0.24953177 0.24953177 0.75057394\n",
      "C 0.00008479 0.33283360 0.00008479\n",
      "C 0.08339786 0.41672965 0.08339786\n",
      "C 0.00001566 0.50004871 0.16656881\n",
      "C 0.08326671 0.58351323 0.24989560\n",
      "C 0.16713153 0.33408428 0.16713153\n",
      "C 0.25020487 0.41712458 0.25020487\n",
      "C 0.16656881 0.50004871 0.00001566\n",
      "C 0.24989560 0.58351323 0.08326671\n",
      "C 0.00027688 0.33386603 0.33386603\n",
      "C 0.08336456 0.41697495 0.41697495\n",
      "C 0.00001683 0.49998105 0.49998105\n",
      "C 0.08321204 0.58306345 0.58306345\n",
      "C 0.16670944 0.33337114 0.50028918\n",
      "C 0.24993513 0.41657415 0.58357284\n",
      "C 0.16670944 0.50028918 0.33337114\n",
      "C 0.24993513 0.58357284 0.41657415\n",
      "C 0.99985602 0.33340447 0.66654379\n",
      "C 0.08316950 0.41622885 0.75039993\n",
      "C 0.99998161 0.49983489 0.83346972\n",
      "C 0.08322304 0.58335107 0.91669677\n",
      "C 0.16638542 0.33231108 0.83427654\n",
      "C 0.24992639 0.41652052 0.91689915\n",
      "C 0.16665894 0.49975729 0.66687415\n",
      "C 0.24993176 0.58319109 0.75005301\n",
      "C 0.99984338 0.66702816 0.99984338\n",
      "C 0.08281135 0.75072909 0.08281135\n",
      "C 0.99895986 0.83461141 0.16485431\n",
      "C 0.08251591 0.91825974 0.24836734\n",
      "C 0.16622186 0.66720992 0.16622186\n",
      "C 0.24953177 0.75057394 0.24953177\n",
      "C 0.16485431 0.83461141 0.99895986\n",
      "C 0.24836734 0.91825974 0.08251591\n",
      "C 0.99985602 0.66654379 0.33340447\n",
      "C 0.08316950 0.75039993 0.41622885\n",
      "C 0.99998161 0.83346972 0.49983489\n",
      "C 0.08322304 0.91669677 0.58335107\n",
      "C 0.16665894 0.66687415 0.49975729\n",
      "C 0.24993176 0.75005301 0.58319109\n",
      "C 0.16638542 0.83427654 0.33231108\n",
      "C 0.24992639 0.91689915 0.41652052\n",
      "C 0.00008183 0.66614375 0.66614375\n",
      "C 0.08289904 0.74902978 0.74902978\n",
      "C 0.00053850 0.83348289 0.83348289\n",
      "C1 0.08713894 0.91048546 0.91048546\n",
      "C 0.16657469 0.66657430 0.83322405\n",
      "C 0.25035752 0.74957641 0.91684864\n",
      "C 0.16657469 0.83322405 0.66657430\n",
      "C 0.25035752 0.91684864 0.74957641\n",
      "C 0.33283360 0.00008479 0.00008479\n",
      "C 0.41672965 0.08339786 0.08339786\n",
      "C 0.33408428 0.16713153 0.16713153\n",
      "C 0.41712458 0.25020487 0.25020487\n",
      "C 0.50004871 0.00001566 0.16656881\n",
      "C 0.58351323 0.08326671 0.24989560\n",
      "C 0.50004871 0.16656881 0.00001566\n",
      "C 0.58351323 0.24989560 0.08326671\n",
      "C 0.33386603 0.00027688 0.33386603\n",
      "C 0.41697495 0.08336456 0.41697495\n",
      "C 0.33337114 0.16670944 0.50028918\n",
      "C 0.41657415 0.24993513 0.58357284\n",
      "C 0.49998105 0.00001683 0.49998105\n",
      "C 0.58306345 0.08321204 0.58306345\n",
      "C 0.50028918 0.16670944 0.33337114\n",
      "C 0.58357284 0.24993513 0.41657415\n",
      "C 0.33340447 0.99985602 0.66654379\n",
      "C 0.41622885 0.08316950 0.75039993\n",
      "C 0.33231108 0.16638542 0.83427654\n",
      "C 0.41652052 0.24992639 0.91689915\n",
      "C 0.49983489 0.99998161 0.83346972\n",
      "C 0.58335107 0.08322304 0.91669677\n",
      "C 0.49975729 0.16665894 0.66687415\n",
      "C 0.58319109 0.24993176 0.75005301\n",
      "C 0.33386603 0.33386603 0.00027688\n",
      "C 0.41697495 0.41697495 0.08336456\n",
      "C 0.33337114 0.50028918 0.16670944\n",
      "C 0.41657415 0.58357284 0.24993513\n",
      "C 0.50028918 0.33337114 0.16670944\n",
      "C 0.58357284 0.41657415 0.24993513\n",
      "C 0.49998105 0.49998105 0.00001683\n",
      "C 0.58306345 0.58306345 0.08321204\n",
      "C 0.33347050 0.33347050 0.33347050\n",
      "C 0.41673247 0.41673247 0.41673247\n",
      "C 0.33329711 0.50005503 0.50005503\n",
      "C 0.41664684 0.58334948 0.58334948\n",
      "C 0.50005503 0.33329711 0.50005503\n",
      "C 0.58334948 0.41664684 0.58334948\n",
      "C 0.50005503 0.50005503 0.33329711\n",
      "C 0.58334948 0.58334948 0.41664684\n",
      "C 0.33309935 0.33309935 0.66702679\n",
      "C 0.41654604 0.41654604 0.75010768\n",
      "C 0.33320910 0.49986185 0.83336463\n",
      "C 0.41672856 0.58316778 0.91673106\n",
      "C 0.49986185 0.33320910 0.83336463\n",
