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 -36.73404414421 -0.88 11.0 354ms
2 -36.67597664890 + -1.24 -1.47 1.0 92.0ms
3 +20.99499526877 + 1.76 -0.18 7.0 216ms
4 -36.00924729617 1.76 -0.93 6.0 221ms
5 -35.45218695340 + -0.25 -0.99 4.0 175ms
6 -35.63768230182 -0.73 -1.03 4.0 165ms
7 -36.73564802716 0.04 -1.80 3.0 136ms
8 -36.73837828570 -2.56 -1.91 2.0 116ms
9 -36.73708142925 + -2.89 -2.03 2.0 114ms
10 -36.74207902392 -2.30 -2.21 1.0 95.4ms
11 -36.74213255168 -4.27 -2.29 1.0 91.5ms
12 -36.74219379585 -4.21 -2.39 2.0 114ms
13 -36.74245347953 -3.59 -2.74 1.0 97.6ms
14 -36.74233757005 + -3.94 -2.78 2.0 129ms
15 -36.73856270185 + -2.42 -2.26 3.0 154ms
16 -36.74233661347 -2.42 -2.81 3.0 139ms
17 -36.74237260939 -4.44 -2.97 2.0 99.5ms
18 -36.73467576266 + -2.11 -2.12 4.0 165ms
19 -36.74251353933 -2.11 -3.63 3.0 149ms
20 -36.74251394794 -6.39 -3.61 2.0 131ms
21 -36.74251452137 -6.24 -4.10 1.0 92.8ms
22 -36.74251457664 -7.26 -4.18 2.0 113ms
23 -36.74251468280 -6.97 -4.51 1.0 99.0ms
24 -36.74251476124 -7.11 -4.58 2.0 127ms
25 -36.74251476724 -8.22 -4.94 1.0 97.1ms
26 -36.74251477004 -8.55 -4.94 2.0 117ms
27 -36.74251477196 -8.72 -5.29 1.0 92.7ms
28 -36.74251476653 + -8.27 -5.12 3.0 140ms
29 -36.74251477243 -8.23 -5.50 3.0 131ms
30 -36.74251477138 + -8.98 -5.43 2.0 118ms
31 -36.74251477223 -9.07 -5.59 3.0 132ms
32 -36.74251477283 -9.23 -5.64 2.0 138ms
33 -36.74251477296 -9.87 -6.07 2.0 109ms
34 -36.74251477300 -10.51 -6.22 1.0 97.4ms
35 -36.74251477303 -10.44 -6.60 2.0 132ms
36 -36.74251477304 -11.30 -6.88 2.0 109ms
37 -36.74251477304 -12.22 -7.20 2.0 132ms
38 -36.74251477304 + -13.67 -7.16 2.0 119ms
39 -36.74251477304 -12.94 -7.35 2.0 109ms
40 -36.74251477304 -13.19 -7.54 1.0 98.4ms
41 -36.74251477304 + -14.15 -7.71 1.0 101ms
42 -36.74251477304 + -14.15 -7.80 2.0 109ms
43 -36.74251477304 + -14.15 -7.72 2.0 114ms
44 -36.74251477304 -13.85 -8.04 2.0 123ms
45 -36.74251477304 -13.85 -8.55 2.0 128ms
46 -36.74251477304 -13.85 -8.91 2.0 132ms
47 -36.74251477304 + -13.67 -8.04 4.0 167ms
48 -36.74251477304 -13.85 -8.91 4.0 157ms
49 -36.74251477304 + -Inf -9.14 1.0 94.2ms
50 -36.74251477304 + -Inf -9.52 3.0 128ms
51 -36.74251477304 + -Inf -9.69 2.0 133ms
52 -36.74251477304 + -Inf -9.89 1.0 93.5ms
53 -36.74251477304 + -14.15 -9.97 2.0 106ms
54 -36.74251477304 -14.15 -10.31 1.0 92.8ms
55 -36.74251477304 + -14.15 -9.73 4.0 154ms
56 -36.74251477304 -14.15 -10.39 3.0 140ms
57 -36.74251477304 + -13.85 -10.64 2.0 124ms
58 -36.74251477304 + -Inf -11.22 1.0 103ms
59 -36.74251477304 -13.85 -10.55 4.0 178ms
60 -36.74251477304 + -Inf -11.23 3.0 150ms
61 -36.74251477304 + -Inf -11.69 2.0 110ms
62 -36.74251477304 + -Inf -11.83 2.0 133ms
63 -36.74251477304 + -Inf -11.66 2.0 116ms
64 -36.74251477304 + -13.85 -12.02 2.0 102mswhile 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 -36.73174177943 -0.88 11.0 359ms
2 -36.73979149242 -2.09 -1.36 1.0 88.2ms
3 -36.73894763856 + -3.07 -1.61 3.0 119ms
4 -36.74224053186 -2.48 -2.25 1.0 93.1ms
5 -36.74241027244 -3.77 -2.49 3.0 135ms
6 -36.74243710889 -4.57 -2.41 5.0 115ms
7 -36.74249397482 -4.25 -2.92 1.0 94.4ms
8 -36.74251252781 -4.73 -3.33 1.0 90.6ms
9 -36.74251390386 -5.86 -3.47 4.0 129ms
10 -36.74251465584 -6.12 -3.96 1.0 116ms
11 -36.74251472360 -7.17 -4.40 3.0 118ms
12 -36.74251472688 -8.48 -4.59 3.0 138ms
13 -36.74251476044 -7.47 -4.94 3.0 118ms
14 -36.74251477283 -7.91 -5.64 1.0 97.5ms
15 -36.74251477262 + -9.70 -5.59 4.0 148ms
16 -36.74251477302 -9.41 -5.96 2.0 238ms
17 -36.74251477303 -10.92 -6.25 3.0 783ms
18 -36.74251477304 -11.06 -6.62 1.0 96.5ms
19 -36.74251477304 -12.70 -6.96 2.0 130ms
20 -36.74251477304 -12.75 -7.28 3.0 114ms
21 -36.74251477304 -14.15 -7.52 4.0 140ms
22 -36.74251477304 + -Inf -7.78 2.0 122ms
23 -36.74251477304 + -Inf -7.98 2.0 119ms
24 -36.74251477304 + -Inf -8.22 2.0 120ms
25 -36.74251477304 + -14.15 -8.57 2.0 136ms
26 -36.74251477304 -14.15 -9.00 3.0 158ms
27 -36.74251477304 + -Inf -9.13 3.0 117ms
28 -36.74251477304 + -13.85 -9.61 1.0 94.0ms
29 -36.74251477304 + -Inf -9.97 4.0 140ms
30 -36.74251477304 + -Inf -10.16 5.0 119ms
31 -36.74251477304 -13.85 -10.31 2.0 134ms
32 -36.74251477304 + -Inf -11.04 2.0 108ms
33 -36.74251477304 + -Inf -11.10 3.0 137ms
34 -36.74251477304 + -14.15 -11.49 1.0 93.3ms
35 -36.74251477304 -14.15 -11.62 3.0 139ms
36 -36.74251477304 + -Inf -11.91 1.0 93.6ms
37 -36.74251477304 + -13.85 -12.14 3.0 120msGiven 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))44.024487774020315The 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_min46.24420984666This 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))4.723595629002867Since 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).