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 -36.73415397211 -0.88 11.0 1.25s
2 -36.67151264190 + -1.20 -1.50 1.0 224ms
3 +22.04240708384 + 1.77 -0.18 7.0 256ms
4 -36.69719356238 1.77 -1.28 6.0 190ms
5 -36.28769769216 + -0.39 -1.22 3.0 116ms
6 -36.19922193654 + -1.05 -1.19 3.0 115ms
7 -36.72689098574 -0.28 -1.72 3.0 147ms
8 -36.73766505760 -1.97 -2.08 2.0 95.6ms
9 -36.74097419590 -2.48 -2.15 2.0 112ms
10 -36.74116117503 -3.73 -2.16 2.0 113ms
11 -36.74201564380 -3.07 -2.35 2.0 103ms
12 -36.74188270742 + -3.88 -2.51 2.0 104ms
13 -36.74242608410 -3.26 -2.93 1.0 92.4ms
14 -36.74245272093 -4.57 -2.92 3.0 123ms
15 -36.74062704908 + -2.74 -2.43 3.0 121ms
16 -36.74245420918 -2.74 -3.25 3.0 150ms
17 -36.74245656927 -5.63 -3.31 1.0 84.8ms
18 -36.73985710541 + -2.59 -2.36 4.0 151ms
19 -36.74247981238 -2.58 -3.77 3.0 129ms
20 -36.74247647882 + -5.48 -3.52 2.0 114ms
21 -36.74247954978 -5.51 -3.88 2.0 114ms
22 -36.74248060515 -5.98 -4.27 2.0 95.5ms
23 -36.74248045094 + -6.81 -4.03 2.0 114ms
24 -36.74248064943 -6.70 -4.45 2.0 115ms
25 -36.74248065993 -7.98 -4.70 1.0 84.4ms
26 -36.74248067220 -7.91 -5.35 1.0 84.2ms
27 -36.74248064740 + -7.61 -4.85 4.0 141ms
28 -36.74248067116 -7.62 -5.43 3.0 122ms
29 -36.74248067250 -8.87 -5.57 2.0 105ms
30 -36.74248066965 + -8.54 -5.31 3.0 122ms
31 -36.74248067188 -8.65 -5.58 3.0 120ms
32 -36.74248067267 -9.10 -6.41 2.0 105ms
33 -36.74248067257 + -10.03 -5.92 4.0 145ms
34 -36.74248067268 -9.98 -6.77 3.0 127ms
35 -36.74248067268 -12.35 -6.87 2.0 124ms
36 -36.74248067268 -12.44 -6.91 2.0 116ms
37 -36.74248067268 -12.66 -7.44 1.0 85.6ms
38 -36.74248067268 + -13.19 -7.19 3.0 138ms
39 -36.74248067268 -13.19 -7.62 2.0 109ms
40 -36.74248067268 + -13.67 -7.62 2.0 121ms
41 -36.74248067268 + -13.00 -7.45 3.0 121ms
42 -36.74248067268 -12.89 -7.79 2.0 106ms
43 -36.74248067268 + -Inf -7.89 3.0 121ms
44 -36.74248067268 -13.67 -8.44 2.0 99.0ms
45 -36.74248067268 + -14.15 -8.76 2.0 114ms
46 -36.74248067268 -14.15 -8.52 2.0 106ms
47 -36.74248067268 + -Inf -8.68 3.0 120ms
48 -36.74248067268 + -Inf -8.86 2.0 113ms
49 -36.74248067268 + -Inf -9.30 1.0 84.9ms
50 -36.74248067268 -13.85 -9.82 3.0 124ms
51 -36.74248067268 + -13.85 -9.93 2.0 121ms
52 -36.74248067268 + -Inf -9.94 2.0 94.4ms
53 -36.74248067268 + -Inf -9.81 2.0 98.7ms
54 -36.74248067268 + -Inf -10.13 3.0 123ms
55 -36.74248067268 + -Inf -9.52 3.0 132ms
56 -36.74248067268 + -Inf -10.06 3.0 121ms
57 -36.74248067268 -14.15 -10.69 2.0 105ms
58 -36.74248067268 + -14.15 -10.65 3.0 128ms
59 -36.74248067268 -13.85 -11.01 2.0 106ms
60 -36.74248067268 + -13.67 -11.01 2.0 114ms
61 -36.74248067268 -14.15 -11.05 2.0 106ms
62 -36.74248067268 + -Inf -11.46 2.0 101ms
63 -36.74248067268 + -Inf -11.85 2.0 95.6ms
64 -36.74248067268 -13.85 -11.55 3.0 128ms
65 -36.74248067268 + -13.67 -12.13 2.0 114ms
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 -36.73400962593 -0.88 11.0 927ms
2 -36.74055572393 -2.18 -1.37 1.0 631ms
3 -36.74203637902 -2.83 -2.04 2.0 110ms
4 -36.74211580679 -4.10 -2.08 4.0 94.2ms
5 -36.74244686733 -3.48 -2.74 1.0 116ms
6 -36.74245929785 -4.91 -2.65 3.0 114ms
7 -36.74247877275 -4.71 -3.24 1.0 81.2ms
8 -36.74248022407 -5.84 -3.58 1.0 85.6ms
9 -36.74247994996 + -6.56 -3.45 3.0 115ms
10 -36.74248064379 -6.16 -4.12 1.0 82.3ms
11 -36.74248065151 -8.11 -4.25 3.0 117ms
12 -36.74248066843 -7.77 -4.92 1.0 84.1ms
13 -36.74248067239 -8.40 -5.17 5.0 126ms
14 -36.74248067252 -9.90 -5.33 1.0 90.6ms
15 -36.74248067259 -10.15 -5.62 2.0 106ms
16 -36.74248067268 -10.07 -6.01 2.0 120ms
17 -36.74248067268 -12.02 -6.43 2.0 89.2ms
18 -36.74248067268 -11.48 -6.70 3.0 111ms
19 -36.74248067268 -12.67 -7.05 2.0 96.2ms
20 -36.74248067268 -13.30 -7.36 2.0 113ms
21 -36.74248067268 + -Inf -7.58 3.0 104ms
22 -36.74248067268 + -Inf -7.82 1.0 84.3ms
23 -36.74248067268 -14.15 -7.93 2.0 105ms
24 -36.74248067268 + -Inf -8.22 1.0 90.4ms
25 -36.74248067268 + -Inf -8.61 3.0 97.5ms
26 -36.74248067268 -13.85 -9.10 2.0 119ms
27 -36.74248067268 + -13.55 -9.40 2.0 89.4ms
28 -36.74248067268 -13.85 -9.59 3.0 105ms
29 -36.74248067268 -14.15 -9.98 2.0 120ms
30 -36.74248067268 + -14.15 -10.53 2.0 97.3ms
31 -36.74248067268 + -Inf -10.53 2.0 120ms
32 -36.74248067268 + -Inf -10.81 1.0 84.3ms
33 -36.74248067268 + -Inf -11.26 1.0 84.1ms
34 -36.74248067268 + -Inf -11.71 3.0 126ms
35 -36.74248067268 -14.15 -11.87 3.0 118ms
36 -36.74248067268 + -Inf -12.05 2.0 104ms
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).δVρ
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))44.02449666170174The 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_min46.244219182459815This 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))4.7235759985838675Since 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).