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 -36.73381335966 -0.88 11.0 352ms
2 -36.64462503839 + -1.05 -1.43 1.0 96.0ms
3 +31.37221362818 + 1.83 -0.15 7.0 222ms
4 -36.58479363786 1.83 -1.10 7.0 237ms
5 -33.75456415744 + 0.45 -0.83 4.0 160ms
6 -36.60902259688 0.46 -1.43 4.0 157ms
7 -36.73397290960 -0.90 -1.78 2.0 114ms
8 -36.73752624236 -2.45 -2.07 2.0 114ms
9 -36.74144229148 -2.41 -2.04 2.0 128ms
10 -36.74240588272 -3.02 -2.39 1.0 100ms
11 -36.74234773612 + -4.24 -2.40 1.0 94.1ms
12 -36.74234502399 + -5.57 -2.77 1.0 95.4ms
13 -36.74246660703 -3.92 -2.94 1.0 101ms
14 -36.74242715645 + -4.40 -2.98 1.0 95.5ms
15 -36.74216349741 + -3.58 -2.73 2.0 129ms
16 -36.74237636404 -3.67 -2.93 2.0 124ms
17 -36.74201723054 + -3.44 -2.73 3.0 138ms
18 -36.74247657827 -3.34 -3.48 2.0 123ms
19 -36.74247330560 + -5.49 -3.50 2.0 135ms
20 -36.74247595297 -5.58 -3.47 2.0 124ms
21 -36.74244531280 + -4.51 -3.28 3.0 128ms
22 -36.74247967914 -4.46 -3.94 2.0 121ms
23 -36.74248065122 -6.01 -4.35 2.0 111ms
24 -36.74248058258 + -7.16 -4.48 3.0 138ms
25 -36.74248063795 -7.26 -4.74 2.0 127ms
26 -36.74248067007 -7.49 -5.20 2.0 131ms
27 -36.74248067101 -9.03 -5.36 2.0 107ms
28 -36.74248067067 + -9.47 -5.26 2.0 130ms
29 -36.74248060776 + -7.20 -4.64 3.0 152ms
30 -36.74248067259 -7.19 -5.94 3.0 147ms
31 -36.74248067262 -10.52 -6.01 3.0 151ms
32 -36.74248067263 -10.76 -5.99 2.0 122ms
33 -36.74248067267 -10.41 -6.42 1.0 95.2ms
34 -36.74248067264 + -10.50 -6.21 3.0 143ms
35 -36.74248067268 -10.40 -6.85 2.0 121ms
36 -36.74248067268 -11.77 -7.17 2.0 135ms
37 -36.74248067268 + -12.94 -7.09 3.0 118ms
38 -36.74248067268 -12.63 -7.41 2.0 126ms
39 -36.74248067268 -12.85 -7.91 2.0 130ms
40 -36.74248067268 + -13.67 -7.71 3.0 134ms
41 -36.74248067268 -14.15 -7.96 2.0 123ms
42 -36.74248067268 -14.15 -8.22 2.0 111ms
43 -36.74248067268 -14.15 -8.48 2.0 130ms
44 -36.74248067268 -14.15 -8.61 3.0 116ms
45 -36.74248067268 + -14.15 -8.74 3.0 134ms
46 -36.74248067268 + -Inf -9.15 1.0 100ms
47 -36.74248067268 + -Inf -9.36 3.0 146ms
48 -36.74248067268 + -Inf -9.42 2.0 116ms
49 -36.74248067268 + -Inf -10.11 2.0 106ms
50 -36.74248067268 + -Inf -9.86 3.0 149ms
51 -36.74248067268 + -14.15 -10.18 2.0 135ms
52 -36.74248067268 -14.15 -10.22 2.0 111ms
53 -36.74248067268 + -Inf -10.54 2.0 110ms
54 -36.74248067268 -13.85 -10.42 2.0 132ms
55 -36.74248067268 + -13.85 -10.09 3.0 142ms
56 -36.74248067268 + -Inf -10.99 3.0 138ms
57 -36.74248067268 -14.15 -10.99 2.0 115ms
58 -36.74248067268 + -14.15 -11.46 2.0 112ms
59 -36.74248067268 + -Inf -11.37 3.0 146ms
60 -36.74248067268 + -14.15 -11.53 3.0 138ms
61 -36.74248067268 -14.15 -12.09 2.0 110mswhile 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 -36.73296667142 -0.88 12.0 351ms
2 -36.73987986524 -2.16 -1.36 1.0 90.3ms
3 -36.74036519327 -3.31 -1.71 4.0 129ms
4 -36.74199710039 -2.79 -2.04 1.0 90.9ms
5 -36.74240565455 -3.39 -2.58 4.0 107ms
6 -36.74244219454 -4.44 -2.51 3.0 139ms
7 -36.74247752057 -4.45 -3.06 1.0 93.4ms
8 -36.74247962380 -5.68 -3.39 1.0 97.8ms
9 -36.74247954376 + -7.10 -3.36 5.0 126ms
10 -36.74248061003 -5.97 -4.04 1.0 98.2ms
11 -36.74248064742 -7.43 -4.18 4.0 138ms
12 -36.74248066750 -7.70 -4.62 1.0 100ms
13 -36.74248067221 -8.33 -5.06 3.0 126ms
14 -36.74248067252 -9.50 -5.26 3.0 131ms
15 -36.74248067260 -10.10 -5.46 2.0 120ms
16 -36.74248067267 -10.20 -5.83 2.0 106ms
17 -36.74248067268 -10.97 -6.07 3.0 135ms
18 -36.74248067268 -11.44 -6.46 2.0 106ms
19 -36.74248067268 -11.96 -7.15 3.0 127ms
20 -36.74248067268 -13.45 -7.35 7.0 161ms
21 -36.74248067268 -13.85 -7.58 2.0 107ms
22 -36.74248067268 + -Inf -7.86 1.0 100ms
23 -36.74248067268 + -13.85 -8.42 2.0 111ms
24 -36.74248067268 -13.67 -8.27 4.0 144ms
25 -36.74248067268 + -14.15 -8.53 1.0 97.2ms
26 -36.74248067268 -13.85 -8.87 2.0 121ms
27 -36.74248067268 + -13.85 -9.29 1.0 95.3ms
28 -36.74248067268 -14.15 -9.55 4.0 136ms
29 -36.74248067268 + -13.67 -9.84 1.0 95.8ms
30 -36.74248067268 -14.15 -10.07 4.0 137ms
31 -36.74248067268 -13.85 -10.68 1.0 95.3ms
32 -36.74248067268 + -13.67 -10.88 4.0 149ms
33 -36.74248067268 -13.85 -11.21 2.0 101ms
34 -36.74248067268 + -14.15 -11.54 6.0 135ms
35 -36.74248067268 -13.67 -11.65 2.0 128ms
36 -36.74248067268 + -14.15 -12.11 1.0 99.2msGiven 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))44.024488980277866The 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_min46.24421111373726This 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))4.723535012295406Since 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).