      "C 0.58316778 0.41672856 0.91673106\n",
      "C 0.49997743 0.49997743 0.66672237\n",
      "C 0.58336242 0.58336242 0.75004203\n",
      "C 0.33340447 0.66654379 0.99985602\n",
      "C 0.41622885 0.75039993 0.08316950\n",
      "C 0.33231108 0.83427654 0.16638542\n",
      "C 0.41652052 0.91689915 0.24992639\n",
      "C 0.49975729 0.66687415 0.16665894\n",
      "C 0.58319109 0.75005301 0.24993176\n",
      "C 0.49983489 0.83346972 0.99998161\n",
      "C 0.58335107 0.91669677 0.08322304\n",
      "C 0.33309935 0.66702679 0.33309935\n",
      "C 0.41654604 0.75010768 0.41654604\n",
      "C 0.33320910 0.83336463 0.49986185\n",
      "C 0.41672856 0.91673106 0.58316778\n",
      "C 0.49997743 0.66672237 0.49997743\n",
      "C 0.58336242 0.75004203 0.58336242\n",
      "C 0.49986185 0.83336463 0.33320910\n",
      "C 0.58316778 0.91673106 0.41672856\n",
      "C 0.33323057 0.66669044 0.66669044\n",
      "C 0.41664480 0.74999600 0.74999600\n",
      "C 0.33322293 0.83331221 0.83331221\n",
      "C 0.41644973 0.91672376 0.91672376\n",
      "C 0.49995303 0.66666581 0.83331752\n",
      "C 0.58318248 0.74995219 0.91666536\n",
      "C 0.49995303 0.83331752 0.66666581\n",
      "C 0.58318248 0.91666536 0.74995219\n",
      "C 0.66702816 0.99984338 0.99984338\n",
      "C 0.75072909 0.08281135 0.08281135\n",
      "C 0.66720992 0.16622186 0.16622186\n",
      "C 0.75057394 0.24953177 0.24953177\n",
      "C 0.83461141 0.99895986 0.16485431\n",
      "C 0.91825974 0.08251591 0.24836734\n",
      "C 0.83461141 0.16485431 0.99895986\n",
      "C 0.91825974 0.24836734 0.08251591\n",
      "C 0.66654379 0.99985602 0.33340447\n",
      "C 0.75039993 0.08316950 0.41622885\n",
      "C 0.66687415 0.16665894 0.49975729\n",
      "C 0.75005301 0.24993176 0.58319109\n",
      "C 0.83346972 0.99998161 0.49983489\n",
      "C 0.91669677 0.08322304 0.58335107\n",
      "C 0.83427654 0.16638542 0.33231108\n",
      "C 0.91689915 0.24992639 0.41652052\n",
      "C 0.66614375 0.00008183 0.66614375\n",
      "C 0.74902978 0.08289904 0.74902978\n",
      "C 0.66657430 0.16657469 0.83322405\n",
      "C 0.74957641 0.25035752 0.91684864\n",
      "C 0.83348289 0.00053850 0.83348289\n",
      "C2 0.91048546 0.08713894 0.91048546\n",
      "C 0.83322405 0.16657469 0.66657430\n",
      "C 0.91684864 0.25035752 0.74957641\n",
      "C 0.66654379 0.33340447 0.99985602\n",
      "C 0.75039993 0.41622885 0.08316950\n",
      "C 0.66687415 0.49975729 0.16665894\n",
      "C 0.75005301 0.58319109 0.24993176\n",
      "C 0.83427654 0.33231108 0.16638542\n",
      "C 0.91689915 0.41652052 0.24992639\n",
      "C 0.83346972 0.49983489 0.99998161\n",
      "C 0.91669677 0.58335107 0.08322304\n",
      "C 0.66702679 0.33309935 0.33309935\n",
      "C 0.75010768 0.41654604 0.41654604\n",
      "C 0.66672237 0.49997743 0.49997743\n",
      "C 0.75004203 0.58336242 0.58336242\n",
      "C 0.83336463 0.33320910 0.49986185\n",
      "C 0.91673106 0.41672856 0.58316778\n",
      "C 0.83336463 0.49986185 0.33320910\n",
      "C 0.91673106 0.58316778 0.41672856\n",
      "C 0.66669044 0.33323057 0.66669044\n",
      "C 0.74999600 0.41664480 0.74999600\n",
      "C 0.66666581 0.49995303 0.83331752\n",
      "C 0.74995219 0.58318248 0.91666536\n",
      "C 0.83331221 0.33322293 0.83331221\n",
      "C 0.91672376 0.41644973 0.91672376\n",
      "C 0.83331752 0.49995303 0.66666581\n",
      "C 0.91666536 0.58318248 0.74995219\n",
      "C 0.66614375 0.66614375 0.00008183\n",
      "C 0.74902978 0.74902978 0.08289904\n",
      "C 0.66657430 0.83322405 0.16657469\n",
      "C 0.74957641 0.91684864 0.25035752\n",
      "C 0.83322405 0.66657430 0.16657469\n",
      "C 0.91684864 0.74957641 0.25035752\n",
      "C 0.83348289 0.83348289 0.00053850\n",
      "C3 0.91048546 0.91048546 0.08713894\n",
      "C 0.66669044 0.66669044 0.33323057\n",
      "C 0.74999600 0.74999600 0.41664480\n",
      "C 0.66666581 0.83331752 0.49995303\n",
      "C 0.74995219 0.91666536 0.58318248\n",
      "C 0.83331752 0.66666581 0.49995303\n",
      "C 0.91666536 0.74995219 0.58318248\n",
      "C 0.83331221 0.83331221 0.33322293\n",
      "C 0.91672376 0.91672376 0.41644973\n",
      "C 0.66676564 0.66676564 0.66676564\n",
      "C 0.75009290 0.75009290 0.75009290\n",
