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 -36.73046324682 -0.88 12.0 1.34s
2 -36.57079195268 + -0.80 -1.38 1.0 327ms
3 +46.76090862750 + 1.92 -0.10 8.0 277ms
4 -35.56877114114 1.92 -0.92 7.0 227ms
5 -32.90664278262 + 0.43 -0.76 5.0 180ms
6 -36.41487625992 0.55 -1.25 5.0 160ms
7 -36.69359423743 -0.55 -1.57 2.0 111ms
8 -36.72263803676 -1.54 -1.87 2.0 102ms
9 -36.73796098920 -1.81 -1.89 2.0 121ms
10 -36.74162105453 -2.44 -2.15 2.0 108ms
11 -36.74067821163 + -3.03 -2.19 1.0 89.3ms
12 -36.74171209239 -2.99 -2.42 1.0 95.7ms
13 -36.74237742063 -3.18 -2.66 1.0 89.3ms
14 -36.74240484632 -4.56 -2.73 2.0 104ms
15 -36.73704088765 + -2.27 -2.18 3.0 137ms
16 -36.74171971092 -2.33 -2.58 3.0 129ms
17 -36.74093206979 + -3.10 -2.45 3.0 134ms
18 -36.74240246401 -2.83 -2.83 3.0 119ms
19 -36.74229608737 + -3.97 -2.86 2.0 118ms
20 -36.74246448003 -3.77 -3.39 2.0 98.4ms
21 -36.74247461471 -4.99 -3.34 2.0 126ms
22 -36.74247186328 + -5.56 -3.55 2.0 104ms
23 -36.74247898047 -5.15 -3.93 1.0 94.1ms
24 -36.74247955963 -6.24 -3.99 3.0 132ms
25 -36.74248063428 -5.97 -4.51 2.0 109ms
26 -36.74248065290 -7.73 -4.61 3.0 123ms
27 -36.74248018272 + -6.33 -4.15 3.0 143ms
28 -36.74248017708 + -8.25 -4.20 3.0 128ms
29 -36.74248066834 -6.31 -5.04 3.0 134ms
30 -36.74248062259 + -7.34 -4.69 3.0 139ms
31 -36.74248067237 -7.30 -5.47 3.0 128ms
32 -36.74248067234 + -10.55 -5.43 2.0 122ms
33 -36.74248067255 -9.68 -5.92 1.0 94.5ms
34 -36.74248067261 -10.22 -6.02 3.0 125ms
35 -36.74248067254 + -10.14 -5.87 2.0 118ms
36 -36.74248067264 -10.00 -6.19 2.0 104ms
37 -36.74248067264 -11.32 -6.17 3.0 127ms
38 -36.74248067268 -10.44 -6.72 2.0 109ms
39 -36.74248067268 + -11.54 -6.53 3.0 131ms
40 -36.74248067268 -11.54 -6.91 1.0 94.1ms
41 -36.74248067268 -12.01 -6.93 2.0 114ms
42 -36.74248067268 + -12.24 -6.98 1.0 94.3ms
43 -36.74248067268 -12.00 -7.36 3.0 114ms
44 -36.74248067268 -12.72 -7.75 2.0 126ms
45 -36.74248067268 -14.15 -8.06 2.0 98.2ms
46 -36.74248067268 + -Inf -8.13 3.0 134ms
47 -36.74248067268 -14.15 -8.51 2.0 100ms
48 -36.74248067268 + -14.15 -8.31 2.0 118ms
49 -36.74248067268 -14.15 -8.58 3.0 122ms
50 -36.74248067268 + -Inf -8.77 2.0 109ms
51 -36.74248067268 + -Inf -9.17 2.0 96.3ms
52 -36.74248067268 -14.15 -9.24 2.0 127ms
53 -36.74248067268 + -Inf -9.33 2.0 104ms
54 -36.74248067268 + -14.15 -8.90 3.0 133ms
55 -36.74248067268 -13.85 -9.33 3.0 128ms
56 -36.74248067268 + -14.15 -9.28 3.0 124ms
57 -36.74248067268 + -14.15 -9.97 2.0 104ms
