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.oncvpsp3.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 -36.73428021615 -0.88 11.0 398ms
2 -36.71524842247 + -1.72 -1.57 1.0 91.5ms
3 -7.304051169376 + 1.47 -0.33 6.0 211ms
4 -36.46068681068 1.46 -1.09 8.0 224ms
5 -36.53554558489 -1.13 -1.32 4.0 153ms
6 -36.19382764659 + -0.47 -1.20 4.0 166ms
7 -36.72215678971 -0.28 -1.83 3.0 163ms
8 -36.73982008731 -1.75 -2.02 2.0 116ms
9 -36.73963951969 + -3.74 -2.16 2.0 135ms
10 -36.74089717462 -2.90 -2.18 2.0 130ms
11 -36.74183900087 -3.03 -2.37 1.0 96.3ms
12 -36.74203236118 -3.71 -3.03 1.0 101ms
13 -36.74207082424 -4.41 -3.21 3.0 146ms
14 -36.74208208335 -4.95 -3.47 2.0 116ms
15 -36.74203697229 + -4.35 -3.22 3.0 158ms
16 -36.74205913442 -4.65 -3.19 4.0 153ms
17 -36.74191071209 + -3.83 -2.92 3.0 149ms
18 -36.74205674756 -3.84 -3.35 4.0 153ms
19 -36.74207306843 -4.79 -3.51 3.0 134ms
20 -36.74208361005 -4.98 -4.35 2.0 112ms
21 -36.74208363290 -7.64 -4.33 4.0 175ms
22 -36.74208365929 -7.58 -4.21 2.0 139ms
23 -36.74208375297 -7.03 -4.71 2.0 130ms
24 -36.74208375688 -8.41 -5.02 2.0 137ms
25 -36.74208376444 -8.12 -5.28 1.0 101ms
26 -36.74208376446 -10.57 -5.41 2.0 136ms
27 -36.74208376434 + -9.92 -5.65 1.0 96.2ms
28 -36.74208376469 -9.45 -5.99 1.0 103ms
29 -36.74208376173 + -8.53 -5.31 4.0 172ms
30 -36.74208376458 -8.55 -5.88 4.0 189ms
31 -36.74208376478 -9.70 -6.43 3.0 122ms
32 -36.74208376478 + -11.56 -6.47 3.0 149ms
33 -36.74208376478 -11.89 -6.52 5.0 139ms
34 -36.74208376479 -11.08 -6.90 2.0 111ms
35 -36.74208376479 -11.94 -7.15 2.0 103ms
36 -36.74208376479 + -14.15 -7.33 3.0 143ms
37 -36.74208376479 -12.97 -7.42 1.0 103ms
38 -36.74208376479 -13.85 -7.79 1.0 96.6ms
39 -36.74208376479 + -Inf -8.06 2.0 135ms
40 -36.74208376479 + -13.85 -7.78 2.0 138ms
41 -36.74208376479 + -13.85 -7.73 4.0 149ms
42 -36.74208376479 + -13.85 -7.61 4.0 156ms
43 -36.74208376479 -13.37 -8.46 3.0 147ms
44 -36.74208376479 -13.85 -9.13 2.0 107ms
45 -36.74208376479 + -13.85 -8.81 4.0 174ms
46 -36.74208376479 + -Inf -9.09 3.0 159ms
47 -36.74208376479 + -Inf -9.78 2.0 125ms
48 -36.74208376479 -13.85 -9.61 3.0 143ms
49 -36.74208376479 + -Inf -9.72 2.0 121ms
50 -36.74208376479 + -13.85 -9.94 1.0 96.4ms
51 -36.74208376479 + -Inf -10.39 1.0 101ms
52 -36.74208376479 -14.15 -10.36 2.0 139ms
53 -36.74208376479 + -Inf -10.88 1.0 96.2ms
54 -36.74208376479 + -14.15 -10.44 3.0 152ms
55 -36.74208376479 + -Inf -10.32 4.0 221ms
56 -36.74208376479 + -Inf -11.03 4.0 170ms
57 -36.74208376479 -13.85 -11.21 2.0 116ms
58 -36.74208376479 + -13.85 -11.31 2.0 130ms
59 -36.74208376479 -14.15 -11.24 3.0 147ms
60 -36.74208376479 + -Inf -11.59 5.0 137ms
61 -36.74208376479 + -14.15 -12.25 2.0 135mswhile 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 -36.73415368139 -0.88 10.0 366ms
2 -36.74003466815 -2.23 -1.36 1.0 97.0ms
3 -36.73944685822 + -3.23 -1.71 4.0 132ms
4 -36.74182517334 -2.62 -2.23 1.0 98.1ms
5 -36.74184538592 -4.69 -2.49 4.0 150ms
6 -36.74203763190 -3.72 -2.54 3.0 108ms
7 -36.74205089467 -4.88 -2.90 1.0 126ms
8 -36.74207764017 -4.57 -3.09 2.0 109ms
9 -36.74208169789 -5.39 -3.46 2.0 105ms
10 -36.74208345541 -5.76 -4.09 6.0 137ms
11 -36.74208352826 -7.14 -4.25 3.0 154ms
12 -36.74208374879 -6.66 -4.55 2.0 121ms
13 -36.74208373402 + -7.83 -4.78 2.0 215ms
14 -36.74208376446 -7.52 -5.61 2.0 117ms
15 -36.74208376460 -9.85 -5.85 6.0 837ms
16 -36.74208376473 -9.89 -6.11 3.0 123ms
17 -36.74208376477 -10.42 -6.38 2.0 137ms
18 -36.74208376479 -10.71 -6.95 2.0 118ms
19 -36.74208376479 -13.00 -7.19 4.0 150ms
20 -36.74208376479 -12.94 -7.50 2.0 108ms
21 -36.74208376479 -13.45 -7.90 6.0 144ms
22 -36.74208376479 + -13.85 -8.05 2.0 151ms
23 -36.74208376479 + -Inf -8.47 1.0 109ms
24 -36.74208376479 + -Inf -8.79 5.0 128ms
25 -36.74208376479 -13.85 -9.25 2.0 117ms
26 -36.74208376479 + -13.85 -9.53 4.0 159ms
27 -36.74208376479 -13.85 -9.86 1.0 105ms
28 -36.74208376479 + -13.67 -10.07 3.0 145ms
29 -36.74208376479 -13.85 -10.56 2.0 106ms
30 -36.74208376479 + -Inf -10.86 3.0 144ms
31 -36.74208376479 + -Inf -11.15 9.0 157ms
32 -36.74208376479 -14.15 -11.81 2.0 107ms
33 -36.74208376479 + -Inf -11.92 4.0 156ms
34 -36.74208376479 + -14.15 -12.15 2.0 130msGiven 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 - 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))44.024488544619835The 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_min46.244210656113275This 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))4.723583969848129Since 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).