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 -36.73543013722 -0.88 12.0 1.33s
2 -36.70332878684 + -1.49 -1.52 1.0 323ms
3 +1.712125743151 + 1.58 -0.27 7.0 240ms
4 -35.84007036865 1.57 -0.91 6.0 194ms
5 -36.06765782101 -0.64 -1.12 4.0 150ms
6 -35.61965037397 + -0.35 -1.03 4.0 135ms
7 -36.73437953961 0.05 -1.79 3.0 132ms
8 -36.73871632519 -2.36 -1.96 2.0 101ms
9 -36.74131954528 -2.58 -2.21 2.0 129ms
10 -36.74201395864 -3.16 -2.26 2.0 120ms
11 -36.74236140435 -3.46 -2.47 2.0 101ms
12 -36.74234468274 + -4.78 -2.46 2.0 110ms
13 -36.74245699104 -3.95 -3.00 1.0 89.9ms
14 -36.74246672478 -5.01 -3.02 2.0 116ms
15 -36.74200819554 + -3.34 -2.72 2.0 113ms
16 -36.74238394689 -3.43 -2.92 3.0 128ms
17 -36.74239370274 -5.01 -3.06 2.0 108ms
18 -36.74223929647 + -3.81 -2.88 3.0 122ms
19 -36.74247662084 -3.62 -3.59 2.0 115ms
20 -36.74248045620 -5.42 -4.04 2.0 126ms
21 -36.74248054620 -7.05 -4.31 2.0 121ms
22 -36.74248045615 + -7.05 -4.22 2.0 111ms
23 -36.74248063780 -6.74 -4.58 2.0 94.4ms
24 -36.74248061746 + -7.69 -4.35 3.0 127ms
25 -36.74248067071 -7.27 -5.11 2.0 108ms
26 -36.74248067168 -9.01 -5.14 2.0 126ms
27 -36.74248066159 + -8.00 -5.00 3.0 111ms
28 -36.74248067245 -7.96 -5.61 2.0 102ms
29 -36.74248067266 -9.68 -6.06 3.0 127ms
30 -36.74248067246 + -9.70 -5.88 3.0 130ms
31 -36.74248067199 + -9.33 -5.58 3.0 132ms
32 -36.74248067262 -9.20 -6.08 2.0 113ms
33 -36.74248067251 + -9.97 -5.92 3.0 132ms
34 -36.74248067268 -9.77 -6.71 2.0 114ms
35 -36.74248067268 -13.15 -6.75 2.0 147ms
36 -36.74248067268 + -12.47 -6.64 2.0 113ms
37 -36.74248067268 -11.66 -7.24 2.0 102ms
38 -36.74248067268 -12.66 -7.46 3.0 126ms
39 -36.74248067268 + -11.82 -6.90 3.0 133ms
40 -36.74248067268 -11.81 -7.50 4.0 142ms
41 -36.74248067268 + -13.25 -7.32 2.0 117ms
42 -36.74248067268 -13.67 -7.62 2.0 111ms
43 -36.74248067268 -13.37 -7.85 2.0 102ms
44 -36.74248067268 -14.15 -8.43 2.0 97.4ms
45 -36.74248067268 + -14.15 -7.87 4.0 148ms
46 -36.74248067268 -14.15 -8.22 3.0 128ms
47 -36.74248067268 + -Inf -8.40 3.0 125ms
48 -36.74248067268 + -Inf -8.83 2.0 104ms
49 -36.74248067268 -13.67 -8.74 2.0 125ms
50 -36.74248067268 + -13.85 -9.33 2.0 104ms
51 -36.74248067268 + -Inf -8.65 4.0 158ms
52 -36.74248067268 -13.85 -9.55 3.0 139ms
53 -36.74248067268 + -14.15 -9.49 2.0 125ms
54 -36.74248067268 + -14.15 -9.54 2.0 101ms
55 -36.74248067268 + -Inf -10.10 2.0 108ms
56 -36.74248067268 + -Inf -10.08 2.0 121ms
57 -36.74248067268 + -Inf -9.78 3.0 131ms
58 -36.74248067268 + -Inf -10.21 3.0 120ms
59 -36.74248067268 + -Inf -9.87 3.0 131ms
60 -36.74248067268 + -Inf -10.89 3.0 128ms
61 -36.74248067268 + -Inf -11.46 2.0 125ms
62 -36.74248067268 + -13.85 -11.46 3.0 131ms
63 -36.74248067268 -13.85 -10.97 3.0 132ms
64 -36.74248067268 + -Inf -11.52 3.0 128ms
65 -36.74248067268 -14.15 -11.86 1.0 93.6ms
66 -36.74248067268 + -13.85 -11.98 3.0 109ms
67 -36.74248067268 -14.15 -12.42 2.0 125ms
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 -36.73119332927 -0.87 12.0 1.02s
2 -36.73891356259 -2.11 -1.37 1.0 1.05s
3 -36.73852135304 + -3.41 -1.81 3.0 114ms
4 -36.74187127839 -2.47 -1.97 2.0 243ms
5 -36.74184071166 + -4.51 -2.38 1.0 88.3ms
6 -36.74245566601 -3.21 -2.61 4.0 1.23s
7 -36.74234956278 + -3.97 -2.78 2.0 120ms
8 -36.74247849919 -3.89 -3.36 1.0 88.9ms
9 -36.74247741211 + -5.96 -3.29 3.0 128ms
10 -36.74248046065 -5.52 -3.75 1.0 89.4ms
11 -36.74248058679 -6.90 -4.09 2.0 95.2ms
12 -36.74248062746 -7.39 -4.53 2.0 123ms
13 -36.74248066693 -7.40 -4.68 1.0 91.8ms
14 -36.74248067184 -8.31 -5.18 1.0 90.3ms
15 -36.74248067178 + -10.22 -5.41 3.0 139ms
16 -36.74248067240 -9.21 -5.61 1.0 105ms
17 -36.74248067268 -9.57 -6.30 2.0 129ms
18 -36.74248067268 -11.23 -6.48 8.0 186ms
19 -36.74248067268 -12.63 -6.78 2.0 119ms
20 -36.74248067268 -12.17 -7.14 2.0 130ms
21 -36.74248067268 + -14.15 -7.28 2.0 138ms
22 -36.74248067268 -13.55 -7.67 1.0 91.7ms
23 -36.74248067268 -14.15 -7.83 3.0 104ms
24 -36.74248067268 + -Inf -8.08 2.0 101ms
25 -36.74248067268 + -Inf -8.41 2.0 96.8ms
26 -36.74248067268 -13.85 -8.83 3.0 129ms
27 -36.74248067268 + -13.85 -9.04 3.0 115ms
28 -36.74248067268 + -Inf -9.40 1.0 91.1ms
29 -36.74248067268 -13.85 -9.72 3.0 129ms
30 -36.74248067268 + -13.85 -10.03 1.0 91.0ms
31 -36.74248067268 + -Inf -10.19 3.0 128ms
32 -36.74248067268 -13.85 -10.57 2.0 105ms
33 -36.74248067268 + -Inf -10.83 3.0 101ms
34 -36.74248067268 + -13.85 -11.42 2.0 123ms
35 -36.74248067268 + -14.15 -11.64 5.0 124ms
36 -36.74248067268 -14.15 -11.83 2.0 96.2ms
37 -36.74248067268 -13.85 -12.14 2.0 123ms
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))44.02448897981028The 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_min46.244211113246095This 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))4.723602957123797Since 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).