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 -36.73111555024 -0.88 12.0 349ms
2 -36.70804713707 + -1.64 -1.57 1.0 88.7ms
3 +0.208959247089 + 1.57 -0.29 6.0 203ms
4 -36.51943731457 1.57 -1.08 6.0 196ms
5 -36.72709433511 -0.68 -1.75 3.0 137ms
6 -35.87754072144 + -0.07 -1.10 4.0 146ms
7 -36.69278085155 -0.09 -1.64 3.0 136ms
8 -36.74165129713 -1.31 -2.24 2.0 106ms
9 -36.73826396914 + -2.47 -1.96 3.0 146ms
10 -36.74010658199 -2.73 -2.12 2.0 119ms
11 -36.74194787654 -2.73 -2.51 2.0 119ms
12 -36.74237879980 -3.37 -2.82 2.0 102ms
13 -36.74246153234 -4.08 -3.00 2.0 111ms
14 -36.74246886037 -5.14 -3.21 4.0 128ms
15 -36.74246811523 + -6.13 -3.34 1.0 94.4ms
16 -36.74217407637 + -3.53 -2.80 3.0 149ms
17 -36.74241699490 -3.61 -3.14 3.0 135ms
18 -36.74238645916 + -4.52 -3.05 3.0 144ms
19 -36.74247851648 -4.04 -3.58 2.0 122ms
20 -36.74248046017 -5.71 -4.21 2.0 116ms
21 -36.74248029967 + -6.79 -4.09 3.0 145ms
22 -36.74248049206 -6.72 -4.20 2.0 113ms
23 -36.74248064988 -6.80 -4.64 1.0 94.5ms
24 -36.74248020900 + -6.36 -4.18 4.0 152ms
25 -36.74248060795 -6.40 -4.61 2.0 119ms
26 -36.74248066324 -7.26 -4.96 2.0 125ms
27 -36.74248067226 -8.04 -5.39 2.0 133ms
28 -36.74248066527 + -8.16 -5.09 3.0 136ms
29 -36.74248066295 + -8.63 -5.06 3.0 141ms
30 -36.74248067172 -8.06 -5.39 3.0 125ms
31 -36.74248067258 -9.07 -5.84 2.0 116ms
32 -36.74248067266 -10.09 -6.16 2.0 127ms
33 -36.74248067266 -11.71 -6.37 1.0 98.5ms
34 -36.74248067268 -10.92 -6.63 2.0 104ms
35 -36.74248067268 -11.43 -6.65 3.0 135ms
36 -36.74248067268 + -11.27 -6.57 2.0 117ms
37 -36.74248067268 -11.15 -7.26 2.0 110ms
38 -36.74248067268 -13.19 -7.53 3.0 144ms
39 -36.74248067268 + -12.03 -7.04 3.0 140ms
40 -36.74248067268 -12.07 -7.52 3.0 136ms
41 -36.74248067268 -13.25 -7.61 2.0 126ms
42 -36.74248067268 -13.19 -8.27 2.0 110ms
43 -36.74248067268 + -13.85 -8.07 3.0 149ms
44 -36.74248067268 -13.85 -8.24 3.0 128ms
45 -36.74248067268 + -14.15 -8.68 2.0 108ms
46 -36.74248067268 -14.15 -8.80 2.0 128ms
47 -36.74248067268 + -Inf -8.87 2.0 124ms
48 -36.74248067268 + -Inf -9.28 2.0 104ms
49 -36.74248067268 + -Inf -8.81 3.0 144ms
50 -36.74248067268 + -Inf -9.60 3.0 137ms
51 -36.74248067268 -13.85 -9.50 3.0 134ms
52 -36.74248067268 + -13.85 -9.61 3.0 135ms
53 -36.74248067268 + -14.15 -9.75 2.0 115ms
54 -36.74248067268 -14.15 -9.72 3.0 127ms
55 -36.74248067268 + -Inf -10.02 2.0 115ms
56 -36.74248067268 + -Inf -9.91 3.0 134ms
57 -36.74248067268 + -Inf -10.67 2.0 114ms
58 -36.74248067268 -13.85 -10.81 3.0 144ms
59 -36.74248067268 + -13.67 -10.81 2.0 115ms
60 -36.74248067268 -14.15 -11.10 1.0 95.4ms
61 -36.74248067268 + -Inf -11.21 3.0 128ms
62 -36.74248067268 + -Inf -11.50 2.0 115ms
63 -36.74248067268 + -Inf -11.04 3.0 138ms
64 -36.74248067268 + -Inf -11.89 3.0 145ms
65 -36.74248067268 + -Inf -11.83 3.0 144ms
66 -36.74248067268 + -Inf -11.63 2.0 122ms
67 -36.74248067268 + -Inf -11.65 3.0 142ms
68 -36.74248067268 + -13.85 -12.18 2.0 109mswhile 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 -36.73400019927 -0.88 12.0 357ms
2 -36.74006532773 -2.22 -1.36 1.0 91.3ms
3 -36.74048891070 -3.37 -1.82 6.0 129ms
4 -36.74209168689 -2.80 -2.14 2.0 104ms
5 -36.74229934395 -3.68 -2.62 2.0 123ms
6 -36.74234643559 -4.33 -2.56 2.0 127ms
7 -36.74243812320 -4.04 -2.84 1.0 98.6ms
8 -36.74246917657 -4.51 -3.36 2.0 104ms
9 -36.74247914588 -5.00 -3.52 2.0 133ms
10 -36.74248039973 -5.90 -3.93 1.0 94.8ms
11 -36.74248055870 -6.80 -4.28 4.0 116ms
12 -36.74248042255 + -6.87 -4.31 3.0 140ms
13 -36.74248067176 -6.60 -5.06 2.0 116ms
14 -36.74248067122 + -9.27 -5.36 4.0 144ms
15 -36.74248067261 -8.86 -5.64 2.0 135ms
16 -36.74248067265 -10.36 -6.00 2.0 117ms
17 -36.74248067268 -10.58 -6.38 1.0 94.7ms
18 -36.74248067268 -11.20 -6.66 2.0 133ms
19 -36.74248067268 -12.01 -7.18 1.0 94.8ms
20 -36.74248067268 + -13.45 -7.36 3.0 149ms
21 -36.74248067268 -13.19 -7.64 2.0 106ms
22 -36.74248067268 -13.67 -8.14 2.0 135ms
23 -36.74248067268 + -14.15 -8.47 2.0 103ms
24 -36.74248067268 -14.15 -8.53 3.0 141ms
25 -36.74248067268 + -Inf -9.06 1.0 96.2ms
26 -36.74248067268 + -14.15 -9.27 3.0 136ms
27 -36.74248067268 -13.67 -9.62 2.0 130ms
28 -36.74248067268 + -13.67 -10.15 2.0 117ms
29 -36.74248067268 -13.67 -10.34 2.0 130ms
30 -36.74248067268 + -13.67 -10.89 2.0 109ms
31 -36.74248067268 + -Inf -10.84 3.0 145ms
32 -36.74248067268 -13.67 -11.13 1.0 95.0ms
33 -36.74248067268 + -13.85 -11.85 2.0 115ms
34 -36.74248067268 + -14.15 -11.76 3.0 144ms
35 -36.74248067268 -14.15 -12.17 1.0 101msGiven 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))44.02448898019606The 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_min46.244211113651325This 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))4.723598093706142Since 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).