In this example, we show how to define custom solvers. Our system will again be silicon, because we are not very imaginative
using DFTK, LinearAlgebra
a = 10.26
lattice = a / 2 * [[0 1 1.];
[1 0 1.];
[1 1 0.]]
Si = ElementPsp(:Si, psp=load_psp("hgh/lda/Si-q4"))
atoms = [Si, Si]
positions = [ones(3)/8, -ones(3)/8]
# We take very (very) crude parameters
model = model_LDA(lattice, atoms, positions)
basis = PlaneWaveBasis(model; Ecut=5, kgrid=[1, 1, 1]);
We define our custom fix-point solver: simply a damped fixed-point
function my_fp_solver(f, x0, max_iter; tol)
mixing_factor = .7
x = x0
fx = f(x)
for n = 1:max_iter
inc = fx - x
if norm(inc) < tol
break
end
x = x + mixing_factor * inc
fx = f(x)
end
(fixpoint=x, converged=norm(fx-x) < tol)
end;
Our eigenvalue solver just forms the dense matrix and diagonalizes it explicitly (this only works for very small systems)
function my_eig_solver(A, X0; maxiter, tol, kwargs...)
n = size(X0, 2)
A = Array(A)
E = eigen(A)
λ = E.values[1:n]
X = E.vectors[:, 1:n]
(λ=λ, X=X, residual_norms=[], iterations=0, converged=true, n_matvec=0)
end;
Finally we also define our custom mixing scheme. It will be a mixture
of simple mixing (for the first 2 steps) and than default to Kerker mixing.
In the mixing interface δF
is (ρ_\text{out} - ρ_\text{in})
, i.e.
the difference in density between two subsequent SCF steps and the mix
function returns δρ
, which is added to ρ_\text{in}
to yield ρ_\text{next}
,
the density for the next SCF step.
struct MyMixing
n_simple # Number of iterations for simple mixing
end
MyMixing() = MyMixing(2)
function DFTK.mix_density(mixing::MyMixing, basis, δF; n_iter, kwargs...)
if n_iter <= mixing.n_simple
return δF # Simple mixing -> Do not modify update at all
else
# Use the default KerkerMixing from DFTK
DFTK.mix_density(KerkerMixing(), basis, δF; kwargs...)
end
end
That's it! Now we just run the SCF with these solvers
scfres = self_consistent_field(basis;
tol=1e-8,
solver=my_fp_solver,
eigensolver=my_eig_solver,
mixing=MyMixing());
n Energy log10(ΔE) log10(Δρ) Diag --- --------------- --------- --------- ---- 1 -7.236023732203 -0.50 0.0 2 -7.249611449036 -1.87 -0.92 0.0 3 -7.251175102701 -2.81 -1.34 0.0 4 -7.251296007223 -3.92 -1.65 0.0 5 -7.251327328326 -4.50 -1.96 0.0 6 -7.251335597614 -5.08 -2.25 0.0 7 -7.251337862375 -5.64 -2.54 0.0 8 -7.251338511713 -6.19 -2.83 0.0 9 -7.251338707003 -6.71 -3.10 0.0 10 -7.251338768389 -7.21 -3.37 0.0 11 -7.251338788417 -7.70 -3.62 0.0 12 -7.251338795146 -8.17 -3.87 0.0
Note that the default convergence criterion is on the difference of
energy from one step to the other; when this gets below tol
, the
"driver" self_consistent_field
artificially makes the fixpoint
solver think it's converged by forcing f(x) = x
. You can customize
this with the is_converged
keyword argument to
self_consistent_field
.