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 -36.73387182399 -0.88 10.0 1.28s
2 -36.72772449276 + -2.21 -1.65 1.0 223ms
3 -20.35924589203 + 1.21 -0.46 5.0 214ms
4 -36.68056377869 1.21 -1.31 5.0 152ms
5 -36.71679096100 -1.44 -1.71 2.0 104ms
6 -36.12778134235 + -0.23 -1.17 3.0 140ms
7 -36.72465444147 -0.22 -1.86 3.0 117ms
8 -36.74181253142 -1.77 -2.33 2.0 95.5ms
9 -36.74090084319 + -3.04 -2.05 2.0 112ms
10 -36.74164434858 -3.13 -2.36 2.0 112ms
11 -36.74215199513 -3.29 -2.58 2.0 102ms
12 -36.74236571241 -3.67 -2.87 2.0 102ms
13 -36.74247447367 -3.96 -3.48 2.0 114ms
14 -36.74247780712 -5.48 -3.46 2.0 114ms
15 -36.74241734644 + -4.22 -3.11 3.0 127ms
16 -36.74247684093 -4.23 -3.69 3.0 115ms
17 -36.74247516937 + -5.78 -3.53 2.0 97.0ms
18 -36.74247961816 -5.35 -4.03 2.0 112ms
19 -36.74247101717 + -5.07 -3.58 3.0 128ms
20 -36.74248037503 -5.03 -4.26 3.0 120ms
21 -36.74248065118 -6.56 -4.74 2.0 120ms
22 -36.74248064314 + -8.09 -4.56 2.0 114ms
23 -36.74248066306 -7.70 -4.76 2.0 134ms
24 -36.74248067071 -8.12 -5.26 1.0 84.2ms
25 -36.74248067235 -8.79 -5.34 2.0 103ms
26 -36.74248067248 -9.87 -5.55 2.0 106ms
27 -36.74248067249 -11.12 -5.72 1.0 83.8ms
28 -36.74248067259 -10.00 -5.87 1.0 86.5ms
29 -36.74248067256 + -10.49 -5.91 2.0 97.6ms
30 -36.74248067258 -10.58 -6.00 2.0 96.0ms
31 -36.74248067232 + -9.59 -5.78 3.0 119ms
32 -36.74248067264 -9.49 -6.25 2.0 112ms
33 -36.74248067263 + -10.86 -6.14 2.0 97.0ms
34 -36.74248067268 -10.30 -6.88 3.0 105ms
35 -36.74248067268 -12.89 -7.04 2.0 120ms
36 -36.74248067268 -12.41 -7.32 2.0 105ms
37 -36.74248067268 -13.67 -7.62 2.0 120ms
38 -36.74248067268 -13.37 -7.94 2.0 97.7ms
39 -36.74248067268 -14.15 -8.10 2.0 91.9ms
40 -36.74248067268 + -14.15 -8.56 2.0 112ms
41 -36.74248067268 -13.67 -8.53 2.0 114ms
42 -36.74248067268 + -13.85 -9.00 2.0 92.1ms
43 -36.74248067268 -13.85 -8.88 3.0 127ms
44 -36.74248067268 + -14.15 -8.39 3.0 119ms
45 -36.74248067268 + -14.15 -8.97 3.0 121ms
46 -36.74248067268 + -Inf -9.05 2.0 112ms
47 -36.74248067268 + -Inf -9.39 3.0 103ms
48 -36.74248067268 + -Inf -9.42 2.0 119ms
49 -36.74248067268 + -Inf -10.15 2.0 94.1ms
50 -36.74248067268 + -Inf -10.11 3.0 122ms
51 -36.74248067268 + -Inf -10.36 2.0 102ms
52 -36.74248067268 + -Inf -10.29 3.0 118ms
53 -36.74248067268 -13.85 -10.33 2.0 105ms
54 -36.74248067268 + -13.85 -10.82 2.0 98.4ms
55 -36.74248067268 + -14.15 -11.22 2.0 113ms
56 -36.74248067268 -14.15 -10.36 4.0 143ms
57 -36.74248067268 + -Inf -11.59 4.0 140ms
58 -36.74248067268 + -14.15 -11.28 3.0 122ms
59 -36.74248067268 -13.85 -11.99 3.0 125ms
60 -36.74248067268 -14.15 -12.27 2.0 92.1ms
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 -36.73448682822 -0.88 10.0 937ms
2 -36.74044842976 -2.22 -1.36 1.0 632ms
3 -36.74046675772 -4.74 -1.76 3.0 90.2ms
4 -36.74216129156 -2.77 -2.26 2.0 122ms
5 -36.74232815682 -3.78 -2.40 3.0 123ms
6 -36.74245369192 -3.90 -2.65 1.0 81.2ms
7 -36.74244542899 + -5.08 -3.03 1.0 81.4ms
8 -36.74247378033 -4.55 -3.13 2.0 124ms
9 -36.74247803697 -5.37 -3.55 1.0 83.6ms
10 -36.74248051801 -5.61 -4.03 3.0 96.0ms
11 -36.74248012760 + -6.41 -4.09 3.0 125ms
12 -36.74248064422 -6.29 -4.53 2.0 90.0ms
13 -36.74248066158 -7.76 -4.87 3.0 130ms
14 -36.74248067178 -7.99 -5.19 2.0 93.5ms
15 -36.74248067261 -9.08 -5.57 2.0 120ms
16 -36.74248067252 + -10.07 -5.82 1.0 84.9ms
17 -36.74248067264 -9.91 -6.16 1.0 85.0ms
18 -36.74248067268 -10.47 -6.48 2.0 120ms
19 -36.74248067268 -11.61 -6.72 1.0 85.2ms
20 -36.74248067268 -12.20 -7.17 3.0 110ms
21 -36.74248067268 -13.85 -7.44 2.0 120ms
22 -36.74248067268 -14.15 -7.62 2.0 100ms
23 -36.74248067268 + -14.15 -7.86 1.0 91.1ms
24 -36.74248067268 + -Inf -8.08 1.0 85.4ms
25 -36.74248067268 -14.15 -8.66 2.0 104ms
26 -36.74248067268 -14.15 -8.97 2.0 120ms
27 -36.74248067268 + -13.85 -9.23 1.0 85.1ms
28 -36.74248067268 -14.15 -9.63 3.0 105ms
29 -36.74248067268 -14.15 -9.76 2.0 115ms
30 -36.74248067268 + -Inf -9.96 1.0 84.9ms
31 -36.74248067268 + -14.15 -10.41 2.0 110ms
32 -36.74248067268 + -Inf -10.91 2.0 114ms
33 -36.74248067268 + -14.15 -11.28 2.0 99.3ms
34 -36.74248067268 -14.15 -11.56 3.0 119ms
35 -36.74248067268 + -Inf -11.78 2.0 96.0ms
36 -36.74248067268 -13.85 -12.33 1.0 91.1ms
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))44.02448898014948The 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_min46.2442111136024This 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))4.7235928595984635Since 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).