      "C 0.66679851 0.83348831 0.83348831\n",
      "C 0.75027490 0.91667191 0.91667191\n",
      "C 0.83348831 0.66679851 0.83348831\n",
      "C 0.91667191 0.75027490 0.91667191\n",
      "C 0.83348831 0.83348831 0.66679851\n",
      "C 0.91667191 0.91667191 0.75027490\n"
     ]
    }
   ],
   "source": [
    "%%bash\n",
    "cat pw_singlet_1.in\n",
    "\n",
    "mpirun -n 4 pw.x -i pw_singlet_1.in > pw_singlet_1.out"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, we run PyZFS to compute the $\\mathbf{d}$ tensor."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%%bash\n",
    "mpirun -n 4 pyzfs --wfcfmt qeh5 --prefix singlet_1 --memory low > zfs_singlet_1.out\n",
    "mv zfs.xml zfs_singlet_1.xml"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The same is done for the other two singlet configurations that have the spin up electron localized on `C2` or `C3`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "&CONTROL\n",
      "calculation = 'scf'\n",
      "pseudo_dir = './'\n",
      "prefix = 'singlet_2'\n",
      "/\n",
      "&SYSTEM\n",
      "ibrav = 0\n",
      "ntyp = 5\n",
      "nat = 215\n",
      "tot_charge = -1.0\n",
      "tot_magnetization = 0.0\n",
      "starting_magnetization(1) = -0.125 ! C1\n",
      "starting_magnetization(2) =  0.25  ! C2\n",
      "starting_magnetization(3) = -0.125 ! C3\n",
      "starting_magnetization(4) =  0.0   ! C\n",
      "starting_magnetization(5) =  0.0   ! N\n",
      "nspin = 2\n",
      "nbnd = 628\n",
      "ecutwfc = 60.0\n",
      "/\n",
      "&ELECTRONS\n",
      "diago_full_acc = .true.\n",
      "mixing_beta = 0.2\n",
      "/\n",
      "ATOMIC_SPECIES\n",
      "C1  12.01099968  C_ONCV_PBE-1.2.upf\n",
      "C2  12.01099968  C_ONCV_PBE-1.2.upf\n",
      "C3  12.01099968  C_ONCV_PBE-1.2.upf\n",
      "C   12.01099968  C_ONCV_PBE-1.2.upf\n",
      "N   14.00699997  N_ONCV_PBE-1.2.upf\n",
      "CELL_PARAMETERS angstrom\n",
      "10.704  0.000  0.000\n",
      " 0.000 10.704  0.000\n",
      " 0.000  0.000 10.704\n",
      "K_POINTS gamma\n",
      "ATOMIC_POSITIONS crystal\n",
      "N 0.09152860 0.09152860 0.09152860\n",
      "C 0.00406095 0.16656435 0.16656435\n",
      "C 0.08375956 0.25132100 0.25132100\n",
      "C 0.16656435 0.00406095 0.16656435\n",
      "C 0.25132100 0.08375956 0.25132100\n",
      "C 0.16656435 0.16656435 0.00406095\n",
      "C 0.25132100 0.25132100 0.08375956\n",
      "C 0.00008479 0.00008479 0.33283360\n",
      "C 0.08339786 0.08339786 0.41672965\n",
      "C 0.00001566 0.16656881 0.50004871\n",
      "C 0.08326671 0.24989560 0.58351323\n",
      "C 0.16656881 0.00001566 0.50004871\n",
      "C 0.24989560 0.08326671 0.58351323\n",
      "C 0.16713153 0.16713153 0.33408428\n",
      "C 0.25020487 0.25020487 0.41712458\n",
      "C 0.99984338 0.99984338 0.66702816\n",
      "C 0.08281135 0.08281135 0.75072909\n",
      "C 0.99895986 0.16485431 0.83461141\n",
      "C 0.08251591 0.24836734 0.91825974\n",
      "C 0.16485431 0.99895986 0.83461141\n",
      "C 0.24836734 0.08251591 0.91825974\n",
      "C 0.16622186 0.16622186 0.66720992\n",
      "C 0.24953177 0.24953177 0.75057394\n",
      "C 0.00008479 0.33283360 0.00008479\n",
      "C 0.08339786 0.41672965 0.08339786\n",
      "C 0.00001566 0.50004871 0.16656881\n",
      "C 0.08326671 0.58351323 0.24989560\n",
      "C 0.16713153 0.33408428 0.16713153\n",
      "C 0.25020487 0.41712458 0.25020487\n",
      "C 0.16656881 0.50004871 0.00001566\n",
      "C 0.24989560 0.58351323 0.08326671\n",
      "C 0.00027688 0.33386603 0.33386603\n",
      "C 0.08336456 0.41697495 0.41697495\n",
      "C 0.00001683 0.49998105 0.49998105\n",
      "C 0.08321204 0.58306345 0.58306345\n",
      "C 0.16670944 0.33337114 0.50028918\n",
      "C 0.24993513 0.41657415 0.58357284\n",
      "C 0.16670944 0.50028918 0.33337114\n",
      "C 0.24993513 0.58357284 0.41657415\n",
      "C 0.99985602 0.33340447 0.66654379\n",
      "C 0.08316950 0.41622885 0.75039993\n",
      "C 0.99998161 0.49983489 0.83346972\n",
      "C 0.08322304 0.58335107 0.91669677\n",
      "C 0.16638542 0.33231108 0.83427654\n",
      "C 0.24992639 0.41652052 0.91689915\n",
      "C 0.16665894 0.49975729 0.66687415\n",
      "C 0.24993176 0.58319109 0.75005301\n",
      "C 0.99984338 0.66702816 0.99984338\n",
