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, load_psp("hgh/lda/Si-q4"))
atoms = [Si, Si]
positions = [ones(3)/8, -ones(3)/8]
# We take very (very) crude parameters
model = model_DFT(lattice, atoms, positions; functionals=LDA())
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, info0; maxiter)
mixing_factor = .7
x = x0
info = info0
for n = 1:maxiter
fx, info = f(x, info)
if info.converged || info.timedout
break
end
x = x + mixing_factor * (fx - x)
end
(; fixpoint=x, info)
end;Note that the fixpoint map f operates on an auxiliary variable info for state bookkeeping. Early termination criteria are flagged from inside the function f using boolean flags info.converged and info.timedout. For control over these criteria, see the is_converged and maxtime keyword arguments of self_consistent_field.
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
endThat'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 292ms
2 -7.226157343674 -0.87 -0.65 0.0 356ms
3 -7.249782177445 -1.63 -1.17 0.0 83.3ms
4 -7.250986294028 -2.92 -1.48 0.0 82.8ms
5 -7.251258312426 -3.57 -1.78 0.0 87.9ms
6 -7.251319808416 -4.21 -2.07 0.0 108ms
7 -7.251334099764 -4.84 -2.35 0.0 78.7ms
8 -7.251337569219 -5.46 -2.62 0.0 190ms
9 -7.251338457710 -6.05 -2.88 0.0 71.7ms
10 -7.251338698748 -6.62 -3.14 0.0 77.5ms
11 -7.251338767946 -7.16 -3.39 0.0 71.6ms
12 -7.251338788853 -7.68 -3.64 0.0 82.3ms
13 -7.251338795449 -8.18 -3.88 0.0 82.4ms
14 -7.251338797603 -8.67 -4.12 0.0 84.9msNote that the default convergence criterion is the difference in density. When this gets below tol, the fixed-point solver terminates. You can also customize this with the is_converged keyword argument to self_consistent_field, as shown below.
Customizing the convergence criterion
Here is an example of a defining a custom convergence criterion and specifying it using the is_converged callback keyword to self_consistent_field.
function my_convergence_criterion(info)
tol = 1e-10
length(info.history_Etot) < 2 && return false
ΔE = (info.history_Etot[end-1] - info.history_Etot[end])
ΔE < tol
end
scfres2 = self_consistent_field(basis;
solver=my_fp_solver,
is_converged=my_convergence_criterion,
eigensolver=my_eig_solver,
mixing=MyMixing());n Energy log10(ΔE) log10(Δρ) Diag Δtime
--- --------------- --------- --------- ---- ------
1 -7.090073978406 -0.39 0.0 168ms
2 -7.226157343674 -0.87 -0.65 0.0 235ms
3 -7.249782177445 -1.63 -1.17 0.0 70.8ms
4 -7.250986294028 -2.92 -1.48 0.0 78.2ms
5 -7.251258312426 -3.57 -1.78 0.0 83.7ms
6 -7.251319808416 -4.21 -2.07 0.0 85.4ms
7 -7.251334099764 -4.84 -2.35 0.0 85.0ms
8 -7.251337569219 -5.46 -2.62 0.0 145ms
9 -7.251338457710 -6.05 -2.88 0.0 72.1ms
10 -7.251338698748 -6.62 -3.14 0.0 74.4ms
11 -7.251338767946 -7.16 -3.39 0.0 83.7ms
12 -7.251338788853 -7.68 -3.64 0.0 83.4ms
13 -7.251338795449 -8.18 -3.88 0.0 124ms
14 -7.251338797603 -8.67 -4.12 0.0 71.6ms
15 -7.251338798325 -9.14 -4.36 0.0 70.7ms
16 -7.251338798572 -9.61 -4.59 0.0 82.0ms
17 -7.251338798658 -10.07 -4.82 0.0 85.5ms