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 -36.73301621213 -0.88 12.0 1.34s
2 -36.33731195137 + -0.40 -1.22 1.0 247ms
3 +130.4090063397 + 2.22 0.06 17.0 405ms
4 -30.45218072249 2.21 -0.60 10.0 331ms
5 -33.89108219820 0.54 -0.81 4.0 184ms
6 -30.98609835876 + 0.46 -0.66 5.0 201ms
7 -36.73992501035 0.76 -1.79 3.0 160ms
8 -36.73648158798 + -2.46 -1.81 1.0 96.8ms
9 -36.74126555853 -2.32 -2.08 1.0 102ms
10 -36.74187892285 -3.21 -2.19 2.0 134ms
11 -36.74115420441 + -3.14 -2.21 1.0 103ms
12 -36.74192649933 -3.11 -2.44 1.0 99.1ms
13 -36.74101029843 + -3.04 -2.34 2.0 120ms
14 -36.74194134683 -3.03 -2.55 1.0 103ms
15 -36.74101287636 + -3.03 -2.45 2.0 124ms
16 -36.71963785322 + -1.67 -1.89 4.0 167ms
17 -36.74225635321 -1.65 -2.76 3.0 156ms
18 -36.74191288452 + -3.46 -2.66 3.0 144ms
19 -36.74233337472 -3.38 -2.94 3.0 140ms
20 -36.74234796007 -4.84 -2.92 2.0 137ms
21 -36.74247541968 -3.89 -3.34 2.0 121ms
22 -36.74248041254 -5.30 -4.17 1.0 104ms
23 -36.74248010784 + -6.52 -3.84 4.0 177ms
24 -36.74248045086 -6.46 -4.11 2.0 124ms
25 -36.74248061078 -6.80 -4.30 1.0 98.9ms
26 -36.74248064293 -7.49 -4.44 1.0 105ms
27 -36.74248052357 + -6.92 -4.27 2.0 117ms
28 -36.74247930838 + -5.92 -4.00 3.0 144ms
29 -36.74248062974 -5.88 -4.68 3.0 150ms
30 -36.74248054050 + -7.05 -4.46 3.0 145ms
31 -36.74248061312 -7.14 -4.58 2.0 124ms
32 -36.74248067152 -7.23 -5.23 2.0 121ms
33 -36.74248067070 + -9.09 -5.39 3.0 149ms
34 -36.74248067235 -8.78 -5.72 2.0 111ms
35 -36.74248067258 -9.63 -5.82 2.0 137ms
36 -36.74248067266 -10.11 -5.97 1.0 118ms
37 -36.74248067267 -10.97 -6.37 2.0 119ms
38 -36.74248067267 -12.07 -6.54 3.0 134ms
39 -36.74248067268 -11.68 -6.60 2.0 141ms
40 -36.74248067268 -11.25 -7.05 2.0 109ms
41 -36.74248067265 + -10.56 -6.34 4.0 180ms
42 -36.74248067268 -10.57 -7.12 3.0 161ms
43 -36.74248067268 -12.70 -7.03 2.0 113ms
44 -36.74248067268 -12.67 -7.56 2.0 121ms
45 -36.74248067268 -13.19 -7.89 2.0 136ms
46 -36.74248067268 + -Inf -7.78 2.0 124ms
47 -36.74248067268 + -13.30 -7.51 3.0 151ms
48 -36.74248067268 -13.25 -7.97 2.0 117ms
49 -36.74248067268 + -Inf -8.15 2.0 241ms
50 -36.74248067268 + -14.15 -8.24 2.0 137ms
51 -36.74248067268 + -Inf -8.74 1.0 1.26s
52 -36.74248067268 + -Inf -8.03 3.0 159ms
53 -36.74248067268 -13.67 -8.87 4.0 164ms
54 -36.74248067268 + -Inf -8.52 3.0 145ms
55 -36.74248067268 + -13.85 -8.78 3.0 144ms
56 -36.74248067268 + -Inf -9.38 2.0 109ms
57 -36.74248067268 -14.15 -9.28 3.0 138ms
58 -36.74248067268 + -14.15 -9.31 2.0 126ms
59 -36.74248067268 + -Inf -9.74 2.0 122ms
60 -36.74248067268 + -13.85 -10.00 2.0 156ms
61 -36.74248067268 -13.85 -10.35 1.0 117ms
62 -36.74248067268 + -Inf -10.35 2.0 156ms
63 -36.74248067268 -13.85 -10.92 2.0 123ms
64 -36.74248067268 + -13.85 -10.58 3.0 178ms
65 -36.74248067268 -13.85 -10.73 3.0 161ms