      "C 0.08281135 0.75072909 0.08281135\n",
      "C 0.99895986 0.83461141 0.16485431\n",
      "C 0.08251591 0.91825974 0.24836734\n",
      "C 0.16622186 0.66720992 0.16622186\n",
      "C 0.24953177 0.75057394 0.24953177\n",
      "C 0.16485431 0.83461141 0.99895986\n",
      "C 0.24836734 0.91825974 0.08251591\n",
      "C 0.99985602 0.66654379 0.33340447\n",
      "C 0.08316950 0.75039993 0.41622885\n",
      "C 0.99998161 0.83346972 0.49983489\n",
      "C 0.08322304 0.91669677 0.58335107\n",
      "C 0.16665894 0.66687415 0.49975729\n",
      "C 0.24993176 0.75005301 0.58319109\n",
      "C 0.16638542 0.83427654 0.33231108\n",
      "C 0.24992639 0.91689915 0.41652052\n",
      "C 0.00008183 0.66614375 0.66614375\n",
      "C 0.08289904 0.74902978 0.74902978\n",
      "C 0.00053850 0.83348289 0.83348289\n",
      "C1 0.08713894 0.91048546 0.91048546\n",
      "C 0.16657469 0.66657430 0.83322405\n",
      "C 0.25035752 0.74957641 0.91684864\n",
      "C 0.16657469 0.83322405 0.66657430\n",
      "C 0.25035752 0.91684864 0.74957641\n",
      "C 0.33283360 0.00008479 0.00008479\n",
      "C 0.41672965 0.08339786 0.08339786\n",
      "C 0.33408428 0.16713153 0.16713153\n",
      "C 0.41712458 0.25020487 0.25020487\n",
      "C 0.50004871 0.00001566 0.16656881\n",
      "C 0.58351323 0.08326671 0.24989560\n",
      "C 0.50004871 0.16656881 0.00001566\n",
      "C 0.58351323 0.24989560 0.08326671\n",
      "C 0.33386603 0.00027688 0.33386603\n",
      "C 0.41697495 0.08336456 0.41697495\n",
      "C 0.33337114 0.16670944 0.50028918\n",
      "C 0.41657415 0.24993513 0.58357284\n",
      "C 0.49998105 0.00001683 0.49998105\n",
      "C 0.58306345 0.08321204 0.58306345\n",
      "C 0.50028918 0.16670944 0.33337114\n",
      "C 0.58357284 0.24993513 0.41657415\n",
      "C 0.33340447 0.99985602 0.66654379\n",
      "C 0.41622885 0.08316950 0.75039993\n",
      "C 0.33231108 0.16638542 0.83427654\n",
      "C 0.41652052 0.24992639 0.91689915\n",
      "C 0.49983489 0.99998161 0.83346972\n",
      "C 0.58335107 0.08322304 0.91669677\n",
      "C 0.49975729 0.16665894 0.66687415\n",
      "C 0.58319109 0.24993176 0.75005301\n",
      "C 0.33386603 0.33386603 0.00027688\n",
      "C 0.41697495 0.41697495 0.08336456\n",
      "C 0.33337114 0.50028918 0.16670944\n",
      "C 0.41657415 0.58357284 0.24993513\n",
      "C 0.50028918 0.33337114 0.16670944\n",
      "C 0.58357284 0.41657415 0.24993513\n",
      "C 0.49998105 0.49998105 0.00001683\n",
      "C 0.58306345 0.58306345 0.08321204\n",
      "C 0.33347050 0.33347050 0.33347050\n",
      "C 0.41673247 0.41673247 0.41673247\n",
      "C 0.33329711 0.50005503 0.50005503\n",
      "C 0.41664684 0.58334948 0.58334948\n",
      "C 0.50005503 0.33329711 0.50005503\n",
      "C 0.58334948 0.41664684 0.58334948\n",
      "C 0.50005503 0.50005503 0.33329711\n",
      "C 0.58334948 0.58334948 0.41664684\n",
      "C 0.33309935 0.33309935 0.66702679\n",
      "C 0.41654604 0.41654604 0.75010768\n",
      "C 0.33320910 0.49986185 0.83336463\n",
      "C 0.41672856 0.58316778 0.91673106\n",
      "C 0.49986185 0.33320910 0.83336463\n",
      "C 0.58316778 0.41672856 0.91673106\n",
      "C 0.49997743 0.49997743 0.66672237\n",
      "C 0.58336242 0.58336242 0.75004203\n",
      "C 0.33340447 0.66654379 0.99985602\n",
      "C 0.41622885 0.75039993 0.08316950\n",
      "C 0.33231108 0.83427654 0.16638542\n",
      "C 0.41652052 0.91689915 0.24992639\n",
      "C 0.49975729 0.66687415 0.16665894\n",
      "C 0.58319109 0.75005301 0.24993176\n",
      "C 0.49983489 0.83346972 0.99998161\n",
      "C 0.58335107 0.91669677 0.08322304\n",
      "C 0.33309935 0.66702679 0.33309935\n",
      "C 0.41654604 0.75010768 0.41654604\n",
      "C 0.33320910 0.83336463 0.49986185\n",
      "C 0.41672856 0.91673106 0.58316778\n",
      "C 0.49997743 0.66672237 0.49997743\n",
      "C 0.58336242 0.75004203 0.58336242\n",
      "C 0.49986185 0.83336463 0.33320910\n",
      "C 0.58316778 0.91673106 0.41672856\n",
      "C 0.33323057 0.66669044 0.66669044\n",
      "C 0.41664480 0.74999600 0.74999600\n",
      "C 0.33322293 0.83331221 0.83331221\n",
      "C 0.41644973 0.91672376 0.91672376\n",
      "C 0.49995303 0.66666581 0.83331752\n",
      "C 0.58318248 0.74995219 0.91666536\n",
