tutorials/21-compare-radial-atomic-potentials.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Compute and compare different radial potentials"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: Precompiling JAC [830ae420-d14d-11e8-2f94-6b071437414d]\n",
      "└ @ Base loading.jl:1342\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Welcome to JAC:  A community approach to the computation of atomic structures, cascades and time evolutions [(C) Copyright by Stephan Fritzsche, Jena (2018-2021)].\n"
     ]
    }
   ],
   "source": [
    "using Plots\n",
    "using JAC"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "We shall here describe how different atomic (radial) potentials can be calculated and applied within the JAC toolbox.\n",
    "\n",
    "Any (radial) potential is handled in JAC by the data struct `Radial.Potential` that (may) contain a name, the radial function $Z(r)$ as well as the radial grid on which the potential is defined. The potential function `Zr` typically contains the effective charge $Z(r)$ as seen by an electron, so that the full radial potential is given by $V = -\\:Z(r) / r$. This convention has been frequently applied in atomic physics and has also been adopted in JAC.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\texttt{struct  Radial.Potential}  ... defines a struct for a local radial potential.\n",
       "\n",
       "\\begin{verbatim}\n",
       "+ name           ::String            ... A name for the potential.\n",
       "+ Zr             ::Array{Float64,1}  ... radial potential function Z(r) = - r * V(r) as usual in atomic structure theory.\n",
       "+ grid           ::RadialGrid        ... radial grid on which the potential is defined.\n",
       "\\end{verbatim}\n",
       "\\rule{\\textwidth}{1pt}\n",
       "\\texttt{Radial.Potential()}  ... constructor to define an 'empty' instance of the radial potential.\n",
       "\n"
      ],
      "text/markdown": [
       "`struct  Radial.Potential`  ... defines a struct for a local radial potential.\n",
       "\n",
       "```\n",
       "+ name           ::String            ... A name for the potential.\n",
       "+ Zr             ::Array{Float64,1}  ... radial potential function Z(r) = - r * V(r) as usual in atomic structure theory.\n",
       "+ grid           ::RadialGrid        ... radial grid on which the potential is defined.\n",
       "```\n",
       "\n",
       "---\n",
       "\n",
       "`Radial.Potential()`  ... constructor to define an 'empty' instance of the radial potential.\n"
      ],
      "text/plain": [
       "  \u001b[36mstruct Radial.Potential\u001b[39m ... defines a struct for a local radial potential.\n",
       "\n",
       "\u001b[36m  + name           ::String            ... A name for the potential.\u001b[39m\n",
       "\u001b[36m  + Zr             ::Array{Float64,1}  ... radial potential function Z(r) = - r * V(r) as usual in atomic structure theory.\u001b[39m\n",
       "\u001b[36m  + grid           ::RadialGrid        ... radial grid on which the potential is defined.\u001b[39m\n",
       "\n",
       "  ────────────────────────────────────────────────────────────────────────────\n",
       "\n",
       "  \u001b[36mRadial.Potential()\u001b[39m ... constructor to define an 'empty' instance of the\n",
       "  radial potential."
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "? Radial.Potential"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "Perhaps, the simplest choice of a radial potential is the **nuclear** potential which, of course, depends on the nuclear model. Let us first simply consider a $\\;^{12}$C $\\;$ carbon atom with a Fermi-distributed nuclear charge and a standard radial grid.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Fermi nuclear model for Z = 6.0 with mass = 12.18, radius R = 2.4934845703596404 fm and nuclear spin I = 0, dipole moment mu = 0.0 and quadrupole moment Q = 0.0."
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "grid     = Radial.Grid(true)\n",
    "nucModel = Nuclear.Model(6., \"Fermi\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "Then, the corresponding nuclear potential is generated by simply calling:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "nuclear-potential: Fermi-distributed (radial) potential ... defined on 392 grid points ...\n",
       "Zr:    [0.0, 0.015619818207380716, 0.035905804451892086, 0.060430845202356614, 0.08495453726302814, 0.10523736995823374, 0.11777879390742439, 0.12582486891780031, 0.14610064533238984, 0.17888509371708336  …  0..31112937694872056, 0.32313641267043025, 0.35185818903226784, 0.39828287685804126, 0.4543674906380097, 0.5103989493317181, 0.5566993479831148, 0.5853078885893802, 0.6022668932973185, 0.6428176395098482]  ...  [6.0, 6.000000000000002, 6.000000000000001, 6.000000000000002, 6.000000000000001, 6.0, 6.000000000000001, 6.0, 6.000000000000002, 6.000000000000002  …  6.000000000000002, 6.000000000000001, 6.000000000000002, 6.000000000000002, 6.000000000000001, 6.000000000000001, 6.0, 6.000000000000001, 6.000000000000001, 6.000000000000001]\n",
       "Radial grid:  rnt = 2.0e-6,  h = 0.05,  hp = 0.0,  NoPoints = 392,  ntL = 69,  ntS = 71, orderL = 7,  orderS = 8,  nsL = 62,  nsS = 63,  mesh = JAC.Radial.MeshGL(), ...  \n",
       "r:    [1.780504510281598e-8, 9.042759120177923e-8, 2.0787030685460475e-7]  ...  [564.4302351890452, 595.097239730825, 614.0606632064549]\n",
