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 -36.73171714185 -0.88 12.0 1.37s
2 -36.57701299322 + -0.81 -1.38 1.0 327ms
┌ Warning: Eigensolver not converged
│ n_iter =
│ 1-element Vector{Int64}:
│ 23
└ @ DFTK ~/work/DFTK.jl/DFTK.jl/src/scf/self_consistent_field.jl:102
3 +57.19557722483 + 1.97 -0.07 23.0 411ms
4 -36.22912695249 1.97 -1.04 10.0 278ms
5 -35.53293845754 + -0.16 -1.00 4.0 153ms
6 -35.03760776234 + -0.31 -0.93 6.0 204ms
7 -36.71255013900 0.22 -1.58 3.0 126ms
8 -36.73755597859 -1.60 -2.05 2.0 91.8ms
9 -36.73812168523 -3.25 -1.91 2.0 136ms
10 -36.74192109165 -2.42 -2.29 2.0 113ms
11 -36.74099149525 + -3.03 -2.18 2.0 111ms
12 -36.74238586690 -2.86 -2.59 1.0 89.2ms
13 -36.74234302266 + -4.37 -2.43 2.0 119ms
14 -36.74244985271 -3.97 -2.81 1.0 89.2ms
15 -36.73749798383 + -2.31 -2.20 3.0 136ms
16 -36.74227832157 -2.32 -2.83 4.0 141ms
17 -36.74117300093 + -2.96 -2.47 3.0 132ms
18 -36.73717612984 + -2.40 -2.19 4.0 142ms
19 -36.74241050712 -2.28 -2.98 3.0 142ms
20 -36.74246681771 -4.25 -2.99 2.0 103ms
21 -36.74245317604 + -4.87 -3.06 2.0 105ms
22 -36.74247959449 -4.58 -3.62 1.0 89.6ms
23 -36.74247360356 + -5.22 -3.38 3.0 140ms
24 -36.74247977008 -5.21 -3.89 2.0 105ms
25 -36.74248056774 -6.10 -4.18 2.0 115ms
26 -36.74248041578 + -6.82 -4.24 2.0 121ms
27 -36.74248062086 -6.69 -4.58 1.0 94.0ms
28 -36.74247925486 + -5.86 -3.98 3.0 128ms
29 -36.74248063888 -5.86 -4.76 3.0 142ms
30 -36.74247626075 + -5.36 -3.75 4.0 151ms
31 -36.74248044867 -5.38 -4.28 4.0 157ms
32 -36.74248067080 -6.65 -5.25 3.0 128ms
33 -36.74248067214 -8.87 -5.44 2.0 126ms
34 -36.74248067220 -10.25 -5.64 1.0 89.3ms
35 -36.74248067250 -9.51 -5.80 2.0 108ms
36 -36.74248067260 -10.03 -5.95 2.0 99.1ms
37 -36.74248067268 -10.09 -6.56 2.0 126ms
38 -36.74248067268 -11.63 -6.61 2.0 122ms
39 -36.74248067268 -11.77 -6.99 1.0 94.0ms
40 -36.74248067268 + -11.47 -6.74 3.0 127ms
41 -36.74248067268 -11.44 -7.08 3.0 132ms
42 -36.74248067268 + -11.79 -6.90 3.0 122ms
43 -36.74248067268 -11.69 -7.63 3.0 127ms
44 -36.74248067268 -13.67 -7.72 2.0 121ms
45 -36.74248067268 + -Inf -7.71 2.0 108ms
46 -36.74248067268 -14.15 -8.38 1.0 89.0ms
47 -36.74248067268 -14.15 -8.14 3.0 144ms
48 -36.74248067268 + -14.15 -8.25 3.0 127ms
49 -36.74248067268 + -14.15 -8.92 2.0 108ms
50 -36.74248067268 -13.67 -8.97 3.0 129ms
51 -36.74248067268 + -Inf -9.35 1.0 93.6ms
52 -36.74248067268 + -14.15 -9.48 1.0 89.3ms
53 -36.74248067268 + -Inf -9.41 3.0 123ms
54 -36.74248067268 + -Inf -9.69 2.0 103ms
55 -36.74248067268 -14.15 -9.71 2.0 123ms
56 -36.74248067268 + -14.15 -9.81 3.0 116ms
