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 -36.73104066971 -0.88 11.0 338ms
2 -36.41915499678 + -0.51 -1.26 1.0 89.6ms
3 +120.0165312252 + 2.19 0.04 17.0 334ms
4 -32.67490232501 2.18 -0.67 10.0 279ms
5 -30.46012523345 + 0.35 -0.64 5.0 174ms
6 -34.27772509235 0.58 -0.81 5.0 164ms
7 -36.63193601652 0.37 -1.45 4.0 156ms
8 -36.70848927710 -1.12 -1.64 2.0 107ms
9 -36.73789523093 -1.53 -1.96 2.0 121ms
10 -36.74075382940 -2.54 -2.06 3.0 129ms
11 -36.73864753409 + -2.68 -2.08 2.0 108ms
12 -36.74048612583 -2.74 -2.26 1.0 91.2ms
13 -36.74059293731 -3.97 -2.33 1.0 94.5ms
14 -36.74182463996 -2.91 -2.51 1.0 94.3ms
15 -36.74163159031 + -3.71 -2.49 1.0 92.1ms
16 -36.70992472228 + -1.50 -1.80 3.0 144ms
17 -36.74015334767 -1.52 -2.37 4.0 153ms
18 -36.74238805595 -2.65 -2.78 2.0 110ms
19 -36.73970324761 + -2.57 -2.33 3.0 136ms
20 -36.74245433018 -2.56 -3.06 3.0 131ms
21 -36.74247372796 -4.71 -3.03 2.0 128ms
22 -36.74250563735 -4.50 -3.46 2.0 103ms
23 -36.74250989020 -5.37 -3.67 3.0 129ms
24 -36.74251351495 -5.44 -3.74 2.0 125ms
25 -36.74251426271 -6.13 -3.97 1.0 95.2ms
26 -36.74251469943 -6.36 -4.29 2.0 110ms
27 -36.74251475170 -7.28 -4.62 2.0 101ms
28 -36.74251466724 + -7.07 -4.50 3.0 156ms
29 -36.74251468541 -7.74 -4.56 3.0 134ms
30 -36.74251475946 -7.13 -4.96 2.0 104ms
31 -36.74251463715 + -6.91 -4.49 3.0 136ms
32 -36.74251477287 -6.87 -5.53 3.0 131ms
33 -36.74251477271 + -9.78 -5.49 3.0 137ms
34 -36.74251476692 + -8.24 -5.05 3.0 137ms
35 -36.74251476528 + -8.79 -5.10 3.0 136ms
36 -36.74251477102 -8.24 -5.32 3.0 147ms
37 -36.74251477296 -8.71 -5.92 2.0 107ms
38 -36.74251477299 -10.58 -6.09 3.0 118ms
39 -36.74251477301 -10.62 -6.30 2.0 120ms
40 -36.74251477303 -10.69 -6.57 3.0 137ms
41 -36.74251477303 + -11.74 -6.51 2.0 100ms
42 -36.74251477304 -11.14 -6.81 2.0 110ms
43 -36.74251477302 + -10.69 -6.38 3.0 134ms
44 -36.74251477304 -10.71 -6.83 3.0 162ms
45 -36.74251477304 -11.87 -7.34 2.0 106ms
46 -36.74251477304 -13.67 -7.33 2.0 128ms
47 -36.74251477304 + -12.07 -7.02 3.0 133ms
48 -36.74251477304 -12.16 -7.26 3.0 134ms
49 -36.74251477304 -12.64 -8.07 2.0 103ms
50 -36.74251477304 + -13.85 -8.11 3.0 142ms
51 -36.74251477304 + -Inf -7.88 2.0 121ms
52 -36.74251477304 + -Inf -7.93 2.0 117ms
53 -36.74251477304 -14.15 -8.13 3.0 277ms
54 -36.74251477304 -14.15 -8.63 2.0 725ms
55 -36.74251477304 + -13.85 -8.53 3.0 147ms
56 -36.74251477304 + -Inf -8.02 3.0 145ms
57 -36.74251477304 + -Inf -8.91 3.0 135ms
58 -36.74251477304 + -Inf -9.08 3.0 118ms
59 -36.74251477304 + -Inf -9.30 2.0 120ms
60 -36.74251477304 -13.85 -9.55 2.0 125ms
61 -36.74251477304 + -Inf -9.65 2.0 149ms
62 -36.74251477304 + -14.15 -9.31 3.0 150ms
63 -36.74251477304 -14.15 -9.57 2.0 120ms
64 -36.74251477304 + -14.15 -9.93 2.0 110ms
65 -36.74251477304 -14.15 -9.93 2.0 118ms
66 -36.74251477304 + -Inf -10.51 2.0 102ms