       "wr:   [4.530145586285996e-8, 9.785739473901763e-8, 1.3358660616640612e-7]  ...  [34.882547080539766, 25.552825070730517, 11.829256033233948]\n",
       "tS:   [0.0, 0.0, 0.0]  ...  [618.7099715607404, 618.7099715607404, 618.7099715607404]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nucPotential = Nuclear.nuclearPotential(nucModel, grid)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "As seen from the output, the effective charge Z(r) vanishes near to the origin but quickly increases to the full charge $Z = 6.\\;$ at larger $r$-values, i.e. near to the surface and outside of the nucleus\n",
    "\n",
    "To calculate **atomic** potentials, we also need to know the electron density around the nucleus. Most often, this density refers to a particular atomic level. Therefore, we first need to compute some of the low-lying levels of, for instance, neutral carbon. We shall here just compute the levels of the $1s^2 2s^2 2p^2$ ground-state configuration. We use the *simplest* approximation for these computation but will explain in some later tutorials how we could control and refine such simple structure calculations, if needed.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "... in performSCF ...\n",
      ">> include Configuration: 1s_1/2^2 2s_1/2^2 2p_1/2^0 2p_3/2^2 \n",
      ">> include Configuration: 1s_1/2^2 2s_1/2^2 2p_1/2^1 2p_3/2^1 \n",
      ">> include Configuration: 1s_1/2^2 2s_1/2^2 2p_1/2^2 2p_3/2^0 \n",
      "(Re-) Define a new standard subshell list.\n",
      "Start SCF process with hydrogenic orbitals.\n",
      ">> (Re-) Define a storage array for various B-spline matrices:\n",
      "Nuclear model = Fermi nuclear model for Z = 6.0 with mass = 12..18, radius R = 2.4934845703596404 fm and nuclear spin I = 0, dipole moment mu = 0.0 and quadrupole moment Q = 0.0. \n",
      "Generate hydrogenic orbital for subshell 1s_1/2 \n",
      "  -----------------------------------------------------------------------------\n",
      "   Index    Subshell     Energies [a.u.]    Dirac-E  [a.u.]     Delta-E / |E|    \n",
      "  -----------------------------------------------------------------------------\n",
      "      1      1s_1/2      -1.80086298e+01    -1.80086350e+01    +2.86262559e-07    \n",
      "      2      2s_1/2      -4.50269922e+00    -4.50269857e+00    -1.45559221e-07    \n",
      "      3      3s_1/2      -2.00098002e+00    -2.00095940e+00    -1.03076960e-05    \n",
      "      4      4s_1/2      -1.12563225e+00    -1.12543844e+00    -1.72180085e-04    \n",
      "      5      5s_1/2      -7.21146347e-01    -7.20234830e-01    -1.26398373e-03    \n",
      "      6      6s_1/2      -5.32451642e-01    -5.00139888e-01    -6.06848607e-02    \n",
      "      7      7s_1/2      -4.90629779e-01    -3.67436826e-01    -2.51091470e-01    \n",
      "      8      8s_1/2      -3.59967650e-01    -2.81311119e-01    -2.18509998e-01    \n",
      "      9      9s_1/2      -2.64672687e-01    -2.22265641e-01    -1.60224489e-01    \n",
      "     10     10s_1/2      -1.92971517e-01    -1.80031939e-01    -6.70543411e-02    \n",
      "     11     11s_1/2      -1.39221907e-01    -1.48784504e-01    +6.86860039e-02    \n",
      "     12     12s_1/2      -9.97470087e-02    -1.25018733e-01    +2.53358214e-01    \n",
      "     13     13s_1/2      -7.12392898e-02    -1.06523685e-01    +4.95294033e-01    \n",
      "     14     14s_1/2      -5.09329579e-02    -9.18486436e-02    +8.03324358e-01    \n",
      "     15     15s_1/2      -3.66769635e-02    -8.00097188e-02    +1.18147063e+00    \n",
      "     16     16s_1/2      -2.68413869e-02    -7.03205343e-02    +1.61985472e+00    \n",
      "     17     17s_1/2      -2.01786016e-02    -6.22904546e-02    +2.08695597e+00    \n",
      "     18     18s_1/2      -1.57303103e-02    -5.55612291e-02    +2.53211272e+00    \n",
      "     19     19s_1/2      -1.27870521e-02    -4.98663309e-02    +2.89975192e+00    \n",
      "     20     20s_1/2      -1.08896963e-02    -4.50041540e-02    +3.13272811e+00    \n",
      "     21     21s_1/2      +3.29838682e-01    -4.08199215e-02    +1.12375723e+00    \n",
      "     22     22s_1/2      +3.68762540e+00    -3.71932146e-02    +1.01008595e+00    \n",
      "     23     23s_1/2      +1.30100858e+01    -3.40292102e-02    +1.00261560e+00    \n",
      "     24     24s_1/2      +1.91723432e+01    -3.12524195e-02    +1.00163008e+00    \n",
      "     25     25s_1/2      +3.54517685e+01    -2.88021434e-02    +1.00081243e+00    \n",
      "     26     26s_1/2      +8.56121472e+01    -2.66291266e-02    +1.00031104e+00    \n",
      "     27     27s_1/2      +1.92982410e+02    -2.46930634e-02    +1.00012795e+00    \n",
      "     28     28s_1/2      +4.16439613e+02    -2.29607143e-02    +1.00005514e+00    \n",
      "     29     29s_1/2      +8.70655802e+02    -2.14044706e-02    +1.00002458e+00    \n",
      "     30     30s_1/2      +1.77004041e+03    -2.00012468e-02    +1.00001130e+00    \n",
      "     31     31s_1/2      +3.48867571e+03    -1.87316200e-02    +1.00000537e+00    \n",
      "     32     32s_1/2      +6.61716498e+03    -1.75791539e-02    +1.00000266e+00    \n",
      "     33     33s_1/2      +1.19816413e+04    -1.65298645e-02    +1.00000138e+00    \n",
      "     34     34s_1/2      +2.06414211e+04    -1.55717933e-02    +1.00000075e+00    \n",
      "     35     35s_1/2      +3.39351998e+04    -1.46946655e-02    +1.00000043e+00    \n",
      "     36     36s_1/2      +5.36599896e+04    -1.38896135e-02    +1.00000026e+00    \n",
