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 -36.73205891477 -0.88 12.0 344ms
2 -36.64951214682 + -1.08 -1.45 1.0 88.7ms
3 +29.16564636111 + 1.82 -0.15 8.0 220ms
4 -36.56628370270 1.82 -1.07 7.0 215ms
5 -35.68939754137 + -0.06 -1.02 3.0 126ms
6 -36.15712893327 -0.33 -1.17 5.0 162ms
7 -36.72550489487 -0.25 -1.76 3.0 128ms
8 -36.73740172148 -1.92 -2.06 2.0 100ms
9 -36.73959954041 -2.66 -1.99 2.0 120ms
10 -36.74105126513 -2.84 -2.10 2.0 117ms
11 -36.74129865240 -3.61 -2.33 1.0 92.4ms
12 -36.74217964356 -3.06 -2.53 1.0 91.0ms
13 -36.74249555008 -3.50 -2.85 2.0 103ms
14 -36.74238357871 + -3.95 -2.80 2.0 125ms
15 -36.73552992005 + -2.16 -2.15 3.0 141ms
16 -36.74217358421 -2.18 -2.73 3.0 138ms
17 -36.73881664344 + -2.47 -2.23 3.0 134ms
18 -36.74251114611 -2.43 -3.45 3.0 134ms
19 -36.74251164651 -6.30 -3.58 3.0 139ms
20 -36.74251132519 + -6.49 -3.74 2.0 104ms
21 -36.74251444790 -5.51 -3.90 2.0 101ms
22 -36.74251437484 + -7.14 -3.95 2.0 114ms
23 -36.74251467747 -6.52 -4.39 2.0 113ms
24 -36.74251414721 + -6.28 -4.15 3.0 131ms
25 -36.74251476245 -6.21 -4.73 3.0 126ms
26 -36.74251476635 -8.41 -4.91 2.0 108ms
27 -36.74251476805 -8.77 -5.15 2.0 105ms
28 -36.74251476030 + -8.11 -4.95 3.0 135ms
29 -36.74251476325 -8.53 -4.96 2.0 118ms
30 -36.74251476913 -8.23 -5.25 2.0 118ms
31 -36.74251477211 -8.53 -5.53 2.0 106ms
32 -36.74251477299 -9.06 -6.01 3.0 119ms
33 -36.74251477274 + -9.60 -5.79 3.0 136ms
34 -36.74251477283 -10.04 -5.89 3.0 126ms
35 -36.74251477303 -9.72 -6.37 2.0 105ms
36 -36.74251477304 -11.01 -6.65 3.0 125ms
37 -36.74251477303 + -11.61 -6.57 2.0 126ms
38 -36.74251477304 -11.53 -6.99 2.0 105ms
39 -36.74251477304 + -12.87 -7.01 2.0 109ms
40 -36.74251477304 -11.99 -7.28 2.0 109ms
41 -36.74251477304 + -12.59 -7.18 2.0 106ms
42 -36.74251477304 + -11.88 -6.89 3.0 129ms
43 -36.74251477304 -11.96 -7.11 3.0 132ms
44 -36.74251477304 -12.33 -7.53 3.0 126ms
45 -36.74251477304 -13.25 -7.87 2.0 123ms
46 -36.74251477304 + -14.15 -7.80 3.0 123ms
47 -36.74251477304 -13.85 -8.07 2.0 108ms
48 -36.74251477304 + -14.15 -8.68 3.0 116ms
49 -36.74251477304 + -Inf -8.32 3.0 149ms
50 -36.74251477304 -13.85 -8.48 3.0 135ms
51 -36.74251477304 + -Inf -9.27 2.0 109ms
52 -36.74251477304 + -Inf -9.34 3.0 131ms
53 -36.74251477304 + -13.85 -9.74 2.0 106ms
54 -36.74251477304 -14.15 -9.16 3.0 141ms
55 -36.74251477304 -14.15 -9.54 3.0 133ms
56 -36.74251477304 + -Inf -9.68 3.0 132ms
57 -36.74251477304 + -Inf -9.99 2.0 105ms
58 -36.74251477304 + -14.15 -9.68 3.0 246ms
59 -36.74251477304 -14.15 -10.13 3.0 715ms
60 -36.74251477304 + -14.15 -10.30 2.0 106ms
61 -36.74251477304 + -Inf -10.41 2.0 124ms
62 -36.74251477304 + -14.15 -10.49 2.0 107ms
63 -36.74251477304 + -Inf -11.12 1.0 93.4ms
64 -36.74251477304 -14.15 -11.20 3.0 150ms
65 -36.74251477304 + -14.15 -11.65 2.0 114ms
66 -36.74251477304 -14.15 -11.44 3.0 138ms
67 -36.74251477304 + -14.15 -11.61 3.0 149ms
68 -36.74251477304 -13.85 -12.00 2.0 100mswhile 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 -36.73337055375 -0.88 11.0 323ms
2 -36.74028360221 -2.16 -1.37 1.0 87.3ms
3 -36.74208696636 -2.74 -2.12 3.0 130ms
4 -36.74208058581 + -5.20 -2.05 6.0 123ms
5 -36.74245948546 -3.42 -2.74 1.0 89.0ms
6 -36.74249449186 -4.46 -2.93 3.0 125ms
7 -36.74251092131 -4.78 -3.14 1.0 93.5ms
8 -36.74251390090 -5.53 -3.68 1.0 91.0ms
9 -36.74251450181 -6.22 -3.68 4.0 138ms
10 -36.74251465709 -6.81 -3.91 1.0 95.6ms
11 -36.74251476533 -6.97 -4.58 1.0 91.9ms
12 -36.74251477106 -8.24 -4.85 6.0 145ms
13 -36.74251477219 -8.94 -5.17 2.0 98.3ms
14 -36.74251477274 -9.26 -5.39 3.0 134ms
15 -36.74251477300 -9.58 -5.84 2.0 101ms
16 -36.74251477302 -10.68 -6.11 3.0 134ms
17 -36.74251477304 -10.85 -6.70 2.0 102ms
18 -36.74251477304 + -12.79 -6.84 4.0 144ms
19 -36.74251477304 -11.86 -7.19 2.0 111ms
20 -36.74251477304 -13.19 -7.46 3.0 134ms
21 -36.74251477304 -14.15 -7.66 2.0 98.1ms
22 -36.74251477304 + -Inf -8.00 1.0 96.2ms
23 -36.74251477304 + -13.85 -8.26 5.0 117ms
24 -36.74251477304 -13.85 -8.64 2.0 102ms
25 -36.74251477304 + -Inf -8.99 3.0 134ms
26 -36.74251477304 + -Inf -9.41 1.0 95.9ms
27 -36.74251477304 + -14.15 -9.23 3.0 134ms
28 -36.74251477304 + -14.15 -9.85 2.0 110ms
29 -36.74251477304 + -Inf -10.11 3.0 130ms
30 -36.74251477304 -13.85 -10.58 1.0 95.9ms
31 -36.74251477304 + -14.15 -10.83 2.0 129ms
32 -36.74251477304 -14.15 -11.09 2.0 110ms
33 -36.74251477304 + -Inf -11.43 1.0 96.0ms
34 -36.74251477304 + -14.15 -11.67 2.0 128ms
35 -36.74251477304 -14.15 -12.05 1.0 96.4msGiven 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))44.02448834509092The 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_min46.24421044652408This 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))4.723588772107268Since 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).