Analysing SCF convergence
The goal of this example is to explain the differing convergence behaviour of SCF algorithms depending on the choice of the mixing. For this we look at the eigenpairs of the Jacobian governing the SCF convergence, that is
\[1 - α P^{-1} \varepsilon^\dagger \qquad \text{with} \qquad \varepsilon^\dagger = (1-\chi_0 K).\]
where $α$ is the damping $P^{-1}$ is the mixing preconditioner (e.g. KerkerMixing, LdosMixing) and $\varepsilon^\dagger$ is the dielectric operator.
We thus investigate the largest and smallest eigenvalues of $(P^{-1} \varepsilon^\dagger)$ and $\varepsilon^\dagger$. The ratio of largest to smallest eigenvalue of this operator is the condition number
\[\kappa = \frac{\lambda_\text{max}}{\lambda_\text{min}},\]
which can be related to the rate of convergence of the SCF. The smaller the condition number, the faster the convergence. For more details on SCF methods, see Self-consistent field methods.
For our investigation we consider a crude aluminium setup:
using AtomsBuilder
using DFTK
system_Al = bulk(:Al; cubic=true) * (4, 1, 1)FlexibleSystem(Al₁₆, periodicity = TTT):
cell_vectors : [ 16.2 0 0;
0 4.05 0;
0 0 4.05]u"Å"
and we discretise:
using PseudoPotentialData
pseudopotentials = PseudoFamily("dojo.nc.sr.lda.v0_4_1.standard.upf")
model_Al = model_DFT(system_Al; functionals=LDA(), temperature=1e-3,
symmetries=false, pseudopotentials)
basis_Al = PlaneWaveBasis(model_Al; Ecut=7, kgrid=[1, 1, 1]);On aluminium (a metal) already for moderate system sizes (like the 8 layers we consider here) the convergence without mixing / preconditioner is slow:
# Note: DFTK uses the self-adapting LdosMixing() by default, so to truly disable
# any preconditioning, we need to supply `mixing=SimpleMixing()` explicitly.
scfres_Al = self_consistent_field(basis_Al; tol=1e-12, mixing=SimpleMixing());n Energy log10(ΔE) log10(Δρ) Diag Δtime
--- --------------- --------- --------- ---- ------
1 -36.73221949253 -0.88 13.0 362ms
2 -36.58600182625 + -0.84 -1.37 1.0 94.2ms
3 +43.37654535133 + 1.90 -0.11 22.0 309ms
4 -36.52967798537 1.90 -1.07 11.0 323ms
5 -36.65098831795 -0.92 -1.42 2.0 128ms
6 -35.47002956280 + 0.07 -1.01 4.0 160ms
7 -36.64274469298 0.07 -1.48 3.0 151ms
8 -36.73965999663 -1.01 -2.02 2.0 111ms
9 -36.74072817836 -2.97 -2.13 2.0 134ms
10 -36.73928790844 + -2.84 -2.01 2.0 115ms
11 -36.74203387662 -2.56 -2.41 2.0 123ms
12 -36.74220522244 -3.77 -2.52 1.0 96.3ms
13 -36.74230800765 -3.99 -2.74 2.0 113ms
14 -36.74246514974 -3.80 -2.97 2.0 136ms
15 -36.73826113196 + -2.38 -2.25 3.0 154ms
16 -36.74238927144 -2.38 -2.94 3.0 151ms
17 -36.74245625367 -4.17 -3.19 2.0 118ms
18 -36.74038987620 + -2.68 -2.40 4.0 167ms
19 -36.74225694389 -2.73 -2.81 3.0 154ms
20 -36.74247890011 -3.65 -3.73 3.0 142ms
21 -36.74247167557 + -5.14 -3.50 3.0 153ms
22 -36.74247958568 -5.10 -3.73 2.0 128ms
23 -36.74248020966 -6.20 -4.05 2.0 120ms
24 -36.74248054887 -6.47 -4.41 1.0 95.9ms
25 -36.74248066711 -6.93 -4.86 2.0 135ms
26 -36.74248067204 -8.31 -5.24 2.0 108ms
27 -36.74248066974 + -8.64 -5.29 3.0 146ms
28 -36.74248066506 + -8.33 -5.02 3.0 132ms
29 -36.74248067243 -8.13 -5.72 2.0 117ms
30 -36.74248066734 + -8.29 -5.18 4.0 172ms
31 -36.74248067253 -8.28 -5.81 3.0 144ms
32 -36.74248067260 -10.16 -6.01 2.0 131ms
33 -36.74248067262 -10.74 -6.12 2.0 118ms
34 -36.74248067263 -11.17 -6.01 3.0 135ms
35 -36.74248067268 -10.26 -6.69 1.0 101ms
36 -36.74248067268 + -11.29 -6.43 3.0 148ms
37 -36.74248067268 -11.38 -6.44 2.0 130ms
38 -36.74248067268 -11.85 -6.94 2.0 115ms