      "     37     37s_1/2      +8.23126120e+04    -1.31489512e-02    +1.00000016e+00    \n",
      "     38     38s_1/2      +1.23496165e+05    -1.24659907e-02    +1.00000010e+00    \n",
      "      :       :    \n",
      "     55     55s_1/2      +6.10652378e+07    -5.95061791e-03    +1.00000000e+00    \n",
      "     56     56s_1/2      +9.25632000e+07    -5.73998987e-03    +1.00000000e+00    \n",
      "     57     57s_1/2      +1.41114179e+08    -5.54035017e-03    +1.00000000e+00    \n",
      "     58     58s_1/2      +2.38291809e+08    -5.35094755e-03    +1.00000000e+00    \n",
      "     59     59s_1/2      +4.17648909e+08    -5.17109386e-03    +1.00000000e+00    \n",
      "     60     60s_1/2      +2.97166129e+09    -5.00015784e-03    +1.00000000e+00    \n",
      "  -----------------------------------------------------------------------------\n",
      ">> Use hydrogenic orbital from this symmetry block also for 1s_1/2.\n",
      ">> Use hydrogenic orbital from this symmetry block also for 2s_1/2.\n",
      "Generate hydrogenic orbital for subshell 2p_1/2 \n",
      "  -----------------------------------------------------------------------------\n",
      "   Index    Subshell     Energies [a.u.]    Dirac-E  [a.u.]     Delta-E / |E|    \n",
      "  -----------------------------------------------------------------------------\n",
      "      1      2p_1/2      -4.50269892e+00    -4.50269857e+00    -7.83932749e-08    \n",
      "      2      3p_1/2      -2.00096708e+00    -2.00095940e+00    -3.84008694e-06    \n",
      "      3      4p_1/2      -1.12549935e+00    -1.12543844e+00    -5.41160266e-05    \n",
      "      4      5p_1/2      -7.20399106e-01    -7.20234830e-01    -2.28034672e-04    \n",
      "      5      6p_1/2      -5.00852349e-01    -5.00139888e-01    -1.42249772e-03    \n",
      "      6      7p_1/2      -3.91466347e-01    -3.67436826e-01    -6.13833611e-02    \n",
      "      7      8p_1/2      -3.49578293e-01    -2.81311119e-01    -1.95284361e-01    \n",
      "      8      9p_1/2      -2.61796944e-01    -2.22265641e-01    -1.50999864e-01    \n",
      "      9     10p_1/2      -1.91431501e-01    -1.80031939e-01    -5.95490388e-02    \n",
      "     10     11p_1/2      -1.38431431e-01    -1.48784504e-01    +7.47884542e-02    \n",
      "     11     12p_1/2      -9.93620315e-02    -1.25018733e-01    +2.58214339e-01    \n",
      "     12     13p_1/2      -7.10422386e-02    -1.06523685e-01    +4.99441559e-01    \n",
      "     13     14p_1/2      -5.08329419e-02    -9.18486436e-02    +8.06872474e-01    \n",
      "     14     15p_1/2      -3.66279560e-02    -8.00097188e-02    +1.18438940e+00    \n",
      "     15     16p_1/2      -2.68157676e-02    -7.03205343e-02    +1.62235769e+00    \n",
      "     16     17p_1/2      -2.01654332e-02    -6.22904546e-02    +2.08897180e+00    \n",
      "     17     18p_1/2      -1.57260830e-02    -5.55612291e-02    +2.53306217e+00    \n",
      "     18     19p_1/2      -1.27874185e-02    -4.98663309e-02    +2.89964018e+00    \n",
      "     19     20p_1/2      -1.08829100e-02    -4.50041540e-02    +3.13530516e+00    \n",
      "     20     21p_1/2      +3.90210774e-01    -4.08199215e-02    +1.10460993e+00    \n",
      "     21     22p_1/2      +3.31828208e+00    -3.71932146e-02    +1.01120858e+00    \n",
      "     22     23p_1/2      +1.08803165e+01    -3.40292102e-02    +1.00312759e+00    \n",
      "     23     24p_1/2      +1.91723678e+01    -3.12524195e-02    +1.00163008e+00    \n",
      "     24     25p_1/2      +2.82338607e+01    -2.88021434e-02    +1.00102013e+00    \n",
      "     25     26p_1/2      +6.59855270e+01    -2.66291266e-02    +1.00040356e+00    \n",
      "     26     27p_1/2      +1.45688122e+02    -2.46930634e-02    +1.00016949e+00    \n",
      "     27     28p_1/2      +3.10534289e+02    -2.29607143e-02    +1.00007394e+00    \n",
      "     28     29p_1/2      +6.45435890e+02    -2.14044706e-02    +1.00003316e+00    \n",
      "     29     30p_1/2      +1.31210880e+03    -2.00012468e-02    +1.00001524e+00    \n",
      "     30     31p_1/2      +2.60238354e+03    -1.87316200e-02    +1.00000720e+00    \n",
      "     31     32p_1/2      +4.99988603e+03    -1.75791539e-02    +1.00000352e+00    \n",
      "     32     33p_1/2      +9.21998254e+03    -1.65298645e-02    +1.00000179e+00    \n",
      "     33     34p_1/2      +1.62106308e+04    -1.55717933e-02    +1.00000096e+00    \n",
      "     34     35p_1/2      +2.71671718e+04    -1.46946655e-02    +1.00000054e+00    \n",
      "     35     36p_1/2      +4.36438294e+04    -1.38896135e-02    +1.00000032e+00    \n",
      "     36     37p_1/2      +6.77773165e+04    -1.31489512e-02    +1.00000019e+00    \n",
      "     37     38p_1/2      +1.02597736e+05    -1.24659907e-02    +1.00000012e+00    \n",
      "     38     39p_1/2      +1.52435772e+05    -1.18348904e-02    +1.00000008e+00    \n",
      "      :       :    \n",
      "     55     56p_1/2      +7.62996154e+07    -5.73998987e-03    +1.00000000e+00    \n",
      "     56     57p_1/2      +1.16057956e+08    -5.54035017e-03    +1.00000000e+00    \n",
      "     57     58p_1/2      +1.84137715e+08    -5.35094755e-03    +1.00000000e+00    \n",
      "     58     59p_1/2      +3.17842715e+08    -5.17109386e-03    +1.00000000e+00    \n",
      "     59     60p_1/2      +6.64007971e+08    -5.00015784e-03    +1.00000000e+00    \n",
      "     60     61p_1/2      +7.14213563e+11    -4.83755953e-03    +1.00000000e+00    \n",
