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
He = ElementPsp(:He, psp=load_psp("hgh/lda/He-q2"))
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[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
1.7741838413278463
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[1])]
ψ_dual = [Complex.(T.(real(ψ[1])), T.(imag(ψ[1])))]
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], 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 -2.770460250802 NaN 2.95e-01 0.80 8.0 2 -2.772045139710 -1.58e-03 5.23e-02 0.80 1.0 3 -2.772083287792 -3.81e-05 1.17e-03 0.80 2.0 4 -2.772083416240 -1.28e-07 1.88e-04 0.80 2.0 5 -2.772083417796 -1.56e-09 2.90e-05 0.80 2.0 Polarizability via ForwardDiff: 1.7725476339733843 Polarizability via finite difference: 1.7741838413278463