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 -36.73326805754 -0.88 11.0 1.36s
2 -36.72873784727 + -2.34 -1.63 1.0 373ms
3 -21.94122041041 + 1.17 -0.47 7.0 179ms
4 -36.04800101493 1.15 -1.07 5.0 167ms
5 -36.37279285292 -0.49 -1.26 4.0 152ms
6 -36.38876548263 -1.80 -1.27 3.0 129ms
7 -36.72861401349 -0.47 -1.78 3.0 128ms
8 -36.73657653818 -2.10 -1.90 2.0 105ms
9 -36.74220999211 -2.25 -2.53 1.0 91.6ms
10 -36.74199623433 + -3.67 -2.35 3.0 130ms
11 -36.74227498365 -3.55 -2.44 2.0 103ms
12 -36.74240363654 -3.89 -2.82 1.0 88.7ms
13 -36.74245483587 -4.29 -3.03 2.0 112ms
14 -36.74247946504 -4.61 -3.44 2.0 110ms
15 -36.74237422158 + -3.98 -3.01 3.0 135ms
16 -36.74247644453 -3.99 -3.66 3.0 127ms
17 -36.74243657207 + -4.40 -3.21 3.0 131ms
18 -36.74232172036 + -3.94 -2.96 3.0 136ms
19 -36.74247864199 -3.80 -3.74 3.0 238ms
20 -36.74248021680 -5.80 -3.99 1.0 88.7ms
21 -36.74248054057 -6.49 -4.14 2.0 1.24s
22 -36.74248051292 + -7.56 -4.35 2.0 97.9ms
23 -36.74248062831 -6.94 -4.65 2.0 120ms
24 -36.74248067012 -7.38 -4.88 2.0 103ms
25 -36.74248067204 -8.72 -5.35 1.0 88.4ms
26 -36.74248067145 + -9.23 -5.46 3.0 129ms
27 -36.74248067025 + -8.92 -5.32 3.0 127ms
28 -36.74248067187 -8.79 -5.57 3.0 121ms
29 -36.74248067197 -10.02 -5.57 2.0 113ms
30 -36.74248067243 -9.34 -5.81 3.0 118ms
31 -36.74248067259 -9.81 -6.04 1.0 104ms
32 -36.74248067267 -10.10 -6.22 2.0 142ms
33 -36.74248067268 -10.90 -6.62 2.0 117ms
34 -36.74248067268 -11.51 -6.78 2.0 123ms
35 -36.74248067268 + -11.30 -6.62 3.0 141ms
36 -36.74248067268 -11.54 -6.80 2.0 104ms
37 -36.74248067268 -11.60 -7.05 2.0 113ms
38 -36.74248067268 -12.29 -7.47 2.0 98.1ms
39 -36.74248067268 + -12.14 -7.04 3.0 135ms
40 -36.74248067268 -12.14 -7.53 3.0 128ms
41 -36.74248067268 + -13.30 -7.58 2.0 111ms
42 -36.74248067268 -13.15 -7.92 2.0 97.8ms
43 -36.74248067268 + -13.15 -7.53 3.0 128ms
44 -36.74248067268 + -13.15 -7.42 3.0 137ms
45 -36.74248067268 -12.85 -8.30 3.0 131ms
46 -36.74248067268 -13.67 -8.34 2.0 121ms
47 -36.74248067268 + -14.15 -8.59 2.0 105ms
48 -36.74248067268 + -14.15 -8.90 2.0 103ms
49 -36.74248067268 + -13.85 -9.08 2.0 126ms
50 -36.74248067268 -13.55 -9.25 2.0 103ms
51 -36.74248067268 + -14.15 -9.32 2.0 112ms
52 -36.74248067268 + -14.15 -9.42 3.0 123ms
53 -36.74248067268 + -Inf -9.52 3.0 112ms
54 -36.74248067268 + -14.15 -10.05 1.0 93.1ms
55 -36.74248067268 -13.85 -9.47 4.0 143ms
56 -36.74248067268 + -Inf -9.96 3.0 153ms
57 -36.74248067268 + -14.15 -10.20 2.0 103ms
58 -36.74248067268 + -Inf -10.68 2.0 108ms
59 -36.74248067268 -13.85 -10.42 3.0 135ms
60 -36.74248067268 + -13.85 -10.91 2.0 122ms
61 -36.74248067268 + -Inf -11.15 2.0 98.4ms
62 -36.74248067268 + -Inf -10.88 2.0 126ms
63 -36.74248067268 + -14.15 -11.03 2.0 109ms
64 -36.74248067268 -14.15 -11.46 2.0 108ms
65 -36.74248067268 + -Inf -11.57 3.0 122ms
66 -36.74248067268 + -14.15 -11.91 2.0 108ms
67 -36.74248067268 + -14.15 -11.60 3.0 135ms
68 -36.74248067268 -13.85 -12.08 3.0 126ms
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 -36.73261686493 -0.88 10.0 1.02s
2 -36.73967956398 -2.15 -1.36 1.0 1.07s
3 -36.74047257475 -3.10 -1.64 2.0 119ms
4 -36.74231967226 -2.73 -2.29 1.0 110ms
5 -36.74228904701 + -4.51 -2.67 7.0 133ms
6 -36.74238227807 -4.03 -2.53 2.0 119ms
7 -36.74244942963 -4.17 -2.99 1.0 88.4ms
8 -36.74246720027 -4.75 -3.19 2.0 108ms
9 -36.74247984490 -4.90 -3.48 2.0 122ms
10 -36.74248016310 -6.50 -3.79 1.0 94.6ms
11 -36.74248065943 -6.30 -4.41 1.0 90.6ms
12 -36.74248066716 -8.11 -4.69 5.0 140ms
13 -36.74248067157 -8.36 -4.93 2.0 123ms
14 -36.74248067175 -9.76 -5.39 2.0 102ms
15 -36.74248067266 -9.04 -5.90 5.0 114ms
16 -36.74248067267 -11.31 -5.85 3.0 142ms
17 -36.74248067268 -11.07 -6.32 1.0 95.8ms
18 -36.74248067268 -11.48 -6.62 3.0 121ms
19 -36.74248067268 -12.14 -6.98 2.0 128ms
20 -36.74248067268 -12.66 -7.43 2.0 96.8ms
21 -36.74248067268 -14.15 -7.49 2.0 129ms
22 -36.74248067268 + -14.15 -7.86 1.0 91.3ms
23 -36.74248067268 + -Inf -8.35 3.0 127ms
24 -36.74248067268 -14.15 -8.62 5.0 113ms
25 -36.74248067268 + -Inf -9.02 2.0 128ms
26 -36.74248067268 + -Inf -9.33 2.0 110ms
27 -36.74248067268 -13.85 -9.81 2.0 97.0ms
28 -36.74248067268 + -13.85 -9.68 3.0 137ms
29 -36.74248067268 + -Inf -10.07 2.0 96.7ms
30 -36.74248067268 + -14.15 -10.27 2.0 121ms
31 -36.74248067268 -13.85 -10.87 1.0 91.3ms
32 -36.74248067268 -14.15 -11.02 3.0 143ms
33 -36.74248067268 + -14.15 -11.31 2.0 96.8ms
34 -36.74248067268 -14.15 -11.71 2.0 128ms
35 -36.74248067268 + -Inf -12.20 2.0 102ms
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))44.024488980656656The 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_min46.24421111413515This 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))4.723540830580944Since 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).