forwarddiff.jl

# # 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](@ref).

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

# 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")