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; 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)
    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

1.7735580915002023

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   Δtime
---   ---------------   ---------   ---------   ----   ------
  1   -2.770804865403                   -0.52    9.0    162ms
  2   -2.772062015679       -2.90       -1.32    1.0    105ms
  3   -2.772083028599       -4.68       -2.44    1.0    106ms
  4   -2.772083344983       -6.50       -3.14    1.0    134ms
  5   -2.772083417707       -7.14       -4.44    2.0    124ms
  6   -2.772083417761      -10.27       -4.54    1.0    110ms
  7   -2.772083417807      -10.34       -5.54    1.0    126ms
  8   -2.772083417811      -11.47       -6.23    2.0    159ms
  9   -2.772083417811      -13.32       -6.66    1.0    123ms
 10   -2.772083417811      -13.69       -7.31    1.0    132ms
 11   -2.772083417811   +  -13.83       -7.72    2.0    142ms
 12   -2.772083417811      -14.03       -8.81    1.0    116ms

Polarizability via ForwardDiff:       1.7725349669914756
Polarizability via finite difference: 1.7735580915002023