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 -36.73355362441 -0.88 11.0 1.31s
2 -36.72290183353 + -1.97 -1.63 1.0 287ms
3 -12.54940907667 + 1.38 -0.37 7.0 189ms
4 -36.63281882430 1.38 -1.40 6.0 178ms
5 -36.66552334200 -1.49 -1.51 2.0 140ms
6 -36.13148384819 + -0.27 -1.17 3.0 124ms
7 -36.71562043246 -0.23 -1.77 3.0 121ms
8 -36.74020836879 -1.61 -2.09 2.0 107ms
9 -36.74029106409 -4.08 -2.23 2.0 121ms
10 -36.74159463574 -2.88 -2.16 2.0 118ms
11 -36.74243243075 -3.08 -2.72 2.0 106ms
12 -36.74246206072 -4.53 -2.98 2.0 117ms
13 -36.74247494952 -4.89 -3.35 2.0 122ms
14 -36.74247195699 + -5.52 -3.32 2.0 105ms
15 -36.74238115619 + -4.04 -3.06 3.0 129ms
16 -36.74244409569 -4.20 -3.20 3.0 133ms
17 -36.74240783237 + -4.44 -3.08 2.0 114ms
18 -36.74229300276 + -3.94 -2.93 3.0 133ms
19 -36.74247704019 -3.74 -3.70 3.0 136ms
20 -36.74248064201 -5.44 -4.46 2.0 106ms
21 -36.74248060697 + -7.46 -4.55 3.0 152ms
22 -36.74248066668 -7.22 -4.78 2.0 95.1ms
23 -36.74248066954 -8.54 -5.08 2.0 110ms
24 -36.74248067139 -8.73 -5.27 2.0 100ms
25 -36.74248067223 -9.08 -5.40 2.0 212ms
26 -36.74248067232 -10.04 -5.60 1.0 90.0ms
27 -36.74248066810 + -8.37 -5.16 3.0 925ms
28 -36.74248067093 -8.55 -5.38 2.0 109ms
29 -36.74248067239 -8.84 -5.64 2.0 112ms
30 -36.74248067111 + -8.89 -5.46 3.0 128ms
31 -36.74248067165 -9.27 -5.54 3.0 128ms
32 -36.74248067267 -8.99 -6.23 2.0 105ms
33 -36.74248067259 + -10.12 -5.84 3.0 133ms
34 -36.74248067267 -10.09 -6.58 3.0 129ms
35 -36.74248067268 -11.24 -6.80 1.0 91.5ms
36 -36.74248067268 -11.68 -7.34 3.0 113ms
37 -36.74248067268 -13.25 -7.37 3.0 133ms
38 -36.74248067268 -13.85 -7.41 1.0 93.8ms
39 -36.74248067268 -13.25 -7.78 1.0 110ms
40 -36.74248067268 + -13.85 -7.70 3.0 135ms
41 -36.74248067268 -13.67 -8.03 2.0 98.5ms
42 -36.74248067268 + -13.45 -7.55 3.0 130ms
43 -36.74248067268 -13.37 -7.95 3.0 121ms
44 -36.74248067268 + -Inf -8.24 2.0 103ms
45 -36.74248067268 + -Inf -8.18 3.0 134ms
46 -36.74248067268 + -13.85 -8.65 2.0 105ms
47 -36.74248067268 -13.67 -9.01 2.0 121ms
48 -36.74248067268 + -Inf -9.18 2.0 98.1ms
49 -36.74248067268 + -14.15 -8.70 3.0 135ms
50 -36.74248067268 -13.85 -9.17 3.0 131ms
51 -36.74248067268 + -13.85 -9.55 1.0 89.8ms
52 -36.74248067268 + -Inf -9.47 2.0 127ms
53 -36.74248067268 -14.15 -10.17 2.0 104ms
54 -36.74248067268 -14.15 -10.33 3.0 122ms
55 -36.74248067268 + -13.85 -9.97 3.0 133ms
56 -36.74248067268 -13.85 -10.12 3.0 131ms
57 -36.74248067268 + -14.15 -10.32 2.0 119ms
58 -36.74248067268 + -14.15 -10.55 3.0 111ms
59 -36.74248067268 -13.85 -10.66 3.0 119ms
60 -36.74248067268 + -13.85 -11.32 2.0 100ms
61 -36.74248067268 -14.15 -11.02 4.0 150ms
62 -36.74248067268 + -14.15 -11.54 3.0 110ms
63 -36.74248067268 + -14.15 -11.95 2.0 118ms
64 -36.74248067268 + -Inf -11.51 4.0 144ms
65 -36.74248067268 -14.15 -12.15 3.0 134ms
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 -36.73247713700 -0.88 11.0 1.01s
2 -36.74003576504 -2.12 -1.36 1.0 616ms
3 -36.74092711023 -3.05 -1.73 3.0 116ms
4 -36.74227450815 -2.87 -2.21 1.0 119ms
5 -36.74240345954 -3.89 -2.56 4.0 112ms
6 -36.74239834782 + -5.29 -2.41 2.0 121ms
7 -36.74247299928 -4.13 -3.10 1.0 98.7ms
8 -36.74247685219 -5.41 -3.16 2.0 95.1ms
9 -36.74247894626 -5.68 -3.34 1.0 95.2ms
10 -36.74248053827 -5.80 -4.06 1.0 90.8ms
11 -36.74248064537 -6.97 -4.20 3.0 141ms
12 -36.74248066774 -7.65 -4.43 1.0 96.1ms
13 -36.74248066374 + -8.40 -4.44 2.0 108ms
14 -36.74248067077 -8.15 -4.77 1.0 96.0ms
15 -36.74248067230 -8.81 -5.11 2.0 116ms
16 -36.74248067255 -9.61 -5.35 2.0 102ms
17 -36.74248067265 -9.99 -5.89 4.0 121ms
18 -36.74248067267 -10.75 -6.10 2.0 124ms
19 -36.74248067268 -11.28 -6.26 1.0 97.0ms
20 -36.74248067268 -11.58 -6.67 2.0 106ms
21 -36.74248067268 -13.00 -6.83 2.0 120ms
22 -36.74248067268 -13.03 -6.95 3.0 108ms
23 -36.74248067268 -13.00 -7.39 1.0 92.7ms
24 -36.74248067268 -14.15 -7.83 3.0 126ms
25 -36.74248067268 -13.67 -8.05 2.0 123ms
26 -36.74248067268 + -13.85 -8.12 2.0 121ms
27 -36.74248067268 + -Inf -8.69 1.0 94.6ms
28 -36.74248067268 -14.15 -8.77 3.0 139ms
29 -36.74248067268 + -14.15 -9.22 1.0 96.1ms
30 -36.74248067268 -14.15 -9.59 3.0 130ms
31 -36.74248067268 -14.15 -9.81 1.0 95.2ms
32 -36.74248067268 + -13.85 -10.12 2.0 101ms
33 -36.74248067268 + -Inf -10.59 3.0 114ms
34 -36.74248067268 -14.15 -10.87 2.0 120ms
35 -36.74248067268 + -14.15 -11.21 2.0 117ms
36 -36.74248067268 -14.15 -11.35 2.0 128ms
37 -36.74248067268 + -Inf -11.76 1.0 96.2ms
38 -36.74248067268 + -14.15 -12.24 2.0 124ms
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))44.02448915171343The 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_min46.24421129381663This 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))4.723577767037165Since 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).