      "C 0.49995303 0.83331752 0.66666581\n",
      "C 0.58318248 0.91666536 0.74995219\n",
      "C 0.66702816 0.99984338 0.99984338\n",
      "C 0.75072909 0.08281135 0.08281135\n",
      "C 0.66720992 0.16622186 0.16622186\n",
      "C 0.75057394 0.24953177 0.24953177\n",
      "C 0.83461141 0.99895986 0.16485431\n",
      "C 0.91825974 0.08251591 0.24836734\n",
      "C 0.83461141 0.16485431 0.99895986\n",
      "C 0.91825974 0.24836734 0.08251591\n",
      "C 0.66654379 0.99985602 0.33340447\n",
      "C 0.75039993 0.08316950 0.41622885\n",
      "C 0.66687415 0.16665894 0.49975729\n",
      "C 0.75005301 0.24993176 0.58319109\n",
      "C 0.83346972 0.99998161 0.49983489\n",
      "C 0.91669677 0.08322304 0.58335107\n",
      "C 0.83427654 0.16638542 0.33231108\n",
      "C 0.91689915 0.24992639 0.41652052\n",
      "C 0.66614375 0.00008183 0.66614375\n",
      "C 0.74902978 0.08289904 0.74902978\n",
      "C 0.66657430 0.16657469 0.83322405\n",
      "C 0.74957641 0.25035752 0.91684864\n",
      "C 0.83348289 0.00053850 0.83348289\n",
      "C2 0.91048546 0.08713894 0.91048546\n",
      "C 0.83322405 0.16657469 0.66657430\n",
      "C 0.91684864 0.25035752 0.74957641\n",
      "C 0.66654379 0.33340447 0.99985602\n",
      "C 0.75039993 0.41622885 0.08316950\n",
      "C 0.66687415 0.49975729 0.16665894\n",
      "C 0.75005301 0.58319109 0.24993176\n",
      "C 0.83427654 0.33231108 0.16638542\n",
      "C 0.91689915 0.41652052 0.24992639\n",
      "C 0.83346972 0.49983489 0.99998161\n",
      "C 0.91669677 0.58335107 0.08322304\n",
      "C 0.66702679 0.33309935 0.33309935\n",
      "C 0.75010768 0.41654604 0.41654604\n",
      "C 0.66672237 0.49997743 0.49997743\n",
      "C 0.75004203 0.58336242 0.58336242\n",
      "C 0.83336463 0.33320910 0.49986185\n",
      "C 0.91673106 0.41672856 0.58316778\n",
      "C 0.83336463 0.49986185 0.33320910\n",
      "C 0.91673106 0.58316778 0.41672856\n",
      "C 0.66669044 0.33323057 0.66669044\n",
      "C 0.74999600 0.41664480 0.74999600\n",
      "C 0.66666581 0.49995303 0.83331752\n",
      "C 0.74995219 0.58318248 0.91666536\n",
      "C 0.83331221 0.33322293 0.83331221\n",
      "C 0.91672376 0.41644973 0.91672376\n",
      "C 0.83331752 0.49995303 0.66666581\n",
      "C 0.91666536 0.58318248 0.74995219\n",
      "C 0.66614375 0.66614375 0.00008183\n",
      "C 0.74902978 0.74902978 0.08289904\n",
      "C 0.66657430 0.83322405 0.16657469\n",
      "C 0.74957641 0.91684864 0.25035752\n",
      "C 0.83322405 0.66657430 0.16657469\n",
      "C 0.91684864 0.74957641 0.25035752\n",
      "C 0.83348289 0.83348289 0.00053850\n",
      "C3 0.91048546 0.91048546 0.08713894\n",
      "C 0.66669044 0.66669044 0.33323057\n",
      "C 0.74999600 0.74999600 0.41664480\n",
      "C 0.66666581 0.83331752 0.49995303\n",
      "C 0.74995219 0.91666536 0.58318248\n",
      "C 0.83331752 0.66666581 0.49995303\n",
      "C 0.91666536 0.74995219 0.58318248\n",
      "C 0.83331221 0.83331221 0.33322293\n",
      "C 0.91672376 0.91672376 0.41644973\n",
      "C 0.66676564 0.66676564 0.66676564\n",
      "C 0.75009290 0.75009290 0.75009290\n",
      "C 0.66679851 0.83348831 0.83348831\n",
      "C 0.75027490 0.91667191 0.91667191\n",
      "C 0.83348831 0.66679851 0.83348831\n",
      "C 0.91667191 0.75027490 0.91667191\n",
      "C 0.83348831 0.83348831 0.66679851\n",
      "C 0.91667191 0.91667191 0.75027490\n"
     ]
    }
   ],
   "source": [
    "%%bash\n",
    "cat pw_singlet_2.in\n",
    "\n",
    "mpirun -n 4 pw.x -i pw_singlet_2.in > pw_singlet_2.out\n",
    "mpirun -n 4 pyzfs --wfcfmt qeh5 --prefix singlet_2 --memory low > zfs_singlet_2.out\n",
    "mv zfs.xml zfs_singlet_2.xml"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "&CONTROL\n",
      "calculation = 'scf'\n",
      "pseudo_dir = './'\n",
      "prefix = 'singlet_3'\n",
      "/\n",
      "&SYSTEM\n",
      "ibrav = 0\n",
      "ntyp = 5\n",
      "nat = 215\n",
      "tot_charge = -1.0\n",
      "tot_magnetization = 0.0\n",
      "starting_magnetization(1) = -0.125 ! C1\n",
      "starting_magnetization(2) = -0.125 ! C2\n",
      "starting_magnetization(3) =  0.25  ! C3\n",
      "starting_magnetization(4) =  0.0   ! C\n",
      "starting_magnetization(5) =  0.0   ! N\n",
      "nspin = 2\n",
      "nbnd = 628\n",