57 -36.74248067268 + -14.15 -9.95 2.0 98.0ms
58 -36.74248067268 -13.85 -10.36 2.0 103ms
59 -36.74248067268 + -14.15 -10.40 2.0 121ms
60 -36.74248067268 + -14.15 -10.52 2.0 106ms
61 -36.74248067268 -14.15 -11.00 2.0 97.4ms
62 -36.74248067268 -14.15 -10.63 3.0 135ms
63 -36.74248067268 + -Inf -11.05 3.0 140ms
64 -36.74248067268 + -Inf -11.04 2.0 118ms
65 -36.74248067268 + -Inf -11.10 2.0 103ms
66 -36.74248067268 + -14.15 -11.36 2.0 108ms
67 -36.74248067268 + -Inf -11.66 1.0 88.5ms
68 -36.74248067268 -13.67 -11.45 3.0 136ms
69 -36.74248067268 + -13.55 -11.84 2.0 113ms
70 -36.74248067268 + -Inf -11.91 2.0 108ms
71 -36.74248067268 + -Inf -12.23 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 -36.73246762145 -0.88 10.0 1.00s
2 -36.73979400779 -2.14 -1.36 1.0 1.04s
3 -36.74014902877 -3.45 -1.71 4.0 153ms
4 -36.74220560790 -2.69 -2.17 1.0 87.5ms
5 -36.74238728684 -3.74 -2.58 5.0 107ms
6 -36.74240240936 -4.82 -2.41 2.0 128ms
7 -36.74247707940 -4.13 -3.19 1.0 88.0ms
8 -36.74247856989 -5.83 -3.20 3.0 231ms
9 -36.74247993105 -5.87 -3.43 1.0 89.2ms
10 -36.74248039237 -6.34 -3.80 1.0 1.26s
11 -36.74248059818 -6.69 -4.22 3.0 120ms
12 -36.74248066687 -7.16 -4.44 3.0 129ms
13 -36.74248066042 + -8.19 -4.74 1.0 90.5ms
14 -36.74248067242 -7.92 -5.24 1.0 90.8ms
15 -36.74248067263 -9.68 -5.47 6.0 144ms
16 -36.74248067267 -10.43 -5.79 1.0 93.2ms
17 -36.74248067266 + -11.56 -6.17 2.0 131ms
18 -36.74248067268 -10.78 -6.49 2.0 128ms
19 -36.74248067268 + -12.50 -6.74 3.0 153ms
20 -36.74248067268 -12.07 -7.09 1.0 111ms
21 -36.74248067268 -12.77 -7.50 5.0 156ms
22 -36.74248067268 + -13.55 -7.56 3.0 153ms
23 -36.74248067268 -13.55 -7.84 1.0 111ms
24 -36.74248067268 + -Inf -8.06 2.0 139ms
25 -36.74248067268 + -Inf -8.41 2.0 139ms
26 -36.74248067268 -13.85 -9.05 1.0 91.3ms
27 -36.74248067268 + -Inf -9.21 3.0 132ms
28 -36.74248067268 + -13.85 -9.53 1.0 91.0ms
29 -36.74248067268 -14.15 -10.00 3.0 130ms
30 -36.74248067268 + -Inf -10.38 3.0 114ms
31 -36.74248067268 + -Inf -10.37 3.0 134ms
32 -36.74248067268 -14.15 -10.84 2.0 105ms
33 -36.74248067268 + -Inf -10.92 3.0 133ms
34 -36.74248067268 + -Inf -11.30 1.0 90.1ms
35 -36.74248067268 + -13.85 -11.49 3.0 121ms
36 -36.74248067268 -14.15 -11.79 2.0 129ms
37 -36.74248067268 + -14.15 -12.10 2.0 106ms
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))44.02448906836615The 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_min46.24421120626697This 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))4.723581700102194Since 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).