{
"cells": [
{
"cell_type": "markdown",
"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",
"using PseudoPotentialData\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",
" pseudopotentials = PseudoFamily(\"cp2k.nc.sr.lda.v0_1.semicore.gth\")\n",
" atoms = [ElementPsp(:He, pseudopotentials)]\n",
" positions = [[1/2, 1/2, 1/2]]\n",
"\n",
" model = model_DFT(lattice, atoms, positions;\n",
" functionals=[: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 Δtime\n",
"--- --------------- --------- --------- ---- ------\n",
" 1 -2.770859815902 -0.53 9.0 177ms\n",
" 2 -2.772140634348 -2.89 -1.31 1.0 114ms\n",
" 3 -2.772169899236 -4.53 -2.57 1.0 110ms\n",
" 4 -2.772170706547 -6.09 -3.53 2.0 127ms\n",
" 5 -2.772170722269 -7.80 -4.12 1.0 114ms\n",
" 6 -2.772170722951 -9.17 -4.93 1.0 161ms\n",
" 7 -2.772170723014 -10.20 -5.41 2.0 496ms\n",
" 8 -2.772170723015 -12.04 -6.25 1.0 114ms\n",
" 9 -2.772170723015 -14.31 -6.79 1.0 139ms\n",
" 10 -2.772170723015 -14.12 -7.47 1.0 115ms\n",
" 11 -2.772170723015 -14.07 -8.22 2.0 136ms\n",
"n Energy log10(ΔE) log10(Δρ) Diag Δtime\n",
"--- --------------- --------- --------- ---- ------\n",
" 1 -2.770755627493 -0.53 8.0 155ms\n",
" 2 -2.772053846938 -2.89 -1.31 1.0 109ms\n",
" 3 -2.772082926712 -4.54 -2.55 1.0 108ms\n",
" 4 -2.772083368510 -6.35 -3.45 1.0 111ms\n",
" 5 -2.772083416412 -7.32 -3.96 2.0 127ms\n",
" 6 -2.772083417755 -8.87 -5.19 1.0 117ms\n",
" 7 -2.772083417807 -10.28 -5.28 2.0 131ms\n",
" 8 -2.772083417811 -11.50 -6.30 1.0 121ms\n",
" 9 -2.772083417811 -13.23 -6.56 2.0 539ms\n",
" 10 -2.772083417811 -14.18 -7.40 1.0 114ms\n",
" 11 -2.772083417811 + -14.10 -8.00 1.0 116ms\n",
" 12 -2.772083417811 -13.82 -8.66 2.0 154ms\n"
]
},
{
"output_type": "execute_result",
"data": {
"text/plain": "1.7735579730013822"
},
"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. This leads to a density-functional perturbation\n",
"theory problem, which is automatically set up and solved in the background."
],
"metadata": {}
},
{
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"n Energy log10(ΔE) log10(Δρ) Diag Δtime\n",
"--- --------------- --------- --------- ---- ------\n",
" 1 -2.770657324401 -0.53 9.0 180ms\n",
" 2 -2.772053993343 -2.85 -1.30 1.0 106ms\n",
" 3 -2.772082856401 -4.54 -2.65 1.0 110ms\n",
" 4 -2.772083415438 -6.25 -4.03 2.0 127ms\n",
" 5 -2.772083417616 -8.66 -4.45 2.0 128ms\n",
" 6 -2.772083417803 -9.73 -5.42 1.0 600ms\n",
" 7 -2.772083417810 -11.12 -5.99 2.0 148ms\n",
" 8 -2.772083417811 -13.01 -6.50 1.0 152ms\n",
" 9 -2.772083417811 -13.64 -7.01 2.0 145ms\n",
" 10 -2.772083417811 + -15.35 -7.92 1.0 116ms\n",
" 11 -2.772083417811 + -14.35 -8.56 2.0 130ms\n",
"Solving response problem\n",
"[ Info: GMRES linsolve starts with norm of residual = 4.19e+00\n",
"[ Info: GMRES linsolve in iteration 1; step 1: normres = 2.49e-01\n",
"[ Info: GMRES linsolve in iteration 1; step 2: normres = 3.76e-03\n",
"[ Info: GMRES linsolve in iteration 1; step 3: normres = 2.84e-04\n",
"[ Info: GMRES linsolve in iteration 1; step 4: normres = 4.67e-06\n",
"[ Info: GMRES linsolve in iteration 1; step 5: normres = 1.08e-08\n",
"┌ Info: GMRES linsolve converged at iteration 1, step 6:\n",
"│ * norm of residual = 7.68e-10\n",
"└ * number of operations = 8\n",
"\n",
"Polarizability via ForwardDiff: 1.7725349649275122\n",
"Polarizability via finite difference: 1.7735579730013822\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
}
],
"nbformat_minor": 3,
"metadata": {
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "1.11.4"
},
"kernelspec": {
"name": "julia-1.11",
"display_name": "Julia 1.11.4",
"language": "julia"
}
},
"nbformat": 4
}