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 -36.73236537861 -0.88 12.0 1.27s
2 -36.72913387965 + -2.49 -1.59 1.0 242ms
3 -25.78846525899 + 1.04 -0.52 5.0 209ms
4 -32.35621007314 0.82 -0.71 5.0 183ms
5 -36.45878627397 0.61 -1.20 3.0 149ms
6 -35.33651127387 + 0.05 -0.98 3.0 139ms
7 -36.72046830986 0.14 -1.78 3.0 130ms
8 -36.73635221147 -1.80 -1.80 2.0 116ms
9 -36.74154077164 -2.28 -2.12 2.0 104ms
10 -36.74161805910 -4.11 -2.35 1.0 95.1ms
11 -36.74239205602 -3.11 -2.65 2.0 125ms
12 -36.74221484890 + -3.75 -2.48 2.0 108ms
13 -36.74239080961 -3.75 -2.99 1.0 92.7ms
14 -36.74247345588 -4.08 -3.22 2.0 110ms
15 -36.74246220735 + -4.95 -3.35 2.0 129ms
16 -36.74247799634 -4.80 -3.62 2.0 101ms
17 -36.74242265117 + -4.26 -3.18 3.0 137ms
18 -36.74234901979 + -4.13 -3.01 3.0 146ms
19 -36.74247715760 -3.89 -3.78 3.0 132ms
20 -36.74248056082 -5.47 -4.39 2.0 205ms
21 -36.74248048149 + -7.10 -4.23 3.0 141ms
22 -36.74248052906 -7.32 -4.38 2.0 1.00s
23 -36.74248065303 -6.91 -4.78 1.0 93.0ms
24 -36.74248066066 -8.12 -4.74 3.0 136ms
25 -36.74248066652 -8.23 -4.98 2.0 118ms
26 -36.74248067009 -8.45 -5.08 2.0 100ms
27 -36.74248066698 + -8.51 -4.82 2.0 114ms
28 -36.74248067216 -8.29 -5.49 2.0 104ms
29 -36.74248067133 + -9.08 -5.44 3.0 132ms
30 -36.74248066905 + -8.64 -5.28 3.0 130ms
31 -36.74248067076 -8.77 -5.40 3.0 151ms
32 -36.74248067260 -8.74 -6.01 2.0 141ms
33 -36.74248067268 -10.12 -6.42 2.0 140ms
34 -36.74248067265 + -10.51 -6.11 3.0 160ms
35 -36.74248067268 -10.46 -6.82 2.0 119ms
36 -36.74248067268 -12.19 -7.13 2.0 126ms
37 -36.74248067268 -12.97 -7.38 2.0 110ms
38 -36.74248067268 -13.07 -7.46 3.0 109ms
39 -36.74248067268 + -Inf -7.59 1.0 92.4ms
40 -36.74248067268 -13.55 -7.73 2.0 131ms
41 -36.74248067268 + -13.15 -7.53 3.0 133ms
42 -36.74248067268 -13.19 -7.77 3.0 125ms
43 -36.74248067268 -14.15 -7.85 3.0 120ms
44 -36.74248067268 -13.85 -8.38 2.0 102ms
45 -36.74248067268 -14.15 -8.43 3.0 136ms
46 -36.74248067268 + -14.15 -8.97 2.0 115ms
47 -36.74248067268 -14.15 -8.91 3.0 152ms
48 -36.74248067268 -14.15 -9.28 2.0 108ms
49 -36.74248067268 + -Inf -9.07 3.0 131ms
50 -36.74248067268 + -Inf -9.68 2.0 108ms
51 -36.74248067268 + -13.85 -9.47 3.0 145ms
52 -36.74248067268 -14.15 -9.77 2.0 110ms
53 -36.74248067268 + -14.15 -10.05 2.0 129ms
54 -36.74248067268 + -14.15 -10.30 2.0 101ms
55 -36.74248067268 -14.15 -10.27 3.0 138ms
56 -36.74248067268 + -Inf -10.52 2.0 121ms
57 -36.74248067268 + -Inf -10.44 3.0 128ms
58 -36.74248067268 + -Inf -11.05 2.0 106ms
59 -36.74248067268 -13.85 -10.73 3.0 145ms
60 -36.74248067268 + -13.85 -11.44 3.0 133ms
61 -36.74248067268 + -Inf -10.74 4.0 164ms
62 -36.74248067268 -14.15 -11.93 4.0 163ms
63 -36.74248067268 + -14.15 -11.67 3.0 132ms
64 -36.74248067268 + -Inf -11.83 3.0 137ms
65 -36.74248067268 -13.85 -12.13 1.0 96.3ms
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 -36.73245619493 -0.88 12.0 920ms
2 -36.73974055435 -2.14 -1.36 1.0 611ms
3 -36.74185523301 -2.67 -2.10 3.0 148ms
4 -36.74212254955 -3.57 -2.07 6.0 128ms
5 -36.74237761804 -3.59 -2.55 1.0 91.0ms
6 -36.74245167953 -4.13 -2.82 2.0 107ms
7 -36.74247026182 -4.73 -2.97 3.0 117ms
8 -36.74247972378 -5.02 -3.59 1.0 92.8ms
9 -36.74247982705 -6.99 -3.47 4.0 154ms
10 -36.74248048793 -6.18 -3.82 2.0 98.4ms
11 -36.74248054989 -7.21 -4.22 2.0 104ms
12 -36.74248062785 -7.11 -4.60 5.0 126ms
13 -36.74248067151 -7.36 -4.96 3.0 133ms
14 -36.74248067211 -9.22 -5.07 2.0 104ms
15 -36.74248067254 -9.36 -5.41 3.0 113ms
16 -36.74248067266 -9.91 -5.80 2.0 107ms
17 -36.74248067268 -10.88 -6.00 3.0 138ms
18 -36.74248067268 -11.57 -6.26 2.0 104ms
19 -36.74248067268 -11.69 -6.60 3.0 113ms
20 -36.74248067268 -12.18 -6.94 2.0 104ms
21 -36.74248067268 -13.03 -7.19 4.0 127ms
22 -36.74248067268 -13.85 -7.57 5.0 115ms
23 -36.74248067268 -13.85 -7.57 2.0 224ms
24 -36.74248067268 -13.85 -8.12 1.0 94.9ms
25 -36.74248067268 + -Inf -8.27 2.0 1.02s
26 -36.74248067268 + -14.15 -8.62 2.0 102ms
27 -36.74248067268 -13.85 -9.09 1.0 95.0ms
28 -36.74248067268 + -13.85 -9.29 4.0 141ms
29 -36.74248067268 + -Inf -9.47 2.0 100ms
30 -36.74248067268 -14.15 -9.87 2.0 101ms
31 -36.74248067268 + -14.15 -10.08 3.0 134ms
32 -36.74248067268 + -Inf -10.44 2.0 102ms
33 -36.74248067268 + -Inf -10.70 3.0 154ms
34 -36.74248067268 -14.15 -10.91 1.0 117ms
35 -36.74248067268 + -13.67 -11.16 2.0 123ms
36 -36.74248067268 -13.85 -11.46 3.0 138ms
37 -36.74248067268 + -Inf -11.81 3.0 160ms
38 -36.74248067268 + -Inf -12.15 2.0 115ms
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))44.02449232835648The 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_min46.24421463062656This 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))4.723589650916229Since 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).