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 -36.73301774187 -0.88 11.0 1.49s
2 -36.72131360111 + -1.93 -1.59 1.0 340ms
3 -12.72540395414 + 1.38 -0.38 5.0 220ms
4 -36.50839913486 1.38 -1.06 5.0 175ms
5 -36.66765952307 -0.80 -1.45 3.0 138ms
6 -36.38820765917 + -0.55 -1.29 3.0 142ms
7 -36.70096840398 -0.50 -1.71 3.0 128ms
8 -36.74154064261 -1.39 -2.22 2.0 102ms
9 -36.74100359741 + -3.27 -2.20 3.0 136ms
10 -36.74042266181 + -3.24 -2.06 2.0 116ms
11 -36.74203561811 -2.79 -2.54 2.0 109ms
12 -36.74238676694 -3.45 -2.86 2.0 108ms
13 -36.74247285658 -4.07 -3.32 2.0 115ms
14 -36.74246619670 + -5.18 -3.14 2.0 129ms
15 -36.74181251357 + -3.18 -2.66 4.0 144ms
16 -36.74245803876 -3.19 -3.28 3.0 145ms
17 -36.74245696399 + -5.97 -3.19 2.0 118ms
18 -36.74246224481 -5.28 -3.41 2.0 110ms
19 -36.74247680314 -4.84 -3.59 2.0 103ms
20 -36.74248056636 -5.42 -4.29 2.0 112ms
21 -36.74248031273 + -6.60 -4.11 4.0 148ms
22 -36.74248055023 -6.62 -4.23 2.0 111ms
23 -36.74248056960 -7.71 -4.39 2.0 104ms
24 -36.74248066174 -7.04 -4.75 2.0 104ms
25 -36.74248066472 -8.53 -4.95 1.0 90.3ms
26 -36.74248067116 -8.19 -5.10 3.0 119ms
27 -36.74248066762 + -8.45 -5.15 2.0 123ms
28 -36.74248066727 + -9.46 -5.18 3.0 129ms
29 -36.74248067244 -8.29 -5.73 2.0 100ms
30 -36.74248066998 + -8.61 -5.34 4.0 152ms
31 -36.74248067198 -8.70 -5.61 3.0 130ms
32 -36.74248067259 -9.21 -5.95 2.0 111ms
33 -36.74248067256 + -10.53 -5.85 2.0 115ms
34 -36.74248067267 -9.98 -6.26 1.0 95.8ms
35 -36.74248067267 -11.68 -6.45 3.0 121ms
36 -36.74248067267 -11.55 -6.21 2.0 129ms
37 -36.74248067267 + -11.37 -6.44 2.0 106ms
38 -36.74248067268 -10.96 -6.75 1.0 90.9ms
39 -36.74248067268 -11.86 -6.77 2.0 113ms
40 -36.74248067268 + -11.74 -6.72 1.0 90.3ms
41 -36.74248067268 -11.73 -6.99 2.0 111ms
42 -36.74248067268 + -11.92 -6.89 3.0 126ms
43 -36.74248067268 -11.69 -7.46 2.0 112ms
44 -36.74248067268 -13.07 -7.59 2.0 124ms
45 -36.74248067268 + -13.55 -7.67 2.0 111ms
46 -36.74248067268 -13.37 -8.04 1.0 91.1ms
47 -36.74248067268 + -13.67 -7.78 3.0 139ms
48 -36.74248067268 -13.67 -8.33 3.0 114ms
49 -36.74248067268 + -13.55 -7.88 3.0 136ms
50 -36.74248067268 -13.85 -8.60 3.0 129ms
51 -36.74248067268 -13.85 -8.66 2.0 122ms
52 -36.74248067268 + -13.85 -8.79 1.0 90.4ms
53 -36.74248067268 -14.15 -8.62 3.0 133ms
54 -36.74248067268 + -Inf -8.48 3.0 135ms
55 -36.74248067268 -13.85 -9.16 3.0 246ms
56 -36.74248067268 + -13.85 -9.15 2.0 123ms
57 -36.74248067268 + -Inf -9.39 2.0 1.18s
58 -36.74248067268 + -13.85 -9.71 2.0 105ms
59 -36.74248067268 -13.85 -10.24 2.0 100ms
60 -36.74248067268 + -Inf -10.55 3.0 132ms
61 -36.74248067268 -13.85 -10.34 3.0 133ms