      "  -----------------------------------------------------------------------------\n",
      ">> Use hydrogenic orbital from this symmetry block also for 2p_1/2.\n",
      "Generate hydrogenic orbital for subshell 2p_3/2 \n",
      "  -----------------------------------------------------------------------------\n",
      "   Index    Subshell     Energies [a.u.]    Dirac-E  [a.u.]     Delta-E / |E|    \n",
      "  -----------------------------------------------------------------------------\n",
      "      1      2p_3/2      -4.50053984e+00    -4.50053930e+00    -1.21204113e-07    \n",
      "      2      3p_3/2      -2.00033051e+00    -2.00031959e+00    -5.45866748e-06    \n",
      "      3      4p_3/2      -1.12535888e+00    -1.12516853e+00    -1.69142245e-04    \n",
      "      4      5p_3/2      -7.21995656e-01    -7.20096642e-01    -2.63022887e-03    \n",
      "      5      6p_3/2      -5.97444139e-01    -5.00059921e-01    -1.63001377e-01    \n",
      "      6      7p_3/2      -4.94424293e-01    -3.67386470e-01    -2.56940900e-01    \n",
      "      7      8p_3/2      -3.59177671e-01    -2.81277385e-01    -2.16885100e-01    \n",
      "      8      9p_3/2      -2.63788396e-01    -2.22241949e-01    -1.57499145e-01    \n",
      "      9     10p_3/2      -1.92359345e-01    -1.80014668e-01    -6.41750838e-02    \n",
      "     10     11p_3/2      -1.38863305e-01    -1.48771528e-01    +7.13523482e-02    \n",
      "     11     12p_3/2      -9.95487050e-02    -1.25008738e-01    +2.55754539e-01    \n",
      "     12     13p_3/2      -7.11317033e-02    -1.06515824e-01    +4.97445149e-01    \n",
      "     13     14p_3/2      -5.08747570e-02    -9.18423497e-02    +8.05263654e-01    \n",
      "     14     15p_3/2      -3.66451477e-02    -8.00046017e-02    +1.18322497e+00    \n",
      "     15     16p_3/2      -2.68235056e-02    -7.03163180e-02    +1.62144400e+00    \n",
      "     16     17p_3/2      -2.01680501e-02    -6.22869395e-02    +2.08839670e+00    \n",
      "     17     18p_3/2      -1.57236469e-02    -5.55582679e-02    +2.53342124e+00    \n",
      "     18     19p_3/2      -1.27824952e-02    -4.98638131e-02    +2.90094519e+00    \n",
      "     19     20p_3/2      -1.08863013e-02    -4.50019952e-02    +3.13381865e+00    \n",
      "     20     21p_3/2      +1.17220425e-01    -4.08180568e-02    +1.34821625e+00    \n",
      "     21     22p_3/2      +2.67231245e+00    -3.71915928e-02    +1.01391738e+00    \n",
      "     22     23p_3/2      +9.51398956e+00    -3.40277908e-02    +1.00357661e+00    \n",
      "     23     24p_3/2      +1.91723464e+01    -3.12511703e-02    +1.00163001e+00    \n",
      "     24     25p_3/2      +2.54154161e+01    -2.88010381e-02    +1.00113321e+00    \n",
      "     25     26p_3/2      +6.02401276e+01    -2.66281441e-02    +1.00044203e+00    \n",
      "     26     27p_3/2      +1.34078135e+02    -2.46921860e-02    +1.00018416e+00    \n",
      "     27     28p_3/2      +2.87290295e+02    -2.29599276e-02    +1.00007992e+00    \n",
      "     28     29p_3/2      +5.99491189e+02    -2.14037625e-02    +1.00003570e+00    \n",
      "     29     30p_3/2      +1.22310790e+03    -2.00006072e-02    +1.00001635e+00    \n",
      "     30     31p_3/2      +2.43548384e+03    -1.87310403e-02    +1.00000769e+00    \n",
      "     31     32p_3/2      +4.70170709e+03    -1.75786269e-02    +1.00000374e+00    \n",
      "     32     33p_3/2      +8.71938440e+03    -1.65293840e-02    +1.00000190e+00    \n",
      "     33     34p_3/2      +1.54233129e+04    -1.55713539e-02    +1.00000101e+00    \n",
      "     34     35p_3/2      +2.59959054e+04    -1.46942628e-02    +1.00000057e+00    \n",
      "     35     36p_3/2      +4.19697738e+04    -1.38892433e-02    +1.00000033e+00    \n",
      "     36     37p_3/2      +6.54462322e+04    -1.31486103e-02    +1.00000020e+00    \n",
      "     37     38p_3/2      +9.94044320e+04    -1.24656760e-02    +1.00000013e+00    \n",
      "     38     39p_3/2      +1.48105634e+05    -1.18345992e-02    +1.00000008e+00    \n",
      "      :       :    \n",
      "     55     56p_3/2      +7.48071802e+07    -5.73989154e-03    +1.00000000e+00    \n",
      "     56     57p_3/2      +1.13532922e+08    -5.54025693e-03    +1.00000000e+00    \n",
      "     57     58p_3/2      +1.79227271e+08    -5.35085904e-03    +1.00000000e+00    \n",
      "     58     59p_3/2      +3.04742643e+08    -5.17100978e-03    +1.00000000e+00    \n",
      "     59     60p_3/2      +6.10785151e+08    -5.00007789e-03    +1.00000000e+00    \n",
      "     60     61p_3/2      +2.99760494e+09    -4.83748345e-03    +1.00000000e+00    \n",
      "  -----------------------------------------------------------------------------\n",
      ">> Use hydrogenic orbital from this symmetry block also for 2p_3/2.\n",
      "(Re-) Define the standard grid with 392 grid points.\n",
      ">>  for various B-spline matrices:\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Iteration 1 for symmetries ... \n",
      "  1s_1/2::  en [a.u.] = -7.5200227e+00;   self-cons'cy = 4.1086e-01  [1.0000e+02 for sym-block kappa = -1]\n",
      "  2s_1/2::  en [a.u.] = -1.1797900e-02;   self-cons'cy = 9.9477e-01  [1.0000e+02 for sym-block kappa = -1]\n",
      "  2p_1/2::  en [a.u.] = -2.6580562e-01;   self-cons'cy = 8.8852e-01  [1.0000e+02 for sym-block kappa = 1]\n",
      "  2p_3/2::  en [a.u.] = -3.7295300e-01;   self-cons'cy = 8.4695e-01  [1.0000e+02 for sym-block kappa = -2]\n",