66 -36.74248067268 + -13.67 -10.89 2.0 123ms
67 -36.74248067268 -14.15 -10.57 3.0 138ms
68 -36.74248067268 + -Inf -11.46 3.0 131ms
69 -36.74248067268 + -Inf -11.48 2.0 131ms
70 -36.74248067268 + -14.15 -11.40 3.0 132ms
71 -36.74248067268 -14.15 -11.27 3.0 145ms
72 -36.74248067268 + -Inf -11.61 3.0 139ms
73 -36.74248067268 + -Inf -12.16 2.0 112ms
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 -36.73497906879 -0.88 11.0 920ms
2 -36.74033098270 -2.27 -1.36 1.0 658ms
3 -36.73953914904 + -3.10 -1.63 3.0 134ms
4 -36.74237475495 -2.55 -2.34 1.0 93.2ms
5 -36.74243061982 -4.25 -2.67 6.0 120ms
6 -36.74243743834 -5.17 -2.49 3.0 152ms
7 -36.74247756053 -4.40 -3.06 1.0 94.9ms
8 -36.74247539420 + -5.66 -2.99 1.0 95.7ms
9 -36.74247690567 -5.82 -3.10 1.0 104ms
10 -36.74248039902 -5.46 -3.86 1.0 96.4ms
11 -36.74248061884 -6.66 -4.02 5.0 138ms
12 -36.74248063117 -7.91 -4.02 2.0 118ms
13 -36.74248064688 -7.80 -4.06 1.0 98.2ms
14 -36.74248064408 + -8.55 -4.04 1.0 103ms
15 -36.74248058493 + -7.23 -3.87 1.0 98.2ms
16 -36.74248061239 -7.56 -3.95 1.0 103ms
17 -36.74248063615 -7.62 -4.06 1.0 103ms
18 -36.74248064715 -7.96 -4.14 1.0 98.2ms
19 -36.74248065152 -8.36 -4.18 1.0 103ms
20 -36.74248066170 -7.99 -4.33 1.0 98.2ms
21 -36.74248067206 -7.98 -5.04 1.0 103ms
22 -36.74248067195 + -9.94 -4.97 2.0 128ms
23 -36.74248067218 -9.64 -5.07 1.0 98.8ms
24 -36.74248067223 -10.32 -5.10 1.0 103ms
25 -36.74248067266 -9.36 -5.60 1.0 98.2ms
26 -36.74248067267 -10.98 -5.65 1.0 103ms
27 -36.74248067268 -11.23 -5.96 1.0 103ms
28 -36.74248067268 -11.47 -6.07 4.0 114ms
29 -36.74248067267 + -11.39 -5.93 2.0 117ms
30 -36.74248067267 + -11.64 -5.92 1.0 103ms
31 -36.74248067267 -11.98 -5.93 1.0 98.7ms
32 -36.74248067267 -12.44 -5.94 1.0 103ms
33 -36.74248067268 -11.14 -6.43 1.0 98.6ms
34 -36.74248067268 -12.35 -6.85 2.0 119ms
35 -36.74248067268 -12.26 -7.13 2.0 123ms
36 -36.74248067268 -13.55 -7.55 2.0 119ms
37 -36.74248067268 + -Inf -7.70 2.0 138ms
38 -36.74248067268 -14.15 -8.16 2.0 118ms
39 -36.74248067268 + -Inf -8.39 2.0 134ms
40 -36.74248067268 + -Inf -8.84 1.0 103ms
41 -36.74248067268 + -Inf -9.15 3.0 136ms
42 -36.74248067268 -14.15 -9.24 3.0 139ms
43 -36.74248067268 + -14.15 -9.92 1.0 103ms
44 -36.74248067268 -13.85 -10.05 6.0 170ms
45 -36.74248067268 + -Inf -10.52 2.0 104ms
46 -36.74248067268 + -13.85 -10.86 3.0 147ms
47 -36.74248067268 + -Inf -11.21 2.0 108ms
48 -36.74248067268 -13.85 -11.37 2.0 125ms
49 -36.74248067268 + -13.85 -11.75 2.0 245ms
50 -36.74248067268 + -Inf -12.00 2.0 1.24s
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))44.02449363196138The 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_min46.24421599995943This 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))4.723578692113507Since 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).