Notebook

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:

In [1]:

using AtomsBuilder
using DFTK

system_Al = bulk(:Al; cubic=true) * (4, 1, 1)

Out[1]:

FlexibleSystem(Al₁₆, periodicity = TTT):
    cell_vectors      : [    16.2        0        0;
                                0     4.05        0;
                                0        0     4.05]u"Å"

and we discretise:

In [2]:

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:

In [3]:

# 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   -36.73372409678                   -0.88   11.0    480ms
  2   -36.54927769127   +   -0.73       -1.33    1.0    157ms
  3   +61.16604959193   +    1.99       -0.06   22.0    379ms
  4   -35.82323484490        1.99       -0.96   10.0    333ms
  5   -25.75016816170   +    1.00       -0.54    5.0    185ms
  6   -36.36466582549        1.03       -1.19    4.0    171ms
  7   -36.69438399901       -0.48       -1.57    3.0    170ms
  8   -36.71542333898       -1.68       -1.73    2.0    113ms
  9   -36.73973626349       -1.61       -2.06    2.0    108ms
 10   -36.73920968223   +   -3.28       -2.06    3.0    131ms
 11   -36.74133415779       -2.67       -2.28    2.0    110ms
 12   -36.74137938893       -4.34       -2.30    2.0    118ms
 13   -36.74152077867       -3.85       -2.42    1.0   99.0ms
 14   -36.74221927551       -3.16       -2.62    1.0   98.8ms
 15   -36.74132123693   +   -3.05       -2.47    2.0    124ms
 16   -36.73596244634   +   -2.27       -2.11    3.0    174ms
 17   -36.73736756507       -2.85       -2.21    3.0    157ms
 18   -36.74144784101       -2.39       -2.53    3.0    146ms
 19   -36.74150148859       -4.27       -2.50    3.0    147ms
 20   -36.74244708404       -3.02       -2.94    2.0    118ms
 21   -36.74247259177       -4.59       -3.09    2.0    136ms
 22   -36.74251298891       -4.39       -3.68    2.0    117ms
 23   -36.74251206447   +   -6.03       -3.76    2.0    159ms
 24   -36.74251434697       -5.64       -4.10    1.0   99.0ms
 25   -36.74251450683       -6.80       -4.01    2.0    127ms
 26   -36.74251455560       -7.31       -4.33    2.0    115ms
 27   -36.74251474605       -6.72       -4.43    2.0    124ms
 28   -36.74251453687   +   -6.68       -4.36    2.0    127ms
 29   -36.74251476405       -6.64       -4.97    2.0    117ms
 30   -36.74251476752       -8.46       -5.13    3.0    147ms
 31   -36.74251477028       -8.56       -5.20    3.0    178ms
 32   -36.74251477111       -9.08       -5.33    1.0   95.7ms
 33   -36.74251477251       -8.85       -5.66    2.0    117ms
 34   -36.74251477235   +   -9.79       -5.55    3.0    148ms
 35   -36.74251477230   +  -10.25       -5.56    3.0    139ms
 36   -36.74251477302       -9.14       -6.29    2.0    117ms
 37   -36.74251477290   +   -9.91       -5.98    3.0    148ms
 38   -36.74251477303       -9.89       -6.59    3.0    146ms
 39   -36.74251477300   +  -10.46       -6.15    3.0    176ms
 40   -36.74251477303      -10.45       -6.60    3.0    135ms
 41   -36.74251477304      -11.42       -6.98    2.0    117ms
 42   -36.74251477304      -12.58       -7.06    3.0    147ms
 43   -36.74251477304      -12.56       -7.12    1.0   98.3ms
 44   -36.74251477304      -12.89       -7.36    2.0    117ms
 45   -36.74251477304      -12.77       -7.77    1.0   98.4ms
 46   -36.74251477304   +  -12.87       -7.39    3.0    155ms
 47   -36.74251477304      -13.30       -7.40    3.0    173ms
 48   -36.74251477304   +    -Inf       -7.55    3.0    177ms
 49   -36.74251477304      -13.15       -7.81    3.0    143ms
 50   -36.74251477304      -13.67       -8.21    2.0    114ms
 51   -36.74251477304      -14.15       -8.18    3.0    140ms
 52   -36.74251477304   +  -14.15       -8.16    2.0    115ms
 53   -36.74251477304   +    -Inf       -8.22    3.0    123ms
 54   -36.74251477304      -14.15       -8.28    2.0    149ms
 55   -36.74251477304   +    -Inf       -8.90    1.0   99.3ms
 56   -36.74251477304   +    -Inf       -8.68    3.0    156ms
 57   -36.74251477304   +    -Inf       -8.67    2.0    119ms
 58   -36.74251477304   +    -Inf       -9.29    2.0    117ms
 59   -36.74251477304   +  -13.85       -8.60    3.0    166ms
 60   -36.74251477304      -14.15       -9.27    4.0    167ms
 61   -36.74251477304   +    -Inf       -8.59    4.0    169ms
 62   -36.74251477304      -14.15       -9.10    3.0    180ms
 63   -36.74251477304   +  -13.85       -9.24    2.0    129ms
 64   -36.74251477304      -14.15      -10.09    2.0    114ms
 65   -36.74251477304      -14.15       -9.70    4.0    176ms
 66   -36.74251477304   +  -13.85      -10.20    3.0    146ms
 67   -36.74251477304   +    -Inf      -10.24    1.0    101ms
 68   -36.74251477304   +    -Inf      -10.57    1.0    163ms
 69   -36.74251477304      -14.15      -10.49    3.0    692ms
 70   -36.74251477304   +  -14.15      -10.76    2.0    115ms
 71   -36.74251477304   +    -Inf      -10.80    2.0    115ms
 72   -36.74251477304      -13.85      -10.98    2.0    108ms
 73   -36.74251477304   +  -13.85      -11.19    2.0    138ms
 74   -36.74251477304      -13.85      -10.63    3.0    168ms
 75   -36.74251477304   +    -Inf      -11.35    3.0    165ms
 76   -36.74251477304   +  -13.85      -11.02    3.0    161ms
 77   -36.74251477304      -13.85      -11.74    3.0    159ms
 78   -36.74251477304   +  -14.15      -11.93    2.0    138ms
 79   -36.74251477304   +    -Inf      -12.23    2.0    121ms

