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
using PseudoPotentialData

# 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
    pseudopotentials = PseudoFamily("cp2k.nc.sr.lda.v0_1.semicore.gth")
    atoms     = [ElementPsp(:He, pseudopotentials)]
    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

1.7735580371297435

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   -2.770754693356                   -0.52    8.0    268ms
  2   -2.772054943810       -2.89       -1.32    1.0    718ms
  3   -2.772083013662       -4.55       -2.53    1.0    110ms
  4   -2.772083364885       -6.45       -3.39    1.0    111ms
  5   -2.772083416109       -7.29       -3.92    2.0    132ms
  6   -2.772083417773       -8.78       -5.02    1.0    114ms
  7   -2.772083417808      -10.45       -5.34    2.0    136ms
  8   -2.772083417811      -11.65       -6.13    1.0    119ms
  9   -2.772083417811      -13.34       -6.81    1.0    124ms
 10   -2.772083417811      -14.07       -7.47    2.0    139ms
 11   -2.772083417811      -14.45       -7.59    1.0    161ms
 12   -2.772083417811   +  -14.40       -8.29    1.0    125ms
Solving response problem
[ Info: GMRES linsolve starts with norm of residual = 4.19e+00
[ Info: GMRES linsolve in iteration 1; step 1: normres = 2.49e-01
[ Info: GMRES linsolve in iteration 1; step 2: normres = 3.76e-03
[ Info: GMRES linsolve in iteration 1; step 3: normres = 2.84e-04
[ Info: GMRES linsolve in iteration 1; step 4: normres = 4.67e-06
[ Info: GMRES linsolve in iteration 1; step 5: normres = 1.08e-08
[ Info: GMRES linsolve in iteration 1; step 6: normres = 8.59e-09
┌ Info: GMRES linsolve converged at iteration 2, step 1:
│ * norm of residual = 7.22e-10
└ * number of operations = 9

Polarizability via ForwardDiff:       1.772534984562939
Polarizability via finite difference: 1.7735580371297435