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 -36.73273333971 -0.88 11.0 1.10s
2 -36.65008377884 + -1.08 -1.47 1.0 202ms
3 +31.39938771671 + 1.83 -0.14 8.0 180ms
4 -36.66450523062 1.83 -1.29 7.0 220ms
5 -35.92017090489 + -0.13 -1.11 3.0 97.3ms
6 -36.50839181391 -0.23 -1.37 4.0 118ms
7 -36.71819455280 -0.68 -1.82 2.0 118ms
8 -36.73909988706 -1.68 -1.95 2.0 93.1ms
9 -36.73792532512 + -2.93 -2.01 2.0 83.9ms
10 -36.74192678791 -2.40 -2.47 1.0 72.5ms
11 -36.74239574016 -3.33 -2.47 3.0 88.1ms
12 -36.74243426014 -4.41 -2.84 2.0 76.8ms
13 -36.74239649202 + -4.42 -2.89 3.0 91.9ms
14 -36.74247511022 -4.10 -3.10 1.0 73.1ms
15 -36.74150623774 + -3.01 -2.56 3.0 112ms
16 -36.74232339568 -3.09 -2.78 3.0 112ms
17 -36.74207525903 + -3.61 -2.74 3.0 101ms
18 -36.74245864937 -3.42 -3.35 3.0 107ms
19 -36.74246439835 -5.24 -3.44 3.0 107ms
20 -36.74247902124 -4.83 -3.85 2.0 81.0ms
21 -36.74248053792 -5.82 -4.09 2.0 92.8ms
22 -36.74248062323 -7.07 -4.45 2.0 83.8ms
23 -36.74248066823 -7.35 -4.77 2.0 76.0ms
24 -36.74248066226 + -8.22 -4.96 2.0 107ms
25 -36.74248066145 + -9.09 -4.79 2.0 82.8ms
26 -36.74248066985 -8.08 -5.27 1.0 74.7ms
27 -36.74248067058 -9.13 -5.27 3.0 104ms
28 -36.74248067231 -8.76 -5.61 2.0 93.5ms
29 -36.74248067031 + -8.70 -5.32 3.0 101ms
30 -36.74248067266 -8.63 -6.10 3.0 91.9ms
31 -36.74248067264 + -10.72 -6.16 3.0 99.2ms
32 -36.74248067234 + -9.53 -5.78 3.0 103ms
33 -36.74248067268 -9.48 -6.35 3.0 101ms
34 -36.74248067265 + -10.66 -6.22 3.0 101ms
35 -36.74248067268 -10.57 -6.81 2.0 74.2ms
36 -36.74248067268 + -12.28 -6.72 3.0 104ms
37 -36.74248067268 -11.64 -6.99 2.0 78.0ms
38 -36.74248067268 + -11.89 -6.84 3.0 97.9ms
39 -36.74248067268 -11.79 -7.16 3.0 93.6ms
40 -36.74248067268 -12.64 -7.31 1.0 67.2ms
41 -36.74248067268 -12.69 -7.63 2.0 78.8ms
42 -36.74248067268 + -13.67 -7.57 3.0 98.5ms
43 -36.74248067268 -13.19 -8.01 2.0 77.4ms
44 -36.74248067268 + -Inf -8.26 3.0 92.5ms
45 -36.74248067268 + -Inf -8.65 2.0 81.4ms
46 -36.74248067268 + -13.85 -8.32 3.0 107ms
47 -36.74248067268 -13.67 -8.97 2.0 90.6ms
48 -36.74248067268 + -13.85 -8.34 4.0 112ms
49 -36.74248067268 -13.85 -9.39 3.0 115ms
50 -36.74248067268 + -14.15 -9.02 4.0 113ms
51 -36.74248067268 -14.15 -9.91 3.0 98.6ms
52 -36.74248067268 + -13.85 -10.01 2.0 93.7ms
53 -36.74248067268 -13.85 -9.97 2.0 208ms
54 -36.74248067268 + -Inf -9.94 3.0 129ms
55 -36.74248067268 + -14.15 -10.55 2.0 820ms
56 -36.74248067268 + -Inf -10.40 3.0 112ms
57 -36.74248067268 + -14.15 -10.91 2.0 93.1ms
58 -36.74248067268 + -Inf -10.87 3.0 109ms
59 -36.74248067268 -14.15 -11.41 2.0 90.3ms
60 -36.74248067268 + -Inf -11.47 2.0 96.8ms
61 -36.74248067268 -14.15 -11.26 2.0 106ms
62 -36.74248067268 + -Inf -11.74 3.0 108ms
63 -36.74248067268 + -14.15 -12.14 2.0 93.8ms
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 -36.73268481829 -0.88 11.0 774ms
2 -36.74008501051 -2.13 -1.36 1.0 556ms
3 -36.74040261777 -3.50 -1.70 3.0 111ms
4 -36.74224145814 -2.74 -2.16 1.0 76.9ms
5 -36.74243357423 -3.72 -2.66 4.0 91.8ms
6 -36.74242404522 + -5.02 -2.47 3.0 109ms
7 -36.74247755649 -4.27 -3.15 1.0 112ms
8 -36.74247762875 -7.14 -3.13 4.0 80.8ms
9 -36.74247914816 -5.82 -3.29 1.0 66.1ms
10 -36.74248050134 -5.87 -4.07 1.0 66.9ms
11 -36.74248064345 -6.85 -4.21 6.0 109ms
12 -36.74248066205 -7.73 -4.33 1.0 69.1ms
13 -36.74248066799 -8.23 -4.52 2.0 105ms
14 -36.74248067124 -8.49 -4.84 1.0 68.5ms
15 -36.74248067243 -8.93 -5.33 3.0 82.1ms
16 -36.74248067265 -9.64 -5.68 3.0 102ms
17 -36.74248067267 -10.82 -5.88 5.0 86.5ms
18 -36.74248067268 -10.93 -6.32 2.0 98.0ms
19 -36.74248067268 -11.54 -6.47 2.0 96.0ms
20 -36.74248067268 -12.44 -6.99 1.0 74.0ms
21 -36.74248067268 -13.37 -7.41 3.0 97.0ms
22 -36.74248067268 -13.85 -7.41 3.0 104ms
23 -36.74248067268 -13.85 -7.62 1.0 70.2ms
24 -36.74248067268 + -14.15 -8.18 2.0 86.0ms
25 -36.74248067268 + -14.15 -8.23 3.0 97.3ms
26 -36.74248067268 -13.67 -8.70 1.0 77.6ms
27 -36.74248067268 + -13.67 -8.94 3.0 105ms
28 -36.74248067268 -14.15 -9.14 1.0 94.0ms
29 -36.74248067268 -14.15 -9.43 2.0 74.2ms
30 -36.74248067268 + -Inf -10.10 3.0 93.8ms
31 -36.74248067268 + -13.85 -10.33 3.0 103ms
32 -36.74248067268 -13.85 -10.60 3.0 80.5ms
33 -36.74248067268 + -13.85 -10.60 2.0 87.6ms
34 -36.74248067268 -13.85 -11.15 1.0 70.0ms
35 -36.74248067268 + -Inf -11.33 4.0 81.3ms
36 -36.74248067268 + -Inf -11.82 2.0 81.1ms
37 -36.74248067268 + -14.15 -12.14 3.0 96.6ms
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))44.02448898017208The 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_min46.244211113626136This 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))4.723583877505836Since 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).