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 -36.73310011428 -0.88 11.0 342ms
2 -36.72007153232 + -1.89 -1.61 1.0 92.4ms
3 -11.17691986914 + 1.41 -0.36 6.0 186ms
4 -36.61714186035 1.41 -1.31 5.0 179ms
5 -36.57404200945 + -1.37 -1.41 3.0 134ms
6 -36.21211956888 + -0.44 -1.20 4.0 154ms
7 -36.71177207545 -0.30 -1.70 3.0 128ms
8 -36.74167755545 -1.52 -2.20 2.0 100ms
9 -36.74216498520 -3.31 -2.26 2.0 123ms
10 -36.74169520554 + -3.33 -2.23 2.0 115ms
11 -36.74215670802 -3.34 -2.50 2.0 118ms
12 -36.74236122615 -3.69 -2.78 1.0 92.2ms
13 -36.74247823685 -3.93 -3.07 1.0 96.1ms
14 -36.74248524169 -5.15 -2.99 2.0 130ms
15 -36.74248894391 -5.43 -3.28 2.0 104ms
16 -36.74246837762 + -4.69 -3.16 2.0 149ms
17 -36.74250406842 -4.45 -3.46 2.0 127ms
18 -36.74251033676 -5.20 -3.68 1.0 91.7ms
19 -36.74250819288 + -5.67 -3.66 3.0 136ms
20 -36.74251074075 -5.59 -3.70 2.0 117ms
21 -36.74251450379 -5.42 -4.31 2.0 105ms
22 -36.74251467313 -6.77 -4.19 3.0 136ms
23 -36.74251469362 -7.69 -4.52 2.0 111ms
24 -36.74251465823 + -7.45 -4.47 3.0 130ms
25 -36.74251476455 -6.97 -4.91 2.0 114ms
26 -36.74251476666 -8.68 -5.03 2.0 124ms
27 -36.74251474140 + -7.60 -4.74 3.0 135ms
28 -36.74251476586 -7.61 -4.91 3.0 134ms
29 -36.74251476879 -8.53 -5.16 2.0 103ms
30 -36.74251476980 -9.00 -5.24 2.0 120ms
31 -36.74251477054 -9.13 -5.35 2.0 104ms
32 -36.74251477176 -8.91 -5.46 3.0 124ms
33 -36.74251477266 -9.04 -5.72 2.0 114ms
34 -36.74251477299 -9.49 -5.99 2.0 137ms
35 -36.74251477297 + -10.76 -6.08 2.0 106ms
36 -36.74251477300 -10.59 -6.14 2.0 121ms
37 -36.74251477304 -10.42 -6.93 2.0 104ms
38 -36.74251477303 + -11.59 -6.69 4.0 151ms
39 -36.74251477304 -11.68 -6.82 3.0 137ms
40 -36.74251477304 -11.91 -7.20 2.0 112ms
41 -36.74251477304 -13.37 -7.29 2.0 97.1ms
42 -36.74251477304 + -12.36 -7.12 2.0 124ms
43 -36.74251477304 -12.70 -7.27 3.0 119ms
44 -36.74251477304 -12.54 -7.68 2.0 100ms
45 -36.74251477304 -13.67 -7.85 2.0 131ms
46 -36.74251477304 + -13.19 -7.53 3.0 130ms
47 -36.74251477304 -13.30 -8.07 2.0 102ms
48 -36.74251477304 + -14.15 -7.88 3.0 130ms
49 -36.74251477304 -14.15 -8.28 2.0 111ms
50 -36.74251477304 -14.15 -8.12 2.0 108ms
51 -36.74251477304 + -Inf -8.73 2.0 134ms
52 -36.74251477304 + -Inf -8.66 3.0 139ms
53 -36.74251477304 + -Inf -8.91 2.0 107ms
54 -36.74251477304 + -13.85 -8.69 3.0 128ms
55 -36.74251477304 -13.85 -9.03 3.0 125ms
56 -36.74251477304 + -14.15 -9.13 2.0 111ms
57 -36.74251477304 -14.15 -9.64 2.0 105ms
58 -36.74251477304 + -Inf -9.59 3.0 137ms