39 -36.74248067268 -12.22 -7.36 1.0 101ms
40 -36.74248067268 + -12.67 -7.18 2.0 132ms
41 -36.74248067268 -12.48 -7.82 2.0 117ms
42 -36.74248067268 + -12.58 -7.31 4.0 161ms
43 -36.74248067268 -13.37 -7.34 4.0 174ms
44 -36.74248067268 + -13.37 -7.27 3.0 142ms
45 -36.74248067268 -12.67 -7.60 4.0 163ms
46 -36.74248067268 -13.37 -7.91 3.0 134ms
47 -36.74248067268 -13.85 -8.64 2.0 118ms
48 -36.74248067268 + -Inf -8.56 3.0 148ms
49 -36.74248067268 + -Inf -8.46 2.0 118ms
50 -36.74248067268 -13.85 -9.05 2.0 105ms
51 -36.74248067268 + -13.85 -9.23 3.0 122ms
52 -36.74248067268 -14.15 -9.21 2.0 131ms
53 -36.74248067268 + -14.15 -9.65 2.0 117ms
54 -36.74248067268 -13.85 -9.49 2.0 115ms
55 -36.74248067268 + -13.85 -9.55 2.0 118ms
56 -36.74248067268 + -Inf -9.87 2.0 113ms
57 -36.74248067268 + -Inf -9.45 3.0 143ms
58 -36.74248067268 -13.85 -9.97 3.0 150ms
59 -36.74248067268 + -13.85 -9.63 3.0 144ms
60 -36.74248067268 + -Inf -10.06 3.0 139ms
61 -36.74248067268 -14.15 -10.98 1.0 100ms
62 -36.74248067268 + -14.15 -10.79 4.0 179ms
63 -36.74248067268 -13.85 -11.09 2.0 118ms
64 -36.74248067268 + -13.85 -11.28 2.0 113ms
65 -36.74248067268 -13.85 -11.35 1.0 100ms
66 -36.74248067268 + -13.85 -11.52 1.0 96.0ms
67 -36.74248067268 + -Inf -11.49 2.0 129ms
68 -36.74248067268 -13.85 -11.44 3.0 135ms
69 -36.74248067268 + -13.85 -11.47 2.0 116ms
70 -36.74248067268 -13.85 -11.77 2.0 124ms
71 -36.74248067268 + -13.85 -11.45 3.0 144ms
72 -36.74248067268 -14.15 -12.20 3.0 134mswhile when using the Kerker preconditioner it is much faster:
scfres_Al = self_consistent_field(basis_Al; tol=1e-12, mixing=KerkerMixing());n Energy log10(ΔE) log10(Δρ) Diag Δtime
--- --------------- --------- --------- ---- ------
1 -36.73427987294 -0.88 11.0 347ms
2 -36.74020615367 -2.23 -1.36 1.0 94.1ms
3 -36.74032953200 -3.91 -1.74 2.0 114ms
4 -36.74226263702 -2.71 -2.25 2.0 94.5ms
5 -36.74237065937 -3.97 -2.65 3.0 134ms
6 -36.74237872658 -5.09 -2.42 3.0 109ms
7 -36.74244693850 -4.17 -2.88 1.0 95.9ms
8 -36.74247245844 -4.59 -3.41 1.0 91.9ms
9 -36.74248019386 -5.11 -3.66 3.0 135ms
10 -36.74248048921 -6.53 -4.03 2.0 97.9ms
11 -36.74248055944 -7.15 -4.19 5.0 120ms
12 -36.74248066285 -6.99 -4.80 2.0 109ms
13 -36.74248067197 -8.04 -5.15 3.0 138ms
14 -36.74248067235 -9.41 -5.41 5.0 118ms
15 -36.74248067249 -9.85 -5.63 2.0 122ms
16 -36.74248067267 -9.74 -6.10 2.0 105ms
17 -36.74248067268 -11.20 -6.45 2.0 133ms
18 -36.74248067268 + -12.33 -6.65 1.0 94.8ms
19 -36.74248067268 -11.84 -7.11 2.0 114ms
20 -36.74248067268 -13.07 -7.08 3.0 133ms
21 -36.74248067268 -13.85 -7.39 1.0 100ms
22 -36.74248067268 -14.15 -7.58 2.0 99.4ms
23 -36.74248067268 -13.67 -7.87 1.0 99.1ms
24 -36.74248067268 -14.15 -7.98 3.0 133ms
25 -36.74248067268 + -14.15 -8.35 1.0 98.6ms
26 -36.74248067268 + -Inf -8.66 3.0 115ms
27 -36.74248067268 + -Inf -8.90 3.0 138ms
28 -36.74248067268 + -Inf -9.36 1.0 94.2ms
29 -36.74248067268 + -Inf -9.44 4.0 142ms
30 -36.74248067268 + -Inf -9.88 1.0 94.2ms
31 -36.74248067268 -14.15 -10.14 3.0 119ms
32 -36.74248067268 + -Inf -10.62 3.0 138ms
33 -36.74248067268 + -14.15 -10.89 6.0 124ms
34 -36.74248067268 + -Inf -11.02 2.0 127ms
35 -36.74248067268 + -Inf -11.22 1.0 99.2ms
36 -36.74248067268 + -Inf -11.46 1.0 94.1ms
37 -36.74248067268 + -Inf -11.67 1.0 98.3ms
38 -36.74248067268 -13.85 -12.22 2.0 120msGiven this scfres_Al we construct functions representing $\varepsilon^\dagger$ and $P^{-1}$:
# Function, which applies P^{-1} for the case of KerkerMixing
Pinv_Kerker(δρ) = DFTK.mix_density(KerkerMixing(), basis_Al, δρ)
