{
 "cells": [
  {
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   "source": [
    "# Polarizability using automatic differentiation\n",
    "\n",
    "Simple example for computing properties using (forward-mode)\n",
    "automatic differentiation.\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)\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   -2.770831422053                   -0.53    8.0\n",
      "  2   -2.772140124772       -2.88       -1.31    1.0\n",
      "  3   -2.772170111817       -4.52       -2.58    1.0\n",
      "  4   -2.772170715344       -6.22       -3.71    2.0\n",
      "  5   -2.772170722345       -8.15       -4.11    2.0\n",
      "  6   -2.772170723002       -9.18       -4.99    1.0\n",
      "  7   -2.772170723014      -10.93       -5.77    1.0\n",
      "  8   -2.772170723015      -11.94       -6.30    2.0\n",
      "  9   -2.772170723015      -14.03       -6.69    1.0\n",
      " 10   -2.772170723015      -14.45       -7.09    1.0\n",
      " 11   -2.772170723015   +  -15.35       -7.90    2.0\n",
      " 12   -2.772170723015      -13.92       -8.55    1.0\n",
      "n     Energy            log10(ΔE)   log10(Δρ)   Diag\n",
      "---   ---------------   ---------   ---------   ----\n",
      "  1   -2.770789244638                   -0.52    9.0\n",
      "  2   -2.772060985538       -2.90       -1.32    1.0\n",
      "  3   -2.772083109637       -4.66       -2.47    1.0\n",
      "  4   -2.772083345932       -6.63       -3.17    1.0\n",
      "  5   -2.772083416701       -7.15       -3.98    2.0\n",
      "  6   -2.772083417755       -8.98       -4.99    1.0\n",
      "  7   -2.772083417811      -10.25       -5.66    2.0\n",
      "  8   -2.772083417811      -13.94       -6.21    1.0\n",
      "  9   -2.772083417811      -13.16       -6.94    2.0\n",
      " 10   -2.772083417811   +  -13.97       -6.85    1.0\n",
      " 11   -2.772083417811      -13.92       -7.41    1.0\n",
      " 12   -2.772083417811   +  -14.35       -7.54    1.0\n",
      " 13   -2.772083417811   +  -14.51       -7.80    1.0\n",
      " 14   -2.772083417811      -14.88       -7.92    1.0\n",
      " 15   -2.772083417811      -14.07       -8.47    1.0\n"
     ]
    },
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "1.773558037674741"
     },
     "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   -2.770771102104                   -0.52    9.0\n",
      "  2   -2.772060864673       -2.89       -1.32    1.0\n",
      "  3   -2.772083027561       -4.65       -2.44    1.0\n",
      "  4   -2.772083338213       -6.51       -3.13    1.0\n",
      "  5   -2.772083417490       -7.10       -4.22    2.0\n",
      "  6   -2.772083417702       -9.68       -4.55    1.0\n",
      "  7   -2.772083417809       -9.97       -5.69    1.0\n",
      "  8   -2.772083417811      -11.93       -6.07    2.0\n",
      "  9   -2.772083417811      -13.42       -6.43    1.0\n",
      " 10   -2.772083417811      -13.89       -8.25    1.0\n",
      "\n",
      "Polarizability via ForwardDiff:       1.7725349319705532\n",
      "Polarizability via finite difference: 1.773558037674741\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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