# 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
atoms = [He => [[1/2; 1/2; 1/2]]]  # Helium at the center of the box

model = model_DFT(lattice, atoms, [: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
1.7736540306159703

## Forward-mode implicit differentiation

Right now DFTK has no out-of-the-box support for implicit differentiation through the SCF. However one can easily work around this as follows. We keep both a non-dual basis and a basis including duals for easy bookkeeping (but redundant computation ...).

function self_consistent_field_dual(basis::PlaneWaveBasis, basis_dual::PlaneWaveBasis{T};
kwargs...) where T <: ForwardDiff.Dual
scfres = self_consistent_field(basis; kwargs...)
ψ = DFTK.select_occupied_orbitals(basis, scfres.ψ)
filled_occ = DFTK.filled_occupation(basis.model)
n_spin = basis.model.n_spin_components
n_bands = div(basis.model.n_electrons, n_spin * filled_occ)
occupation = [filled_occ * ones(n_bands) for _ in basis.kpoints]

# promote everything eagerly to Dual numbers
occupation_dual = [T.(occupation)]
ψ_dual = [Complex.(T.(real(ψ)), T.(imag(ψ)))]
ρ_dual = compute_density(basis_dual, ψ_dual, occupation_dual)

_, δH = energy_hamiltonian(basis_dual, ψ_dual, occupation_dual; ρ=ρ_dual)
δHψ = δH * ψ_dual
δHψ = [ForwardDiff.partials.(δHψ, 1)]
δψ = DFTK.solve_ΩplusK(basis, ψ, -δHψ, occupation)
δρ = DFTK.compute_δρ(basis, ψ, δψ, occupation)
ρ = ForwardDiff.value.(ρ_dual)
ψ, ρ, δψ, δρ
end;

This function is now used in the following to provide a dual version for the compute_dipole function:

function compute_dipole(ε::ForwardDiff.Dual; tol=1e-8, kwargs...)
T = ForwardDiff.tagtype(ε)
basis = make_basis(ForwardDiff.value(ε); kwargs...)
basis_dual = make_basis(ε; kwargs...)
ψ, ρ, δψ, δρ = self_consistent_field_dual(basis, basis_dual; tol)
ρ_dual = ForwardDiff.Dual{T}.(ρ, δρ)
dipole(basis_dual, ρ_dual)
end;

This setup allows to compute the polarizability via automatic differentiation:

polarizability = ForwardDiff.derivative(compute_dipole, 0.0)
println()
println("Polarizability via ForwardDiff:       $polarizability") println("Polarizability via finite difference:$polarizability_fd")
n     Energy            Eₙ-Eₙ₋₁     ρout-ρin   α      Diag
---   ---------------   ---------   --------   ----   ----
1   -2.770710889987         NaN   2.97e-01   0.80    8.0
2   -2.772051175209   -1.34e-03   5.00e-02   0.80    1.0
3   -2.772082439602   -3.13e-05   2.18e-03   0.80    1.0
4   -2.772083413387   -9.74e-07   1.58e-04   0.80    2.0
5   -2.772083417468   -4.08e-09   5.38e-05   0.80    2.0

Polarizability via ForwardDiff:       1.7724804728391526
Polarizability via finite difference: 1.7736540306159703