      "\n",
      "Iteration 2 for symmetries ... \n",
      "  1s_1/2::  en [a.u.] = -1.0847252e+01;   self-cons'cy = 1.8115e-01  [1.0159e+00 for sym-block kappa = -1]\n",
      "  2s_1/2::  en [a.u.] = -9.2555310e-01;   self-cons'cy = 9.7483e-01  [1.0159e+00 for sym-block kappa = -1]\n",
      "  2p_1/2::  en [a.u.] = -8.0888174e-02;   self-cons'cy = 5.3337e-01  [1.0041e+00 for sym-block kappa = 1]\n",
      "  2p_3/2::  en [a.u.] = -9.3223634e-02;   self-cons'cy = 6.0005e-01  [1.0022e+00 for sym-block kappa = -2]\n",
      "\n",
      "Iteration 3 for symmetries ... \n",
      "  1s_1/2::  en [a.u.] = -9.8287734e+00;   self-cons'cy = 4.9259e-02  [1.0035e+00 for sym-block kappa = -1]\n",
      "  2s_1/2::  en [a.u.] = -4.6270785e-01;   self-cons'cy = 3.3340e-01  [1.0035e+00 for sym-block kappa = -1]\n",
      "  2p_1/2::  en [a.u.] = -2.2971903e-01;   self-cons'cy = 4.7916e-01  [4.7916e-01 for sym-block kappa = 1]\n",
      "  2p_3/2::  en [a.u.] = -1.9401151e-01;   self-cons'cy = 3.5089e-01  [3.5089e-01 for sym-block kappa = -2]\n",
      "\n",
      "Iteration 4 for symmetries ... \n",
      "  1s_1/2::  en [a.u.] = -9.8304649e+00;   self-cons'cy = 8.6039e-05  [9.8777e-01 for sym-block kappa = -1]\n",
      "  2s_1/2::  en [a.u.] = -4.3109895e-01;   self-cons'cy = 3.5364e-02  [9.8777e-01 for sym-block kappa = -1]\n",
      "  2p_1/2::  en [a.u.] = -1.3711325e-01;   self-cons'cy = 2.5245e-01  [2.5245e-01 for sym-block kappa = 1]\n",
      "  2p_3/2::  en [a.u.] = -1.5266588e-01;   self-cons'cy = 1.1926e-01  [1.1926e-01 for sym-block kappa = -2]\n",
      "\n",
      "Iteration 5 for symmetries ... \n",
      "  1s_1/2::  en [a.u.] = -9.9219025e+00;   self-cons'cy = 4.6292e-03  [9.5162e-01 for sym-block kappa = -1]\n",
      "  2s_1/2::  en [a.u.] = -4.7104704e-01;   self-cons'cy = 4.4281e-02  [9.5162e-01 for sym-block kappa = -1]\n",
      "  2p_1/2::  en [a.u.] = -1.6104540e-01;   self-cons'cy = 8.0266e-02  [8.0266e-02 for sym-block kappa = 1]\n",
      "  2p_3/2::  en [a.u.] = -1.5586505e-01;   self-cons'cy = 1.0369e-02  [1.1648e-01 for sym-block kappa = -2]\n",
      "\n",
      "Iteration 6 for symmetries ... \n",
      "  1s_1/2::  en [a.u.] = -9.8785041e+00;   self-cons'cy = 2.1918e-03  [3.7779e-01 for sym-block kappa = -1]\n",
      "  2s_1/2::  en [a.u.] = -4.5450906e-01;   self-cons'cy = 1.7868e-02  [3.7779e-01 for sym-block kappa = -1]\n",
      "  2p_1/2::  en [a.u.] = -1.5900895e-01;   self-cons'cy = 6.3628e-03  [8.1785e-02 for sym-block kappa = 1]\n",
      "  2p_3/2::  en [a.u.] = -1.5898834e-01;   self-cons'cy = 9.9198e-03  [9.9198e-03 for sym-block kappa = -2]\n",
      "\n",
      "Iteration 7 for symmetries ... \n",
      "  1s_1/2::  en [a.u.] = -9.8876018e+00;   self-cons'cy = 4.6027e-04  [1.1083e-02 for sym-block kappa = -1]\n",
      "  2s_1/2::  en [a.u.] = -4.5724602e-01;   self-cons'cy = 3.0019e-03  [1.1083e-02 for sym-block kappa = -1]\n",
      "  2p_1/2::  en [a.u.] = -1.5681238e-01;   self-cons'cy = 6.9551e-03  [6.9551e-03 for sym-block kappa = 1]\n",
      "  2p_3/2::  en [a.u.] = -1.5694876e-01;   self-cons'cy = 6.4556e-03  [6.4556e-03 for sym-block kappa = -2]\n",
      "\n",
      "Iteration 8 for symmetries ... \n",
      "  1s_1/2::  en [a.u.] = -9.8887669e+00;   self-cons'cy = 5.8911e-05  [4.0212e-02 for sym-block kappa = -1]\n",
      "  2s_1/2::  en [a.u.] = -4.5802066e-01;   self-cons'cy = 8.4636e-04  [4.0212e-02 for sym-block kappa = -1]\n",
      "  2p_1/2::  en [a.u.] = -1.5813636e-01;   self-cons'cy = 4.2038e-03  [6.0731e-03 for sym-block kappa = 1]\n",
      "  2p_3/2::  en [a.u.] = -1.5756222e-01;   self-cons'cy = 1.9505e-03  [1.9505e-03 for sym-block kappa = -2]\n",
      "\n",
      "Iteration 9 for symmetries ... \n",
      "  1s_1/2::  en [a.u.] = -9.8869866e+00;   self-cons'cy = 9.0023e-05  [1.9405e-02 for sym-block kappa = -1]\n",
      "  2s_1/2::  en [a.u.] = -4.5727195e-01;   self-cons'cy = 8.1801e-04  [1.9405e-02 for sym-block kappa = -1]\n",
      "  2p_1/2::  en [a.u.] = -1.5776479e-01;   self-cons'cy = 1.1762e-03  [8.3256e-03 for sym-block kappa = 1]\n",
      "  2p_3/2::  en [a.u.] = -1.5751684e-01;   self-cons'cy = 1.4404e-04  [3.3950e-04 for sym-block kappa = -2]\n",
      "\n",
      "Iteration 10 for symmetries ... \n",
      "  1s_1/2::  en [a.u.] = -9.8877264e+00;   self-cons'cy = 3.7412e-05  [4.4621e-03 for sym-block kappa = -1]\n",
      "  2s_1/2::  en [a.u.] = -4.5754534e-01;   self-cons'cy = 2.9885e-04  [4.4621e-03 for sym-block kappa = -1]\n",
      "  2p_1/2::  en [a.u.] = -1.5777735e-01;   self-cons'cy = 3.9821e-05  [8.7345e-03 for sym-block kappa = 1]\n",
      "  2p_3/2::  en [a.u.] = -1.5746184e-01;   self-cons'cy = 1.7462e-04  [2.5264e-04 for sym-block kappa = -2]\n",
      "\n",
      "Iteration 11 for symmetries ... \n",
      "  1s_1/2::  en [a.u.] = -9.8875880e+00;   self-cons'cy = 6.9996e-06  [5.2930e-04 for sym-block kappa = -1]\n",
      "  2s_1/2::  en [a.u.] = -4.5750596e-01;   self-cons'cy = 4.3037e-05  [5.2930e-04 for sym-block kappa = -1]\n",
      "  2p_1/2::  en [a.u.] = -1.5781938e-01;   self-cons'cy = 1.3316e-04  [4.9473e-03 for sym-block kappa = 1]\n",
      "  2p_3/2::  en [a.u.] = -1.5749715e-01;   self-cons'cy = 1.1212e-04  [2.4216e-03 for sym-block kappa = -2]\n",
      "\n",
      "Iteration 12 for symmetries ... \n",
      "  1s_1/2::  en [a.u.] = -9.8875664e+00;   self-cons'cy = 1.0915e-06  [5.9867e-03 for sym-block kappa = -1]\n",
