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 -36.73353772120 -0.88 11.0 366ms
2 -36.56843886683 + -0.78 -1.37 1.0 90.8ms
┌ Warning: Eigensolver not converged
│ n_iter =
│ 1-element Vector{Int64}:
│ 25
└ @ DFTK ~/work/DFTK.jl/DFTK.jl/src/scf/self_consistent_field.jl:80
3 +57.53981112654 + 1.97 -0.07 25.0 370ms
4 -35.78533753727 1.97 -0.93 9.0 296ms
5 -32.53726523139 + 0.51 -0.74 4.0 178ms
6 -36.03089790688 0.54 -1.09 4.0 170ms
7 -36.68727972633 -0.18 -1.55 4.0 158ms
8 -36.74034581706 -1.28 -2.01 2.0 104ms
9 -36.73903212481 + -2.88 -1.93 2.0 133ms
10 -36.74119891035 -2.66 -2.15 2.0 123ms
11 -36.73981824947 + -2.86 -2.13 2.0 120ms
12 -36.74182568047 -2.70 -2.44 1.0 103ms
13 -36.74227957692 -3.34 -2.65 1.0 97.0ms
14 -36.74236918579 -4.05 -2.75 3.0 120ms
15 -36.73799799044 + -2.36 -2.22 3.0 141ms
16 -36.73588752502 + -2.68 -2.13 3.0 155ms
17 -36.73159084097 + -2.37 -2.03 4.0 168ms
18 -36.74228755233 -1.97 -2.82 3.0 159ms
19 -36.74104365253 + -2.91 -2.44 3.0 141ms
20 -36.74247021911 -2.85 -3.28 3.0 145ms
21 -36.74247606384 -5.23 -3.30 2.0 137ms
22 -36.74247723428 -5.93 -3.73 1.0 97.4ms
23 -36.74247403319 + -5.49 -3.35 3.0 148ms
24 -36.74248055776 -5.19 -4.21 2.0 131ms
25 -36.74248065253 -7.02 -4.36 2.0 110ms
26 -36.74248064358 + -8.05 -4.60 2.0 111ms
27 -36.74248064655 -8.53 -4.76 2.0 132ms
28 -36.74248062034 + -7.58 -4.68 3.0 129ms
29 -36.74248063511 -7.83 -4.69 3.0 132ms
30 -36.74248064890 -7.86 -4.86 2.0 119ms
31 -36.74248067140 -7.65 -5.47 2.0 106ms
32 -36.74248066821 + -8.50 -5.20 4.0 166ms
33 -36.74248067204 -8.42 -5.60 3.0 140ms
34 -36.74248067257 -9.28 -5.90 2.0 119ms
35 -36.74248067262 -10.29 -5.81 2.0 124ms
36 -36.74248067267 -10.29 -6.27 2.0 118ms
37 -36.74248067268 -11.18 -6.42 2.0 133ms
38 -36.74248067267 + -12.04 -6.49 1.0 102ms
39 -36.74248067268 -11.16 -6.72 2.0 114ms
40 -36.74248067268 + -11.23 -6.60 3.0 128ms
41 -36.74248067268 -11.64 -6.63 3.0 130ms
42 -36.74248067268 -11.55 -6.78 2.0 117ms
43 -36.74248067268 -11.83 -7.50 1.0 95.9ms
44 -36.74248067268 + -12.56 -7.18 3.0 158ms
45 -36.74248067268 -12.83 -7.36 3.0 140ms
46 -36.74248067268 -12.97 -7.55 2.0 118ms
47 -36.74248067268 -13.19 -7.56 2.0 124ms
48 -36.74248067268 -13.67 -8.00 1.0 97.3ms
49 -36.74248067268 + -Inf -8.08 2.0 138ms
50 -36.74248067268 + -13.85 -7.85 2.0 124ms
51 -36.74248067268 -13.85 -8.64 3.0 131ms
52 -36.74248067268 + -14.15 -8.86 3.0 142ms
53 -36.74248067268 -13.67 -8.80 3.0 148ms
54 -36.74248067268 + -14.15 -8.95 2.0 114ms
55 -36.74248067268 + -Inf -9.34 2.0 110ms
56 -36.74248067268 + -14.15 -9.40 3.0 134ms
57 -36.74248067268 + -14.15 -8.87 3.0 146ms
