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 -36.73316158809 -0.88 11.0 1.34s
2 -36.63011169611 + -0.99 -1.43 1.0 252ms
3 +37.16030281076 + 1.87 -0.13 7.0 267ms
4 -36.48829654671 1.87 -1.03 9.0 371ms
5 -36.66018198585 -0.76 -1.41 3.0 132ms
6 -36.70869232930 -1.31 -1.65 2.0 1.12s
7 -36.05820460338 + -0.19 -1.15 3.0 130ms
8 -36.73635159248 -0.17 -2.04 3.0 140ms
9 -36.74160055200 -2.28 -2.03 2.0 124ms
10 -36.74164584657 -4.34 -2.19 1.0 91.1ms
11 -36.74201411823 -3.43 -2.42 1.0 91.7ms
12 -36.74196151377 + -4.28 -2.56 1.0 92.4ms
13 -36.74243358440 -3.33 -2.91 1.0 92.8ms
14 -36.74244424302 -4.97 -3.05 2.0 118ms
15 -36.74183499188 + -3.22 -2.56 3.0 137ms
16 -36.74247603733 -3.19 -3.52 3.0 134ms
17 -36.74247710104 -5.97 -3.52 3.0 130ms
18 -36.74247759248 -6.31 -3.72 2.0 108ms
19 -36.74247400484 + -5.45 -3.64 3.0 127ms
20 -36.74247209549 + -5.72 -3.59 3.0 133ms
21 -36.74248008469 -5.10 -4.14 2.0 117ms
22 -36.74248016750 -7.08 -4.13 3.0 147ms
23 -36.74248064856 -6.32 -4.74 2.0 109ms
24 -36.74248066538 -7.77 -5.00 2.0 126ms
25 -36.74248065893 + -8.19 -4.88 2.0 112ms
26 -36.74248067024 -7.95 -5.17 1.0 93.2ms
27 -36.74248067225 -8.70 -5.59 3.0 126ms
28 -36.74248067162 + -9.20 -5.39 2.0 126ms
29 -36.74248067249 -9.06 -5.88 2.0 107ms
30 -36.74248067266 -9.78 -6.07 3.0 128ms
31 -36.74248067185 + -9.10 -5.57 4.0 154ms
32 -36.74248067245 -9.22 -5.83 3.0 135ms
33 -36.74248067260 -9.83 -6.07 2.0 118ms
34 -36.74248067268 -10.10 -6.58 2.0 123ms
35 -36.74248067268 -11.69 -6.79 2.0 127ms
36 -36.74248067268 -12.10 -7.11 2.0 109ms
37 -36.74248067268 + -13.55 -6.97 2.0 126ms
38 -36.74248067268 -12.36 -7.40 2.0 114ms
39 -36.74248067268 + -13.45 -7.48 3.0 121ms
40 -36.74248067268 -13.19 -7.60 2.0 109ms
41 -36.74248067268 -14.15 -7.95 1.0 97.4ms
42 -36.74248067268 + -13.67 -7.70 3.0 140ms
43 -36.74248067268 -13.45 -8.29 3.0 134ms
44 -36.74248067268 + -Inf -8.33 2.0 107ms
45 -36.74248067268 -14.15 -8.74 2.0 108ms
46 -36.74248067268 + -Inf -8.28 3.0 147ms
47 -36.74248067268 + -Inf -8.99 3.0 135ms
48 -36.74248067268 + -13.85 -8.83 3.0 126ms
49 -36.74248067268 -14.15 -9.44 3.0 119ms
50 -36.74248067268 + -Inf -9.29 3.0 147ms
51 -36.74248067268 -14.15 -9.66 2.0 127ms
52 -36.74248067268 -14.15 -9.39 3.0 138ms
53 -36.74248067268 + -Inf -9.77 3.0 137ms
54 -36.74248067268 + -Inf -10.13 2.0 110ms
55 -36.74248067268 + -Inf -10.17 2.0 131ms
56 -36.74248067268 + -13.85 -10.43 2.0 112ms
57 -36.74248067268 + -Inf -10.85 2.0 104ms
58 -36.74248067268 -14.15 -10.36 3.0 166ms
59 -36.74248067268 -14.15 -11.06 4.0 154ms
60 -36.74248067268 + -Inf -10.92 2.0 121ms
61 -36.74248067268 + -13.85 -11.60 2.0 113ms
62 -36.74248067268 + -Inf -11.34 3.0 137ms
63 -36.74248067268 -14.15 -11.49 3.0 137ms
64 -36.74248067268 + -Inf -11.20 3.0 129ms
65 -36.74248067268 + -13.67 -11.97 3.0 133ms
66 -36.74248067268 -13.85 -12.34 2.0 130ms
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 -36.73401554671 -0.88 11.0 969ms
2 -36.74020815422 -2.21 -1.36 1.0 662ms
3 -36.74116672210 -3.02 -1.88 3.0 120ms
4 -36.74209216711 -3.03 -2.09 2.0 109ms
5 -36.74244311421 -3.45 -2.86 2.0 104ms
6 -36.74245845612 -4.81 -2.78 4.0 155ms
7 -36.74247532500 -4.77 -3.01 1.0 96.4ms
8 -36.74247995255 -5.33 -3.54 1.0 197ms
9 -36.74248038250 -6.37 -3.79 2.0 107ms
10 -36.74248064216 -6.59 -4.11 2.0 1.07s
11 -36.74248065228 -7.99 -4.33 3.0 116ms
12 -36.74248066818 -7.80 -4.58 5.0 117ms
13 -36.74248066948 -8.89 -4.67 2.0 128ms
14 -36.74248067170 -8.65 -5.05 1.0 94.9ms
15 -36.74248067211 -9.39 -5.31 2.0 120ms
16 -36.74248067261 -9.30 -5.52 1.0 95.0ms
17 -36.74248067265 -10.36 -5.85 2.0 104ms
18 -36.74248067268 -10.57 -6.33 3.0 126ms
19 -36.74248067268 -11.59 -6.71 2.0 129ms
20 -36.74248067268 -12.79 -6.97 5.0 137ms
21 -36.74248067268 + -13.45 -6.89 2.0 143ms
22 -36.74248067268 -12.87 -7.27 1.0 119ms
23 -36.74248067268 -13.25 -7.55 1.0 120ms
24 -36.74248067268 + -Inf -8.03 3.0 123ms
25 -36.74248067268 + -Inf -8.31 3.0 125ms
26 -36.74248067268 + -Inf -8.38 2.0 129ms
27 -36.74248067268 -13.85 -8.63 1.0 94.8ms
28 -36.74248067268 + -13.85 -9.21 2.0 110ms
29 -36.74248067268 + -13.85 -9.41 3.0 132ms
30 -36.74248067268 -13.67 -9.79 2.0 100ms
31 -36.74248067268 + -14.15 -10.14 3.0 134ms
32 -36.74248067268 + -14.15 -10.51 2.0 100ms
33 -36.74248067268 -14.15 -10.66 6.0 135ms
34 -36.74248067268 -14.15 -10.88 2.0 115ms
35 -36.74248067268 -14.15 -11.29 1.0 94.7ms
36 -36.74248067268 + -13.85 -11.73 3.0 135ms
37 -36.74248067268 -13.85 -12.05 2.0 109ms
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))44.02448898065439The 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_min46.24421111413277This 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))4.723591319359145Since 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).