Polarizability using automatic differentiation
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, load_psp("hgh/lda/He-q2"))]
positions = [[1/2, 1/2, 1/2]]
model = model_DFT(lattice, atoms, positions;
functionals=[: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
1.7735583091065321
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 -2.770776068627 -0.52 9.0 230ms
2 -2.772059473223 -2.89 -1.33 1.0 128ms
3 -2.772082869679 -4.63 -2.41 1.0 109ms
4 -2.772083327069 -6.34 -3.09 1.0 119ms
5 -2.772083417750 -7.04 -4.72 2.0 149ms
6 -2.772083417778 -10.54 -4.72 2.0 137ms
7 -2.772083417809 -10.51 -5.76 1.0 119ms
8 -2.772083417811 -11.92 -6.94 2.0 138ms
9 -2.772083417811 + -14.27 -7.25 1.0 120ms
10 -2.772083417811 -14.75 -8.25 1.0 127ms
Polarizability via ForwardDiff: 1.7725349600657292
Polarizability via finite difference: 1.7735583091065321