      "ecutwfc = 60.0\n",
      "/\n",
      "&ELECTRONS\n",
      "diago_full_acc = .true.\n",
      "mixing_beta = 0.2\n",
      "/\n",
      "ATOMIC_SPECIES\n",
      "C1  12.01099968  C_ONCV_PBE-1.2.upf\n",
      "C2  12.01099968  C_ONCV_PBE-1.2.upf\n",
      "C3  12.01099968  C_ONCV_PBE-1.2.upf\n",
      "C   12.01099968  C_ONCV_PBE-1.2.upf\n",
      "N   14.00699997  N_ONCV_PBE-1.2.upf\n",
      "CELL_PARAMETERS angstrom\n",
      "10.704  0.000  0.000\n",
      " 0.000 10.704  0.000\n",
      " 0.000  0.000 10.704\n",
      "K_POINTS gamma\n",
      "ATOMIC_POSITIONS crystal\n",
      "N 0.09152860 0.09152860 0.09152860\n",
      "C 0.00406095 0.16656435 0.16656435\n",
      "C 0.08375956 0.25132100 0.25132100\n",
      "C 0.16656435 0.00406095 0.16656435\n",
      "C 0.25132100 0.08375956 0.25132100\n",
      "C 0.16656435 0.16656435 0.00406095\n",
      "C 0.25132100 0.25132100 0.08375956\n",
      "C 0.00008479 0.00008479 0.33283360\n",
      "C 0.08339786 0.08339786 0.41672965\n",
      "C 0.00001566 0.16656881 0.50004871\n",
      "C 0.08326671 0.24989560 0.58351323\n",
      "C 0.16656881 0.00001566 0.50004871\n",
      "C 0.24989560 0.08326671 0.58351323\n",
      "C 0.16713153 0.16713153 0.33408428\n",
      "C 0.25020487 0.25020487 0.41712458\n",
      "C 0.99984338 0.99984338 0.66702816\n",
      "C 0.08281135 0.08281135 0.75072909\n",
      "C 0.99895986 0.16485431 0.83461141\n",
      "C 0.08251591 0.24836734 0.91825974\n",
      "C 0.16485431 0.99895986 0.83461141\n",
      "C 0.24836734 0.08251591 0.91825974\n",
      "C 0.16622186 0.16622186 0.66720992\n",
      "C 0.24953177 0.24953177 0.75057394\n",
      "C 0.00008479 0.33283360 0.00008479\n",
      "C 0.08339786 0.41672965 0.08339786\n",
      "C 0.00001566 0.50004871 0.16656881\n",
      "C 0.08326671 0.58351323 0.24989560\n",
      "C 0.16713153 0.33408428 0.16713153\n",
      "C 0.25020487 0.41712458 0.25020487\n",
      "C 0.16656881 0.50004871 0.00001566\n",
      "C 0.24989560 0.58351323 0.08326671\n",
      "C 0.00027688 0.33386603 0.33386603\n",
      "C 0.08336456 0.41697495 0.41697495\n",
      "C 0.00001683 0.49998105 0.49998105\n",
      "C 0.08321204 0.58306345 0.58306345\n",
      "C 0.16670944 0.33337114 0.50028918\n",
      "C 0.24993513 0.41657415 0.58357284\n",
      "C 0.16670944 0.50028918 0.33337114\n",
      "C 0.24993513 0.58357284 0.41657415\n",
      "C 0.99985602 0.33340447 0.66654379\n",
      "C 0.08316950 0.41622885 0.75039993\n",
      "C 0.99998161 0.49983489 0.83346972\n",
      "C 0.08322304 0.58335107 0.91669677\n",
      "C 0.16638542 0.33231108 0.83427654\n",
      "C 0.24992639 0.41652052 0.91689915\n",
      "C 0.16665894 0.49975729 0.66687415\n",
      "C 0.24993176 0.58319109 0.75005301\n",
      "C 0.99984338 0.66702816 0.99984338\n",
      "C 0.08281135 0.75072909 0.08281135\n",
      "C 0.99895986 0.83461141 0.16485431\n",
      "C 0.08251591 0.91825974 0.24836734\n",
      "C 0.16622186 0.66720992 0.16622186\n",
      "C 0.24953177 0.75057394 0.24953177\n",
      "C 0.16485431 0.83461141 0.99895986\n",
      "C 0.24836734 0.91825974 0.08251591\n",
      "C 0.99985602 0.66654379 0.33340447\n",
      "C 0.08316950 0.75039993 0.41622885\n",
      "C 0.99998161 0.83346972 0.49983489\n",
      "C 0.08322304 0.91669677 0.58335107\n",
      "C 0.16665894 0.66687415 0.49975729\n",
      "C 0.24993176 0.75005301 0.58319109\n",
      "C 0.16638542 0.83427654 0.33231108\n",
      "C 0.24992639 0.91689915 0.41652052\n",
      "C 0.00008183 0.66614375 0.66614375\n",
      "C 0.08289904 0.74902978 0.74902978\n",
      "C 0.00053850 0.83348289 0.83348289\n",
      "C1 0.08713894 0.91048546 0.91048546\n",
      "C 0.16657469 0.66657430 0.83322405\n",
      "C 0.25035752 0.74957641 0.91684864\n",
      "C 0.16657469 0.83322405 0.66657430\n",
      "C 0.25035752 0.91684864 0.74957641\n",
      "C 0.33283360 0.00008479 0.00008479\n",
      "C 0.41672965 0.08339786 0.08339786\n",
      "C 0.33408428 0.16713153 0.16713153\n",
      "C 0.41712458 0.25020487 0.25020487\n",
      "C 0.50004871 0.00001566 0.16656881\n",
      "C 0.58351323 0.08326671 0.24989560\n",
      "C 0.50004871 0.16656881 0.00001566\n",
      "C 0.58351323 0.24989560 0.08326671\n",
      "C 0.33386603 0.00027688 0.33386603\n",
      "C 0.41697495 0.08336456 0.41697495\n",
