# 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, X, residual_norms=[], n_iter=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-4,
solver=my_fp_solver,
eigensolver=my_eig_solver,
mixing=MyMixing());
```

```
n Energy log10(ΔE) log10(Δρ) Diag Δtime
--- --------------- --------- --------- ---- ------
1 -7.090073978406 -0.39 0.0 225ms
2 -7.226157343674 -0.87 -0.65 0.0 187ms
3 -7.249782177445 -1.63 -1.17 0.0 42.9ms
4 -7.250986294028 -2.92 -1.48 0.0 53.3ms
5 -7.251258312426 -3.57 -1.78 0.0 64.0ms
6 -7.251319808416 -4.21 -2.07 0.0 50.0ms
7 -7.251334099764 -4.84 -2.35 0.0 51.2ms
8 -7.251337569219 -5.46 -2.62 0.0 40.4ms
9 -7.251338457710 -6.05 -2.88 0.0 121ms
10 -7.251338698748 -6.62 -3.14 0.0 38.0ms
11 -7.251338767946 -7.16 -3.39 0.0 39.5ms
12 -7.251338788853 -7.68 -3.64 0.0 48.9ms
13 -7.251338795449 -8.18 -3.88 0.0 48.5ms
14 -7.251338797603 -8.67 -4.12 0.0 49.8ms
```

Note that the default convergence criterion is the difference in density. 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`

.