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 -36.73301524072 -0.88 11.0 359ms
2 -36.70343786808 + -1.53 -1.51 1.0 88.9ms
3 +1.965313475302 + 1.59 -0.27 8.0 218ms
4 -36.16207306313 1.58 -1.00 6.0 205ms
5 -36.34876611119 -0.73 -1.23 3.0 146ms
6 -36.09440328021 + -0.59 -1.16 5.0 159ms
7 -36.71836827922 -0.20 -1.69 3.0 137ms
8 -36.74108686745 -1.64 -2.20 2.0 107ms
9 -36.74172048848 -3.20 -2.18 2.0 131ms
10 -36.74158327441 + -3.86 -2.30 2.0 104ms
11 -36.74196980542 -3.41 -2.36 2.0 119ms
12 -36.74240417375 -3.36 -2.82 2.0 114ms
13 -36.74245738760 -4.27 -3.24 2.0 116ms
14 -36.74245346498 + -5.41 -2.99 3.0 148ms
15 -36.74236482828 + -4.05 -3.02 3.0 135ms
16 -36.74244962723 -4.07 -3.22 3.0 140ms
17 -36.74244849489 + -5.95 -3.28 1.0 94.0ms
18 -36.74097355697 + -2.83 -2.48 3.0 151ms
19 -36.74247465675 -2.82 -3.54 4.0 150ms
20 -36.74248054199 -5.23 -4.21 2.0 112ms
21 -36.74248062469 -7.08 -4.35 2.0 127ms
22 -36.74248065016 -7.59 -4.58 1.0 94.0ms
23 -36.74248058048 + -7.16 -4.26 2.0 126ms
24 -36.74248066906 -7.05 -4.91 2.0 121ms
25 -36.74248065223 + -7.77 -4.80 2.0 133ms
26 -36.74248066878 -7.78 -5.03 1.0 93.6ms
27 -36.74248066621 + -8.59 -4.92 3.0 131ms
28 -36.74248067207 -8.23 -5.49 2.0 104ms
29 -36.74248066943 + -8.58 -5.26 3.0 140ms
30 -36.74248067241 -8.53 -5.83 3.0 128ms
31 -36.74248066654 + -8.23 -5.18 4.0 157ms
32 -36.74248067258 -8.22 -5.98 3.0 144ms
33 -36.74248067266 -10.07 -6.20 2.0 109ms
34 -36.74248067268 -10.76 -6.56 2.0 107ms
35 -36.74248067266 + -10.84 -6.19 3.0 134ms
36 -36.74248067268 -10.78 -6.85 2.0 126ms
37 -36.74248067268 + -11.99 -6.62 3.0 135ms
38 -36.74248067268 -11.70 -7.24 2.0 126ms
39 -36.74248067268 + -12.38 -6.96 3.0 137ms
40 -36.74248067268 -12.33 -7.36 2.0 124ms
41 -36.74248067268 -13.07 -7.98 2.0 104ms
42 -36.74248067268 -14.15 -8.19 3.0 145ms
43 -36.74248067268 + -Inf -7.97 3.0 135ms
44 -36.74248067268 + -13.07 -7.58 4.0 156ms
45 -36.74248067268 -13.07 -8.17 3.0 144ms
46 -36.74248067268 + -Inf -8.49 2.0 109ms
47 -36.74248067268 -13.85 -8.64 2.0 132ms
48 -36.74248067268 + -13.85 -9.03 2.0 108ms
49 -36.74248067268 + -Inf -9.31 1.0 98.7ms
50 -36.74248067268 + -14.15 -9.50 2.0 127ms
51 -36.74248067268 -13.67 -9.25 2.0 124ms
52 -36.74248067268 + -Inf -9.76 2.0 115ms
53 -36.74248067268 + -13.85 -9.81 2.0 106ms
54 -36.74248067268 -14.15 -9.93 1.0 98.8ms
55 -36.74248067268 + -14.15 -9.76 3.0 145ms
56 -36.74248067268 + -Inf -9.65 3.0 139ms
57 -36.74248067268 + -Inf -10.72 3.0 128ms
58 -36.74248067268 + -Inf -10.97 2.0 132ms
59 -36.74248067268 + -Inf -11.41 2.0 103ms
60 -36.74248067268 + -Inf -11.01 3.0 144ms
61 -36.74248067268 + -Inf -11.48 3.0 126ms
62 -36.74248067268 -13.85 -11.63 3.0 130ms
63 -36.74248067268 + -13.85 -11.89 2.0 106ms
64 -36.74248067268 -14.15 -11.81 2.0 127ms
65 -36.74248067268 -14.15 -11.70 2.0 125ms
66 -36.74248067268 + -Inf -11.86 3.0 136ms
67 -36.74248067268 + -13.67 -12.65 1.0 99.4mswhile 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 -36.73437726849 -0.88 11.0 355ms
2 -36.74013516049 -2.24 -1.36 1.0 92.4ms
3 -36.74042018915 -3.55 -1.75 3.0 119ms
4 -36.74220615994 -2.75 -2.24 1.0 97.4ms
5 -36.74233761669 -3.88 -2.58 3.0 127ms
6 -36.74243475132 -4.01 -2.58 3.0 139ms
7 -36.74245520640 -4.69 -2.83 1.0 94.0ms
8 -36.74247560891 -4.69 -3.15 2.0 107ms
9 -36.74247809057 -5.61 -3.32 1.0 94.4ms
10 -36.74248036857 -5.64 -3.91 4.0 112ms
11 -36.74248055863 -6.72 -4.30 5.0 132ms
12 -36.74248063817 -7.10 -4.63 3.0 140ms
13 -36.74248067096 -7.48 -5.04 2.0 112ms
14 -36.74248067129 -9.48 -5.36 2.0 134ms
15 -36.74248067264 -8.87 -5.88 2.0 113ms
16 -36.74248067266 -10.68 -6.10 3.0 145ms
17 -36.74248067268 -10.73 -6.54 2.0 112ms
18 -36.74248067268 -11.48 -6.88 4.0 129ms
19 -36.74248067268 -12.20 -7.18 2.0 136ms
20 -36.74248067268 -14.15 -7.44 2.0 103ms
21 -36.74248067268 -13.19 -7.74 2.0 118ms
22 -36.74248067268 + -Inf -8.10 6.0 140ms
23 -36.74248067268 + -13.85 -8.30 2.0 136ms
24 -36.74248067268 -13.85 -8.62 1.0 97.1ms
25 -36.74248067268 + -13.85 -8.70 2.0 128ms
26 -36.74248067268 -13.85 -9.14 1.0 97.0ms
27 -36.74248067268 + -14.15 -9.64 2.0 136ms
28 -36.74248067268 -14.15 -10.07 2.0 132ms
29 -36.74248067268 + -14.15 -10.22 2.0 118ms
30 -36.74248067268 + -Inf -10.69 2.0 105ms
31 -36.74248067268 -13.85 -10.90 3.0 143ms
32 -36.74248067268 + -13.67 -11.21 1.0 97.1ms
33 -36.74248067268 + -14.15 -11.35 4.0 149ms
34 -36.74248067268 -13.85 -11.83 2.0 113ms
35 -36.74248067268 + -Inf -12.15 2.0 251msGiven 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))44.02448898021258The 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_min46.24421111366868This 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))4.7235817657097305Since 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).