while when using the Kerker preconditioner it is much faster:

In [4]:

scfres_Al = self_consistent_field(basis_Al; tol=1e-12, mixing=KerkerMixing());

n     Energy            log10(ΔE)   log10(Δρ)   Diag   Δtime
---   ---------------   ---------   ---------   ----   ------
  1   -36.73113129843                   -0.88   11.0    412ms
  2   -36.73922462098       -2.09       -1.36    1.0    657ms
  3   -36.73897761884   +   -3.61       -1.61    3.0    128ms
  4   -36.74219400837       -2.49       -2.16    2.0    112ms
  5   -36.74231044516       -3.93       -2.40    5.0    114ms
  6   -36.74244084393       -3.88       -2.45    2.0    103ms
  7   -36.74248499908       -4.36       -3.12    1.0   90.2ms
  8   -36.74251177551       -4.57       -3.22    3.0    151ms
  9   -36.74251295375       -5.93       -3.45    1.0   92.0ms
 10   -36.74251418318       -5.91       -3.76    2.0    101ms
 11   -36.74251471578       -6.27       -4.21    6.0    124ms
 12   -36.74251476845       -7.28       -4.66    2.0    129ms
 13   -36.74251476124   +   -8.14       -4.66    2.0    107ms
 14   -36.74251477267       -7.94       -5.35    1.0   96.0ms
 15   -36.74251477297       -9.52       -5.53    3.0    140ms
 16   -36.74251477301      -10.40       -5.71    2.0   98.3ms
 17   -36.74251477292   +  -10.04       -5.84    2.0    104ms
 18   -36.74251477304       -9.95       -6.58    1.0    114ms
 19   -36.74251477304      -12.10       -6.66    8.0    172ms
 20   -36.74251477304   +  -11.98       -6.82    1.0   95.9ms
 21   -36.74251477304      -12.13       -7.08    2.0    131ms
 22   -36.74251477304      -12.28       -7.57    2.0    106ms
 23   -36.74251477304   +  -14.15       -7.74    3.0    139ms
 24   -36.74251477304      -14.15       -7.94    1.0   96.1ms
 25   -36.74251477304   +    -Inf       -8.52    2.0    103ms
 26   -36.74251477304   +  -13.85       -8.41    3.0    143ms
 27   -36.74251477304      -13.85       -8.65    1.0    112ms
 28   -36.74251477304   +  -13.85       -9.20    2.0    105ms
 29   -36.74251477304   +    -Inf       -9.49    3.0    144ms
 30   -36.74251477304      -13.85       -9.91    1.0   96.2ms
 31   -36.74251477304   +    -Inf      -10.16    3.0    142ms
 32   -36.74251477304   +    -Inf      -10.56    1.0   96.0ms
 33   -36.74251477304   +  -14.15      -10.86    2.0    111ms
 34   -36.74251477304   +    -Inf      -11.31    3.0    128ms
 35   -36.74251477304   +    -Inf      -11.36    2.0    131ms
 36   -36.74251477304      -14.15      -11.80    2.0    104ms
 37   -36.74251477304   +  -14.15      -12.08    4.0    132ms

Given this scfres_Al we construct functions representing $\varepsilon^\dagger$ and $P^{-1}$ :

In [5]:

# 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
end

Out[5]:

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.

In [6]:

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))

┌ Info: Arnoldi eigsolve finished after 1 iterations:
│ * 7 eigenvalues converged
│ * norm of residuals = (8.18e-06, 2.11e-08, 6.48e-09, 1.67e-05, 9.36e-06, 2.68e-05, 2.73e-05)
└ * number of operations = 15

Out[6]:

44.02448779307413

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

In [7]:

λ_Simple_min = 0.952

Out[7]:

0.952

This makes the condition number around 30:

In [8]:

cond_Simple = λ_Simple_max / λ_Simple_min

Out[8]:

46.24420986667451

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:

In [9]:

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))

Out[9]:

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:

In [10]:

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))

┌ Info: Arnoldi eigsolve finished after 1 iterations:
│ * 3 eigenvalues converged
│ * norm of residuals = (2.12e-04, 4.21e-04, 2.33e-04)
└ * number of operations = 12

Out[10]:

Clearly the charge-sloshing mode is no longer dominating.

The largest eigenvalue is now

In [11]:

maximum(real.(λ_Kerker))

Out[11]:

4.7236110498662836

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).