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 -36.73403065700 -0.88 10.0 1.11s
2 -36.72601062664 + -2.10 -1.63 1.0 253ms
3 -15.74770484286 + 1.32 -0.41 5.0 172ms
4 -36.70775038044 1.32 -1.38 6.0 141ms
5 -36.73462880826 -1.57 -1.87 2.0 83.7ms
6 -36.52490693647 + -0.68 -1.39 3.0 111ms
7 -36.66489764143 -0.85 -1.60 5.0 123ms
8 -36.73957983304 -1.13 -2.14 2.0 89.1ms
9 -36.73996308164 -3.42 -2.02 2.0 101ms
10 -36.74150255525 -2.81 -2.29 2.0 87.0ms
11 -36.74224763383 -3.13 -2.58 2.0 86.4ms
12 -36.74240365860 -3.81 -2.90 2.0 81.0ms
13 -36.74244635033 -4.37 -3.04 2.0 104ms
14 -36.74247893258 -4.49 -3.49 1.0 70.9ms
15 -36.74246364744 + -4.82 -3.41 2.0 103ms
16 -36.74234374620 + -3.92 -2.98 3.0 103ms
17 -36.74247935331 -3.87 -3.73 3.0 114ms
18 -36.74248061974 -5.90 -4.22 2.0 75.3ms
19 -36.74247844715 + -5.66 -3.87 3.0 110ms
20 -36.74247586833 + -5.59 -3.72 4.0 242ms
21 -36.74248063123 -5.32 -4.45 3.0 111ms
22 -36.74248063970 -8.07 -4.69 2.0 1.26s
23 -36.74248066899 -7.53 -4.99 2.0 78.8ms
24 -36.74248066859 + -9.40 -5.11 2.0 98.5ms
25 -36.74248067197 -8.47 -5.11 1.0 72.1ms
26 -36.74248067249 -9.29 -5.45 1.0 73.6ms
27 -36.74248067250 -11.14 -5.77 2.0 77.9ms
28 -36.74248066758 + -8.31 -5.21 3.0 110ms
29 -36.74248067255 -8.30 -5.91 3.0 104ms
30 -36.74248067266 -9.96 -6.24 1.0 72.8ms
31 -36.74248067264 + -10.71 -6.08 3.0 106ms
32 -36.74248067259 + -10.30 -6.07 3.0 110ms
33 -36.74248067262 -10.49 -6.03 3.0 123ms
34 -36.74248067268 -10.23 -6.76 2.0 101ms
35 -36.74248067268 -12.27 -6.92 3.0 124ms
36 -36.74248067268 -12.94 -6.78 2.0 120ms
37 -36.74248067268 -12.15 -6.98 2.0 90.5ms
38 -36.74248067268 -12.33 -7.21 1.0 73.2ms
39 -36.74248067268 + -12.45 -7.08 3.0 102ms
40 -36.74248067267 + -10.88 -6.49 4.0 121ms
41 -36.74248067268 -10.87 -7.58 3.0 116ms
42 -36.74248067268 -13.85 -7.63 2.0 89.4ms
43 -36.74248067268 -14.15 -8.11 1.0 74.3ms
44 -36.74248067268 + -12.77 -7.40 3.0 117ms
45 -36.74248067268 -12.73 -8.50 3.0 121ms
46 -36.74248067268 + -14.15 -8.25 3.0 116ms
47 -36.74248067268 + -Inf -8.83 3.0 105ms
48 -36.74248067268 + -Inf -9.29 2.0 102ms
49 -36.74248067268 + -Inf -9.33 3.0 95.5ms
50 -36.74248067268 + -Inf -9.92 2.0 83.9ms
51 -36.74248067268 + -Inf -9.77 3.0 122ms
52 -36.74248067268 + -Inf -9.57 3.0 120ms
53 -36.74248067268 + -Inf -10.08 2.0 99.4ms
54 -36.74248067268 -14.15 -9.91 3.0 114ms
55 -36.74248067268 + -14.15 -10.00 2.0 86.9ms
56 -36.74248067268 -13.85 -10.30 3.0 94.9ms
57 -36.74248067268 + -Inf -10.51 3.0 94.2ms
58 -36.74248067268 + -13.85 -10.76 2.0 103ms
59 -36.74248067268 -13.85 -10.70 2.0 102ms
60 -36.74248067268 + -13.85 -10.52 3.0 110ms
61 -36.74248067268 + -Inf -11.47 3.0 102ms
62 -36.74248067268 + -Inf -11.53 3.0 124ms
63 -36.74248067268 + -Inf -11.75 1.0 79.3ms
64 -36.74248067268 + -Inf -11.85 1.0 73.5ms
65 -36.74248067268 + -Inf -11.50 3.0 109ms
66 -36.74248067268 + -Inf -12.22 3.0 109ms
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 -36.73321760182 -0.88 11.0 846ms
2 -36.73982242432 -2.18 -1.37 1.0 861ms
3 -36.74037362238 -3.26 -1.75 2.0 100ms
4 -36.74215626295 -2.75 -2.13 1.0 95.6ms
5 -36.74229799557 -3.85 -2.63 3.0 85.6ms
6 -36.74243466318 -3.86 -2.51 7.0 114ms
7 -36.74245850725 -4.62 -3.04 1.0 81.7ms
8 -36.74247757956 -4.72 -3.17 1.0 73.0ms
9 -36.74247738053 + -6.70 -3.36 2.0 88.4ms
10 -36.74248045292 -5.51 -3.98 2.0 88.1ms
11 -36.74248065360 -6.70 -4.21 4.0 115ms
12 -36.74248066589 -7.91 -4.54 4.0 86.9ms
13 -36.74248067187 -8.22 -5.06 2.0 95.3ms
14 -36.74248067248 -9.22 -5.21 3.0 107ms
15 -36.74248067262 -9.85 -5.48 1.0 79.8ms
16 -36.74248067263 -11.09 -5.50 3.0 101ms
17 -36.74248067267 -10.33 -5.97 1.0 80.1ms
18 -36.74248067268 -11.08 -6.33 3.0 102ms
19 -36.74248067268 -12.02 -6.85 4.0 92.3ms
20 -36.74248067268 -12.83 -7.20 3.0 113ms
21 -36.74248067268 -14.15 -7.58 3.0 94.1ms
22 -36.74248067268 -14.15 -7.58 6.0 103ms
23 -36.74248067268 -13.85 -7.96 1.0 74.8ms
24 -36.74248067268 + -Inf -8.12 2.0 93.4ms
25 -36.74248067268 + -14.15 -8.59 1.0 74.9ms
26 -36.74248067268 -14.15 -8.93 3.0 120ms
27 -36.74248067268 + -14.15 -9.11 1.0 74.6ms
28 -36.74248067268 + -Inf -9.39 2.0 104ms
29 -36.74248067268 -13.85 -9.72 2.0 102ms
30 -36.74248067268 + -13.85 -10.27 2.0 93.5ms
31 -36.74248067268 + -Inf -10.47 3.0 107ms
32 -36.74248067268 + -Inf -10.82 2.0 84.0ms
33 -36.74248067268 + -Inf -11.13 3.0 86.9ms
34 -36.74248067268 -13.85 -11.42 3.0 106ms
35 -36.74248067268 + -Inf -11.69 3.0 107ms
36 -36.74248067268 + -13.85 -11.81 2.0 94.3ms
37 -36.74248067268 + -Inf -12.05 1.0 74.1ms
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))44.02449141685063The 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_min46.24421367316243This 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))4.72365361850425Since 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).