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 -36.73538931630 -0.88 11.0 347ms
2 -36.73822349993 -2.55 -1.65 1.0 94.1ms
3 -33.76418002761 + 0.47 -0.81 4.0 160ms
4 -34.98147220594 0.09 -0.91 4.0 156ms
5 -36.70112594611 0.24 -1.52 3.0 144ms
6 -36.71955464655 -1.73 -1.86 2.0 103ms
7 -36.40747295958 + -0.51 -1.31 4.0 152ms
8 -36.74111058664 -0.48 -2.23 3.0 134ms
9 -36.74117087401 -4.22 -2.09 2.0 123ms
10 -36.74157013744 -3.40 -2.22 2.0 124ms
11 -36.74236945370 -3.10 -2.73 2.0 113ms
12 -36.74241749457 -4.32 -2.71 2.0 113ms
13 -36.74250986915 -4.03 -3.33 1.0 95.6ms
14 -36.74250785609 + -5.70 -3.39 3.0 136ms
15 -36.74237872580 + -3.89 -2.98 3.0 136ms
16 -36.74247226870 -4.03 -3.21 3.0 136ms
17 -36.74251429726 -4.38 -3.80 2.0 106ms
18 -36.74251227494 + -5.69 -3.70 3.0 129ms
19 -36.74251341709 -5.94 -3.94 2.0 140ms
20 -36.74250219795 + -4.95 -3.52 3.0 141ms
21 -36.74251446570 -4.91 -4.13 3.0 136ms
22 -36.74251465799 -6.72 -4.43 2.0 130ms
23 -36.74251475149 -7.03 -4.71 1.0 92.0ms
24 -36.74251477033 -7.72 -5.17 3.0 120ms
25 -36.74251477024 + -10.09 -5.17 3.0 139ms
26 -36.74251477195 -8.77 -5.48 1.0 91.8ms
27 -36.74251477057 + -8.86 -5.30 3.0 151ms
28 -36.74251477302 -8.61 -6.01 2.0 111ms
29 -36.74251477298 + -10.40 -6.04 3.0 141ms
30 -36.74251477300 -10.73 -6.17 2.0 111ms
31 -36.74251477303 -10.60 -6.25 2.0 107ms
32 -36.74251477283 + -9.72 -5.90 3.0 130ms
33 -36.74251477302 -9.72 -6.44 3.0 135ms
34 -36.74251477304 -10.91 -7.12 2.0 101ms
35 -36.74251477304 + -11.89 -6.74 3.0 145ms
36 -36.74251477304 -11.96 -7.06 2.0 146ms
37 -36.74251477303 + -11.60 -6.78 3.0 131ms
38 -36.74251477304 -11.56 -7.54 3.0 127ms
39 -36.74251477304 + -12.66 -7.29 3.0 144ms
40 -36.74251477304 -13.11 -7.41 3.0 131ms
41 -36.74251477304 -13.45 -7.45 2.0 111ms
42 -36.74251477304 -13.11 -7.51 2.0 120ms
43 -36.74251477304 -13.45 -7.82 2.0 106ms
44 -36.74251477304 + -13.55 -7.72 3.0 150ms
45 -36.74251477304 -13.37 -8.01 2.0 112ms
46 -36.74251477304 + -13.85 -8.23 2.0 117ms
47 -36.74251477304 -13.85 -8.85 2.0 129ms
48 -36.74251477304 + -13.85 -8.90 3.0 139ms
49 -36.74251477304 -13.67 -9.49 1.0 91.3ms
50 -36.74251477304 + -14.15 -9.15 2.0 129ms
51 -36.74251477304 + -14.15 -9.32 3.0 132ms
52 -36.74251477304 -14.15 -9.44 1.0 96.5ms
53 -36.74251477304 + -13.85 -9.33 3.0 139ms
54 -36.74251477304 + -Inf -10.12 2.0 107ms
55 -36.74251477304 -13.85 -10.18 3.0 145ms
56 -36.74251477304 + -Inf -9.92 3.0 135ms
57 -36.74251477304 + -Inf -10.64 2.0 106ms
58 -36.74251477304 + -14.15 -10.51 2.0 130ms
59 -36.74251477304 -14.15 -10.64 2.0 121ms
60 -36.74251477304 + -14.15 -11.08 2.0 100ms
61 -36.74251477304 -14.15 -11.14 2.0 115ms
62 -36.74251477304 + -Inf -11.25 2.0 128ms
63 -36.74251477304 + -14.15 -11.49 1.0 96.3ms
64 -36.74251477304 -14.15 -11.60 2.0 129ms
65 -36.74251477304 + -14.15 -11.18 3.0 144ms
66 -36.74251477304 + -14.15 -11.19 3.0 146ms
67 -36.74251477304 + -Inf -12.02 3.0 137mswhile 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 -36.73417006400 -0.88 10.0 359ms
2 -36.73994903878 -2.24 -1.36 1.0 93.1ms
3 -36.74001698069 -4.17 -1.66 3.0 118ms
4 -36.74232880203 -2.64 -2.25 1.0 165ms
5 -36.74244148624 -3.95 -2.46 3.0 128ms
6 -36.74246812135 -4.57 -2.48 5.0 718ms
7 -36.74250202371 -4.47 -3.03 1.0 90.8ms
8 -36.74251273961 -4.97 -3.24 2.0 99.0ms
9 -36.74251412272 -5.86 -3.57 2.0 98.1ms
10 -36.74251470185 -6.24 -4.16 2.0 105ms
11 -36.74251474537 -7.36 -4.46 6.0 145ms
12 -36.74251476270 -7.76 -4.88 2.0 140ms
13 -36.74251476253 + -9.77 -4.98 3.0 197ms
14 -36.74251477276 -7.99 -5.63 2.0 132ms
15 -36.74251477262 + -9.85 -5.63 3.0 143ms
16 -36.74251477301 -9.41 -6.09 1.0 93.2ms
17 -36.74251477302 -10.79 -6.31 3.0 136ms
18 -36.74251477304 -10.85 -6.93 1.0 93.0ms
19 -36.74251477304 -12.02 -7.22 3.0 149ms
20 -36.74251477304 -12.92 -7.37 2.0 128ms
21 -36.74251477304 -13.85 -7.78 1.0 92.7ms
22 -36.74251477304 + -Inf -7.90 2.0 137ms
23 -36.74251477304 + -Inf -8.34 1.0 94.0ms
24 -36.74251477304 + -Inf -8.85 3.0 122ms
25 -36.74251477304 -13.85 -9.09 3.0 142ms
26 -36.74251477304 + -Inf -9.31 1.0 93.9ms
27 -36.74251477304 + -13.85 -9.47 2.0 109ms
28 -36.74251477304 + -Inf -9.87 1.0 98.2ms
29 -36.74251477304 -14.15 -10.25 2.0 132ms
30 -36.74251477304 -14.15 -10.47 2.0 104ms
31 -36.74251477304 + -14.15 -10.61 2.0 106ms
32 -36.74251477304 -14.15 -10.98 1.0 98.1ms
33 -36.74251477304 + -14.15 -11.19 4.0 148ms
34 -36.74251477304 + -Inf -11.73 2.0 103ms
35 -36.74251477304 + -14.15 -11.52 3.0 147ms
36 -36.74251477304 + -Inf -12.18 2.0 114msGiven 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))44.024488232797744The 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_min46.24421032856906This 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))4.723588044107179Since 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).