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
1.773557857904911
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 -2.770819205559 -0.52 9.0 226ms
2 -2.772062458408 -2.91 -1.32 1.0 124ms
3 -2.772083042392 -4.69 -2.44 1.0 100ms
4 -2.772083344701 -6.52 -3.14 1.0 102ms
5 -2.772083417609 -7.14 -4.32 2.0 118ms
6 -2.772083417744 -9.87 -4.67 1.0 108ms
7 -2.772083417809 -10.18 -5.83 1.0 108ms
8 -2.772083417811 -11.85 -6.44 2.0 124ms
9 -2.772083417811 -13.74 -7.16 1.0 107ms
10 -2.772083417811 -14.15 -8.52 2.0 129ms
Solving response problem
Iter Restart Krydim log10(res) avg(CG) Δtime Comment
---- ------- ------ ---------- ------- ------ ---------------
13.0 183ms Non-interacting
1 0 1 -0.60 10.0 655ms
2 0 2 -2.42 8.0 117ms
3 0 3 -3.55 6.0 111ms
4 0 4 -5.33 5.0 107ms
5 0 5 -7.85 2.0 93.6ms
6 0 6 -10.40 1.0 86.0ms
7 1 1 -8.08 12.0 222ms Restart
8 1 2 -9.75 1.0 88.2ms
13.0 130ms Final orbitals
Polarizability via ForwardDiff: 1.7725349643443338
Polarizability via finite difference: 1.773557857904911