59 -36.74251477304 + -Inf -9.90 2.0 113ms
60 -36.74251477304 + -13.85 -10.11 1.0 122ms
61 -36.74251477304 -13.85 -10.26 1.0 93.2ms
62 -36.74251477304 + -Inf -9.95 3.0 135ms
63 -36.74251477304 + -Inf -10.25 2.0 142ms
64 -36.74251477304 + -Inf -10.42 2.0 123ms
65 -36.74251477304 + -Inf -10.92 1.0 95.8ms
66 -36.74251477304 -14.15 -10.48 3.0 140ms
67 -36.74251477304 + -14.15 -10.79 3.0 137ms
68 -36.74251477304 -13.85 -11.29 2.0 110ms
69 -36.74251477304 + -13.67 -10.76 3.0 163ms
70 -36.74251477304 + -Inf -11.28 3.0 136ms
71 -36.74251477304 + -14.15 -11.89 2.0 111ms
72 -36.74251477304 -13.85 -12.03 3.0 133mswhile 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 -36.73099827249 -0.88 12.0 342ms
2 -36.73935128157 -2.08 -1.36 1.0 87.3ms
3 -36.74161160119 -2.65 -1.93 3.0 152ms
4 -36.74215700279 -3.26 -2.06 1.0 93.1ms
5 -36.74241488519 -3.59 -2.54 4.0 104ms
6 -36.74247167982 -4.25 -2.56 4.0 134ms
7 -36.74250835397 -4.44 -3.03 1.0 89.1ms
8 -36.74251318381 -5.32 -3.49 1.0 95.7ms
9 -36.74251324494 -7.21 -3.29 4.0 151ms
10 -36.74251466149 -5.85 -3.96 1.0 91.8ms
11 -36.74251474238 -7.09 -4.32 1.0 96.8ms
12 -36.74251474048 + -8.72 -4.48 3.0 128ms
13 -36.74251476978 -7.53 -4.71 2.0 153ms
14 -36.74251476392 + -8.23 -4.84 2.0 107ms
15 -36.74251477259 -8.06 -5.11 1.0 97.2ms
16 -36.74251477294 -9.45 -5.42 4.0 113ms
17 -36.74251477298 -10.39 -5.87 2.0 103ms
18 -36.74251477302 -10.39 -6.04 3.0 136ms
19 -36.74251477304 -10.91 -6.46 1.0 97.0ms
20 -36.74251477303 + -11.26 -6.37 4.0 146ms
21 -36.74251477304 -11.39 -6.63 1.0 100ms
22 -36.74251477304 -11.71 -7.05 2.0 117ms
23 -36.74251477304 -13.55 -7.43 3.0 127ms
24 -36.74251477304 -13.25 -7.64 3.0 138ms
25 -36.74251477304 + -Inf -8.09 1.0 98.3ms
26 -36.74251477304 -13.85 -8.75 3.0 128ms
27 -36.74251477304 + -14.15 -8.81 3.0 142ms
28 -36.74251477304 + -Inf -9.29 2.0 113ms
29 -36.74251477304 + -Inf -9.44 6.0 153ms
30 -36.74251477304 -14.15 -9.90 1.0 93.4ms
31 -36.74251477304 + -Inf -10.32 4.0 149ms
32 -36.74251477304 + -Inf -10.54 3.0 136ms
33 -36.74251477304 + -Inf -10.94 1.0 97.9ms
34 -36.74251477304 + -13.85 -11.14 3.0 132ms
35 -36.74251477304 -13.85 -11.50 1.0 226ms
36 -36.74251477304 + -Inf -11.72 4.0 783ms
37 -36.74251477304 + -Inf -12.10 2.0 108msGiven 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))44.02448777207383The 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_min46.244209844615376This 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))4.7235965411322836Since 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).