# Function which applies ε† = 1 - χ0 K
function epsilon(δρ)
δV = apply_kernel(basis_Al, δρ; ρ=scfres_Al.ρ)
χ0δV = apply_χ0(scfres_Al, δV).δρ
δρ - χ0δV
endepsilon (generic function with 1 method)With these functions available we can now compute the desired eigenvalues. For simplicity we only consider the first few largest ones.
using KrylovKit
λ_Simple, X_Simple = eigsolve(epsilon, randn(size(scfres_Al.ρ)), 3, :LM;
tol=1e-3, eager=true, verbosity=2)
λ_Simple_max = maximum(real.(λ_Simple))44.02450835507706The smallest eigenvalue is a bit more tricky to obtain, so we will just assume
λ_Simple_min = 0.9520.952This makes the condition number around 30:
cond_Simple = λ_Simple_max / λ_Simple_min46.24423146541708This does not sound large compared to the condition numbers you might know from linear systems.
However, this is sufficient to cause a notable slowdown, which would be even more pronounced if we did not use Anderson, since we also would need to drastically reduce the damping (try it!).
Having computed the eigenvalues of the dielectric matrix we can now also look at the eigenmodes, which are responsible for the bad convergence behaviour. The largest eigenmode for example:
using Statistics
using Plots
mode_xy = mean(real.(X_Simple[1]), dims=3)[:, :, 1, 1] # Average along z axis
heatmap(mode_xy', c=:RdBu_11, aspect_ratio=1, grid=false,
legend=false, clim=(-0.006, 0.006))
This mode can be physically interpreted as the reason why this SCF converges slowly. For example in this case it displays a displacement of electron density from the centre to the extremal parts of the unit cell. This phenomenon is called charge-sloshing.
We repeat the exercise for the Kerker-preconditioned dielectric operator:
λ_Kerker, X_Kerker = eigsolve(Pinv_Kerker ∘ epsilon,
randn(size(scfres_Al.ρ)), 3, :LM;
tol=1e-3, eager=true, verbosity=2)
mode_xy = mean(real.(X_Kerker[1]), dims=3)[:, :, 1, 1] # Average along z axis
heatmap(mode_xy', c=:RdBu_11, aspect_ratio=1, grid=false,
legend=false, clim=(-0.006, 0.006))
Clearly the charge-sloshing mode is no longer dominating.
The largest eigenvalue is now
maximum(real.(λ_Kerker))4.7235868235609715Since the smallest eigenvalue in this case remains of similar size (it is now around 0.8), this implies that the conditioning improves noticeably when KerkerMixing is used.
Note: Since LdosMixing requires solving a linear system at each application of $P^{-1}$, determining the eigenvalues of $P^{-1} \varepsilon^\dagger$ is slightly more expensive and thus not shown. The results are similar to KerkerMixing, however.
We could repeat the exercise for an insulating system (e.g. a Helium chain). In this case you would notice that the condition number without mixing is actually smaller than the condition number with Kerker mixing. In other words employing Kerker mixing makes the convergence worse. A closer investigation of the eigenvalues shows that Kerker mixing reduces the smallest eigenvalue of the dielectric operator this time, while keeping the largest value unchanged. Overall the conditioning thus workens.
Takeaways:
- For metals the conditioning of the dielectric matrix increases steeply with system size.
- The Kerker preconditioner tames this and makes SCFs on large metallic systems feasible by keeping the condition number of order 1.
- For insulating systems the best approach is to not use any mixing.
- The ideal mixing strongly depends on the dielectric properties of system which is studied (metal versus insulator versus semiconductor).