Simple example for computing properties using (forward-mode) automatic differentiation. For a more classical approach and more details about computing polarizabilities, see Polarizability by linear response.
using DFTK
using LinearAlgebra
using ForwardDiff
# Construct PlaneWaveBasis given a particular electric field strength
# Again we take the example of a Helium atom.
function make_basis(ε::T; a=10., Ecut=30) where {T}
lattice=T(a) * I(3) # lattice is a cube of $a$ Bohrs
# Helium at the center of the box
atoms = [ElementPsp(:He, psp=load_psp("hgh/lda/He-q2"))]
positions = [[1/2, 1/2, 1/2]]
model = model_DFT(lattice, atoms, positions, [:lda_x, :lda_c_vwn];
extra_terms=[ExternalFromReal(r -> -ε * (r[1] - a/2))],
symmetries=false)
PlaneWaveBasis(model; Ecut, kgrid=[1, 1, 1]) # No k-point sampling on isolated system
end
# dipole moment of a given density (assuming the current geometry)
function dipole(basis, ρ)
@assert isdiag(basis.model.lattice)
a = basis.model.lattice[1, 1]
rr = [a * (r[1] - 1/2) for r in r_vectors(basis)]
sum(rr .* ρ) * basis.dvol
end
# Function to compute the dipole for a given field strength
function compute_dipole(ε; tol=1e-8, kwargs...)
scfres = self_consistent_field(make_basis(ε; kwargs...); tol)
dipole(scfres.basis, scfres.ρ)
end;
With this in place we can compute the polarizability from finite differences (just like in the previous example):
polarizability_fd = let
ε = 0.01
(compute_dipole(ε) - compute_dipole(0.0)) / ε
end
n Energy log10(ΔE) log10(Δρ) Diag Δtime --- --------------- --------- --------- ---- ------ 1 -2.770805509700 -0.52 9.0 2 -2.772145785646 -2.87 -1.32 1.0 163ms 3 -2.772170260912 -4.61 -2.46 1.0 166ms 4 -2.772170638205 -6.42 -3.15 1.0 214ms 5 -2.772170722585 -7.07 -4.08 2.0 193ms 6 -2.772170722745 -9.80 -4.21 1.0 172ms 7 -2.772170722984 -9.62 -4.70 1.0 203ms 8 -2.772170723009 -10.60 -5.09 1.0 178ms 9 -2.772170723015 -11.24 -6.22 1.0 200ms 10 -2.772170723015 -12.82 -6.19 2.0 219ms 11 -2.772170723015 -13.24 -6.84 1.0 183ms 12 -2.772170723015 -14.01 -7.62 2.0 235ms 13 -2.772170723015 + -14.21 -7.75 1.0 187ms 14 -2.772170723015 + -15.05 -8.16 1.0 214ms n Energy log10(ΔE) log10(Δρ) Diag Δtime --- --------------- --------- --------- ---- ------ 1 -2.770625943749 -0.53 9.0 2 -2.772045645794 -2.85 -1.30 1.0 171ms 3 -2.772082285558 -4.44 -2.70 1.0 188ms 4 -2.772083404636 -5.95 -3.52 2.0 191ms 5 -2.772083413612 -8.05 -3.74 2.0 191ms 6 -2.772083417735 -8.38 -5.31 1.0 217ms 7 -2.772083417809 -10.13 -5.40 3.0 219ms 8 -2.772083417811 -11.74 -5.94 1.0 178ms 9 -2.772083417811 -13.21 -6.34 1.0 204ms 10 -2.772083417811 -13.43 -6.86 1.0 181ms 11 -2.772083417811 + -13.88 -7.87 2.0 232ms 12 -2.772083417811 -14.12 -7.78 2.0 207ms 13 -2.772083417811 -14.07 -8.51 1.0 197ms
1.773557979689211
We do the same thing using automatic differentiation. Under the hood this uses custom rules to implicitly differentiate through the self-consistent field fixed-point problem.
polarizability = ForwardDiff.derivative(compute_dipole, 0.0)
println()
println("Polarizability via ForwardDiff: $polarizability")
println("Polarizability via finite difference: $polarizability_fd")
n Energy log10(ΔE) log10(Δρ) Diag Δtime --- --------------- --------- --------- ---- ------ 1 -2.770789297183 -0.52 9.0 2 -2.772057410798 -2.90 -1.32 1.0 166ms 3 -2.772083133636 -4.59 -2.52 1.0 171ms 4 -2.772083368767 -6.63 -3.32 1.0 210ms 5 -2.772083414757 -7.34 -3.78 2.0 195ms 6 -2.772083417668 -8.54 -4.67 1.0 174ms 7 -2.772083417798 -9.89 -4.98 2.0 259ms 8 -2.772083417810 -10.93 -5.68 1.0 182ms 9 -2.772083417811 -12.10 -6.09 2.0 216ms 10 -2.772083417811 -13.05 -7.21 1.0 206ms 11 -2.772083417811 + -14.21 -7.78 1.0 184ms 12 -2.772083417811 -14.03 -8.87 2.0 218ms Polarizability via ForwardDiff: 1.7725349677726303 Polarizability via finite difference: 1.773557979689211