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 ASEconvert
using DFTK
using LazyArtifacts
ase_Al = ase.build.bulk("Al"; cubic=true) * pytuple((4, 1, 1))
system_Al = attach_psp(pyconvert(AbstractSystem, ase_Al);
Al=artifact"pd_nc_sr_pbe_standard_0.4.1_upf/Al.upf")FlexibleSystem(Al₁₆, periodic = TTT):
bounding_box : [ 16.2 0 0;
0 4.05 0;
0 0 4.05]u"Å"
.---------------------------------------.
/| |
* | |
|Al Al Al Al |
| | |
| .--Al--------Al--------Al--------Al-----.
|/ Al Al Al Al /
Al--------Al--------Al--------Al--------*
and we discretise:
model_Al = model_LDA(system_Al; temperature=1e-3, symmetries=false)
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 -35.98148248935 -0.86 13.0 341ms
2 -35.96393481880 + -1.76 -1.59 1.0 113ms
3 -5.210587155496 + 1.49 -0.33 6.0 191ms
4 -35.91660264314 1.49 -1.27 8.0 203ms
5 -35.95690016469 -1.39 -1.66 2.0 118ms
6 -35.27740118167 + -0.17 -1.14 5.0 162ms
7 -35.95353871405 -0.17 -1.69 4.0 140ms
8 -35.98932173216 -1.45 -2.39 2.0 106ms
9 -35.98895834444 + -3.44 -2.17 4.0 172ms
10 -35.98837972258 + -3.24 -2.17 2.0 121ms
11 -35.98965508036 -2.89 -2.61 2.0 121ms
12 -35.98973544367 -4.09 -2.83 1.0 92.3ms
13 -35.98975996143 -4.61 -3.19 1.0 117ms
14 -35.98975923204 + -6.14 -3.32 2.0 126ms
15 -35.98723179396 + -2.60 -2.37 4.0 157ms
16 -35.98976458536 -2.60 -3.70 5.0 167ms
17 -35.98976613087 -5.81 -4.10 2.0 97.0ms
18 -35.98975481386 + -4.95 -3.51 4.0 157ms
19 -35.98976641764 -4.94 -4.24 4.0 139ms
20 -35.98976662133 -6.69 -4.35 3.0 123ms
21 -35.98976676579 -6.84 -4.76 2.0 104ms
22 -35.98976676793 -8.67 -4.66 2.0 125ms
23 -35.98976678078 -7.89 -5.00 2.0 126ms
24 -35.98976678269 -8.72 -5.10 1.0 92.3ms
25 -35.98976678421 -8.82 -5.45 1.0 91.9ms
26 -35.98976678436 -9.80 -5.82 2.0 109ms
27 -35.98976678257 + -8.75 -5.41 4.0 153ms
28 -35.98976678395 -8.86 -5.67 4.0 144ms
29 -35.98976678452 -9.24 -6.71 2.0 109ms
30 -35.98976678451 + -11.00 -6.47 5.0 218ms
31 -35.98976678452 -10.97 -6.82 3.0 127ms
32 -35.98976678452 + -11.94 -6.88 2.0 110ms
33 -35.98976678452 + -11.92 -6.71 3.0 127ms
34 -35.98976678452 -11.52 -7.17 2.0 105ms
35 -35.98976678452 + -13.45 -7.29 3.0 159ms
36 -35.98976678452 -12.70 -7.45 2.0 109ms
37 -35.98976678452 -13.45 -7.76 4.0 150ms
38 -35.98976678452 + -Inf -8.02 2.0 127ms
39 -35.98976678452 + -13.67 -7.79 2.0 113ms
40 -35.98976678452 -13.67 -8.49 3.0 126ms
41 -35.98976678452 + -Inf -8.28 2.0 127ms
42 -35.98976678452 + -13.85 -7.93 4.0 146ms
43 -35.98976678452 -14.15 -9.03 4.0 153ms
44 -35.98976678452 + -14.15 -8.97 2.0 126ms
45 -35.98976678452 -13.85 -9.15 2.0 149ms
