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.oncvpsp3.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 -36.73303395601 -0.88 11.0 408ms
2 -36.63513273746 + -1.01 -1.45 1.0 97.7ms
3 +34.37190970911 + 1.85 -0.14 22.0 352ms
4 -36.44543980441 1.85 -1.11 8.0 268ms
5 -36.21769280663 + -0.64 -1.18 4.0 164ms
6 -36.23500221181 -1.76 -1.21 4.0 164ms
7 -36.73330560709 -0.30 -1.86 3.0 149ms
8 -36.73942727430 -2.21 -2.15 2.0 138ms
9 -36.73821824025 + -2.92 -1.95 3.0 151ms
10 -36.74142642011 -2.49 -2.25 2.0 136ms
11 -36.74133932667 + -4.06 -2.51 2.0 110ms
12 -36.74197114922 -3.20 -2.65 1.0 98.0ms
13 -36.74207505550 -3.98 -3.31 1.0 101ms
14 -36.74206301749 + -4.92 -3.26 3.0 158ms
15 -36.74192218058 + -3.85 -2.93 2.0 124ms
16 -36.74193894979 -4.78 -2.94 4.0 172ms
17 -36.74202380816 -4.07 -3.12 3.0 139ms
18 -36.74192768893 + -4.02 -2.97 3.0 175ms
19 -36.74207959410 -3.82 -3.73 3.0 157ms
20 -36.74208341701 -5.42 -3.97 3.0 150ms
21 -36.74208361547 -6.70 -4.15 1.0 98.4ms
22 -36.74208362381 -8.08 -4.17 2.0 134ms
23 -36.74208374839 -6.90 -4.71 1.0 102ms
24 -36.74208375850 -8.00 -4.92 3.0 160ms
25 -36.74208376130 -8.55 -4.83 2.0 131ms
26 -36.74208376338 -8.68 -5.19 1.0 116ms
27 -36.74208376460 -8.92 -5.69 1.0 98.1ms
28 -36.74208376380 + -9.10 -5.50 2.0 141ms
29 -36.74208376476 -9.02 -6.11 5.0 147ms
30 -36.74208376375 + -9.00 -5.56 4.0 180ms
31 -36.74208376468 -9.03 -6.01 4.0 169ms
32 -36.74208376474 -10.19 -6.20 3.0 141ms
33 -36.74208376479 -10.37 -6.73 2.0 127ms
34 -36.74208376478 + -11.50 -6.64 3.0 151ms
35 -36.74208376479 -11.38 -7.00 2.0 122ms
36 -36.74208376479 -12.20 -7.49 2.0 106ms
37 -36.74208376479 + -13.67 -7.41 4.0 158ms
38 -36.74208376479 -13.67 -7.58 2.0 135ms
39 -36.74208376479 + -Inf -7.74 2.0 121ms
40 -36.74208376479 -13.55 -8.35 1.0 105ms
41 -36.74208376479 + -14.15 -8.29 3.0 193ms
42 -36.74208376479 + -14.15 -8.43 2.0 119ms
43 -36.74208376479 -14.15 -8.81 2.0 110ms
44 -36.74208376479 + -14.15 -8.32 4.0 167ms
45 -36.74208376479 + -Inf -9.08 4.0 160ms
46 -36.74208376479 + -Inf -9.01 3.0 141ms
47 -36.74208376479 -13.85 -9.19 3.0 131ms
48 -36.74208376479 + -13.85 -9.75 1.0 103ms
49 -36.74208376479 -14.15 -10.01 3.0 170ms
50 -36.74208376479 -14.15 -9.60 4.0 161ms
51 -36.74208376479 + -13.85 -10.29 3.0 154ms
52 -36.74208376479 + -14.15 -10.44 3.0 152ms
53 -36.74208376479 -14.15 -10.43 3.0 129ms
54 -36.74208376479 -13.85 -11.21 1.0 104ms
55 -36.74208376479 + -13.85 -11.08 3.0 149ms
56 -36.74208376479 -14.15 -11.10 3.0 218ms
57 -36.74208376479 + -Inf -11.40 3.0 145ms
58 -36.74208376479 + -14.15 -11.73 3.0 131ms
59 -36.74208376479 -13.85 -11.87 2.0 142ms
60 -36.74208376479 + -13.85 -11.94 2.0 122ms
61 -36.74208376479 + -Inf -12.07 2.0 123mswhile 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 -36.73400539981 -0.88 12.0 402ms
2 -36.73998293407 -2.22 -1.36 1.0 96.6ms
3 -36.73762451251 + -2.63 -1.56 4.0 140ms
4 -36.74190354996 -2.37 -2.29 1.0 97.4ms
5 -36.74193419304 -4.51 -2.50 9.0 179ms
6 -36.74206317447 -3.89 -2.62 1.0 95.0ms
7 -36.74206015872 + -5.52 -3.08 1.0 102ms
8 -36.74207632176 -4.79 -3.38 4.0 124ms
9 -36.74208325537 -5.16 -3.57 2.0 166ms
10 -36.74208358234 -6.49 -4.12 1.0 103ms
11 -36.74208339031 + -6.72 -4.18 3.0 124ms
12 -36.74208375191 -6.44 -4.70 2.0 112ms
13 -36.74208376448 -7.90 -5.08 8.0 190ms
14 -36.74208376352 + -9.02 -5.15 3.0 149ms
15 -36.74208376450 -9.01 -5.55 1.0 105ms
16 -36.74208376478 -9.56 -6.13 2.0 117ms
17 -36.74208376478 -12.21 -6.42 4.0 179ms
18 -36.74208376479 -11.11 -6.61 3.0 117ms
19 -36.74208376479 + -12.11 -6.86 7.0 145ms
20 -36.74208376479 -11.99 -7.25 2.0 140ms
21 -36.74208376479 + -Inf -7.65 2.0 109ms
22 -36.74208376479 -13.67 -7.94 3.0 148ms
23 -36.74208376479 + -Inf -8.26 2.0 110ms
24 -36.74208376479 + -Inf -8.70 5.0 132ms
25 -36.74208376479 + -14.15 -8.78 3.0 184ms
26 -36.74208376479 + -Inf -9.06 2.0 112ms
27 -36.74208376479 + -Inf -9.81 3.0 127ms
28 -36.74208376479 + -Inf -9.64 8.0 188ms
29 -36.74208376479 -14.15 -9.81 2.0 111ms
30 -36.74208376479 + -Inf -10.31 1.0 104ms
31 -36.74208376479 + -13.85 -10.82 2.0 143ms
32 -36.74208376479 -13.85 -10.90 6.0 158ms
33 -36.74208376479 + -13.85 -11.33 2.0 129ms
34 -36.74208376479 -14.15 -11.50 3.0 149ms
35 -36.74208376479 + -Inf -11.72 2.0 112ms
36 -36.74208376479 + -Inf -12.07 2.0 108msGiven 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 - 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))44.024488533655614The 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_min46.24421064459624This 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))4.723587951332476Since 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).