58 -36.74248067268 -13.85 -9.23 3.0 141ms
59 -36.74248067268 + -14.15 -10.00 2.0 143ms
60 -36.74248067268 + -Inf -9.75 3.0 149ms
61 -36.74248067268 -14.15 -10.14 2.0 119ms
62 -36.74248067268 + -14.15 -10.07 2.0 113ms
63 -36.74248067268 + -Inf -10.46 1.0 103ms
64 -36.74248067268 + -Inf -9.97 3.0 151ms
65 -36.74248067268 + -Inf -10.85 3.0 150ms
66 -36.74248067268 + -Inf -11.28 2.0 105ms
67 -36.74248067268 + -Inf -11.29 2.0 138ms
68 -36.74248067268 + -Inf -11.37 2.0 114ms
69 -36.74248067268 -13.85 -11.51 2.0 113ms
70 -36.74248067268 + -14.15 -11.94 2.0 108ms
71 -36.74248067268 + -14.15 -11.94 3.0 149ms
72 -36.74248067268 -14.15 -11.60 3.0 146ms
73 -36.74248067268 + -14.15 -11.78 3.0 144ms
74 -36.74248067268 + -14.15 -12.23 1.0 97.2mswhile 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 -36.73272568765 -0.88 11.0 360ms
2 -36.73990368965 -2.14 -1.37 1.0 92.5ms
3 -36.73962810277 + -3.56 -1.66 3.0 127ms
4 -36.74228930321 -2.57 -2.30 1.0 93.1ms
5 -36.74240813912 -3.93 -2.52 6.0 155ms
6 -36.74241052431 -5.62 -2.42 1.0 93.3ms
7 -36.74247006169 -4.23 -2.97 1.0 98.1ms
8 -36.74247749014 -5.13 -3.20 2.0 103ms
9 -36.74247942423 -5.71 -3.51 2.0 109ms
10 -36.74248057085 -5.94 -4.07 2.0 106ms
11 -36.74248065835 -7.06 -4.45 3.0 145ms
12 -36.74248066982 -7.94 -4.78 3.0 117ms
13 -36.74248066999 -9.78 -5.11 3.0 142ms
14 -36.74248067260 -8.58 -5.63 2.0 103ms
15 -36.74248067265 -10.26 -5.91 3.0 147ms
16 -36.74248067268 -10.63 -6.30 4.0 113ms
17 -36.74248067268 -11.56 -6.55 3.0 130ms
18 -36.74248067268 -12.07 -6.86 2.0 131ms
19 -36.74248067268 -12.36 -7.29 2.0 108ms
20 -36.74248067268 -13.45 -7.46 3.0 137ms
21 -36.74248067268 -13.45 -7.87 2.0 111ms
22 -36.74248067268 + -14.15 -8.18 1.0 96.9ms
23 -36.74248067268 + -Inf -8.68 3.0 126ms
24 -36.74248067268 -13.85 -8.99 4.0 145ms
25 -36.74248067268 + -14.15 -9.10 5.0 124ms
26 -36.74248067268 + -Inf -9.57 1.0 96.6ms
27 -36.74248067268 -13.85 -9.70 3.0 144ms
28 -36.74248067268 + -Inf -10.19 2.0 106ms
29 -36.74248067268 + -13.85 -10.36 2.0 137ms
30 -36.74248067268 -14.15 -10.69 1.0 96.5ms
31 -36.74248067268 + -14.15 -11.10 2.0 112ms
32 -36.74248067268 + -Inf -11.40 2.0 131ms
33 -36.74248067268 + -Inf -11.53 3.0 123ms
34 -36.74248067268 + -14.15 -11.86 1.0 97.1ms
35 -36.74248067268 -13.67 -12.24 3.0 142msGiven 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))44.02448930222839The 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_min46.24421145192058This 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))4.7235805737320815Since 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).