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 -36.73368191907 -0.88 14.0 378ms
2 -36.63185699033 + -0.99 -1.40 1.0 94.1ms
3 +33.75519766980 + 1.85 -0.14 7.0 222ms
4 -35.87217235891 1.84 -0.85 7.0 333ms
5 -36.25964486154 -0.41 -1.13 3.0 143ms
6 -33.01373080309 + 0.51 -0.78 6.0 1.24s
7 -36.73847730626 0.57 -1.82 4.0 151ms
8 -36.73913901434 -3.18 -1.85 2.0 126ms
9 -36.73614787705 + -2.52 -1.98 2.0 104ms
10 -36.74091487923 -2.32 -2.17 2.0 98.5ms
11 -36.74230755218 -2.86 -2.42 2.0 101ms
12 -36.74183028902 + -3.32 -2.50 2.0 108ms
13 -36.74237729043 -3.26 -2.72 1.0 100ms
14 -36.74241429730 -4.43 -2.77 2.0 103ms
15 -36.73461784103 + -2.11 -2.11 3.0 135ms
16 -36.74154122446 -2.16 -2.44 4.0 162ms
17 -36.74191042673 -3.43 -2.67 2.0 121ms
18 -36.72721642935 + -1.83 -1.97 3.0 150ms
19 -36.74246399288 -1.82 -3.24 4.0 149ms
20 -36.74245487247 + -5.04 -2.98 2.0 127ms
21 -36.74246994783 -4.82 -3.40 2.0 126ms
22 -36.74247843728 -5.07 -3.69 1.0 94.0ms
23 -36.74247799867 + -6.36 -3.82 3.0 136ms
24 -36.74247997667 -5.70 -4.07 2.0 115ms
25 -36.74248041021 -6.36 -4.20 3.0 127ms
26 -36.74248058330 -6.76 -4.32 1.0 100ms
27 -36.74248060047 -7.77 -4.60 2.0 105ms
28 -36.74248066649 -7.18 -4.89 2.0 135ms
29 -36.74248066607 + -9.37 -5.05 2.0 112ms
30 -36.74248048907 + -6.75 -4.41 3.0 154ms
31 -36.74248067202 -6.74 -5.53 4.0 154ms
32 -36.74248066957 + -8.61 -5.20 3.0 150ms
33 -36.74248067248 -8.54 -5.83 3.0 126ms
34 -36.74248067250 -10.78 -5.64 2.0 128ms
35 -36.74248067262 -9.93 -6.01 2.0 116ms
36 -36.74248067268 -10.22 -6.58 2.0 108ms
37 -36.74248067267 + -11.28 -6.55 3.0 143ms
38 -36.74248067268 -11.31 -6.66 2.0 114ms
39 -36.74248067267 + -11.42 -6.57 2.0 109ms
40 -36.74248067268 -11.19 -6.87 2.0 109ms
41 -36.74248067268 -11.81 -7.03 2.0 119ms
42 -36.74248067268 + -11.62 -6.82 2.0 134ms
43 -36.74248067268 -11.76 -6.95 3.0 126ms
44 -36.74248067268 -12.12 -7.45 2.0 114ms
45 -36.74248067268 -12.97 -7.88 3.0 118ms
46 -36.74248067268 + -Inf -7.82 3.0 142ms
47 -36.74248067268 + -14.15 -7.72 2.0 119ms
48 -36.74248067268 -14.15 -7.94 2.0 125ms
49 -36.74248067268 + -13.25 -7.61 2.0 119ms
50 -36.74248067268 -13.25 -8.07 3.0 141ms
51 -36.74248067268 + -Inf -8.19 1.0 94.6ms
52 -36.74248067268 + -Inf -8.56 2.0 125ms
53 -36.74248067268 -14.15 -8.92 2.0 158ms
54 -36.74248067268 -14.15 -9.26 2.0 102ms
55 -36.74248067268 + -14.15 -8.98 3.0 140ms
56 -36.74248067268 + -Inf -9.51 3.0 129ms
57 -36.74248067268 + -Inf -9.64 1.0 93.6ms
58 -36.74248067268 + -Inf -9.84 2.0 106ms
59 -36.74248067268 + -Inf -10.06 2.0 127ms