      "C 0.33337114 0.16670944 0.50028918\n",
      "C 0.41657415 0.24993513 0.58357284\n",
      "C 0.49998105 0.00001683 0.49998105\n",
      "C 0.58306345 0.08321204 0.58306345\n",
      "C 0.50028918 0.16670944 0.33337114\n",
      "C 0.58357284 0.24993513 0.41657415\n",
      "C 0.33340447 0.99985602 0.66654379\n",
      "C 0.41622885 0.08316950 0.75039993\n",
      "C 0.33231108 0.16638542 0.83427654\n",
      "C 0.41652052 0.24992639 0.91689915\n",
      "C 0.49983489 0.99998161 0.83346972\n",
      "C 0.58335107 0.08322304 0.91669677\n",
      "C 0.49975729 0.16665894 0.66687415\n",
      "C 0.58319109 0.24993176 0.75005301\n",
      "C 0.33386603 0.33386603 0.00027688\n",
      "C 0.41697495 0.41697495 0.08336456\n",
      "C 0.33337114 0.50028918 0.16670944\n",
      "C 0.41657415 0.58357284 0.24993513\n",
      "C 0.50028918 0.33337114 0.16670944\n",
      "C 0.58357284 0.41657415 0.24993513\n",
      "C 0.49998105 0.49998105 0.00001683\n",
      "C 0.58306345 0.58306345 0.08321204\n",
      "C 0.33347050 0.33347050 0.33347050\n",
      "C 0.41673247 0.41673247 0.41673247\n",
      "C 0.33329711 0.50005503 0.50005503\n",
      "C 0.41664684 0.58334948 0.58334948\n",
      "C 0.50005503 0.33329711 0.50005503\n",
      "C 0.58334948 0.41664684 0.58334948\n",
      "C 0.50005503 0.50005503 0.33329711\n",
      "C 0.58334948 0.58334948 0.41664684\n",
      "C 0.33309935 0.33309935 0.66702679\n",
      "C 0.41654604 0.41654604 0.75010768\n",
      "C 0.33320910 0.49986185 0.83336463\n",
      "C 0.41672856 0.58316778 0.91673106\n",
      "C 0.49986185 0.33320910 0.83336463\n",
      "C 0.58316778 0.41672856 0.91673106\n",
      "C 0.49997743 0.49997743 0.66672237\n",
      "C 0.58336242 0.58336242 0.75004203\n",
      "C 0.33340447 0.66654379 0.99985602\n",
      "C 0.41622885 0.75039993 0.08316950\n",
      "C 0.33231108 0.83427654 0.16638542\n",
      "C 0.41652052 0.91689915 0.24992639\n",
      "C 0.49975729 0.66687415 0.16665894\n",
      "C 0.58319109 0.75005301 0.24993176\n",
      "C 0.49983489 0.83346972 0.99998161\n",
      "C 0.58335107 0.91669677 0.08322304\n",
      "C 0.33309935 0.66702679 0.33309935\n",
      "C 0.41654604 0.75010768 0.41654604\n",
      "C 0.33320910 0.83336463 0.49986185\n",
      "C 0.41672856 0.91673106 0.58316778\n",
      "C 0.49997743 0.66672237 0.49997743\n",
      "C 0.58336242 0.75004203 0.58336242\n",
      "C 0.49986185 0.83336463 0.33320910\n",
      "C 0.58316778 0.91673106 0.41672856\n",
      "C 0.33323057 0.66669044 0.66669044\n",
      "C 0.41664480 0.74999600 0.74999600\n",
      "C 0.33322293 0.83331221 0.83331221\n",
      "C 0.41644973 0.91672376 0.91672376\n",
      "C 0.49995303 0.66666581 0.83331752\n",
      "C 0.58318248 0.74995219 0.91666536\n",
      "C 0.49995303 0.83331752 0.66666581\n",
      "C 0.58318248 0.91666536 0.74995219\n",
      "C 0.66702816 0.99984338 0.99984338\n",
      "C 0.75072909 0.08281135 0.08281135\n",
      "C 0.66720992 0.16622186 0.16622186\n",
      "C 0.75057394 0.24953177 0.24953177\n",
      "C 0.83461141 0.99895986 0.16485431\n",
      "C 0.91825974 0.08251591 0.24836734\n",
      "C 0.83461141 0.16485431 0.99895986\n",
      "C 0.91825974 0.24836734 0.08251591\n",
      "C 0.66654379 0.99985602 0.33340447\n",
      "C 0.75039993 0.08316950 0.41622885\n",
      "C 0.66687415 0.16665894 0.49975729\n",
      "C 0.75005301 0.24993176 0.58319109\n",
      "C 0.83346972 0.99998161 0.49983489\n",
      "C 0.91669677 0.08322304 0.58335107\n",
      "C 0.83427654 0.16638542 0.33231108\n",
      "C 0.91689915 0.24992639 0.41652052\n",
      "C 0.66614375 0.00008183 0.66614375\n",
      "C 0.74902978 0.08289904 0.74902978\n",
      "C 0.66657430 0.16657469 0.83322405\n",
      "C 0.74957641 0.25035752 0.91684864\n",
      "C 0.83348289 0.00053850 0.83348289\n",
      "C2 0.91048546 0.08713894 0.91048546\n",
      "C 0.83322405 0.16657469 0.66657430\n",
      "C 0.91684864 0.25035752 0.74957641\n",
      "C 0.66654379 0.33340447 0.99985602\n",
      "C 0.75039993 0.41622885 0.08316950\n",
      "C 0.66687415 0.49975729 0.16665894\n",
      "C 0.75005301 0.58319109 0.24993176\n",
      "C 0.83427654 0.33231108 0.16638542\n",
      "C 0.91689915 0.41652052 0.24992639\n",
      "C 0.83346972 0.49983489 0.99998161\n",
