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 -36.73260492819 -0.87 11.0 1.41s
2 -36.70333684273 + -1.53 -1.49 1.0 276ms
3 -1.017380484262 + 1.55 -0.29 6.0 188ms
4 -35.67022167157 1.54 -0.92 6.0 229ms
5 -36.50734545916 -0.08 -1.12 3.0 134ms
6 -36.73710216833 -0.64 -1.92 2.0 112ms
7 -36.63281463994 + -0.98 -1.52 3.0 148ms
8 -36.62001523802 + -1.89 -1.51 5.0 153ms
9 -36.74066781359 -0.92 -2.10 3.0 132ms
10 -36.73930281274 + -2.86 -1.95 2.0 115ms
11 -36.74219486154 -2.54 -2.49 2.0 120ms
12 -36.74224685727 -4.28 -2.56 2.0 109ms
13 -36.74246310531 -3.67 -2.93 1.0 107ms
14 -36.74247231201 -5.04 -3.29 2.0 133ms
15 -36.74200509806 + -3.33 -2.72 3.0 135ms
16 -36.74246878673 -3.33 -3.05 4.0 142ms
17 -36.74243267225 + -4.44 -3.17 2.0 119ms
18 -36.74248020075 -4.32 -3.80 2.0 104ms
19 -36.74248032284 -6.91 -3.98 2.0 127ms
20 -36.74239914687 + -4.09 -3.11 4.0 153ms
21 -36.74248056492 -4.09 -4.40 4.0 158ms
22 -36.74248055076 + -7.85 -4.34 2.0 122ms
23 -36.74248061642 -7.18 -4.66 2.0 111ms
24 -36.74248065513 -7.41 -4.75 2.0 103ms
25 -36.74248066557 -7.98 -5.11 2.0 96.5ms
26 -36.74248067218 -8.18 -5.52 3.0 135ms
27 -36.74248067160 + -9.23 -5.47 3.0 111ms
28 -36.74248067250 -9.05 -5.78 1.0 95.3ms
29 -36.74248067238 + -9.92 -5.71 3.0 121ms
30 -36.74248067249 -9.96 -5.76 2.0 119ms
31 -36.74248067268 -9.72 -6.37 1.0 89.0ms
32 -36.74248067148 + -8.92 -5.51 3.0 151ms
33 -36.74248067260 -8.95 -6.08 4.0 154ms
34 -36.74248067268 -10.11 -6.50 3.0 134ms
35 -36.74248067268 -11.46 -6.99 1.0 89.4ms
36 -36.74248067268 -13.45 -7.20 2.0 131ms
37 -36.74248067268 + -13.55 -7.18 3.0 111ms
38 -36.74248067268 + -11.54 -6.79 3.0 134ms
39 -36.74248067268 -11.51 -7.45 3.0 128ms
40 -36.74248067268 -12.92 -7.72 2.0 110ms
41 -36.74248067268 -13.67 -8.07 2.0 122ms
42 -36.74248067268 + -14.15 -8.30 2.0 102ms
43 -36.74248067268 + -13.67 -7.73 3.0 135ms
44 -36.74248067268 + -Inf -7.82 4.0 151ms
45 -36.74248067268 -13.67 -8.38 3.0 129ms
46 -36.74248067268 -14.15 -8.66 2.0 105ms
47 -36.74248067268 + -Inf -8.37 3.0 127ms
48 -36.74248067268 + -13.85 -9.04 2.0 104ms
49 -36.74248067268 -13.85 -9.26 3.0 137ms
50 -36.74248067268 + -13.85 -9.03 3.0 121ms
51 -36.74248067268 -13.85 -8.89 3.0 135ms
52 -36.74248067268 + -14.15 -9.27 3.0 130ms
53 -36.74248067268 -14.15 -9.91 1.0 95.6ms
54 -36.74248067268 + -14.15 -10.04 3.0 137ms
55 -36.74248067268 -14.15 -10.18 1.0 95.9ms
56 -36.74248067268 + -Inf -10.29 2.0 119ms
57 -36.74248067268 + -14.15 -10.55 1.0 217ms
58 -36.74248067268 + -Inf -10.35 2.0 123ms
59 -36.74248067268 -13.85 -10.76 3.0 1.28s
60 -36.74248067268 + -14.15 -10.26 3.0 144ms
61 -36.74248067268 + -14.15 -11.10 3.0 129ms
62 -36.74248067268 -14.15 -11.67 2.0 99.3ms
63 -36.74248067268 + -13.85 -11.69 3.0 130ms
64 -36.74248067268 + -Inf -11.61 3.0 121ms
65 -36.74248067268 -13.85 -12.01 2.0 104ms
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 -36.73396484906 -0.88 11.0 1.01s
2 -36.73999813294 -2.22 -1.36 1.0 1.08s
3 -36.74159377890 -2.80 -1.87 3.0 138ms
4 -36.74228402142 -3.16 -2.19 1.0 104ms
5 -36.74240780016 -3.91 -2.68 4.0 123ms
6 -36.74242554651 -4.75 -2.49 3.0 147ms
7 -36.74247621813 -4.30 -3.15 1.0 105ms
8 -36.74247926184 -5.52 -3.33 2.0 117ms
9 -36.74247913176 + -6.89 -3.29 2.0 119ms
10 -36.74248056424 -5.84 -4.21 1.0 107ms
11 -36.74248063692 -7.14 -4.33 6.0 206ms
12 -36.74248066297 -7.58 -4.52 1.0 91.3ms
13 -36.74248066881 -8.23 -4.86 1.0 90.9ms
14 -36.74248067253 -8.43 -5.33 2.0 123ms
15 -36.74248067259 -10.19 -5.48 2.0 106ms
16 -36.74248067266 -10.18 -5.94 2.0 96.7ms
17 -36.74248067268 -10.69 -6.40 3.0 192ms
18 -36.74248067268 -11.86 -6.77 5.0 157ms
19 -36.74248067268 + -13.30 -6.92 3.0 110ms
20 -36.74248067268 -12.70 -7.17 2.0 103ms
21 -36.74248067268 -13.19 -7.60 2.0 129ms
22 -36.74248067268 + -Inf -7.83 3.0 135ms
23 -36.74248067268 + -13.85 -7.94 2.0 107ms
24 -36.74248067268 -13.67 -8.28 2.0 100ms
25 -36.74248067268 + -Inf -8.80 3.0 130ms
26 -36.74248067268 + -Inf -9.04 2.0 103ms
27 -36.74248067268 + -Inf -9.03 3.0 129ms
28 -36.74248067268 + -Inf -9.70 1.0 91.0ms
29 -36.74248067268 + -Inf -9.64 4.0 153ms
30 -36.74248067268 + -Inf -10.14 2.0 100ms
31 -36.74248067268 -14.15 -10.46 2.0 128ms
32 -36.74248067268 + -14.15 -10.81 2.0 100ms
33 -36.74248067268 + -14.15 -11.28 4.0 128ms
34 -36.74248067268 + -Inf -11.39 2.0 123ms
35 -36.74248067268 + -14.15 -11.58 1.0 96.5ms
36 -36.74248067268 -13.85 -11.84 2.0 106ms
37 -36.74248067268 + -Inf -12.33 1.0 96.2ms
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))44.02448898157884The 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_min46.24421111510382This 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))4.723584244145614Since 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).