62 -36.74248067268 + -13.85 -10.50 3.0 138ms
63 -36.74248067268 + -13.85 -10.86 1.0 89.0ms
64 -36.74248067268 -13.55 -11.03 3.0 122ms
65 -36.74248067268 + -14.15 -11.36 2.0 98.4ms
66 -36.74248067268 + -14.15 -11.56 2.0 122ms
67 -36.74248067268 -13.85 -11.73 1.0 91.3ms
68 -36.74248067268 + -14.15 -11.28 3.0 138ms
69 -36.74248067268 + -Inf -11.45 4.0 165ms
70 -36.74248067268 + -14.15 -11.87 3.0 146ms
71 -36.74248067268 + -14.15 -11.68 2.0 134ms
72 -36.74248067268 -13.85 -12.35 3.0 135ms
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 -36.73251286090 -0.88 12.0 948ms
2 -36.73977290946 -2.14 -1.36 1.0 1.11s
3 -36.73998734378 -3.67 -1.74 4.0 123ms
4 -36.74210867048 -2.67 -2.15 1.0 88.3ms
5 -36.74232247931 -3.67 -2.66 5.0 109ms
6 -36.74243746626 -3.94 -2.54 3.0 137ms
7 -36.74247098379 -4.47 -3.01 1.0 119ms
8 -36.74247748215 -5.19 -3.13 1.0 91.4ms
9 -36.74247932856 -5.73 -3.27 1.0 91.2ms
10 -36.74248027711 -6.02 -3.97 1.0 91.2ms
11 -36.74248061723 -6.47 -4.16 3.0 138ms
12 -36.74248062777 -7.98 -4.18 1.0 104ms
13 -36.74248066428 -7.44 -4.39 2.0 98.4ms
14 -36.74248066547 -8.93 -4.58 2.0 108ms
15 -36.74248066653 -8.97 -4.68 1.0 93.3ms
16 -36.74248067235 -8.23 -5.37 1.0 98.1ms
17 -36.74248067259 -9.62 -5.64 4.0 141ms
18 -36.74248067268 -10.07 -5.87 4.0 113ms
19 -36.74248067246 + -9.66 -5.48 3.0 138ms
20 -36.74248067268 -9.66 -6.18 2.0 117ms
21 -36.74248067268 -11.34 -6.69 2.0 122ms
22 -36.74248067268 + -13.85 -6.80 2.0 126ms
23 -36.74248067268 -12.59 -7.20 4.0 114ms
24 -36.74248067268 -13.67 -7.47 2.0 102ms
25 -36.74248067268 + -Inf -7.76 2.0 130ms
26 -36.74248067268 + -Inf -7.94 1.0 92.3ms
27 -36.74248067268 -14.15 -8.21 2.0 103ms
28 -36.74248067268 + -14.15 -8.72 2.0 106ms
29 -36.74248067268 + -Inf -8.71 4.0 137ms
30 -36.74248067268 + -Inf -8.95 1.0 97.0ms
31 -36.74248067268 -14.15 -9.05 2.0 117ms
32 -36.74248067268 + -14.15 -9.49 1.0 96.8ms
33 -36.74248067268 -13.85 -9.82 2.0 125ms
34 -36.74248067268 + -13.85 -10.26 1.0 97.0ms
35 -36.74248067268 -13.85 -10.24 3.0 130ms
36 -36.74248067268 + -14.15 -10.66 1.0 97.4ms
37 -36.74248067268 + -13.85 -11.01 3.0 122ms
38 -36.74248067268 -14.15 -11.28 2.0 117ms
39 -36.74248067268 -13.85 -11.39 1.0 96.7ms
40 -36.74248067268 + -14.15 -11.66 3.0 113ms
41 -36.74248067268 + -Inf -12.07 2.0 112ms
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)
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))44.024492373961145The 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_min46.24421467853062This 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))4.723567719007871Since 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).