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 -36.73382328601 -0.88 11.0 1.37s
2 -36.73846737840 -2.33 -1.67 1.0 317ms
3 -34.34098160345 + 0.38 -0.86 4.0 187ms
4 -35.39365550679 0.02 -0.99 4.0 145ms
5 -36.74102393591 0.13 -2.01 4.0 137ms
6 -36.73396052781 + -2.15 -1.82 2.0 123ms
7 -36.65243474662 + -1.09 -1.58 3.0 129ms
8 -36.69032590259 -1.42 -1.69 3.0 129ms
9 -36.74195951483 -1.29 -2.36 3.0 124ms
10 -36.74232043420 -3.44 -2.53 2.0 123ms
11 -36.74245234890 -3.88 -2.86 2.0 101ms
12 -36.74242578436 + -4.58 -2.91 2.0 113ms
13 -36.74247703858 -4.29 -3.49 1.0 88.8ms
14 -36.74247704310 -8.34 -3.36 3.0 132ms
15 -36.74243830878 + -4.41 -3.15 3.0 128ms
16 -36.74178940042 + -3.19 -2.65 3.0 140ms
17 -36.74248016578 -3.16 -3.85 4.0 140ms
18 -36.74248063097 -6.33 -4.44 2.0 107ms
19 -36.74247858322 + -5.69 -3.83 3.0 141ms
20 -36.74247922434 -6.19 -3.98 3.0 140ms
21 -36.74248061076 -5.86 -4.54 3.0 127ms
22 -36.74248066468 -7.27 -4.78 2.0 98.2ms
23 -36.74248067104 -8.20 -5.38 2.0 95.1ms
24 -36.74248067090 + -9.86 -5.28 3.0 139ms
25 -36.74248067161 -9.15 -5.45 1.0 88.7ms
26 -36.74248067257 -9.02 -5.82 1.0 88.5ms
27 -36.74248067263 -10.23 -5.99 3.0 131ms
28 -36.74248066724 + -8.27 -5.19 4.0 148ms
29 -36.74248067258 -8.27 -5.98 3.0 141ms
30 -36.74248067268 -10.02 -6.49 2.0 96.7ms
31 -36.74248067262 + -10.26 -6.06 3.0 133ms
32 -36.74248067268 -10.22 -6.85 3.0 121ms
33 -36.74248067268 -11.93 -7.01 2.0 125ms
34 -36.74248067268 -12.38 -7.29 2.0 100ms
35 -36.74248067268 + -13.67 -7.37 1.0 93.2ms
36 -36.74248067268 + -13.37 -7.49 2.0 102ms
37 -36.74248067268 -13.45 -7.58 2.0 116ms
38 -36.74248067268 + -13.03 -7.43 2.0 103ms
39 -36.74248067268 -12.85 -8.27 2.0 107ms
40 -36.74248067268 + -14.15 -8.16 4.0 148ms
41 -36.74248067268 + -14.15 -7.90 3.0 130ms
42 -36.74248067268 + -Inf -8.01 3.0 126ms
43 -36.74248067268 -14.15 -8.76 2.0 108ms
44 -36.74248067268 -14.15 -8.67 3.0 127ms
45 -36.74248067268 + -14.15 -8.67 2.0 117ms
46 -36.74248067268 -13.85 -8.99 1.0 87.9ms
47 -36.74248067268 + -Inf -9.36 2.0 106ms
48 -36.74248067268 + -14.15 -9.57 3.0 128ms
49 -36.74248067268 + -Inf -9.21 3.0 131ms
50 -36.74248067268 -14.15 -9.76 2.0 109ms
51 -36.74248067268 + -Inf -9.91 2.0 103ms
52 -36.74248067268 + -Inf -9.41 3.0 149ms
53 -36.74248067268 + -14.15 -10.21 3.0 131ms
54 -36.74248067268 + -Inf -10.15 3.0 129ms
55 -36.74248067268 -14.15 -9.99 3.0 131ms
56 -36.74248067268 + -Inf -10.24 2.0 112ms
57 -36.74248067268 + -14.15 -10.58 2.0 97.9ms
58 -36.74248067268 + -14.15 -10.48 2.0 109ms
59 -36.74248067268 + -Inf -11.16 2.0 209ms
60 -36.74248067268 -13.85 -11.08 3.0 146ms