67 -36.74251477304 + -13.85 -10.39 3.0 146ms
68 -36.74251477304 -13.85 -9.84 3.0 142ms
69 -36.74251477304 + -13.85 -10.93 3.0 132ms
70 -36.74251477304 -14.15 -11.03 2.0 129ms
71 -36.74251477304 -14.15 -10.72 3.0 126ms
72 -36.74251477304 -14.15 -10.91 3.0 135ms
73 -36.74251477304 + -14.15 -11.11 2.0 110ms
74 -36.74251477304 + -Inf -11.22 2.0 107ms
75 -36.74251477304 + -13.85 -11.19 2.0 110ms
76 -36.74251477304 + -Inf -11.58 2.0 106ms
77 -36.74251477304 -13.85 -11.43 3.0 130ms
78 -36.74251477304 + -13.85 -12.17 2.0 109mswhile 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 -36.73367121135 -0.88 11.0 369ms
2 -36.74024080701 -2.18 -1.36 1.0 87.5ms
3 -36.74159184267 -2.87 -1.79 3.0 129ms
4 -36.74239357136 -3.10 -2.24 1.0 91.3ms
5 -36.74244095277 -4.32 -2.63 6.0 118ms
6 -36.74244562308 -5.33 -2.45 4.0 131ms
7 -36.74251104887 -4.18 -3.36 1.0 93.0ms
8 -36.74251375047 -5.57 -3.49 3.0 141ms
9 -36.74251394215 -6.72 -3.55 1.0 94.3ms
10 -36.74251449270 -6.26 -3.90 1.0 95.3ms
11 -36.74251467358 -6.74 -4.41 1.0 97.3ms
12 -36.74251477100 -7.01 -4.84 3.0 132ms
13 -36.74251477254 -8.81 -5.29 6.0 130ms
14 -36.74251477284 -9.53 -5.48 2.0 130ms
15 -36.74251477301 -9.77 -6.00 1.0 96.7ms
16 -36.74251477300 + -11.15 -6.07 3.0 129ms
17 -36.74251477304 -10.46 -6.61 1.0 96.6ms
18 -36.74251477304 + -12.50 -6.85 3.0 139ms
19 -36.74251477304 -12.15 -7.24 2.0 102ms
20 -36.74251477304 -13.37 -7.64 3.0 141ms
21 -36.74251477304 -13.67 -8.08 2.0 109ms
22 -36.74251477304 + -Inf -8.36 4.0 137ms
23 -36.74251477304 + -Inf -8.63 5.0 117ms
24 -36.74251477304 + -13.85 -8.98 2.0 105ms
25 -36.74251477304 + -Inf -9.20 2.0 129ms
26 -36.74251477304 -14.15 -9.35 1.0 96.3ms
27 -36.74251477304 -13.85 -9.68 1.0 96.3ms
28 -36.74251477304 + -14.15 -9.81 2.0 103ms
29 -36.74251477304 + -14.15 -10.32 2.0 111ms
30 -36.74251477304 + -13.85 -10.24 3.0 142ms
31 -36.74251477304 -13.67 -10.27 1.0 96.2ms
32 -36.74251477304 + -Inf -10.29 1.0 96.0ms
33 -36.74251477304 + -Inf -10.43 1.0 96.0ms
34 -36.74251477304 + -13.85 -10.41 1.0 96.0ms
35 -36.74251477304 -14.15 -10.43 1.0 93.1ms
36 -36.74251477304 -14.15 -10.44 1.0 95.8ms
37 -36.74251477304 + -13.85 -10.46 1.0 96.1ms
38 -36.74251477304 -13.85 -10.48 1.0 96.0ms
39 -36.74251477304 + -13.85 -10.50 1.0 93.1ms
40 -36.74251477304 -13.85 -10.55 1.0 120ms
41 -36.74251477304 + -14.15 -11.16 1.0 95.7ms
42 -36.74251477304 + -14.15 -11.32 5.0 140ms
43 -36.74251477304 + -Inf -11.56 2.0 112ms
44 -36.74251477304 + -Inf -11.90 2.0 129ms
45 -36.74251477304 -13.85 -12.15 1.0 96.0msGiven 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))44.024489068379445The 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_min46.24421120628093This 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))4.723583891357821Since 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).