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

1.7735578035977049

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. This leads to a density-functional perturbation theory problem, which is automatically set up and solved in the background.

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   -2.770753434384                   -0.53    8.0    191ms
  2   -2.772055665178       -2.89       -1.31    1.0    109ms
  3   -2.772083109003       -4.56       -2.59    1.0    142ms
  4   -2.772083396660       -6.54       -3.69    1.0    120ms
  5   -2.772083417093       -7.69       -4.10    2.0    139ms
  6   -2.772083417784       -9.16       -5.26    1.0    120ms
  7   -2.772083417809      -10.59       -5.54    2.0    221ms
  8   -2.772083417811      -11.96       -6.21    1.0    790ms
  9   -2.772083417811      -13.50       -6.64    1.0    122ms
 10   -2.772083417811   +  -15.35       -7.85    2.0    145ms
 11   -2.772083417811   +  -14.57       -8.15    2.0    151ms

Polarizability via ForwardDiff:       1.7725349844378764
Polarizability via finite difference: 1.7735578035977049