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 -36.73323730013 -0.88 12.0 400ms
2 -36.65612793990 + -1.11 -1.45 1.0 90.0ms
3 +28.98931407855 + 1.82 -0.15 7.0 295ms
4 -36.45545231838 1.82 -1.06 6.0 243ms
5 -36.61025039698 -0.81 -1.35 3.0 146ms
6 -35.56157323675 + 0.02 -1.03 4.0 160ms
7 -36.69864963710 0.06 -1.65 3.0 139ms
8 -36.73670297328 -1.42 -2.01 2.0 118ms
9 -36.73901169936 -2.64 -2.06 2.0 178ms
10 -36.73575513762 + -2.49 -1.97 2.0 121ms
11 -36.74152569886 -2.24 -2.23 2.0 126ms
12 -36.74215551913 -3.20 -2.53 1.0 101ms
13 -36.74242617505 -3.57 -2.75 1.0 96.3ms
14 -36.74240691278 + -4.72 -2.77 2.0 111ms
15 -36.74106349352 + -2.87 -2.47 3.0 143ms
16 -36.74174886310 -3.16 -2.54 3.0 143ms
17 -36.74244445047 -3.16 -3.11 2.0 120ms
18 -36.74218192581 + -3.58 -2.79 3.0 189ms
19 -36.74246834522 -3.54 -3.22 3.0 142ms
20 -36.74250812968 -4.40 -3.48 2.0 114ms
21 -36.74251270876 -5.34 -3.81 1.0 101ms
22 -36.74251404421 -5.87 -3.97 2.0 138ms
23 -36.74251472013 -6.17 -4.28 2.0 115ms
24 -36.74251456482 + -6.81 -4.26 3.0 134ms
25 -36.74251471140 -6.83 -4.52 1.0 123ms
26 -36.74251475521 -7.36 -4.80 3.0 155ms
27 -36.74251471523 + -7.40 -4.46 3.0 151ms
28 -36.74251476670 -7.29 -5.11 2.0 130ms
29 -36.74251477211 -8.27 -5.42 2.0 108ms
30 -36.74251473601 + -7.44 -4.76 4.0 170ms
31 -36.74251467534 + -7.22 -4.56 4.0 177ms
32 -36.74251477268 -7.01 -5.59 3.0 158ms
33 -36.74251476564 + -8.15 -5.11 3.0 183ms
34 -36.74251477290 -8.14 -5.86 3.0 148ms
35 -36.74251477302 -9.90 -6.22 2.0 113ms
36 -36.74251477303 -10.99 -6.47 2.0 139ms
37 -36.74251477304 -11.62 -6.58 2.0 118ms
38 -36.74251477304 + -12.36 -6.82 2.0 121ms
39 -36.74251477303 + -11.00 -6.39 3.0 150ms
40 -36.74251477304 -10.95 -6.95 2.0 173ms
41 -36.74251477304 + -Inf -7.21 2.0 113ms
42 -36.74251477304 + -12.55 -7.10 2.0 140ms
43 -36.74251477304 -12.29 -7.80 2.0 115ms
44 -36.74251477304 + -12.70 -7.36 3.0 167ms
45 -36.74251477304 -12.70 -7.87 3.0 148ms
46 -36.74251477304 + -13.07 -7.50 3.0 147ms
47 -36.74251477304 -13.07 -8.06 3.0 146ms
48 -36.74251477304 + -Inf -7.87 3.0 175ms
49 -36.74251477304 + -Inf -8.61 2.0 120ms
50 -36.74251477304 -14.15 -8.61 3.0 157ms
51 -36.74251477304 -14.15 -8.88 2.0 114ms
52 -36.74251477304 + -Inf -8.80 2.0 126ms
53 -36.74251477304 + -14.15 -9.20 1.0 101ms
54 -36.74251477304 -14.15 -9.37 3.0 144ms
55 -36.74251477304 + -Inf -9.68 1.0 101ms
56 -36.74251477304 + -Inf -9.60 2.0 161ms
57 -36.74251477304 + -13.85 -9.43 2.0 114ms
58 -36.74251477304 -13.85 -9.47 3.0 147ms
59 -36.74251477304 + -Inf -10.05 2.0 119ms
60 -36.74251477304 -14.15 -10.13 3.0 140ms
