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    "# Polarizability using automatic differentiation\n",
    "\n",
    "Simple example for computing properties using (forward-mode)\n",
    "automatic differentation.\n",
    "For a more classical approach and more details about computing polarizabilities,\n",
    "see Polarizability by linear response."
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "using DFTK\n",
    "using LinearAlgebra\n",
    "using ForwardDiff\n",
    "\n",
    "# Construct PlaneWaveBasis given a particular electric field strength\n",
    "# Again we take the example of a Helium atom.\n",
    "function make_basis(ε::T; a=10., Ecut=30) where T\n",
    "    lattice=T(a) * I(3)  # lattice is a cube of ``a`` Bohrs\n",
    "    # Helium at the center of the box\n",
    "    atoms     = [ElementPsp(:He, psp=load_psp(\"hgh/lda/He-q2\"))]\n",
    "    positions = [[1/2, 1/2, 1/2]]\n",
    "\n",
    "    model = model_DFT(lattice, atoms, positions, [:lda_x, :lda_c_vwn];\n",
    "                      extra_terms=[ExternalFromReal(r -> -ε * (r[1] - a/2))],\n",
    "                      symmetries=false)\n",
    "    PlaneWaveBasis(model; Ecut, kgrid=[1, 1, 1])  # No k-point sampling on isolated system\n",
    "end\n",
    "\n",
    "# dipole moment of a given density (assuming the current geometry)\n",
    "function dipole(basis, ρ)\n",
    "    @assert isdiag(basis.model.lattice)\n",
    "    a  = basis.model.lattice[1, 1]\n",
    "    rr = [a * (r[1] - 1/2) for r in r_vectors(basis)]\n",
    "    sum(rr .* ρ) * basis.dvol\n",
    "end\n",
    "\n",
    "# Function to compute the dipole for a given field strength\n",
    "function compute_dipole(ε; tol=1e-8, kwargs...)\n",
    "    scfres = self_consistent_field(make_basis(ε; kwargs...), tol=tol)\n",
    "    dipole(scfres.basis, scfres.ρ)\n",
    "end;"
   ],
   "metadata": {},
   "execution_count": 1
  },
  {
   "cell_type": "markdown",
   "source": [
    "With this in place we can compute the polarizability from finite differences\n",
    "(just like in the previous example):"
   ],
   "metadata": {}
  },
  {
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "n     Energy            log10(ΔE)   log10(Δρ)   Diag\n",
      "---   ---------------   ---------   ---------   ----\n",
      "  1   -2.770913397474                   -0.52    9.0\n",
      "  2   -2.772149612126       -2.91       -1.32    1.0\n",
      "  3   -2.772169910421       -4.69       -2.37    2.0\n",
      "  4   -2.772170722591       -6.09       -4.00    2.0\n",
      "  5   -2.772170723008       -9.38       -5.00    2.0\n",
      "n     Energy            log10(ΔE)   log10(Δρ)   Diag\n",
      "---   ---------------   ---------   ---------   ----\n",
      "  1   -2.770751990144                   -0.53    8.0\n",
      "  2   -2.772047381018       -2.89       -1.31    1.0\n",
      "  3   -2.772082596381       -4.45       -2.61    1.0\n",
      "  4   -2.772083416150       -6.09       -4.09    2.0\n",
      "  5   -2.772083417754       -8.79       -4.64    2.0\n"
     ]
    },
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "1.7732661269077885"
     },
     "metadata": {},
     "execution_count": 2
    }
   ],
   "cell_type": "code",
   "source": [
    "polarizability_fd = let\n",
    "    ε = 0.01\n",
    "    (compute_dipole(ε) - compute_dipole(0.0)) / ε\n",
    "end"
   ],
   "metadata": {},
   "execution_count": 2
  },
  {
   "cell_type": "markdown",
   "source": [
    "We do the same thing using automatic differentiation. Under the hood this uses\n",
    "custom rules to implicitly differentiate through the self-consistent\n",
    "field fixed-point problem."
   ],
   "metadata": {}
  },
  {
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "n     Energy            log10(ΔE)   log10(Δρ)   Diag\n",
      "---   ---------------   ---------   ---------   ----\n",
      "  1   -2.770735900316                   -0.53    8.0\n",
      "  2   -2.772052363233       -2.88       -1.31    1.0\n",
      "  3   -2.772082720727       -4.52       -2.56    1.0\n",
      "  4   -2.772083402296       -6.17       -3.51    2.0\n",
      "  5   -2.772083417177       -7.83       -4.08    2.0\n",
      "  6   -2.772083417807       -9.20       -4.97    2.0\n",
      "\n",
      "Polarizability via ForwardDiff:       1.7725254937958959\n",
      "Polarizability via finite difference: 1.7732661269077885\n"
     ]
    }
   ],
   "cell_type": "code",
   "source": [
    "polarizability = ForwardDiff.derivative(compute_dipole, 0.0)\n",
    "println()\n",
    "println(\"Polarizability via ForwardDiff:       $polarizability\")\n",
    "println(\"Polarizability via finite difference: $polarizability_fd\")"
   ],
   "metadata": {},
   "execution_count": 3
  }
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