      "  2s_1/2::  en [a.u.] = -4.5749229e-01;   self-cons'cy = 1.4948e-05  [5.9867e-03 for sym-block kappa = -1]\n",
      "  2p_1/2::  en [a.u.] = -1.5779823e-01;   self-cons'cy = 6.7015e-05  [7.2270e-03 for sym-block kappa = 1]\n",
      "  2p_3/2::  en [a.u.] = -1.5748475e-01;   self-cons'cy = 3.9357e-05  [1.5148e-03 for sym-block kappa = -2]\n",
      "\n",
      "Iteration 13 for symmetries ... \n",
      "  1s_1/2::  en [a.u.] = -9.8875951e+00;   self-cons'cy = 1.4477e-06  [3.8027e-03 for sym-block kappa = -1]\n",
      "  2s_1/2::  en [a.u.] = -4.5750431e-01;   self-cons'cy = 1.3137e-05  [3.8027e-03 for sym-block kappa = -1]\n",
      "  2p_1/2::  en [a.u.] = -1.5780392e-01;   self-cons'cy = 1.8038e-05  [3.1219e-03 for sym-block kappa = 1]\n",
      "  2p_3/2::  en [a.u.] = -1.5748753e-01;   self-cons'cy = 8.7984e-06  [8.0725e-04 for sym-block kappa = -2]\n",
      "\n",
      "Iteration 14 for symmetries ... \n",
      "  1s_1/2::  en [a.u.] = -9.8875828e+00;   self-cons'cy = 6.2113e-07  [1.9958e-03 for sym-block kappa = -1]\n",
      "  2s_1/2::  en [a.u.] = -4.5749972e-01;   self-cons'cy = 5.0155e-06  [1.9958e-03 for sym-block kappa = -1]\n",
      "  2p_1/2::  en [a.u.] = -1.5780354e-01;   self-cons'cy = 1.2119e-06  [8.4819e-03 for sym-block kappa = 1]\n",
      "  2p_3/2::  en [a.u.] = -1.5748662e-01;   self-cons'cy = 2.8674e-06  [9.0598e-04 for sym-block kappa = -2]\n",
      "\n",
      "Iteration 15 for symmetries ... \n",
      "  1s_1/2::  en [a.u.] = -9.8875852e+00;   self-cons'cy = 1.2060e-07  [2.2848e-03 for sym-block kappa = -1]\n",
      "  2s_1/2::  en [a.u.] = -4.5750044e-01;   self-cons'cy = 7.9212e-07  [2.2848e-03 for sym-block kappa = -1]\n",
      "  2p_1/2::  en [a.u.] = -1.5780288e-01;   self-cons'cy = 2.0969e-06  [2.6339e-03 for sym-block kappa = 1]\n",
      "  2p_3/2::  en [a.u.] = -1.5748337e-01;   self-cons'cy = 1.0315e-05  [3.3724e-04 for sym-block kappa = -2]\n",
      "\n",
      "Iteration 16 for symmetries ... \n",
      "  1s_1/2::  en [a.u.] = -9.8875797e+00;   self-cons'cy = 2.7772e-07  [2.5581e-03 for sym-block kappa = -1]\n",
      "  2s_1/2::  en [a.u.] = -4.5749834e-01;   self-cons'cy = 2.2960e-06  [2.5581e-03 for sym-block kappa = -1]\n",
      "  2p_1/2::  en [a.u.] = -1.5780162e-01;   self-cons'cy = 3.9730e-06  [4.2922e-03 for sym-block kappa = 1]\n",
      "  2p_3/2::  en [a.u.] = -1.5750323e-01;   self-cons'cy = 6.3022e-05  [5.2459e-02 for sym-block kappa = -2]\n",
      "\n",
      "Iteration 17 for symmetries ... \n",
      "  1s_1/2::  en [a.u.] = -9.8876009e+00;   self-cons'cy = 1.0752e-06  [1.4029e-01 for sym-block kappa = -1]\n",
      "  2s_1/2::  en [a.u.] = -4.5750762e-01;   self-cons'cy = 1.0142e-05  [1.4029e-01 for sym-block kappa = -1]\n",
      "  2p_1/2::  en [a.u.] = -1.5780767e-01;   self-cons'cy = 1.9163e-05  [3.9575e-02 for sym-block kappa = 1]\n",
      "  2p_3/2::  en [a.u.] = -1.5747701e-01;   self-cons'cy = 8.3217e-05  [5.1914e-02 for sym-block kappa = -2]\n",
      "\n",
      "Iteration 18 for symmetries ... \n",
      "  1s_1/2::  en [a.u.] = -9.8875905e+00;   self-cons'cy = 5.2704e-07  [1.3699e-01 for sym-block kappa = -1]\n",
      "  2s_1/2::  en [a.u.] = -4.5750316e-01;   self-cons'cy = 4.8734e-06  [1.3699e-01 for sym-block kappa = -1]\n",
      "  2p_1/2::  en [a.u.] = -1.5780667e-01;   self-cons'cy = 3.1817e-06  [3.5417e-02 for sym-block kappa = 1]\n",
      "  2p_3/2::  en [a.u.] = -1.5749228e-01;   self-cons'cy = 4.8467e-05  [1.8564e-03 for sym-block kappa = -2]\n",
      "\n",
      "Iteration 19 for symmetries ... \n",
      "  1s_1/2::  en [a.u.] = -9.8875796e+00;   self-cons'cy = 5.5241e-07  [4.6347e-03 for sym-block kappa = -1]\n",
      "  2s_1/2::  en [a.u.] = -4.5749816e-01;   self-cons'cy = 5.4647e-06  [4.6347e-03 for sym-block kappa = -1]\n",
      "  2p_1/2::  en [a.u.] = -1.5780205e-01;   self-cons'cy = 1.4617e-05  [7.6300e-03 for sym-block kappa = 1]\n",
      "  2p_3/2::  en [a.u.] = -1.5748034e-01;   self-cons'cy = 3.7921e-05  [2.1824e-03 for sym-block kappa = -2]\n",
      "\n",
      "Iteration 20 for symmetries ... \n",
      "  1s_1/2::  en [a.u.] = -9.8875827e+00;   self-cons'cy = 1.5842e-07  [7.9981e-03 for sym-block kappa = -1]\n",
      "  2s_1/2::  en [a.u.] = -4.5749956e-01;   self-cons'cy = 1.5347e-06  [7.9981e-03 for sym-block kappa = -1]\n",
      "  2p_1/2::  en [a.u.] = -1.5780204e-01;   self-cons'cy = 4.8305e-08  [5.9528e-03 for sym-block kappa = 1]\n",
      "  2p_3/2::  en [a.u.] = -1.5749003e-01;   self-cons'cy = 3.0795e-05  [4.5643e-04 for sym-block kappa = -2]\n",
      "\n",
      "Iteration 21 for symmetries ... \n",
      "  1s_1/2::  en [a.u.] = -9.8875807e+00;   self-cons'cy = 1.0191e-07  [1.6140e-03 for sym-block kappa = -1]\n",
      "  2s_1/2::  en [a.u.] = -4.5749856e-01;   self-cons'cy = 1.0993e-06  [1.6140e-03 for sym-block kappa = -1]\n",
      "  2p_1/2::  en [a.u.] = -1.5780148e-01;   self-cons'cy = 1.7703e-06  [6.8175e-03 for sym-block kappa = 1]\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  2p_3/2::  en [a.u.] = -1.5748592e-01;   self-cons'cy = 1.3057e-05  [2.9726e-04 for sym-block kappa = -2]\n",
      "\n",
      "Iteration 22 for symmetries ... \n",
      "  1s_1/2::  en [a.u.] = -9.8875874e+00;   self-cons'cy = 3.3662e-07  [6.8162e-04 for sym-block kappa = -1]\n",
      "  2s_1/2::  en [a.u.] = -4.5750136e-01;   self-cons'cy = 3.0578e-06  [6.8162e-04 for sym-block kappa = -1]\n",
      "  2p_1/2::  en [a.u.] = -1.5780336e-01;   self-cons'cy = 5.9492e-06  [1.8607e-03 for sym-block kappa = 1]\n",
