# Custom solvers

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   -7.195093036597                   -0.46    0.0
2   -7.243134951128       -1.32       -0.81    0.0
3   -7.250778101659       -2.12       -1.27    0.0
4   -7.251207646101       -3.37       -1.59    0.0
5   -7.251307660999       -4.00       -1.89    0.0
6   -7.251331108722       -4.63       -2.19    0.0
7   -7.251336796059       -5.25       -2.47    0.0
8   -7.251338245641       -5.84       -2.75    0.0
9   -7.251338636930       -6.41       -3.01    0.0
10   -7.251338748902       -6.95       -3.27    0.0
11   -7.251338782714       -7.47       -3.53    0.0
12   -7.251338793400       -7.97       -3.78    0.0
13   -7.251338796901       -8.46       -4.02    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 fixed-point solver think it's converged by forcing f(x) = x. You can customize this with the is_converged keyword argument to self_consistent_field.