46 -35.98976678452 + -Inf -8.97 3.0 129ms
47 -35.98976678452 + -14.15 -9.65 3.0 126ms
48 -35.98976678452 -14.15 -10.23 2.0 127ms
49 -35.98976678452 + -Inf -10.10 3.0 136ms
50 -35.98976678452 + -Inf -10.35 2.0 98.1ms
51 -35.98976678452 + -Inf -10.70 2.0 114ms
52 -35.98976678452 + -Inf -10.48 3.0 117ms
53 -35.98976678452 + -14.15 -10.65 3.0 126ms
54 -35.98976678452 + -14.15 -10.92 2.0 127ms
55 -35.98976678452 -14.15 -11.23 3.0 134ms
56 -35.98976678452 -14.15 -11.63 2.0 109ms
57 -35.98976678452 + -Inf -11.63 2.0 127ms
58 -35.98976678452 + -13.85 -11.89 2.0 109ms
59 -35.98976678452 + -Inf -11.79 3.0 135ms
60 -35.98976678452 -13.85 -11.88 3.0 146ms
61 -35.98976678452 -14.15 -12.05 2.0 110mswhile 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 -35.98153549458 -0.86 12.0 355ms
2 -35.98750838449 -2.22 -1.34 1.0 88.3ms
3 -35.98606916490 + -2.84 -1.61 3.0 128ms
4 -35.98952584638 -2.46 -2.23 1.0 89.5ms
5 -35.98948597679 + -4.40 -2.42 7.0 156ms
6 -35.98973160713 -3.61 -2.56 8.0 136ms
7 -35.98971741463 + -4.85 -2.94 1.0 117ms
8 -35.98975262448 -4.45 -3.02 3.0 132ms
9 -35.98975434462 -5.76 -3.36 1.0 90.9ms
10 -35.98976641689 -4.92 -4.04 2.0 104ms
11 -35.98976655456 -6.86 -4.22 3.0 137ms
12 -35.98976675633 -6.70 -4.53 3.0 110ms
13 -35.98976676859 -7.91 -4.82 3.0 110ms
14 -35.98976678427 -7.80 -5.42 7.0 147ms
15 -35.98976678446 -9.72 -5.64 3.0 148ms
16 -35.98976678435 + -9.94 -5.82 2.0 174ms
17 -35.98976678444 -10.03 -6.03 7.0 144ms
18 -35.98976678452 -10.09 -6.38 3.0 137ms
19 -35.98976678452 -11.55 -6.67 1.0 94.4ms
20 -35.98976678452 -13.37 -6.97 5.0 124ms
21 -35.98976678452 -12.41 -7.18 3.0 130ms
22 -35.98976678452 -13.00 -7.38 2.0 131ms
23 -35.98976678452 -13.11 -7.95 2.0 104ms
24 -35.98976678452 + -14.15 -8.05 4.0 194ms
25 -35.98976678452 -14.15 -8.33 1.0 94.7ms
26 -35.98976678452 + -Inf -8.90 6.0 129ms
27 -35.98976678452 + -14.15 -9.48 4.0 146ms
28 -35.98976678452 + -Inf -9.83 6.0 140ms
29 -35.98976678452 + -14.15 -10.24 3.0 141ms
30 -35.98976678452 + -Inf -10.37 3.0 114ms
31 -35.98976678452 -13.85 -11.02 8.0 180ms
32 -35.98976678452 + -Inf -11.25 3.0 140ms
33 -35.98976678452 + -14.15 -11.51 3.0 139ms
34 -35.98976678452 + -14.15 -11.65 2.0 103ms
35 -35.98976678452 -13.85 -12.11 2.0 102msGiven 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 - 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))44.387453262984856The 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_min46.625476116580735This 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))4.685461232752242Since 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).