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 -36.73375588975 -0.88 10.0 1.25s
2 -36.67256747706 + -1.21 -1.51 1.0 285ms
3 +23.87130611918 + 1.78 -0.17 7.0 189ms
4 -36.59890242930 1.78 -1.23 6.0 181ms
5 -36.64311792421 -1.35 -1.51 3.0 107ms
6 -35.89264312765 + -0.12 -1.10 3.0 148ms
7 -36.72386898782 -0.08 -1.77 3.0 115ms
8 -36.74048317805 -1.78 -2.14 2.0 91.6ms
9 -36.73957728556 + -3.04 -2.18 2.0 242ms
10 -36.74108651045 -2.82 -2.15 2.0 95.9ms
11 -36.74147278310 -3.41 -2.35 2.0 93.6ms
12 -36.74210774032 -3.20 -2.56 2.0 1.29s
13 -36.74242900783 -3.49 -2.86 1.0 83.7ms
14 -36.74243343684 -5.35 -2.90 3.0 113ms
15 -36.73963270647 + -2.55 -2.32 3.0 118ms
16 -36.74213789968 -2.60 -2.76 3.0 118ms
17 -36.74247249479 -3.48 -3.42 2.0 105ms
18 -36.74246885945 + -5.44 -3.42 2.0 112ms
19 -36.74239735233 + -4.15 -3.11 3.0 118ms
20 -36.74247810494 -4.09 -3.71 3.0 117ms
21 -36.74247980053 -5.77 -3.92 2.0 96.1ms
22 -36.74248032918 -6.28 -4.02 2.0 114ms
23 -36.74248051828 -6.72 -4.12 2.0 108ms
24 -36.74248065414 -6.87 -4.60 1.0 82.9ms
25 -36.74248064204 + -7.92 -4.41 3.0 130ms
26 -36.74248066971 -7.56 -4.97 2.0 100ms
27 -36.74248037100 + -6.52 -4.30 3.0 126ms
28 -36.74248053880 -6.78 -4.49 3.0 127ms
29 -36.74248067079 -6.88 -5.28 3.0 119ms
30 -36.74248067247 -8.77 -5.66 2.0 112ms
31 -36.74248067228 + -9.70 -5.75 2.0 103ms
32 -36.74248067191 + -9.43 -5.62 3.0 118ms
33 -36.74248067267 -9.12 -6.35 2.0 96.1ms
34 -36.74248067267 -12.11 -6.51 3.0 118ms
35 -36.74248067267 + -11.30 -6.35 2.0 105ms
36 -36.74248067268 -10.88 -7.00 2.0 103ms
37 -36.74248067268 -12.23 -7.19 3.0 121ms
38 -36.74248067268 + -12.42 -6.98 3.0 111ms
39 -36.74248067268 -12.33 -7.55 2.0 94.2ms
40 -36.74248067268 -13.15 -7.84 3.0 119ms
41 -36.74248067268 + -13.07 -7.50 3.0 120ms
42 -36.74248067268 -13.07 -7.83 3.0 118ms
43 -36.74248067268 -13.67 -8.42 2.0 97.3ms
44 -36.74248067268 + -13.85 -8.22 3.0 124ms
45 -36.74248067268 + -Inf -8.38 3.0 117ms
46 -36.74248067268 + -14.15 -8.57 2.0 103ms
47 -36.74248067268 -14.15 -8.59 3.0 118ms
48 -36.74248067268 + -14.15 -9.05 2.0 103ms
49 -36.74248067268 -14.15 -9.10 2.0 113ms
50 -36.74248067268 + -Inf -9.43 1.0 83.4ms
51 -36.74248067268 + -Inf -9.03 3.0 126ms
52 -36.74248067268 + -14.15 -9.69 3.0 117ms
53 -36.74248067268 -14.15 -10.08 2.0 90.7ms
54 -36.74248067268 + -Inf -9.61 3.0 132ms
55 -36.74248067268 -14.15 -10.19 3.0 118ms
56 -36.74248067268 + -Inf -10.29 2.0 103ms
57 -36.74248067268 + -Inf -10.09 3.0 113ms
58 -36.74248067268 -14.15 -11.02 2.0 96.2ms
59 -36.74248067268 + -13.85 -11.08 3.0 125ms
60 -36.74248067268 + -Inf -11.13 2.0 106ms
61 -36.74248067268 + -Inf -11.09 3.0 112ms
62 -36.74248067268 + -Inf -11.34 1.0 89.5ms
63 -36.74248067268 -13.85 -11.64 2.0 112ms
64 -36.74248067268 + -13.55 -11.58 3.0 107ms
65 -36.74248067268 -14.15 -12.40 3.0 99.3ms
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 -36.73449449852 -0.88 10.0 954ms
2 -36.74063216546 -2.21 -1.37 1.0 583ms
3 -36.74164316186 -3.00 -1.96 2.0 109ms
4 -36.74212964094 -3.31 -2.07 4.0 130ms
5 -36.74237684856 -3.61 -2.60 1.0 81.0ms
6 -36.74245429799 -4.11 -2.63 3.0 105ms
7 -36.74247468905 -4.69 -3.16 1.0 87.5ms
8 -36.74247906800 -5.36 -3.43 3.0 125ms
9 -36.74247978383 -6.15 -3.53 2.0 105ms
10 -36.74248018060 -6.40 -3.78 1.0 89.3ms
11 -36.74248061522 -6.36 -4.34 1.0 84.0ms
12 -36.74248065942 -7.35 -4.62 2.0 114ms
13 -36.74248066978 -7.98 -5.04 2.0 95.9ms
14 -36.74248066768 + -8.68 -5.02 3.0 125ms
15 -36.74248067261 -8.31 -5.53 2.0 102ms
16 -36.74248067262 -10.99 -5.85 2.0 114ms
17 -36.74248067266 -10.38 -6.19 2.0 95.5ms
18 -36.74248067268 -10.75 -6.67 2.0 97.7ms
19 -36.74248067268 + -12.83 -6.84 2.0 115ms
20 -36.74248067268 -13.07 -6.97 2.0 113ms
21 -36.74248067268 -12.59 -7.45 1.0 84.8ms
22 -36.74248067268 -13.37 -7.76 3.0 131ms
23 -36.74248067268 -13.67 -8.13 2.0 93.9ms
24 -36.74248067268 + -14.15 -8.39 2.0 114ms
25 -36.74248067268 + -Inf -8.70 1.0 91.2ms
26 -36.74248067268 -14.15 -8.85 3.0 112ms
27 -36.74248067268 + -Inf -9.37 1.0 90.8ms
28 -36.74248067268 -14.15 -9.53 2.0 114ms
29 -36.74248067268 + -14.15 -9.71 1.0 85.0ms
30 -36.74248067268 + -Inf -9.86 1.0 91.1ms
31 -36.74248067268 + -Inf -10.34 1.0 85.0ms
32 -36.74248067268 -13.85 -10.43 3.0 138ms
33 -36.74248067268 + -Inf -10.71 1.0 85.3ms
34 -36.74248067268 + -13.85 -11.25 2.0 108ms
35 -36.74248067268 -14.15 -11.45 3.0 133ms
36 -36.74248067268 + -14.15 -11.85 2.0 95.4ms
37 -36.74248067268 + -Inf -12.05 1.0 90.5ms
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).δVρ
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))44.02448898078408The 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_min46.24421111426899This 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))4.723584908789358Since 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).