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 -36.73298623651 -0.88 12.0 1.51s
2 -36.60968410080 + -0.91 -1.41 1.0 335ms
3 +40.15160177165 + 1.89 -0.12 8.0 306ms
4 -36.40565424266 1.88 -1.05 8.0 256ms
5 -35.92523234301 + -0.32 -1.08 3.0 162ms
6 -35.51142578612 + -0.38 -1.01 4.0 160ms
7 -36.71803394290 0.08 -1.69 3.0 136ms
8 -36.73217375303 -1.85 -1.91 2.0 116ms
9 -36.74180320772 -2.02 -2.12 2.0 109ms
10 -36.74165566132 + -3.83 -2.24 2.0 134ms
11 -36.73906210774 + -2.59 -2.15 2.0 111ms
12 -36.74134616502 -2.64 -2.41 1.0 101ms
13 -36.74240134580 -2.98 -2.76 2.0 103ms
14 -36.74243201688 -4.51 -2.88 4.0 133ms
15 -36.73941948034 + -2.52 -2.28 3.0 139ms
16 -36.74237155862 -2.53 -2.97 4.0 163ms
17 -36.74242789023 -4.25 -3.06 2.0 131ms
18 -36.74237150503 + -4.25 -3.04 3.0 135ms
19 -36.74245144370 -4.10 -3.28 2.0 112ms
20 -36.74247702395 -4.59 -3.70 1.0 100ms
21 -36.74248017857 -5.50 -3.79 3.0 141ms
22 -36.74248042106 -6.62 -4.17 2.0 105ms
23 -36.74247967975 + -6.13 -3.85 3.0 138ms
24 -36.74247880861 + -6.06 -3.88 4.0 162ms
25 -36.74248048055 -5.78 -4.01 2.0 122ms
26 -36.74248032504 + -6.81 -4.25 2.0 115ms
27 -36.74247885070 + -5.83 -3.91 3.0 139ms
28 -36.74248066305 -5.74 -5.00 3.0 152ms
29 -36.74248066872 -8.25 -5.22 3.0 160ms
30 -36.74248066998 -8.90 -5.32 2.0 136ms
31 -36.74248067255 -8.59 -5.72 2.0 112ms
32 -36.74248067071 + -8.73 -5.39 3.0 145ms
33 -36.74248067198 -8.89 -5.52 3.0 139ms
34 -36.74248067245 -9.33 -5.86 2.0 111ms
35 -36.74248067267 -9.66 -6.32 2.0 111ms
36 -36.74248067265 + -10.71 -6.20 3.0 153ms
37 -36.74248067268 -10.53 -6.52 2.0 111ms
38 -36.74248067268 -11.43 -6.91 1.0 99.4ms
39 -36.74248067267 + -10.86 -6.48 3.0 148ms
40 -36.74248067268 -10.87 -7.01 3.0 144ms
41 -36.74248067268 + -12.03 -6.90 2.0 105ms
42 -36.74248067268 -11.88 -7.11 2.0 116ms
43 -36.74248067268 -12.77 -7.31 1.0 94.3ms
44 -36.74248067268 -12.77 -7.67 3.0 134ms
45 -36.74248067268 -13.85 -7.74 2.0 130ms
46 -36.74248067268 + -13.37 -7.61 2.0 116ms
47 -36.74248067268 -13.25 -8.38 2.0 111ms
48 -36.74248067268 + -13.55 -7.64 5.0 177ms
49 -36.74248067268 -13.37 -8.49 3.0 151ms
50 -36.74248067268 + -Inf -8.63 1.0 99.4ms
51 -36.74248067268 + -13.85 -9.20 2.0 111ms
52 -36.74248067268 -14.15 -8.76 3.0 163ms
53 -36.74248067268 + -14.15 -8.64 3.0 149ms
54 -36.74248067268 -13.85 -9.60 3.0 144ms
55 -36.74248067268 + -14.15 -9.39 2.0 130ms
56 -36.74248067268 + -14.15 -9.39 3.0 260ms
57 -36.74248067268 -14.15 -9.98 2.0 111ms
58 -36.74248067268 -14.15 -9.88 3.0 1.35s