      "  2p_3/2::  en [a.u.] = -1.5748640e-01;   self-cons'cy = 1.5071e-06  [4.7217e-04 for sym-block kappa = -2]\n",
      "\n",
      "Iteration 23 for symmetries ... \n",
      "  1s_1/2::  en [a.u.] = -9.8875828e+00;   self-cons'cy = 2.3042e-07  [1.0083e-03 for sym-block kappa = -1]\n",
      "  2s_1/2::  en [a.u.] = -4.5749959e-01;   self-cons'cy = 1.9347e-06  [1.0083e-03 for sym-block kappa = -1]\n",
      "  2p_1/2::  en [a.u.] = -1.5780278e-01;   self-cons'cy = 1.8294e-06  [8.3536e-03 for sym-block kappa = 1]\n",
      "  2p_3/2::  en [a.u.] = -1.5748782e-01;   self-cons'cy = 4.5043e-06  [1.0443e-03 for sym-block kappa = -2]\n",
      "\n",
      "Iteration 24 for symmetries ... \n",
      "  1s_1/2::  en [a.u.] = -9.8875853e+00;   self-cons'cy = 1.2877e-07  [2.4078e-03 for sym-block kappa = -1]\n",
      "  2s_1/2::  en [a.u.] = -4.5750051e-01;   self-cons'cy = 1.0107e-06  [2.4078e-03 for sym-block kappa = -1]\n",
      "  2p_1/2::  en [a.u.] = -1.5780297e-01;   self-cons'cy = 5.8908e-07  [8.7738e-03 for sym-block kappa = 1]\n",
      "  2p_3/2::  en [a.u.] = -1.5748885e-01;   self-cons'cy = 3.2897e-06  [2.4592e-03 for sym-block kappa = -2]\n",
      ">> Maximum number of SCF iterations = 24 is reached at accuracy 3.2897e-06 ... computations proceed.\n",
      "Compute CI matrix of dimension 2 x 2 for the symmetry 0^+ ...   ... done.\n",
      "Compute CI matrix of dimension 2 x 2 for the symmetry 2^+ ...   ... done.\n",
      "Compute CI matrix of dimension 1 x 1 for the symmetry 1^+ ...   ... done.\n",
      "\n",
      "  Eigenenergies:\n",
      "\n",
      "  Level  J Parity          Hartrees                    eV                   [eV]\n",
      "\n",
      "     1    0 +     -1.860330419717123e+02    -5.062216917567124e+03    -5.062216917567124e+03 \n",
      "     2    1 +     -1.860298809535017e+02    -5.062130901880379e+03    -5.062130901880379e+03 \n",
      "     3    2 +     -1.860247005931174e+02    -5.061989937094241e+03    -5.061989937094241e+03 \n",
      "     4    2 +     -1.859731764634554e+02    -5.060587894112663e+03    -5.060587894112663e+03 \n",
      "     5    0 +     -1.858925273993392e+02    -5.058393321296846e+03    -5.058393321296846e+03 \n",
      "\n",
      "  Energy of each level relative to immediately lower level:\n",
      "\n",
      "  Level  J Parity          Hartrees                    eV                   [eV]\n",
      "\n",
      "     2    1 +     3.161018210590782e-03    8.601568674463543e-02    8.601568674463543e-02 \n",
      "     3    2 +     5.180360384258620e-03    1.409647861387768e-01    1.409647861387768e-01 \n",
      "     4    2 +     5.152412966202746e-02    1.402042981577961e+00    1.402042981577961e+00 \n",
      "     5    0 +     8.064906411615880e-02    2.194572815816527e+00    2.194572815816527e+00 \n",
      "\n",
      "  Energy of each level relative to lowest level:\n",
      "\n",
      "  Level  J Parity          Hartrees                    eV                   [eV]\n",
      "\n",
      "     2    1 +     3.161018210590782e-03    8.601568674463543e-02    8.601568674463543e-02 \n",
      "     3    2 +     8.341378594849402e-03    2.269804728834123e-01    2.269804728834123e-01 \n",
      "     4    2 +     5.986550825687686e-02    1.629023454461374e+00    1.629023454461374e+00 \n",
      "     5    0 +     1.405145723730357e-01    3.823596270277901e+00    3.823596270277901e+00 \n"
     ]
    }
   ],
   "source": [
    "basis     = Basics.performSCF([Configuration(\"1s^2 2s^2 2p^2\")], nucModel, grid, AsfSettings())\n",
    "multiplet = Basics.performCI(basis, nucModel, grid, AsfSettings());"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "From this multiplet, we just take the *lowest* level (the ground level) in order to calculate the density for the (electronic) potential. In JAC, we have different choices for generating such an **electronic** potential that just arise from the electronic density, and as seen from ? compute at the REPL:\n",
    "\n",
    "   •    (\"radial potential: core-Hartree\", grid::Radial.Grid, level::Level) \n",
    "       ... to compute a (radial) core-Hartree potential for the given level; \n",
    "           cf. Basics.computePotentialCoreHartree. A potential::RadialPotential \n",
    "           is returned. \n",
    "\n",
    "   •    (\"radial potential: Kohn-Sham\", grid::Radial.Grid, level::Level) \n",
    "       ... to compute a (radial) Kohn-Sham potential for the given level; \n",
    "           cf. Basics.computePotentialKohnSham. A potential::RadialPotential is \n",
    "           returned.\n",
    "\n",
    "   •    (\"radial potential: Dirac-Fock-Slater\", grid::Radial..Grid, level::Level) \n",
    "       ... to compute a (radial) Dirac-Fock-Slater potential for the given level; this \n",
    "           potential is rather simple but includes some undesired self-interaction and \n",
    "           exhibits a wrong asymptotic behaviour. Cf. Basics.computePotentialDFS. \n",
    "           A potential::RadialPotential is returned.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "coreSubshells = Subshell[1s_1/2, 2s_1/2];    subshells = Subshell[1s_1/2, 2s_1/2, 2p_1/2, 2p_3/2]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Kohn-Sham (radial) potential ... defined on 392 grid points ...\n",
