# Polarizability using automatic differentiation

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
positions = [[1/2, 1/2, 1/2]]

model = model_DFT(lattice, atoms, positions, [:lda_x, :lda_c_vwn];
extra_terms=[ExternalFromReal(r -> -ε * (r - 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 - 1/2) 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
1.7736359914220772

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   -2.770768403445                   -0.52    9.0
2   -2.772059671137       -2.89       -1.33    1.0
3   -2.772082340617       -4.64       -2.33    2.0
4   -2.772083417454       -5.97       -3.96    2.0
5   -2.772083417800       -9.46       -4.82    2.0

Polarizability via ForwardDiff:       1.7725483050547999
Polarizability via finite difference: 1.7736359914220772