58 -36.74248067268 + -Inf -10.06 3.0 141ms
59 -36.74248067268 + -Inf -10.22 2.0 104ms
60 -36.74248067268 + -Inf -10.49 1.0 94.4ms
61 -36.74248067268 + -Inf -10.54 2.0 133ms
62 -36.74248067268 + -Inf -10.71 2.0 227ms
63 -36.74248067268 + -Inf -10.88 2.0 122ms
64 -36.74248067268 + -14.15 -10.96 2.0 1.17s
65 -36.74248067268 -13.85 -11.31 1.0 90.6ms
66 -36.74248067268 + -14.15 -11.02 3.0 131ms
67 -36.74248067268 + -Inf -11.92 3.0 129ms
68 -36.74248067268 + -Inf -11.97 2.0 116ms
69 -36.74248067268 + -14.15 -12.37 1.0 88.8ms
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 -36.73080657554 -0.88 11.0 984ms
2 -36.73868027695 -2.10 -1.35 1.0 1.06s
3 -36.73381104206 + -2.31 -1.42 3.0 116ms
4 -36.74219057388 -2.08 -2.34 1.0 87.2ms
5 -36.74216590840 + -4.61 -2.32 6.0 149ms
6 -36.74244928165 -3.55 -2.57 1.0 88.5ms
7 -36.74245349025 -5.38 -2.61 1.0 91.0ms
8 -36.74247501412 -4.67 -3.22 1.0 104ms
9 -36.74247949562 -5.35 -3.38 4.0 139ms
10 -36.74248041548 -6.04 -3.76 2.0 158ms
11 -36.74248052962 -6.94 -4.23 2.0 108ms
12 -36.74248056348 -7.47 -4.47 4.0 136ms
13 -36.74248063671 -7.14 -4.72 3.0 115ms
14 -36.74248067202 -7.45 -5.41 1.0 91.9ms
15 -36.74248067054 + -8.83 -5.33 4.0 148ms
16 -36.74248067253 -8.70 -5.80 1.0 113ms
17 -36.74248067258 -10.31 -5.97 4.0 143ms
18 -36.74248067267 -10.06 -6.31 1.0 90.9ms
19 -36.74248067268 -10.84 -6.65 3.0 129ms
20 -36.74248067268 -12.35 -6.87 1.0 91.2ms
21 -36.74248067268 -12.67 -7.12 5.0 121ms
22 -36.74248067268 -12.56 -7.53 3.0 120ms
23 -36.74248067268 -13.67 -7.77 2.0 123ms
24 -36.74248067268 + -13.85 -8.12 1.0 96.0ms
25 -36.74248067268 + -14.15 -8.24 3.0 105ms
26 -36.74248067268 -14.15 -8.79 2.0 123ms
27 -36.74248067268 -13.85 -9.04 3.0 116ms
28 -36.74248067268 -14.15 -9.25 2.0 117ms
29 -36.74248067268 + -13.85 -9.55 2.0 129ms
30 -36.74248067268 -14.15 -9.77 1.0 91.8ms
31 -36.74248067268 + -14.15 -10.31 2.0 110ms
32 -36.74248067268 + -Inf -10.46 3.0 133ms
33 -36.74248067268 -14.15 -10.61 1.0 95.6ms
34 -36.74248067268 -14.15 -10.99 2.0 106ms
35 -36.74248067268 + -14.15 -11.17 2.0 128ms
36 -36.74248067268 + -14.15 -11.48 1.0 91.8ms
37 -36.74248067268 + -14.15 -12.01 2.0 107ms
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))44.02448914036079The 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_min46.24421128189159This 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))4.723581735498106Since 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).