{
 "cells": [
  {
   "cell_type": "markdown",
   "source": [
    "# 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",
    "    He = ElementPsp(:He, psp=load_psp(\"hgh/lda/He-q2\"))\n",
    "    atoms = [He => [[1/2; 1/2; 1/2]]]  # Helium at the center of the box\n",
    "\n",
    "    model = model_DFT(lattice, atoms, [: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            Eₙ-Eₙ₋₁     ρout-ρin   α      Diag\n",
      "---   ---------------   ---------   --------   ----   ----\n",
      "  1   -2.770678846504         NaN   2.97e-01   0.80    8.0\n",
      "  2   -2.772137366142   -1.46e-03   4.99e-02   0.80    1.0\n",
      "  3   -2.772170583186   -3.32e-05   2.41e-03   0.80    2.0\n",
      "  4   -2.772170722157   -1.39e-07   6.85e-05   0.80    2.0\n",
      "  5   -2.772170722987   -8.30e-10   1.70e-05   0.80    3.0\n",
      "n     Energy            Eₙ-Eₙ₋₁     ρout-ρin   α      Diag\n",
      "---   ---------------   ---------   --------   ----   ----\n",
      "  1   -2.770778324926         NaN   2.98e-01   0.80    8.0\n",
      "  2   -2.772055607034   -1.28e-03   4.87e-02   0.80    1.0\n",
      "  3   -2.772083022243   -2.74e-05   2.78e-03   0.80    1.0\n",
      "  4   -2.772083399965   -3.78e-07   3.20e-04   0.80    2.0\n",
      "  5   -2.772083415187   -1.52e-08   1.46e-04   0.80    2.0\n",
      "  6   -2.772083417777   -2.59e-09   1.32e-05   0.80    2.0\n"
     ]
    },
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": "1.7738451891157154"
     },
     "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": [
    "## Forward-mode implicit differentiation\n",
    "\n",
    "Right now DFTK has no out-of-the-box support for implicit differentiation through the SCF.\n",
    "However one can easily work around this as follows. We keep both a non-dual basis\n",
    "and a basis including duals for easy bookkeeping (but redundant computation ...)."
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "function self_consistent_field_dual(basis::PlaneWaveBasis, basis_dual::PlaneWaveBasis{T};\n",
    "                                    kwargs...) where T <: ForwardDiff.Dual\n",
    "    scfres = self_consistent_field(basis; kwargs...)\n",
    "    ψ = DFTK.select_occupied_orbitals(basis, scfres.ψ)\n",
    "    filled_occ = DFTK.filled_occupation(basis.model)\n",
    "    n_spin = basis.model.n_spin_components\n",
    "    n_bands = div(basis.model.n_electrons, n_spin * filled_occ)\n",
    "    occupation = [filled_occ * ones(n_bands) for _ in basis.kpoints]\n",
    "\n",
    "    # promote everything eagerly to Dual numbers\n",
    "    occupation_dual = [T.(occupation[1])]\n",
    "    ψ_dual = [Complex.(T.(real(ψ[1])), T.(imag(ψ[1])))]\n",
    "    ρ_dual = compute_density(basis_dual, ψ_dual, occupation_dual)\n",
    "\n",
    "    _, δH = energy_hamiltonian(basis_dual, ψ_dual, occupation_dual; ρ=ρ_dual)\n",
    "    δHψ = δH * ψ_dual\n",
    "    δHψ = [ForwardDiff.partials.(δHψ[1], 1)]\n",
    "    δψ = DFTK.solve_ΩplusK(basis, ψ, -δHψ, occupation)\n",
    "    δρ = DFTK.compute_δρ(basis, ψ, δψ, occupation)\n",
    "    ρ = ForwardDiff.value.(ρ_dual)\n",
    "    ψ, ρ, δψ, δρ\n",
    "end;"
   ],
   "metadata": {},
   "execution_count": 3
  },
  {
   "cell_type": "markdown",
   "source": [
    "This function is now used in the following to provide a dual version\n",
    "for the compute_dipole function:"
   ],
   "metadata": {}
  },
  {
   "outputs": [],
   "cell_type": "code",
   "source": [
    "function compute_dipole(ε::ForwardDiff.Dual; tol=1e-8, kwargs...)\n",
    "    T = ForwardDiff.tagtype(ε)\n",
    "    basis = make_basis(ForwardDiff.value(ε); kwargs...)\n",
    "    basis_dual = make_basis(ε; kwargs...)\n",
    "    ψ, ρ, δψ, δρ = self_consistent_field_dual(basis, basis_dual; tol)\n",
    "    ρ_dual = ForwardDiff.Dual{T}.(ρ, δρ)\n",
    "    dipole(basis_dual, ρ_dual)\n",
    "end;"
   ],
   "metadata": {},
   "execution_count": 4
  },
  {
   "cell_type": "markdown",
   "source": [
    "This setup allows to compute the polarizability via automatic differentiation:"
   ],
   "metadata": {}
  },
  {
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "n     Energy            Eₙ-Eₙ₋₁     ρout-ρin   α      Diag\n",
      "---   ---------------   ---------   --------   ----   ----\n",
      "  1   -2.770779143543         NaN   2.98e-01   0.80    8.0\n",
      "  2   -2.772051054186   -1.27e-03   4.85e-02   0.80    1.0\n",
      "  3   -2.772083267433   -3.22e-05   2.97e-03   0.80    2.0\n",
      "  4   -2.772083416404   -1.49e-07   9.22e-05   0.80    2.0\n",
      "  5   -2.772083417729   -1.33e-09   2.89e-05   0.80    2.0\n",
      "\n",
      "Polarizability via ForwardDiff:       1.772564556461622\n",
      "Polarizability via finite difference: 1.7738451891157154\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": 5
  }
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