       "Zr:    [0.0, 8.244855740993904e-5, 0.00010962587388256838, 8.749358831813382e-5, 5.097980506260815e-5, 2.4678201401699168e-5, 1.213322378215754e-5, 5.953426498752406e-6, -1.2660071531622308e-6, 6.913157671878871e-6  …  3.210653174499592e-5, 3.1010863590772863e-5, 2.6459560832301544e-5, 1.434455476562247e-5, -5.421543045549688e-6, -2.551102335625353e-5, -3.0287703356425806e-5, -3.0156511826126174e-5, -3.0663601046824346e-5, -3.386964581742926e-5]  ...  [-5.999099121737457, -5.999340240481848, -5.999518222456781, -5.999708604853091, -5.99908561295028, -5.998748469143674, -5.998751895785086, -5.999009014057045, -5.999295841291825, -5.999531060820585  …  -5.998970054118388, -5.999514364068934, -5.998790713254115, -5.998468381936907, -5.997932073994636, -5.997801855601406, -5.998912038612594, -5.998915289579009, -5.999124696426017, -5.999999999999995]\n",
       "Radial grid:  rnt = 2.0e-6,  h = 0.05,  hp = 0.0,  NoPoints = 392,  ntL = 69,  ntS = 71, orderL = 7,  orderS = 8,  nsL = 62,  nsS = 63,  mesh = JAC.Radial.MeshGL(), ...  \n",
       "r:    [1.780504510281598e-8, 9.042759120177923e-8, 2.0787030685460475e-7]  ...  [564.4302351890452, 595.097239730825, 614.0606632064549]\n",
       "wr:   [4.530145586285996e-8, 9.785739473901763e-8, 1.3358660616640612e-7]  ...  [34.882547080539766, 25.552825070730517, 11.829256033233948]\n",
       "tS:   [0.0, 0.0, 0.0]  ...  [618.7099715607404, 618.7099715607404, 618.7099715607404]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "level  = multiplet.levels[1]\n",
    "potCH  = compute(\"radial potential: core-Hartree\", grid, level)\n",
    "potDFS = compute(\"radial potential: Dirac-Fock-Slater\", grid, level)\n",
    "potKS = compute(\"radial potential: Kohn-Sham\", grid, level)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "All these potentials are represented as *effective charge* $Z(r)$ and can be added to the nuclear potential, for example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "nuclear-potential: Fermi-distributed+DFS (radial) potential ... defined on 392 grid points ...\n",
       "Zr:    [0.0, 0.015702266764790654, 0.03601543032577465, 0.060518338790674746, 0.08500551706809074, 0.10526204815963544, 0.11779092713120655, 0..12583082234429907, 0.14609937932523667, 0.17889200687475523  …  0.3111614834804656, 0.323167423534021, 0.35188464859310015, 0.3982972214128069, 0.4543620690949641, 0.5103734383083619, 0.5566690602797584, 0.5852777320775541, 0.6022362296962717, 0.6427837698640307]  ...  [0.0009008782625432588, 0.0006597595181538907, 0.0004817775432197635, 0.0002913951469105669, 0.0009143870497210216, 0.0012515308563258998, 0.00124810421491528, 0.0009909859429546586, 0.0007041587081770828, 0.00046893917941659424  …  0.0010299458816138696, 0.0004856359310672076, 0.0012092867458868284, 0.001531618063094875, 0.002067926005365095, 0.0021981443985952254, 0.0010879613874061889, 0.0010847104209918967, 0.0008753035739834658, 6.217248937900877e-15]\n",
       "Radial grid:  rnt = 2.0e-6,  h = 0.05,  hp = 0.0,  NoPoints = 392,  ntL = 69,  ntS = 71, orderL = 7,  orderS = 8,  nsL = 62,  nsS = 63,  mesh = JAC.Radial.MeshGL(), ...  \n",
       "r:    [1.780504510281598e-8, 9.042759120177923e-8, 2.0787030685460475e-7]  ...  [564.4302351890452, 595.097239730825, 614.0606632064549]\n",
       "wr:   [4.530145586285996e-8, 9.785739473901763e-8, 1.3358660616640612e-7]  ...  [34.882547080539766, 25.552825070730517, 11.829256033233948]\n",
       "tS:   [0.0, 0.0, 0.0]  ...  [618.7099715607404, 618.7099715607404, 618.7099715607404]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "atomPotDFS = add(nucPotential, potDFS)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "As seen from this output, the effective charge of the overall atomic potential goes to *zero* as required for a neutral atom. We can display these potentials also graphically by using `JAC.plot()`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
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     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nucPotential2 = Nuclear.nuclearPotential(Model(6.,\"uniform\"), grid)\n",
    "nucPotential3 = Nuclear.nuclearPotential(Model(6., \"point\"), grid)\n",
    "atomPotCH = add(nucPotential2, potCH)\n",
    "atomPotKS = add(nucPotential3, potKS)\n",
    "plot(\"radial potentials\", [atomPotDFS,atomPotCH, atomPotKS], grid;N=300)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is directly noticeable, that the functions based on the Kohn-Sham and Dirac-Fock-Slater potential correspond to the expectation for a neutral atom since the curve for large r approaches 0. In contrast, the curve for the Core-Hartree potential converges to a value greater than 0. That is due to the fact that the summation for the Core-Hartree potential runs only over all core orbitals as it is mainly applied for atoms with a single valence electron (alkali atoms). The graph thus shows the effective charge a single valence electron would \"see\". To illustrate the different nuclear potentials, we could also look at the functions at a closer angle, simply by choosing a smaller N (number of grid points to be considered)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
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      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(\"radial potentials\", [atomPotDFS,atomPotCH, atomPotKS], grid;N=75)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The plot shows very well the caracteristics of the different nuclear models implemented in JAC."
   ]
  }
 ],
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