61 -36.74251477304 + -14.15 -9.71 3.0 148ms
62 -36.74251477304 + -Inf -10.34 3.0 148ms
63 -36.74251477304 + -Inf -10.30 2.0 171ms
64 -36.74251477304 -13.85 -10.81 2.0 119ms
65 -36.74251477304 + -13.85 -10.93 3.0 133ms
66 -36.74251477304 + -13.85 -11.16 1.0 96.2ms
67 -36.74251477304 -13.85 -11.48 3.0 142ms
68 -36.74251477304 -14.15 -11.75 2.0 139ms
69 -36.74251477304 + -13.67 -11.49 3.0 156ms
70 -36.74251477304 -13.85 -11.40 3.0 143ms
71 -36.74251477304 + -14.15 -11.60 3.0 183ms
72 -36.74251477304 -14.15 -11.86 2.0 129ms
73 -36.74251477304 + -13.85 -12.11 2.0 113mswhile 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 -36.73331700589 -0.88 12.0 391ms
2 -36.74011291797 -2.17 -1.36 1.0 91.9ms
3 -36.74025920102 -3.83 -1.67 3.0 134ms
4 -36.74231924412 -2.69 -2.27 1.0 132ms
5 -36.74241528551 -4.02 -2.56 6.0 148ms
6 -36.74246308553 -4.32 -2.52 2.0 135ms
7 -36.74249480403 -4.50 -2.86 1.0 98.8ms
8 -36.74250905465 -4.85 -3.10 1.0 95.4ms
9 -36.74251173199 -5.57 -3.34 1.0 100ms
10 -36.74251435631 -5.58 -3.90 1.0 100ms
11 -36.74251469664 -6.47 -4.31 3.0 140ms
12 -36.74251476038 -7.20 -4.54 3.0 161ms
13 -36.74251476666 -8.20 -4.87 5.0 150ms
14 -36.74251476940 -8.56 -5.04 3.0 148ms
15 -36.74251477091 -8.82 -5.30 2.0 116ms
16 -36.74251477284 -8.71 -5.82 1.0 101ms
17 -36.74251477301 -9.77 -6.21 3.0 153ms
18 -36.74251477304 -10.62 -6.55 2.0 140ms
19 -36.74251477304 + -12.17 -6.68 2.0 108ms
20 -36.74251477304 -11.99 -6.97 1.0 143ms
21 -36.74251477304 -12.58 -7.53 2.0 142ms
22 -36.74251477304 -13.67 -7.71 2.0 109ms
23 -36.74251477304 + -14.15 -7.68 3.0 141ms
24 -36.74251477304 -14.15 -7.98 1.0 101ms
25 -36.74251477304 + -14.15 -8.47 2.0 115ms
26 -36.74251477304 + -Inf -8.77 3.0 142ms
27 -36.74251477304 + -Inf -9.10 2.0 135ms
28 -36.74251477304 -14.15 -9.33 2.0 116ms
29 -36.74251477304 + -Inf -9.81 2.0 147ms
30 -36.74251477304 + -14.15 -10.13 3.0 152ms
31 -36.74251477304 -14.15 -10.48 2.0 104ms
32 -36.74251477304 + -14.15 -10.57 3.0 147ms
33 -36.74251477304 -14.15 -11.08 1.0 102ms
34 -36.74251477304 + -14.15 -11.22 3.0 148ms
35 -36.74251477304 + -Inf -11.85 2.0 120ms
36 -36.74251477304 -14.15 -11.92 3.0 200ms
37 -36.74251477304 + -14.15 -12.28 1.0 103msGiven 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 - 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))44.024488054443836The 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_min46.24421014122252This 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))4.723591960232437Since 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).