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 -36.73390212369 -0.88 11.0 1.35s
2 -36.71720869934 + -1.78 -1.56 1.0 292ms
3 -15.94698216106 + 1.32 -0.40 5.0 182ms
4 -35.39310967484 1.29 -0.87 5.0 197ms
5 -36.32930202902 -0.03 -1.22 4.0 160ms
6 -35.66058509126 + -0.17 -1.04 4.0 160ms
7 -36.72726521338 0.03 -1.80 3.0 132ms
8 -36.74024357359 -1.89 -2.09 1.0 95.8ms
9 -36.73749542030 + -2.56 -1.98 2.0 125ms
10 -36.74175246109 -2.37 -2.24 2.0 124ms
11 -36.74149157639 + -3.58 -2.50 1.0 97.2ms
12 -36.74240479735 -3.04 -2.81 1.0 94.1ms
13 -36.74247335448 -4.16 -3.03 2.0 133ms
14 -36.74247412644 -6.11 -3.32 2.0 106ms
15 -36.74247300836 + -5.95 -3.46 2.0 108ms
16 -36.74243120422 + -4.38 -3.18 3.0 126ms
17 -36.74236176483 + -4.16 -3.02 3.0 140ms
18 -36.74239570055 -4.47 -3.10 3.0 134ms
19 -36.74247331998 -4.11 -3.57 3.0 140ms
20 -36.74248023866 -5.16 -4.14 1.0 94.0ms
21 -36.74248053964 -6.52 -4.42 3.0 148ms
22 -36.74248064095 -6.99 -4.55 2.0 132ms
23 -36.74248066233 -7.67 -4.96 2.0 109ms
24 -36.74248066444 -8.68 -4.80 2.0 126ms
25 -36.74248067148 -8.15 -5.33 2.0 116ms
26 -36.74248067220 -9.14 -5.22 2.0 132ms
27 -36.74248067219 + -11.57 -5.57 1.0 93.8ms
28 -36.74248067264 -9.35 -5.83 2.0 114ms
29 -36.74248066957 + -8.51 -5.30 3.0 148ms
30 -36.74248066293 + -8.18 -5.07 4.0 267ms
31 -36.74248067263 -8.01 -6.10 3.0 1.26s
32 -36.74248067268 -10.35 -6.37 2.0 129ms
33 -36.74248067268 -11.39 -6.77 2.0 109ms
34 -36.74248067267 + -10.99 -6.46 3.0 143ms
35 -36.74248067268 -10.93 -7.09 3.0 128ms
36 -36.74248067268 -12.32 -7.54 2.0 108ms
37 -36.74248067268 + -13.19 -7.56 3.0 137ms
38 -36.74248067268 -13.30 -7.55 2.0 118ms
39 -36.74248067268 -13.55 -8.21 2.0 103ms
40 -36.74248067268 + -Inf -8.16 3.0 141ms
41 -36.74248067268 + -Inf -8.54 2.0 109ms
42 -36.74248067268 + -14.15 -8.43 2.0 113ms
43 -36.74248067268 + -14.15 -8.32 3.0 154ms
44 -36.74248067268 -13.85 -8.68 3.0 150ms
45 -36.74248067268 + -Inf -8.92 2.0 130ms
46 -36.74248067268 -13.85 -8.93 2.0 144ms
47 -36.74248067268 + -13.67 -9.36 2.0 110ms
48 -36.74248067268 -13.67 -9.63 3.0 128ms
49 -36.74248067268 + -13.85 -9.68 2.0 128ms
50 -36.74248067268 + -Inf -10.07 1.0 93.3ms
51 -36.74248067268 + -Inf -10.01 3.0 144ms
52 -36.74248067268 -14.15 -10.43 2.0 115ms
53 -36.74248067268 + -14.15 -10.58 2.0 127ms
54 -36.74248067268 + -Inf -10.80 2.0 110ms
55 -36.74248067268 -13.85 -10.22 3.0 144ms
56 -36.74248067268 + -13.85 -11.12 3.0 139ms
57 -36.74248067268 -13.85 -10.82 3.0 143ms
58 -36.74248067268 + -14.15 -11.50 2.0 119ms