      "C 0.91669677 0.58335107 0.08322304\n",
      "C 0.66702679 0.33309935 0.33309935\n",
      "C 0.75010768 0.41654604 0.41654604\n",
      "C 0.66672237 0.49997743 0.49997743\n",
      "C 0.75004203 0.58336242 0.58336242\n",
      "C 0.83336463 0.33320910 0.49986185\n",
      "C 0.91673106 0.41672856 0.58316778\n",
      "C 0.83336463 0.49986185 0.33320910\n",
      "C 0.91673106 0.58316778 0.41672856\n",
      "C 0.66669044 0.33323057 0.66669044\n",
      "C 0.74999600 0.41664480 0.74999600\n",
      "C 0.66666581 0.49995303 0.83331752\n",
      "C 0.74995219 0.58318248 0.91666536\n",
      "C 0.83331221 0.33322293 0.83331221\n",
      "C 0.91672376 0.41644973 0.91672376\n",
      "C 0.83331752 0.49995303 0.66666581\n",
      "C 0.91666536 0.58318248 0.74995219\n",
      "C 0.66614375 0.66614375 0.00008183\n",
      "C 0.74902978 0.74902978 0.08289904\n",
      "C 0.66657430 0.83322405 0.16657469\n",
      "C 0.74957641 0.91684864 0.25035752\n",
      "C 0.83322405 0.66657430 0.16657469\n",
      "C 0.91684864 0.74957641 0.25035752\n",
      "C 0.83348289 0.83348289 0.00053850\n",
      "C3 0.91048546 0.91048546 0.08713894\n",
      "C 0.66669044 0.66669044 0.33323057\n",
      "C 0.74999600 0.74999600 0.41664480\n",
      "C 0.66666581 0.83331752 0.49995303\n",
      "C 0.74995219 0.91666536 0.58318248\n",
      "C 0.83331752 0.66666581 0.49995303\n",
      "C 0.91666536 0.74995219 0.58318248\n",
      "C 0.83331221 0.83331221 0.33322293\n",
      "C 0.91672376 0.91672376 0.41644973\n",
      "C 0.66676564 0.66676564 0.66676564\n",
      "C 0.75009290 0.75009290 0.75009290\n",
      "C 0.66679851 0.83348831 0.83348831\n",
      "C 0.75027490 0.91667191 0.91667191\n",
      "C 0.83348831 0.66679851 0.83348831\n",
      "C 0.91667191 0.75027490 0.91667191\n",
      "C 0.83348831 0.83348831 0.66679851\n",
      "C 0.91667191 0.91667191 0.75027490\n"
     ]
    }
   ],
   "source": [
    "%%bash\n",
    "cat pw_singlet_3.in\n",
    "\n",
    "mpirun -n 4 pw.x -i pw_singlet_3.in > pw_singlet_3.out\n",
    "mpirun -n 4 pyzfs --wfcfmt qeh5 --prefix singlet_3 --memory low > zfs_singlet_3.out\n",
    "mv zfs.xml zfs_singlet_3.xml"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From the XML outputs of PyZFS, we extract the magnetic dipolar interaction $\\mathbf{d}$ tensors of the singlet states."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "d_singlet = {}\n",
    "\n",
    "for conf in range(1, 4):\n",
    "    tree = ET.parse(f\"zfs_singlet_{conf}.xml\")\n",
    "    root = tree.getroot()\n",
    "\n",
    "    D_singlet = root.find(\"DTensor\").text\n",
    "    D_singlet = D_singlet.replace(\"[\", \"\").replace(\"]\", \"\").replace(\"\\n\", \" \")\n",
    "    D_singlet = np.fromstring(D_singlet, sep=\" \").reshape(3, 3)\n",
    "\n",
    "    # PyZFS internally assumes S = 1, so d = D / 2\n",
    "    d_singlet[conf] = D_singlet / 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3 Compute zero-field splitting with correction for spin contamination"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally we build and diagonalize the spin decontaminated spin-spin ZFS tensor $\\tilde{\\mathbf{D}}^{\\mathrm{SS}}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Uncorrected ZFS:\n",
      "D = 3033.97\n",
      "E = -0.68\n",
      "\n",
      "Corrected ZFS:\n",
      "D = 2544.28\n",
      "E = -0.59\n"
     ]
    }
   ],
   "source": [
    "D_corrected = d_triplet - (d_singlet[1] + d_singlet[2] + d_singlet[3]) / 3\n",
    "\n",
    "ev, evc = np.linalg.eig(D_corrected)\n",
    "\n",
    "# For triplet states, compute D and E parameters:\n",
    "# Denote three eigenvalues as Dx, Dy, Dz: |Dz| > |Dx| > |Dy|\n",
    "# D = 3/2 Dz, E = 1/2(Dx - Dy)\n",
    "args = np.abs(ev).argsort()\n",
    "dy, dx, dz = ev[args]\n",
    "ZFS_D_corrected = 1.5 * dz\n",
    "ZFS_E_corrected = 0.5 * (dx - dy)\n",
    "\n",
    "# Summarize results\n",
    "print(\"Uncorrected ZFS:\")\n",
    "print(f\"D = {ZFS_D:.2f}\")\n",
    "print(f\"E = {ZFS_E:.2f}\")\n",
    "print()\n",
    "print(\"Corrected ZFS:\")\n",
    "print(f\"D = {ZFS_D_corrected:.2f}\")\n",
    "print(f\"E = {ZFS_E_corrected:.2f}\")"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.12.12"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}