# Custom solvers¶

In this example, we show how to define custom solvers. Our system will again be silicon, because we are not very imaginative

In [1]:
using DFTK, LinearAlgebra

a = 10.26
lattice = a / 2 * [[0 1 1.];
[1 0 1.];
[1 1 0.]]
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

In [2]:
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)

In [3]:
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.

In [4]:
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

In [5]:
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   -7.223357624888                   -0.49    0.0
2   -7.247632632938       -1.61       -0.88    0.0
3   -7.251051561069       -2.47       -1.32    0.0
4   -7.251268571183       -3.66       -1.63    0.0
5   -7.251321230584       -4.28       -1.93    0.0
6   -7.251334207493       -4.89       -2.23    0.0
7   -7.251337532560       -5.48       -2.52    0.0
8   -7.251338429534       -6.05       -2.80    0.0
9   -7.251338685396       -6.59       -3.07    0.0
10   -7.251338762391       -7.11       -3.33    0.0
11   -7.251338786665       -7.61       -3.59    0.0
12   -7.251338794610       -8.10       -3.84    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.