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
using LazyArtifacts
al_supercell = bulk(:Al; cubic=true) * (4, 1, 1)
system_Al = attach_psp(al_supercell;
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_DFT(system_Al; functionals=LDA(), 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 -35.98030221165 -0.86 12.0 395ms
2 -35.94563716634 + -1.46 -1.53 1.0 91.9ms
3 +7.968857763187 + 1.64 -0.25 7.0 228ms
4 -35.90090928684 1.64 -1.17 9.0 233ms
5 -35.68700922690 + -0.67 -1.28 3.0 143ms
6 -35.30007109112 + -0.41 -1.14 4.0 155ms
7 -35.97387483223 -0.17 -1.82 4.0 156ms
8 -35.98812496510 -1.85 -2.11 2.0 118ms
9 -35.98537913505 + -2.56 -2.09 2.0 130ms
10 -35.98783421416 -2.61 -2.12 2.0 128ms
11 -35.98942964347 -2.80 -2.40 2.0 115ms
12 -35.98946130373 -4.50 -2.76 1.0 102ms
13 -35.98962048581 -3.80 -3.08 1.0 97.5ms
14 -35.98962699496 -5.19 -3.25 4.0 145ms
15 -35.98915640365 + -3.33 -2.73 3.0 146ms
16 -35.98958053758 -3.37 -3.08 4.0 199ms
17 -35.98956923684 + -4.95 -3.10 3.0 140ms
18 -35.98917368079 + -3.40 -2.73 3.0 147ms
19 -35.98962819307 -3.34 -3.63 4.0 156ms
20 -35.98963172448 -5.45 -4.14 2.0 131ms
21 -35.98963187538 -6.82 -4.20 2.0 134ms
22 -35.98963195790 -7.08 -4.56 1.0 101ms
23 -35.98963197843 -7.69 -4.79 3.0 163ms
24 -35.98963198574 -8.14 -5.16 2.0 137ms
25 -35.98963198489 + -9.07 -5.31 2.0 108ms
26 -35.98963198540 -9.29 -5.47 2.0 104ms
27 -35.98963198582 -9.37 -5.75 1.0 101ms
28 -35.98963193318 + -7.28 -4.70 5.0 201ms
29 -35.98963198604 -7.28 -6.01 4.0 189ms
30 -35.98963198567 + -9.43 -5.73 3.0 154ms
31 -35.98963198611 -9.35 -6.35 4.0 162ms
32 -35.98963198612 -11.02 -6.54 2.0 138ms
33 -35.98963198613 -11.21 -6.89 2.0 103ms
34 -35.98963198613 -12.46 -7.13 2.0 109ms
35 -35.98963198613 -13.37 -7.16 3.0 144ms
36 -35.98963198613 -12.85 -7.28 3.0 126ms
37 -35.98963198613 -12.77 -7.61 2.0 107ms
38 -35.98963198613 -14.15 -8.08 1.0 101ms
39 -35.98963198613 + -Inf -7.91 3.0 167ms
40 -35.98963198613 -13.85 -8.27 5.0 141ms
41 -35.98963198613 + -13.67 -7.84 4.0 164ms
42 -35.98963198613 -13.85 -8.56 3.0 147ms
43 -35.98963198613 -13.85 -8.44 3.0 158ms
44 -35.98963198613 + -13.85 -9.09 2.0 115ms
45 -35.98963198613 -13.85 -8.91 4.0 143ms
46 -35.98963198613 + -14.15 -9.38 2.0 123ms
47 -35.98963198613 + -14.15 -9.50 3.0 163ms
48 -35.98963198613 + -Inf -9.84 1.0 97.6ms
49 -35.98963198613 -14.15 -10.17 1.0 101ms
50 -35.98963198613 + -Inf -10.16 2.0 138ms
51 -35.98963198613 + -14.15 -10.42 2.0 120ms
52 -35.98963198613 + -Inf -10.72 2.0 107ms
53 -35.98963198613 -13.85 -10.48 3.0 208ms
54 -35.98963198613 + -13.85 -10.71 4.0 798ms
55 -35.98963198613 -14.15 -10.58 2.0 115ms
56 -35.98963198613 + -Inf -10.38 3.0 162ms
57 -35.98963198613 + -14.15 -11.25 3.0 137ms
58 -35.98963198613 -13.85 -11.62 3.0 149ms
59 -35.98963198613 + -13.85 -11.57 3.0 143ms
60 -35.98963198613 -13.85 -12.07 2.0 104mswhile 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 -35.98280387627 -0.86 10.0 377ms
2 -35.98799533752 -2.28 -1.34 1.0 93.4ms
3 -35.98856402881 -3.25 -1.94 3.0 129ms
4 -35.98919142259 -3.20 -2.04 5.0 154ms
5 -35.98959435790 -3.39 -2.78 1.0 101ms
6 -35.98958547570 + -5.05 -2.68 10.0 189ms
7 -35.98962454826 -4.41 -3.00 1.0 96.2ms
8 -35.98963104025 -5.19 -3.43 1.0 103ms
9 -35.98963091812 + -6.91 -3.37 3.0 125ms
10 -35.98963169043 -6.11 -3.72 1.0 102ms
11 -35.98963191268 -6.65 -4.35 2.0 106ms
12 -35.98963194109 -7.55 -4.59 5.0 170ms
13 -35.98963198479 -7.36 -4.91 6.0 145ms
14 -35.98963198158 + -8.49 -5.17 3.0 120ms
15 -35.98963198608 -8.35 -5.60 3.0 147ms
16 -35.98963198612 -10.39 -6.09 4.0 125ms
17 -35.98963198613 -11.03 -6.38 4.0 158ms
18 -35.98963198613 -11.80 -6.88 7.0 145ms
19 -35.98963198613 -12.89 -7.01 3.0 150ms
20 -35.98963198613 -12.87 -7.33 1.0 109ms
21 -35.98963198613 -14.15 -7.82 5.0 142ms
22 -35.98963198613 -13.85 -8.16 4.0 156ms
23 -35.98963198613 + -Inf -8.37 6.0 140ms
24 -35.98963198613 -14.15 -8.53 3.0 150ms
25 -35.98963198613 + -14.15 -9.10 1.0 104ms
26 -35.98963198613 + -14.15 -9.45 3.0 149ms
27 -35.98963198613 -14.15 -9.74 6.0 157ms
28 -35.98963198613 + -14.15 -10.14 4.0 135ms
29 -35.98963198613 + -Inf -10.35 3.0 148ms
30 -35.98963198613 -14.15 -10.66 2.0 111ms
31 -35.98963198613 -14.15 -11.17 4.0 148ms
32 -35.98963198613 + -13.85 -11.51 6.0 169ms
33 -35.98963198613 -13.85 -11.60 3.0 120ms
34 -35.98963198613 + -13.67 -11.90 2.0 111ms
35 -35.98963198613 -13.67 -12.47 4.0 161msGiven 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))44.42459978718736The 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_min46.664495574776645This 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))4.725308006541669Since 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).