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=[], 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
endThat'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 628ms
3 -7.251174940533 -2.81 -1.34 0.0 94.6ms
4 -7.251295969966 -3.92 -1.65 0.0 92.4ms
5 -7.251327319418 -4.50 -1.95 0.0 94.6ms
6 -7.251335595391 -5.08 -2.25 0.0 94.8ms
7 -7.251337861796 -5.64 -2.54 0.0 97.1ms
8 -7.251338511556 -6.19 -2.83 0.0 103ms
9 -7.251338706958 -6.71 -3.10 0.0 261ms
10 -7.251338768376 -7.21 -3.36 0.0 94.3ms
11 -7.251338788413 -7.70 -3.62 0.0 93.0ms
12 -7.251338795145 -8.17 -3.87 0.0 96.8ms
13 -7.251338797456 -8.64 -4.11 0.0 91.2ms
14 -7.251338798263 -9.09 -4.35 0.0 93.0ms
15 -7.251338798547 -9.55 -4.58 0.0 97.4ms
16 -7.251338798648 -10.00 -4.81 0.0 106ms
17 -7.251338798684 -10.44 -5.04 0.0 251ms
18 -7.251338798697 -10.89 -5.27 0.0 105ms
19 -7.251338798702 -11.33 -5.50 0.0 97.6ms
20 -7.251338798704 -11.78 -5.72 0.0 94.3ms
21 -7.251338798704 -12.22 -5.95 0.0 94.4ms
22 -7.251338798704 -12.66 -6.17 0.0 97.8ms
23 -7.251338798704 -13.11 -6.39 0.0 106ms
24 -7.251338798705 -13.62 -6.62 0.0 247ms
25 -7.251338798705 -13.97 -6.83 0.0 104ms
26 -7.251338798705 -14.21 -7.05 0.0 95.0ms
27 -7.251338798705 -15.05 -7.28 0.0 97.3ms
28 -7.251338798705 -14.75 -7.42 0.0 95.4ms
29 -7.251338798705 + -15.05 -7.66 0.0 96.2ms
30 -7.251338798705 + -13.65 -7.01 0.0 97.7ms
31 -7.251338798705 -13.75 -7.26 0.0 106ms
32 -7.251338798705 -14.75 -7.60 0.0 279ms
33 -7.251338798705 + -Inf -7.88 0.0 100ms
34 -7.251338798705 + -15.05 -7.42 0.0 105ms
35 -7.251338798705 -15.05 -7.67 0.0 101ms
36 -7.251338798705 -14.75 -7.96 0.0 98.1ms
37 -7.251338798705 + -14.75 -7.48 0.0 97.7ms
38 -7.251338798705 -14.75 -7.73 0.0 97.4ms
39 -7.251338798705 + -14.27 -7.32 0.0 101ms
40 -7.251338798705 -14.15 -7.53 0.0 233ms
41 -7.251338798705 + -14.27 -7.50 0.0 103ms
42 -7.251338798705 -14.75 -7.74 0.0 98.8ms
43 -7.251338798705 -14.45 -7.83 0.0 104ms
44 -7.251338798705 + -13.85 -7.15 0.0 103ms
45 -7.251338798705 -13.97 -7.39 0.0 96.7ms
46 -7.251338798705 + -Inf -7.71 0.0 94.9ms
47 -7.251338798705 -14.45 -7.82 0.0 102ms
48 -7.251338798704 + -13.36 -6.81 0.0 226ms
49 -7.251338798705 -13.44 -7.08 0.0 94.8ms
50 -7.251338798705 -14.45 -7.43 0.0 105ms
51 -7.251338798705 -15.05 -7.61 0.0 97.8ms
52 -7.251338798705 + -13.82 -7.08 0.0 105ms
53 -7.251338798705 -13.75 -7.32 0.0 94.2ms
54 -7.251338798705 + -14.75 -7.62 0.0 99.0ms
55 -7.251338798705 + -14.75 -7.46 0.0 104ms
56 -7.251338798705 + -14.75 -7.24 0.0 227ms
57 -7.251338798705 -14.35 -7.49 0.0 102ms
58 -7.251338798705 + -14.57 -7.51 0.0 99.2ms
59 -7.251338798705 -14.57 -7.73 0.0 96.8ms
60 -7.251338798705 + -15.05 -7.84 0.0 101ms
61 -7.251338798705 + -Inf -7.81 0.0 95.3ms
62 -7.251338798705 + -Inf -7.48 0.0 109ms
63 -7.251338798705 + -14.75 -7.57 0.0 102ms
64 -7.251338798705 -15.05 -7.47 0.0 221ms
65 -7.251338798705 + -15.05 -7.70 0.0 100ms
66 -7.251338798705 -14.75 -7.60 0.0 100ms
67 -7.251338798705 + -15.05 -7.73 0.0 97.1ms
68 -7.251338798705 -14.75 -7.84 0.0 99.1ms
69 -7.251338798705 + -14.35 -7.50 0.0 97.5ms
70 -7.251338798705 -14.45 -7.42 0.0 99.1ms
71 -7.251338798705 + -14.75 -7.65 0.0 103ms
72 -7.251338798705 -14.75 -7.86 0.0 231ms
73 -7.251338798705 + -14.10 -7.21 0.0 102ms
74 -7.251338798705 -14.10 -7.44 0.0 99.3ms
75 -7.251338798705 + -Inf -7.67 0.0 98.7ms
76 -7.251338798705 + -Inf -7.71 0.0 102ms
77 -7.251338798705 + -14.75 -7.46 0.0 102ms
78 -7.251338798705 -14.75 -7.74 0.0 98.7ms
79 -7.251338798705 + -14.45 -7.31 0.0 102ms
80 -7.251338798705 -14.57 -7.52 0.0 228ms
81 -7.251338798705 -15.05 -7.81 0.0 97.6ms
82 -7.251338798705 + -Inf -7.90 0.0 98.7ms
83 -7.251338798705 + -15.05 -7.31 0.0 95.2ms
84 -7.251338798705 + -14.57 -7.48 0.0 106ms
85 -7.251338798705 + -Inf -7.40 0.0 97.4ms
86 -7.251338798705 -14.45 -7.60 0.0 105ms
87 -7.251338798705 -14.57 -7.83 0.0 257ms
88 -7.251338798705 + -14.21 -7.33 0.0 96.5ms
89 -7.251338798705 -14.75 -7.51 0.0 98.8ms
90 -7.251338798705 -15.05 -7.65 0.0 108ms
91 -7.251338798705 + -14.75 -7.33 0.0 97.9ms
92 -7.251338798705 + -Inf -7.45 0.0 97.6ms
93 -7.251338798705 -14.35 -7.68 0.0 101ms
94 -7.251338798705 + -14.75 -7.87 0.0 106ms
95 -7.251338798705 + -13.80 -7.08 0.0 229ms
96 -7.251338798705 -13.75 -7.32 0.0 104ms
97 -7.251338798705 + -Inf -7.62 0.0 101ms
98 -7.251338798705 + -14.45 -7.45 0.0 97.9ms
99 -7.251338798705 -14.57 -7.68 0.0 97.7ms
100 -7.251338798705 + -13.97 -7.16 0.0 100ms
101 -7.251338798705 -14.01 -7.41 0.0 97.3ms
┌ Warning: SCF not converged.
└ @ DFTK ~/work/DFTK.jl/DFTK.jl/src/scf/scf_callbacks.jl:38Note 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.