59 -36.74248067268 + -14.15 -11.63 2.0 119ms
60 -36.74248067268 + -Inf -11.88 1.0 98.7ms
61 -36.74248067268 + -14.15 -12.04 3.0 136ms
while 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 -36.73450277450 -0.88 11.0 984ms
2 -36.74044215723 -2.23 -1.36 1.0 655ms
3 -36.74044382176 -5.78 -1.79 5.0 140ms
4 -36.74213801067 -2.77 -2.14 2.0 108ms
5 -36.74235933229 -3.65 -2.62 1.0 91.2ms
6 -36.74239489626 -4.45 -2.42 3.0 144ms
7 -36.74246483295 -4.16 -3.15 1.0 92.6ms
8 -36.74247898368 -4.85 -3.62 6.0 128ms
9 -36.74248002236 -5.98 -3.73 3.0 141ms
10 -36.74248063403 -6.21 -4.04 1.0 98.4ms
11 -36.74248064991 -7.80 -4.03 1.0 94.1ms
12 -36.74248062763 + -7.65 -4.33 1.0 99.1ms
13 -36.74248067030 -7.37 -4.77 3.0 119ms
14 -36.74248067039 -10.06 -4.78 3.0 116ms
15 -36.74248067204 -8.78 -5.03 1.0 100ms
16 -36.74248067083 + -8.92 -5.20 5.0 120ms
17 -36.74248067110 -9.57 -5.20 1.0 99.3ms
18 -36.74248067044 + -9.18 -5.14 1.0 95.3ms
19 -36.74248067025 + -9.72 -5.12 1.0 99.4ms
20 -36.74248066983 + -9.37 -5.10 1.0 100ms
21 -36.74248066886 + -9.01 -5.06 1.0 95.2ms
22 -36.74248066872 + -9.86 -5.05 1.0 99.4ms
23 -36.74248066904 -9.50 -5.06 1.0 95.1ms
24 -36.74248066996 -9.04 -5.11 1.0 100ms
25 -36.74248067261 -8.58 -5.62 2.0 111ms
26 -36.74248067264 -10.53 -5.96 1.0 100ms
27 -36.74248067267 -10.44 -6.33 3.0 122ms
28 -36.74248067268 -11.08 -6.57 2.0 133ms
29 -36.74248067268 + -12.45 -6.80 2.0 112ms
30 -36.74248067268 -11.92 -7.27 1.0 95.0ms
31 -36.74248067268 -14.15 -7.44 3.0 144ms
32 -36.74248067268 -13.19 -7.84 1.0 95.2ms
33 -36.74248067268 -14.15 -8.08 3.0 113ms
34 -36.74248067268 -13.85 -8.05 3.0 143ms
35 -36.74248067268 + -14.15 -8.33 1.0 95.4ms
36 -36.74248067268 -14.15 -8.65 3.0 136ms
37 -36.74248067268 + -13.55 -9.30 1.0 95.0ms
38 -36.74248067268 -13.85 -9.34 4.0 160ms
39 -36.74248067268 + -Inf -9.73 1.0 100ms
40 -36.74248067268 + -Inf -9.90 3.0 139ms
41 -36.74248067268 + -13.85 -10.35 2.0 105ms
42 -36.74248067268 -13.67 -10.83 2.0 133ms
43 -36.74248067268 + -14.15 -11.05 2.0 130ms
44 -36.74248067268 -14.15 -11.27 1.0 100ms
45 -36.74248067268 + -14.15 -11.82 1.0 95.1ms
46 -36.74248067268 + -14.15 -11.83 3.0 160ms
47 -36.74248067268 -14.15 -12.20 1.0 94.9ms
Given 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))44.024488980829986The 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_min46.24421111431722This 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))4.723585068636772Since 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).