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 -36.73230192160 -0.88 10.0 352ms
2 -36.72895426970 + -2.48 -1.64 1.0 96.6ms
3 -21.89228309965 + 1.17 -0.48 5.0 178ms
4 -36.67727166125 1.17 -1.36 5.0 184ms
5 -36.65201768577 + -1.60 -1.52 2.0 118ms
6 -36.11948148432 + -0.27 -1.17 4.0 151ms
7 -36.72298728691 -0.22 -1.77 3.0 134ms
8 -36.74204763909 -1.72 -2.36 2.0 109ms
9 -36.74189914946 + -3.83 -2.15 2.0 129ms
10 -36.74218723873 -3.54 -2.47 2.0 114ms
11 -36.74236930497 -3.74 -2.71 1.0 94.1ms
12 -36.74243014657 -4.22 -2.86 1.0 100ms
13 -36.74245770589 -4.56 -3.13 1.0 95.7ms
14 -36.74247983927 -4.65 -3.64 2.0 122ms
15 -36.74246892441 + -4.96 -3.23 3.0 144ms
16 -36.74247395544 -5.30 -3.55 3.0 133ms
17 -36.74247965319 -5.24 -3.89 2.0 103ms
18 -36.74237034095 + -3.96 -3.05 4.0 166ms
19 -36.74215950442 + -3.68 -2.81 4.0 172ms
20 -36.74247970785 -3.49 -3.91 4.0 168ms
21 -36.74248060221 -6.05 -4.32 2.0 111ms
22 -36.74248065423 -7.28 -4.72 2.0 134ms
23 -36.74248066978 -7.81 -5.00 2.0 108ms
24 -36.74248066950 + -9.55 -4.89 2.0 135ms
25 -36.74248067154 -8.69 -5.13 1.0 95.0ms
26 -36.74248067256 -8.99 -5.64 2.0 105ms
27 -36.74248067209 + -9.33 -5.47 3.0 129ms
28 -36.74248067247 -9.42 -5.82 2.0 117ms
29 -36.74248067263 -9.79 -6.03 2.0 110ms
30 -36.74248067200 + -9.20 -5.63 3.0 144ms
31 -36.74248067144 + -9.26 -5.51 4.0 155ms
32 -36.74248067232 -9.06 -5.76 3.0 137ms
33 -36.74248067267 -9.46 -6.35 3.0 132ms
34 -36.74248067268 -10.97 -6.83 2.0 110ms
35 -36.74248067268 -11.79 -7.05 2.0 134ms
36 -36.74248067268 -13.85 -7.20 1.0 95.1ms
37 -36.74248067268 -12.57 -7.65 2.0 115ms
38 -36.74248067268 -13.85 -7.85 3.0 136ms
39 -36.74248067268 -13.85 -7.85 3.0 122ms
40 -36.74248067268 + -13.85 -8.33 1.0 95.4ms
41 -36.74248067268 -14.15 -8.28 3.0 141ms
42 -36.74248067268 + -13.85 -8.15 2.0 112ms
43 -36.74248067268 -13.85 -8.29 3.0 133ms
44 -36.74248067268 + -Inf -8.47 2.0 120ms
45 -36.74248067268 + -14.15 -8.87 2.0 109ms
46 -36.74248067268 -14.15 -9.04 4.0 135ms
47 -36.74248067268 + -Inf -9.03 3.0 135ms
48 -36.74248067268 + -Inf -9.41 2.0 110ms
49 -36.74248067268 + -Inf -9.45 2.0 134ms
50 -36.74248067268 + -Inf -9.79 2.0 107ms
51 -36.74248067268 + -Inf -10.19 1.0 99.3ms
52 -36.74248067268 + -Inf -10.02 3.0 138ms
53 -36.74248067268 -13.85 -9.99 2.0 139ms
54 -36.74248067268 + -13.85 -9.80 3.0 141ms
55 -36.74248067268 + -Inf -10.32 3.0 131ms
56 -36.74248067268 -14.15 -10.56 2.0 122ms
57 -36.74248067268 + -14.15 -10.91 2.0 107ms
58 -36.74248067268 + -14.15 -11.35 3.0 139ms
59 -36.74248067268 + -Inf -11.41 3.0 123ms
60 -36.74248067268 -14.15 -11.44 3.0 118ms
61 -36.74248067268 -13.85 -11.35 3.0 132ms
62 -36.74248067268 + -Inf -11.96 2.0 110ms
63 -36.74248067268 + -13.67 -11.85 3.0 141ms
64 -36.74248067268 -14.15 -12.05 2.0 110mswhile 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 -36.73271463611 -0.88 12.0 349ms
2 -36.73963885808 -2.16 -1.36 1.0 96.8ms
3 -36.74095778887 -2.88 -2.11 3.0 122ms
4 -36.74177378998 -3.09 -2.05 6.0 152ms
5 -36.74225037130 -3.32 -2.41 1.0 92.6ms
6 -36.74227361629 -4.63 -2.71 2.0 108ms
7 -36.74246554817 -3.72 -2.96 1.0 93.7ms
8 -36.74247940716 -4.86 -3.44 3.0 139ms
9 -36.74248026619 -6.07 -3.62 2.0 100ms
10 -36.74248032352 -7.24 -3.64 2.0 105ms
11 -36.74248058869 -6.58 -4.12 2.0 101ms
12 -36.74248066729 -7.10 -4.49 3.0 144ms
13 -36.74248066834 -8.98 -4.54 5.0 118ms
14 -36.74248066989 -8.81 -4.84 2.0 108ms
15 -36.74248067166 -8.75 -5.11 1.0 96.4ms
16 -36.74248067233 -9.17 -5.31 3.0 124ms
17 -36.74248067266 -9.48 -5.89 2.0 119ms
18 -36.74248067268 -10.83 -6.15 3.0 156ms
19 -36.74248067268 -12.31 -6.37 1.0 101ms
20 -36.74248067268 -11.66 -6.64 1.0 96.5ms
21 -36.74248067268 + -12.36 -6.58 2.0 136ms
22 -36.74248067268 -12.23 -6.93 1.0 96.6ms
23 -36.74248067268 -12.87 -7.41 2.0 117ms
24 -36.74248067268 + -13.55 -7.27 3.0 152ms
25 -36.74248067268 -13.67 -7.50 1.0 101ms
26 -36.74248067268 + -Inf -7.89 2.0 105ms
27 -36.74248067268 + -Inf -8.53 3.0 143ms
28 -36.74248067268 + -Inf -8.70 3.0 150ms
29 -36.74248067268 -13.85 -9.05 2.0 112ms
30 -36.74248067268 + -14.15 -9.47 1.0 101ms
31 -36.74248067268 + -Inf -9.45 3.0 140ms
32 -36.74248067268 + -Inf -9.62 1.0 101ms
33 -36.74248067268 + -Inf -10.15 1.0 96.8ms
34 -36.74248067268 + -13.85 -10.05 3.0 150ms
35 -36.74248067268 -13.85 -10.27 1.0 96.2ms
36 -36.74248067268 + -Inf -10.89 2.0 116ms
37 -36.74248067268 + -13.85 -10.90 3.0 161ms
38 -36.74248067268 -13.85 -11.07 1.0 96.4ms
39 -36.74248067268 + -Inf -11.25 1.0 102ms
40 -36.74248067268 -14.15 -12.12 2.0 124msGiven 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))44.02448898052972The 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_min46.24421111400181This 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))4.723584140231395Since 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).