Simple example for computing properties using (forward-mode) automatic differentation. 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=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 --- --------------- --------- --------- ---- 1 -2.770907117738 -0.52 9.0 2 -2.772149038433 -2.91 -1.32 1.0 3 -2.772170030240 -4.68 -2.39 2.0 4 -2.772170722605 -6.16 -4.07 2.0 ┌ Warning: Negative ρ detected │ min_ρ = -1.237664896679902e-18 └ @ DFTK /home/runner/work/DFTK.jl/DFTK.jl/src/densities.jl:7 5 -2.772170723005 -9.40 -5.00 2.0 n Energy log10(ΔE) log10(Δρ) Diag --- --------------- --------- --------- ---- 1 -2.770681324509 -0.53 8.0 2 -2.772036828676 -2.87 -1.30 1.0 3 -2.772083327254 -4.33 -2.78 2.0 4 -2.772083416680 -7.05 -3.80 2.0 5 -2.772083417796 -8.95 -4.60 2.0
1.77364460650267
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 --- --------------- --------- --------- ---- 1 -2.770741463218 -0.53 8.0 2 -2.772049843356 -2.88 -1.30 1.0 3 -2.772082720625 -4.48 -2.69 1.0 4 -2.772083413349 -6.16 -3.77 2.0 5 -2.772083417381 -8.39 -4.22 2.0 Polarizability via ForwardDiff: 1.7724740060017201 Polarizability via finite difference: 1.77364460650267