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   -36.73260571261                   -0.88   12.0    340ms
  2   -36.73426117503       -2.78       -1.62    1.0   92.6ms
  3   -30.13148225263   +    0.82       -0.65    4.0    161ms
  4   -35.95141590417        0.76       -1.01    4.0    160ms
  5   -36.61536145233       -0.18       -1.46    3.0    134ms
  6   -35.93275514924   +   -0.17       -1.11    4.0    144ms
  7   -36.71642685032       -0.11       -1.66    3.0    141ms
  8   -36.74047107031       -1.62       -2.11    2.0    102ms
  9   -36.73925089459   +   -2.91       -2.08    2.0    126ms
 10   -36.74222963528       -2.53       -2.30    1.0   93.2ms
 11   -36.74215279477   +   -4.11       -2.61    2.0    101ms
 12   -36.74226658969       -3.94       -2.76    2.0    107ms
 13   -36.74245446406       -3.73       -3.17    1.0   96.7ms
 14   -36.74246526339       -4.97       -3.31    2.0    131ms
 15   -36.74245779559   +   -5.13       -3.19    2.0    121ms
 16   -36.74232935071   +   -3.89       -2.95    3.0    138ms
 17   -36.74247757010       -3.83       -3.61    3.0    131ms
 18   -36.74245735104   +   -4.69       -3.37    3.0    129ms
 19   -36.74246813206       -4.97       -3.49    3.0    136ms
 20   -36.74248051488       -4.91       -4.29    2.0    109ms
 21   -36.74247982029   +   -6.16       -3.96    3.0    152ms
 22   -36.74247961720   +   -6.69       -3.91    3.0    134ms
 23   -36.74248065300       -5.98       -4.70    3.0    122ms
 24   -36.74248065410       -8.96       -4.80    2.0    129ms
 25   -36.74248066881       -7.83       -5.10    2.0    104ms
 26   -36.74248066985       -8.98       -5.04    2.0    108ms
 27   -36.74248067209       -8.65       -5.55    1.0   93.1ms
 28   -36.74248064709   +   -7.60       -4.83    4.0    156ms
 29   -36.74248067211       -7.60       -5.65    4.0    158ms
 30   -36.74248066824   +   -8.41       -5.21    3.0    135ms
 31   -36.74248067252       -8.37       -5.82    3.0    132ms
 32   -36.74248067264       -9.90       -5.96    2.0    116ms
 33   -36.74248067000   +   -8.58       -5.33    3.0    142ms
 34   -36.74248067251       -8.60       -5.81    3.0    137ms
 35   -36.74248067268       -9.79       -6.26    2.0    112ms
 36   -36.74248067267   +  -11.00       -6.38    3.0    120ms
 37   -36.74248067268      -10.89       -6.67    1.0   94.0ms
 38   -36.74248067268      -11.60       -6.91    2.0    128ms
 39   -36.74248067268   +  -13.85       -7.23    1.0   94.9ms
 40   -36.74248067268      -12.75       -7.56    2.0    105ms
 41   -36.74248067268   +  -11.85       -6.95    4.0    148ms
 42   -36.74248067268      -11.87       -7.44    3.0    135ms
 43   -36.74248067268      -12.85       -7.94    3.0    129ms
 44   -36.74248067268      -14.15       -8.23    2.0    102ms
 45   -36.74248067268   +  -13.85       -8.01    3.0    133ms
 46   -36.74248067268      -14.15       -8.30    3.0    127ms
 47   -36.74248067268      -14.15       -8.57    2.0    104ms
 48   -36.74248067268   +  -14.15       -8.93    2.0    111ms
 49   -36.74248067268      -13.85       -9.01    2.0    125ms
 50   -36.74248067268   +    -Inf       -9.02    3.0    113ms
 51   -36.74248067268   +    -Inf       -9.35    1.0   98.9ms
 52   -36.74248067268   +    -Inf       -9.71    3.0    118ms
 53   -36.74248067268   +  -13.85       -9.52    3.0    132ms
 54   -36.74248067268      -13.85       -9.72    2.0    121ms
 55   -36.74248067268   +    -Inf       -9.38    3.0    150ms
 56   -36.74248067268   +    -Inf       -9.70    3.0    140ms
 57   -36.74248067268   +  -13.85      -10.08    2.0    109ms
 58   -36.74248067268      -13.85      -10.20    1.0   97.0ms
 59   -36.74248067268   +    -Inf      -10.42    3.0    132ms
 60   -36.74248067268   +  -14.15      -10.57    2.0    138ms
 61   -36.74248067268      -14.15      -10.75    2.0    118ms
 62   -36.74248067268   +    -Inf      -10.84    3.0    137ms
 63   -36.74248067268   +  -14.15      -11.02    1.0    128ms
 64   -36.74248067268   +    -Inf      -11.17    2.0    115ms
 65   -36.74248067268   +    -Inf      -11.20    2.0    109ms
 66   -36.74248067268   +  -14.15      -11.57    1.0    100ms
 67   -36.74248067268      -13.55      -11.12    3.0    156ms
 68   -36.74248067268   +  -13.85      -11.53    3.0    156ms
 69   -36.74248067268   +    -Inf      -12.14    2.0    108ms