61 -36.74248067268 + -14.15 -11.68 2.0 1.15s
62 -36.74248067268 + -14.15 -11.92 3.0 132ms
63 -36.74248067268 -13.85 -11.61 2.0 113ms
64 -36.74248067268 + -Inf -11.56 3.0 126ms
65 -36.74248067268 + -Inf -11.75 3.0 122ms
66 -36.74248067268 + -14.15 -12.34 2.0 112ms
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 -36.73207383021 -0.88 11.0 993ms
2 -36.73991687473 -2.11 -1.37 1.0 1.03s
3 -36.74047470762 -3.25 -1.77 5.0 130ms
4 -36.74194127763 -2.83 -2.04 1.0 96.2ms
5 -36.74234725574 -3.39 -2.74 3.0 116ms
6 -36.74240426307 -4.24 -2.58 3.0 154ms
7 -36.74246651933 -4.21 -3.09 2.0 111ms
8 -36.74247467723 -5.09 -3.36 3.0 129ms
9 -36.74247737894 -5.57 -3.59 2.0 140ms
10 -36.74248049466 -5.51 -3.83 1.0 107ms
11 -36.74248061158 -6.93 -4.11 4.0 148ms
12 -36.74248066790 -7.25 -4.68 2.0 98.2ms
13 -36.74248067124 -8.48 -4.80 3.0 128ms
14 -36.74248067151 -9.58 -4.80 1.0 88.9ms
15 -36.74248066986 + -8.78 -4.74 1.0 89.2ms
16 -36.74248067249 -8.58 -5.20 1.0 89.0ms
17 -36.74248067184 + -9.18 -5.20 3.0 129ms
18 -36.74248067081 + -8.99 -5.23 2.0 94.2ms
19 -36.74248067235 -8.81 -5.56 1.0 88.9ms
20 -36.74248067268 -9.48 -6.06 2.0 104ms
21 -36.74248067267 + -11.32 -6.07 3.0 125ms
22 -36.74248067267 -11.62 -6.12 1.0 94.1ms
23 -36.74248067268 -11.35 -6.23 1.0 89.7ms
24 -36.74248067268 + -13.25 -6.45 1.0 95.6ms
25 -36.74248067268 -11.56 -6.70 1.0 103ms
26 -36.74248067268 -11.89 -6.96 4.0 113ms
27 -36.74248067268 -13.30 -7.21 2.0 98.2ms
28 -36.74248067268 -13.11 -7.33 3.0 125ms
29 -36.74248067268 -13.45 -7.72 1.0 90.4ms
30 -36.74248067268 + -Inf -7.56 3.0 126ms
31 -36.74248067268 + -Inf -7.78 1.0 94.3ms
32 -36.74248067268 -13.85 -8.25 2.0 103ms
33 -36.74248067268 + -Inf -8.88 2.0 115ms
34 -36.74248067268 + -13.85 -8.90 3.0 149ms
35 -36.74248067268 -13.85 -9.13 1.0 89.5ms
36 -36.74248067268 + -Inf -9.38 2.0 103ms
37 -36.74248067268 + -14.15 -9.72 2.0 121ms
38 -36.74248067268 -14.15 -9.92 1.0 94.0ms
39 -36.74248067268 + -14.15 -10.43 2.0 104ms
40 -36.74248067268 + -Inf -10.60 2.0 126ms
41 -36.74248067268 + -14.15 -10.73 3.0 122ms
42 -36.74248067268 + -Inf -11.28 2.0 102ms
43 -36.74248067268 + -Inf -11.13 4.0 142ms
44 -36.74248067268 -14.15 -11.98 2.0 109ms
45 -36.74248067268 -14.15 -12.25 4.0 143ms
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)
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))44.02448900594036The 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_min46.24421114069366This 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))4.723581060131038Since 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).