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
n Energy log10(ΔE) log10(Δρ) Diag --- --------------- --------- --------- ---- 1 -2.770831422053 -0.53 8.0 2 -2.772140124772 -2.88 -1.31 1.0 3 -2.772170111817 -4.52 -2.58 1.0 4 -2.772170715344 -6.22 -3.71 2.0 5 -2.772170722345 -8.15 -4.11 2.0 6 -2.772170723002 -9.18 -4.99 1.0 7 -2.772170723014 -10.93 -5.77 1.0 8 -2.772170723015 -11.94 -6.30 2.0 9 -2.772170723015 -14.03 -6.69 1.0 10 -2.772170723015 -14.45 -7.09 1.0 11 -2.772170723015 + -15.35 -7.90 2.0 12 -2.772170723015 -13.92 -8.55 1.0 n Energy log10(ΔE) log10(Δρ) Diag --- --------------- --------- --------- ---- 1 -2.770789244638 -0.52 9.0 2 -2.772060985538 -2.90 -1.32 1.0 3 -2.772083109637 -4.66 -2.47 1.0 4 -2.772083345932 -6.63 -3.17 1.0 5 -2.772083416701 -7.15 -3.98 2.0 6 -2.772083417755 -8.98 -4.99 1.0 7 -2.772083417811 -10.25 -5.66 2.0 8 -2.772083417811 -13.94 -6.21 1.0 9 -2.772083417811 -13.16 -6.94 2.0 10 -2.772083417811 + -13.97 -6.85 1.0 11 -2.772083417811 -13.92 -7.41 1.0 12 -2.772083417811 + -14.35 -7.54 1.0 13 -2.772083417811 + -14.51 -7.80 1.0 14 -2.772083417811 -14.88 -7.92 1.0 15 -2.772083417811 -14.07 -8.47 1.0
1.773558037674741
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 --- --------------- --------- --------- ---- 1 -2.770771102104 -0.52 9.0 2 -2.772060864673 -2.89 -1.32 1.0 3 -2.772083027561 -4.65 -2.44 1.0 4 -2.772083338213 -6.51 -3.13 1.0 5 -2.772083417490 -7.10 -4.22 2.0 6 -2.772083417702 -9.68 -4.55 1.0 7 -2.772083417809 -9.97 -5.69 1.0 8 -2.772083417811 -11.93 -6.07 2.0 9 -2.772083417811 -13.42 -6.43 1.0 10 -2.772083417811 -13.89 -8.25 1.0 Polarizability via ForwardDiff: 1.7725349319705532 Polarizability via finite difference: 1.773558037674741