# 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 => [ones(3)/8, -ones(3)/8]]
# We take very (very) crude parameters
model = model_LDA(lattice, atoms)
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 Eₙ-Eₙ₋₁ ρout-ρin α Diag --- --------------- --------- -------- ---- ---- 1 -7.227971221753 NaN 3.22e-01 0.80 0.0 2 -7.246548565129 -1.86e-02 1.26e-01 0.80 0.0 3 -7.248998824732 -2.45e-03 4.70e-02 0.80 0.0 4 -7.249162119311 -1.63e-04 2.30e-02 0.80 0.0 5 -7.249202768478 -4.06e-05 1.14e-02 0.80 0.0 6 -7.249213089302 -1.03e-05 5.74e-03 0.80 0.0 7 -7.249215815319 -2.73e-06 2.95e-03 0.80 0.0 8 -7.249216572087 -7.57e-07 1.54e-03 0.80 0.0 9 -7.249216793514 -2.21e-07 8.23e-04 0.80 0.0 10 -7.249216861576 -6.81e-08 4.48e-04 0.80 0.0 11 -7.249216883397 -2.18e-08 2.48e-04 0.80 0.0 12 -7.249216890631 -7.23e-09 1.40e-04 0.80 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`

.