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=[], 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 Δtime --- --------------- --------- --------- ---- ------ 1 -7.235992833898 -0.50 0.0 2 -7.249608665538 -1.87 -0.92 0.0 864ms 3 -7.251174940533 -2.81 -1.34 0.0 103ms 4 -7.251295969966 -3.92 -1.65 0.0 93.1ms 5 -7.251327319418 -4.50 -1.95 0.0 98.9ms 6 -7.251335595391 -5.08 -2.25 0.0 245ms 7 -7.251337861796 -5.64 -2.54 0.0 99.3ms 8 -7.251338511556 -6.19 -2.83 0.0 98.5ms 9 -7.251338706958 -6.71 -3.10 0.0 94.7ms 10 -7.251338768376 -7.21 -3.36 0.0 93.5ms 11 -7.251338788413 -7.70 -3.62 0.0 94.7ms 12 -7.251338795145 -8.17 -3.87 0.0 98.0ms 13 -7.251338797456 -8.64 -4.11 0.0 97.7ms 14 -7.251338798263 -9.09 -4.35 0.0 97.7ms 15 -7.251338798547 -9.55 -4.58 0.0 245ms 16 -7.251338798648 -10.00 -4.81 0.0 97.3ms 17 -7.251338798684 -10.44 -5.04 0.0 96.7ms 18 -7.251338798697 -10.89 -5.27 0.0 100ms 19 -7.251338798702 -11.33 -5.50 0.0 94.7ms 20 -7.251338798704 -11.78 -5.72 0.0 98.4ms 21 -7.251338798704 -12.22 -5.95 0.0 98.5ms 22 -7.251338798704 -12.66 -6.17 0.0 97.1ms 23 -7.251338798704 -13.11 -6.39 0.0 234ms 24 -7.251338798705 -13.62 -6.62 0.0 94.0ms 25 -7.251338798705 -13.97 -6.83 0.0 97.2ms 26 -7.251338798705 -14.21 -7.05 0.0 99.3ms 27 -7.251338798705 -15.05 -7.28 0.0 97.7ms 28 -7.251338798705 -14.75 -7.42 0.0 95.5ms 29 -7.251338798705 + -15.05 -7.66 0.0 103ms 30 -7.251338798705 + -13.65 -7.01 0.0 99.5ms 31 -7.251338798705 -13.75 -7.26 0.0 222ms 32 -7.251338798705 -14.75 -7.60 0.0 93.5ms 33 -7.251338798705 + -Inf -7.88 0.0 96.0ms 34 -7.251338798705 + -15.05 -7.42 0.0 96.5ms 35 -7.251338798705 -15.05 -7.67 0.0 104ms 36 -7.251338798705 -14.75 -7.96 0.0 95.5ms 37 -7.251338798705 + -14.75 -7.48 0.0 101ms 38 -7.251338798705 -14.75 -7.73 0.0 104ms 39 -7.251338798705 + -14.27 -7.32 0.0 105ms 40 -7.251338798705 -14.15 -7.53 0.0 216ms 41 -7.251338798705 + -14.27 -7.50 0.0 97.8ms 42 -7.251338798705 -14.75 -7.74 0.0 105ms 43 -7.251338798705 -14.45 -7.83 0.0 102ms 44 -7.251338798705 + -13.85 -7.15 0.0 98.8ms 45 -7.251338798705 -13.97 -7.39 0.0 100ms 46 -7.251338798705 + -Inf -7.71 0.0 101ms 47 -7.251338798705 -14.45 -7.82 0.0 111ms 48 -7.251338798704 + -13.36 -6.81 0.0 233ms 49 -7.251338798705 -13.44 -7.08 0.0 96.4ms 50 -7.251338798705 -14.45 -7.43 0.0 99.4ms 51 -7.251338798705 -15.05 -7.61 0.0 102ms 52 -7.251338798705 + -13.82 -7.08 0.0 94.7ms 53 -7.251338798705 -13.75 -7.32 0.0 94.3ms 54 -7.251338798705 + -14.75 -7.62 0.0 99.1ms 55 -7.251338798705 + -14.75 -7.46 0.0 100ms 56 -7.251338798705 + -14.75 -7.24 0.0 224ms 57 -7.251338798705 -14.35 -7.49 0.0 93.2ms 58 -7.251338798705 + -14.57 -7.51 0.0 98.1ms 59 -7.251338798705 -14.57 -7.73 0.0 97.5ms 60 -7.251338798705 + -15.05 -7.84 0.0 118ms 61 -7.251338798705 + -Inf -7.81 0.0 104ms 62 -7.251338798705 + -Inf -7.48 0.0 98.9ms 63 -7.251338798705 + -14.75 -7.57 0.0 101ms 64 -7.251338798705 -15.05 -7.47 0.0 98.9ms 65 -7.251338798705 + -15.05 -7.70 0.0 225ms 66 -7.251338798705 -14.75 -7.60 0.0 95.6ms 67 -7.251338798705 + -15.05 -7.73 0.0 93.9ms 68 -7.251338798705 -14.75 -7.84 0.0 98.1ms 69 -7.251338798705 + -14.35 -7.50 0.0 95.3ms 70 -7.251338798705 -14.45 -7.42 0.0 99.3ms 71 -7.251338798705 + -14.75 -7.65 0.0 103ms 72 -7.251338798705 -14.75 -7.86 0.0 101ms 73 -7.251338798705 + -14.10 -7.21 0.0 219ms 74 -7.251338798705 -14.10 -7.44 0.0 96.6ms 75 -7.251338798705 + -Inf -7.67 0.0 111ms 76 -7.251338798705 + -Inf -7.71 0.0 96.0ms 77 -7.251338798705 + -14.75 -7.46 0.0 99.3ms 78 -7.251338798705 -14.75 -7.74 0.0 96.6ms 79 -7.251338798705 + -14.45 -7.31 0.0 103ms 80 -7.251338798705 -14.57 -7.52 0.0 100ms 81 -7.251338798705 -15.05 -7.81 0.0 225ms 82 -7.251338798705 + -Inf -7.90 0.0 95.1ms 83 -7.251338798705 + -15.05 -7.31 0.0 97.9ms 84 -7.251338798705 + -14.57 -7.48 0.0 97.9ms 85 -7.251338798705 + -Inf -7.40 0.0 98.8ms 86 -7.251338798705 -14.45 -7.60 0.0 94.1ms 87 -7.251338798705 -14.57 -7.83 0.0 99.4ms 88 -7.251338798705 + -14.21 -7.33 0.0 101ms 89 -7.251338798705 -14.75 -7.51 0.0 99.3ms 90 -7.251338798705 -15.05 -7.65 0.0 225ms 91 -7.251338798705 + -14.75 -7.33 0.0 98.5ms 92 -7.251338798705 + -Inf -7.45 0.0 102ms 93 -7.251338798705 -14.35 -7.68 0.0 105ms 94 -7.251338798705 + -14.75 -7.87 0.0 103ms 95 -7.251338798705 + -13.80 -7.08 0.0 106ms 96 -7.251338798705 -13.75 -7.32 0.0 105ms 97 -7.251338798705 + -Inf -7.62 0.0 98.8ms 98 -7.251338798705 + -14.45 -7.45 0.0 231ms 99 -7.251338798705 -14.57 -7.68 0.0 93.8ms 100 -7.251338798705 + -13.97 -7.16 0.0 101ms 101 -7.251338798705 -14.01 -7.41 0.0 102ms ┌ Warning: SCF not converged. └ @ DFTK ~/work/DFTK.jl/DFTK.jl/src/scf/scf_callbacks.jl:38
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
.