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   -36.73386796139                   -0.88   12.0    400ms
  2   -36.74014855843       -2.20       -1.36    1.0   95.8ms
  3   -36.73953580366   +   -3.21       -1.62    4.0    145ms
  4   -36.74230116214       -2.56       -2.26    1.0   95.5ms
  5   -36.74240873499       -3.97       -2.56    8.0    161ms
  6   -36.74244636568       -4.42       -2.57    2.0    115ms
  7   -36.74247109434       -4.61       -2.92    1.0    100ms
  8   -36.74247644584       -5.27       -3.13    1.0    100ms
  9   -36.74247886418       -5.62       -3.30    2.0    119ms
 10   -36.74248041227       -5.81       -4.15    1.0   98.1ms
 11   -36.74248062083       -6.68       -4.26    5.0    179ms
 12   -36.74248066666       -7.34       -4.65    2.0    107ms
 13   -36.74248066874       -8.68       -4.98    2.0    138ms
 14   -36.74248067237       -8.44       -5.35    2.0    109ms
 15   -36.74248067261       -9.62       -5.79    3.0    123ms
 16   -36.74248067267      -10.24       -6.05    3.0    150ms
 17   -36.74248067267      -11.35       -6.21    2.0    110ms
 18   -36.74248067268      -10.99       -6.46    1.0    104ms
 19   -36.74248067268      -11.96       -6.82    2.0    105ms
 20   -36.74248067268      -11.98       -7.09    7.0    163ms
 21   -36.74248067268      -13.37       -7.24    2.0    122ms
 22   -36.74248067268      -13.25       -7.54    2.0    132ms
 23   -36.74248067268      -13.67       -7.85    2.0    115ms
 24   -36.74248067268   +    -Inf       -8.20    2.0    137ms
 25   -36.74248067268   +  -13.85       -8.53    2.0    117ms
 26   -36.74248067268      -13.85       -8.63    2.0    142ms
 27   -36.74248067268   +    -Inf       -9.43    1.0    101ms
 28   -36.74248067268   +    -Inf       -9.30    4.0    188ms
 29   -36.74248067268   +    -Inf       -9.74    2.0    119ms
 30   -36.74248067268   +  -13.85      -10.04    2.0    129ms
 31   -36.74248067268      -14.15      -10.46    1.0    104ms
 32   -36.74248067268   +    -Inf      -10.70    2.0    158ms
 33   -36.74248067268      -14.15      -11.06    1.0    126ms
 34   -36.74248067268   +    -Inf      -11.19    3.0    149ms
 35   -36.74248067268   +    -Inf      -11.61    2.0    139ms
 36   -36.74248067268   +  -13.85      -11.82    1.0    106ms
 37   -36.74248067268      -13.85      -12.36    2.0    133ms

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
end
epsilon (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))
44.02449311217027

The smallest eigenvalue is a bit more tricky to obtain, so we will just assume

λ_Simple_min = 0.952
0.952

This makes the condition number around 30:

cond_Simple = λ_Simple_max / λ_Simple_min
46.24421545396037

This 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))
Example block output

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))
Example block output

Clearly the charge-sloshing mode is no longer dominating.

The largest eigenvalue is now

maximum(real.(λ_Kerker))
4.723576776606016

Since 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).