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 -36.73347947989 -0.88 10.0 1.29s
2 -36.64293513790 + -1.04 -1.43 1.0 231ms
3 +33.18892092141 + 1.84 -0.14 8.0 244ms
4 -36.67264026992 1.84 -1.13 7.0 198ms
5 -36.44338824298 + -0.64 -1.28 3.0 118ms
6 -35.57762065118 + -0.06 -1.03 3.0 146ms
7 -36.69797708740 0.05 -1.61 3.0 118ms
8 -36.73833079822 -1.39 -2.06 2.0 96.4ms
9 -36.74090670537 -2.59 -2.18 2.0 117ms
10 -36.74189222094 -3.01 -2.24 1.0 83.1ms
11 -36.74152486506 + -3.43 -2.29 2.0 98.6ms
12 -36.74235158549 -3.08 -2.71 1.0 85.0ms
13 -36.74219348074 + -3.80 -2.73 3.0 122ms
14 -36.74244190006 -3.60 -3.02 2.0 93.2ms
15 -36.74231652901 + -3.90 -2.85 2.0 120ms
16 -36.74244299896 -3.90 -3.14 2.0 101ms
17 -36.74246708008 -4.62 -3.37 2.0 102ms
18 -36.74247162769 -5.34 -3.44 2.0 96.8ms
19 -36.74247738880 -5.24 -3.48 2.0 108ms
20 -36.74239556849 + -4.09 -3.10 3.0 123ms
21 -36.74247828903 -4.08 -3.81 3.0 126ms
22 -36.74247941044 -5.95 -3.81 2.0 119ms
23 -36.74248031221 -6.04 -4.12 1.0 91.0ms
24 -36.74248066055 -6.46 -4.69 1.0 85.5ms
25 -36.74248063785 + -7.64 -4.52 3.0 136ms
26 -36.74248048446 + -6.81 -4.41 3.0 127ms
27 -36.74248066014 -6.76 -4.97 3.0 124ms
28 -36.74248066886 -8.06 -5.19 3.0 110ms
29 -36.74248067217 -8.48 -5.51 2.0 115ms
30 -36.74248067237 -9.69 -5.58 2.0 95.8ms
31 -36.74248067263 -9.59 -5.89 1.0 85.3ms
32 -36.74248066417 + -8.07 -5.05 3.0 142ms
33 -36.74248067263 -8.07 -6.06 3.0 132ms
34 -36.74248067236 + -9.58 -5.71 3.0 128ms
35 -36.74248067264 -9.56 -6.22 2.0 107ms
36 -36.74248067268 -10.42 -6.83 2.0 102ms
37 -36.74248067267 + -10.83 -6.37 3.0 138ms
38 -36.74248067268 -10.97 -6.69 3.0 128ms
39 -36.74248067265 + -10.56 -6.29 3.0 121ms
40 -36.74248067268 -10.49 -7.30 3.0 126ms
41 -36.74248067268 -13.45 -7.41 2.0 115ms
42 -36.74248067268 -13.37 -7.59 2.0 184ms
43 -36.74248067268 -13.25 -7.95 2.0 90.3ms
44 -36.74248067268 + -14.15 -8.15 3.0 872ms
45 -36.74248067268 + -Inf -8.40 2.0 98.5ms
46 -36.74248067268 + -Inf -8.30 3.0 117ms
47 -36.74248067268 + -Inf -8.67 2.0 98.2ms
48 -36.74248067268 + -Inf -8.51 3.0 125ms
49 -36.74248067268 -13.67 -8.95 2.0 111ms
50 -36.74248067268 + -13.85 -9.17 3.0 111ms
51 -36.74248067268 + -Inf -9.26 2.0 118ms
52 -36.74248067268 + -Inf -9.38 3.0 113ms
53 -36.74248067268 -13.85 -9.42 3.0 109ms
54 -36.74248067268 + -13.85 -10.02 2.0 99.0ms
55 -36.74248067268 -14.15 -9.94 2.0 122ms
56 -36.74248067268 + -13.85 -10.48 2.0 118ms
57 -36.74248067268 -13.85 -10.72 3.0 140ms
58 -36.74248067268 + -14.15 -10.81 2.0 104ms
59 -36.74248067268 + -13.85 -10.92 2.0 98.6ms
60 -36.74248067268 -13.55 -10.85 3.0 117ms
61 -36.74248067268 + -13.85 -11.16 1.0 84.9ms
62 -36.74248067268 + -Inf -11.33 3.0 124ms
63 -36.74248067268 + -Inf -11.74 2.0 92.2ms
64 -36.74248067268 + -Inf -11.68 3.0 126ms
65 -36.74248067268 + -Inf -11.86 2.0 98.5ms
66 -36.74248067268 + -Inf -11.94 3.0 107ms
67 -36.74248067268 + -Inf -11.77 2.0 107ms
68 -36.74248067268 + -Inf -12.84 2.0 109ms
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 -36.73446238208 -0.88 11.0 934ms
2 -36.74070585334 -2.20 -1.37 1.0 599ms
3 -36.74193639665 -2.91 -2.17 2.0 124ms
4 -36.74184243098 + -4.03 -1.98 4.0 121ms
5 -36.74244678712 -3.22 -2.72 1.0 82.3ms
6 -36.74244942122 -5.58 -2.73 3.0 132ms
7 -36.74247123645 -4.66 -2.91 1.0 83.0ms
8 -36.74248006784 -5.05 -3.69 1.0 83.5ms
9 -36.74248029365 -6.65 -3.63 3.0 129ms
10 -36.74248061909 -6.49 -3.90 1.0 84.4ms
11 -36.74248065726 -7.42 -4.47 1.0 84.6ms
12 -36.74248066623 -8.05 -4.93 4.0 112ms
13 -36.74248067203 -8.24 -5.18 2.0 119ms
14 -36.74248067167 + -9.44 -5.40 2.0 92.1ms
15 -36.74248067262 -9.02 -5.74 2.0 97.2ms
16 -36.74248067267 -10.31 -6.02 2.0 117ms
17 -36.74248067268 -11.01 -6.26 1.0 90.3ms
18 -36.74248067268 -12.02 -6.48 2.0 103ms
19 -36.74248067268 -11.97 -7.06 1.0 89.7ms
20 -36.74248067268 -13.07 -7.32 4.0 137ms
21 -36.74248067268 + -14.15 -7.38 2.0 90.8ms
22 -36.74248067268 -13.67 -7.72 1.0 90.1ms
23 -36.74248067268 + -Inf -8.25 3.0 105ms
24 -36.74248067268 + -Inf -8.39 2.0 127ms
25 -36.74248067268 + -Inf -8.68 1.0 87.8ms
26 -36.74248067268 -14.15 -8.75 3.0 118ms
27 -36.74248067268 + -14.15 -9.20 1.0 89.3ms
28 -36.74248067268 + -Inf -9.50 3.0 112ms
29 -36.74248067268 + -Inf -9.85 2.0 120ms
30 -36.74248067268 + -Inf -10.14 2.0 91.3ms
31 -36.74248067268 + -Inf -10.65 2.0 121ms
32 -36.74248067268 + -Inf -10.88 2.0 96.6ms
33 -36.74248067268 -14.15 -11.27 2.0 120ms
34 -36.74248067268 + -Inf -11.62 2.0 103ms
35 -36.74248067268 + -14.15 -11.91 2.0 115ms
36 -36.74248067268 -13.85 -12.34 1.0 89.6ms
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).δVρ
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))44.02448902021249The 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_min46.24421115568539This 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))4.7235909684646495Since 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).