59 -36.74248067268 + -13.85 -10.21 2.0 111ms
60 -36.74248067268 -14.15 -10.61 3.0 120ms
61 -36.74248067268 + -14.15 -10.51 2.0 131ms
62 -36.74248067268 + -Inf -10.70 3.0 132ms
63 -36.74248067268 + -Inf -10.93 2.0 105ms
64 -36.74248067268 + -Inf -10.91 3.0 140ms
65 -36.74248067268 -14.15 -10.88 2.0 136ms
66 -36.74248067268 + -14.15 -11.33 2.0 128ms
67 -36.74248067268 + -Inf -11.54 3.0 158ms
68 -36.74248067268 + -Inf -11.35 3.0 156ms
69 -36.74248067268 -14.15 -11.91 2.0 125ms
70 -36.74248067268 + -14.15 -11.72 3.0 165ms
71 -36.74248067268 + -Inf -12.54 2.0 128ms
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 -36.73389760040 -0.87 11.0 1.00s
2 -36.74029209038 -2.19 -1.37 1.0 1.09s
3 -36.74144000385 -2.94 -1.91 3.0 168ms
4 -36.74208764825 -3.19 -2.06 2.0 97.7ms
5 -36.74237033006 -3.55 -2.79 1.0 91.9ms
6 -36.74244372004 -4.13 -2.77 7.0 165ms
7 -36.74247368385 -4.52 -3.14 1.0 93.1ms
8 -36.74247963260 -5.23 -3.56 2.0 105ms
9 -36.74247984474 -6.67 -3.68 2.0 154ms
10 -36.74248051755 -6.17 -3.91 1.0 94.7ms
11 -36.74248060638 -7.05 -4.20 3.0 113ms
12 -36.74248060749 -8.96 -4.47 3.0 120ms
13 -36.74248066414 -7.25 -4.66 2.0 133ms
14 -36.74248067179 -8.12 -4.95 1.0 103ms
15 -36.74248067225 -9.34 -5.16 3.0 131ms
16 -36.74248067257 -9.49 -5.39 1.0 96.5ms
17 -36.74248067268 -9.99 -6.16 3.0 134ms
18 -36.74248067268 -11.29 -6.37 4.0 167ms
19 -36.74248067268 -12.26 -6.65 2.0 103ms
20 -36.74248067268 -12.03 -6.98 2.0 119ms
21 -36.74248067268 -13.85 -7.02 3.0 139ms
22 -36.74248067268 -13.03 -7.31 1.0 96.9ms
23 -36.74248067268 + -Inf -7.51 2.0 108ms
24 -36.74248067268 + -Inf -7.79 2.0 124ms
25 -36.74248067268 -13.85 -8.18 1.0 102ms
26 -36.74248067268 + -13.67 -8.62 3.0 144ms
27 -36.74248067268 -13.55 -8.94 5.0 131ms
28 -36.74248067268 + -Inf -9.26 2.0 138ms
29 -36.74248067268 + -13.67 -9.70 2.0 113ms
30 -36.74248067268 -14.15 -10.00 3.0 144ms
31 -36.74248067268 -14.15 -10.37 2.0 103ms
32 -36.74248067268 + -14.15 -10.57 3.0 130ms
33 -36.74248067268 + -Inf -10.87 2.0 107ms
34 -36.74248067268 + -Inf -11.18 2.0 138ms
35 -36.74248067268 + -Inf -11.53 1.0 103ms
36 -36.74248067268 -13.85 -11.72 3.0 131ms
37 -36.74248067268 + -13.67 -11.93 2.0 139ms
38 -36.74248067268 -14.15 -12.25 1.0 97.6ms
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))44.02448898023293The 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_min46.244211113690056This 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))4.723610401086039Since 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).