60 -36.74248067268 + -Inf -10.20 2.0 114ms
61 -36.74248067268 + -Inf -10.37 2.0 106ms
62 -36.74248067268 + -Inf -10.52 2.0 105ms
63 -36.74248067268 + -Inf -10.38 1.0 93.4ms
64 -36.74248067268 + -Inf -11.07 2.0 108ms
65 -36.74248067268 -14.15 -10.70 3.0 143ms
66 -36.74248067268 + -14.15 -11.43 3.0 154ms
67 -36.74248067268 + -Inf -11.47 2.0 127ms
68 -36.74248067268 + -Inf -11.58 2.0 114ms
69 -36.74248067268 + -Inf -11.92 1.0 93.6ms
70 -36.74248067268 + -Inf -11.85 3.0 139ms
71 -36.74248067268 -13.85 -12.14 2.0 107mswhile 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 -36.73346626760 -0.88 12.0 348ms
2 -36.73991756720 -2.19 -1.36 1.0 96.0ms
3 -36.74029317521 -3.43 -1.76 4.0 127ms
4 -36.74215714240 -2.73 -2.16 2.0 102ms
5 -36.74232935532 -3.76 -2.63 3.0 110ms
6 -36.74241668338 -4.06 -2.51 4.0 142ms
7 -36.74246499889 -4.32 -3.05 1.0 92.8ms
8 -36.74247811535 -4.88 -3.32 5.0 119ms
9 -36.74248013389 -5.69 -3.47 2.0 108ms
10 -36.74248045374 -6.50 -3.85 1.0 98.6ms
11 -36.74248060920 -6.81 -4.04 3.0 109ms
12 -36.74248060613 + -8.51 -4.41 3.0 120ms
13 -36.74248067143 -7.19 -4.82 2.0 131ms
14 -36.74248067175 -9.48 -4.93 2.0 106ms
15 -36.74248067187 -9.92 -5.22 2.0 111ms
16 -36.74248067262 -9.13 -5.57 2.0 107ms
17 -36.74248067266 -10.38 -6.00 5.0 127ms
18 -36.74248067267 -11.54 -6.31 3.0 140ms
19 -36.74248067268 -10.85 -6.81 3.0 116ms
20 -36.74248067268 + -12.12 -6.77 3.0 144ms
21 -36.74248067268 -11.97 -7.19 2.0 115ms
22 -36.74248067268 -12.85 -7.27 1.0 95.8ms
23 -36.74248067268 -14.15 -7.62 1.0 100ms
24 -36.74248067268 -14.15 -7.91 3.0 127ms
25 -36.74248067268 + -Inf -8.15 3.0 140ms
26 -36.74248067268 -14.15 -8.53 1.0 95.9ms
27 -36.74248067268 + -14.15 -8.72 3.0 140ms
28 -36.74248067268 + -14.15 -9.02 1.0 95.7ms
29 -36.74248067268 -13.67 -9.71 3.0 128ms
30 -36.74248067268 + -13.67 -9.85 3.0 134ms
31 -36.74248067268 -14.15 -10.03 4.0 116ms
32 -36.74248067268 -13.67 -10.43 2.0 115ms
33 -36.74248067268 + -13.85 -10.85 2.0 129ms
34 -36.74248067268 + -14.15 -11.08 2.0 116ms
35 -36.74248067268 + -Inf -11.33 2.0 129ms
36 -36.74248067268 -14.15 -11.71 1.0 101ms
37 -36.74248067268 + -Inf -11.99 1.0 95.4ms
38 -36.74248067268 + -Inf -12.47 2.0 128msGiven 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
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))44.02448898011611The 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_min46.244211113567346This 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))4.723582614969774Since 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).