{
"cells": [
{
"cell_type": "markdown",
"source": [
"# Analysing SCF convergence\n",
"\n",
"The goal of this example is to explain the differing convergence behaviour\n",
"of SCF algorithms depending on the choice of the mixing.\n",
"For this we look at the eigenpairs of the Jacobian governing the SCF convergence,\n",
"that is\n",
"$$\n",
"1 - α P^{-1} \\varepsilon^\\dagger \\qquad \\text{with} \\qquad \\varepsilon^\\dagger = (1-\\chi_0 K).\n",
"$$\n",
"where $α$ is the damping $P^{-1}$ is the mixing preconditioner\n",
"(e.g. `KerkerMixing`, `LdosMixing`)\n",
"and $\\varepsilon^\\dagger$ is the dielectric operator.\n",
"\n",
"We thus investigate the largest and smallest eigenvalues of\n",
"$(P^{-1} \\varepsilon^\\dagger)$ and $\\varepsilon^\\dagger$.\n",
"The ratio of largest to smallest eigenvalue of this operator is the condition number\n",
"$$\n",
"\\kappa = \\frac{\\lambda_\\text{max}}{\\lambda_\\text{min}},\n",
"$$\n",
"which can be related to the rate of convergence of the SCF.\n",
"The smaller the condition number, the faster the convergence.\n",
"For more details on SCF methods, see Self-consistent field methods.\n",
"\n",
"For our investigation we consider a crude aluminium setup:"
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "FlexibleSystem(Al₁₆, periodicity = TTT):\n cell_vectors : [ 16.2 0 0;\n 0 4.05 0;\n 0 0 4.05]u\"Å\"\n"
},
"metadata": {},
"execution_count": 1
}
],
"cell_type": "code",
"source": [
"using AtomsBuilder\n",
"using DFTK\n",
"\n",
"system_Al = bulk(:Al; cubic=true) * (4, 1, 1)"
],
"metadata": {},
"execution_count": 1
},
{
"cell_type": "markdown",
"source": [
"and we discretise:"
],
"metadata": {}
},
{
"outputs": [],
"cell_type": "code",
"source": [
"using PseudoPotentialData\n",
"\n",
"pseudopotentials = PseudoFamily(\"dojo.nc.sr.lda.v0_4_1.standard.upf\")\n",
"model_Al = model_DFT(system_Al; functionals=LDA(), temperature=1e-3,\n",
" symmetries=false, pseudopotentials)\n",
"basis_Al = PlaneWaveBasis(model_Al; Ecut=7, kgrid=[1, 1, 1]);"
],
"metadata": {},
"execution_count": 2
},
{
"cell_type": "markdown",
"source": [
"On aluminium (a metal) already for moderate system sizes (like the 8 layers\n",
"we consider here) the convergence without mixing / preconditioner is slow:"
],
"metadata": {}
},
{
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"n Energy log10(ΔE) log10(Δρ) Diag Δtime\n",
"--- --------------- --------- --------- ---- ------\n",
" 1 -36.73372409678 -0.88 11.0 480ms\n",
" 2 -36.54927769127 + -0.73 -1.33 1.0 157ms\n",
" 3 +61.16604959193 + 1.99 -0.06 22.0 379ms\n",
" 4 -35.82323484490 1.99 -0.96 10.0 333ms\n",
" 5 -25.75016816170 + 1.00 -0.54 5.0 185ms\n",
" 6 -36.36466582549 1.03 -1.19 4.0 171ms\n",
" 7 -36.69438399901 -0.48 -1.57 3.0 170ms\n",
" 8 -36.71542333898 -1.68 -1.73 2.0 113ms\n",
" 9 -36.73973626349 -1.61 -2.06 2.0 108ms\n",
" 10 -36.73920968223 + -3.28 -2.06 3.0 131ms\n",
" 11 -36.74133415779 -2.67 -2.28 2.0 110ms\n",
" 12 -36.74137938893 -4.34 -2.30 2.0 118ms\n",
" 13 -36.74152077867 -3.85 -2.42 1.0 99.0ms\n",
" 14 -36.74221927551 -3.16 -2.62 1.0 98.8ms\n",
" 15 -36.74132123693 + -3.05 -2.47 2.0 124ms\n",
" 16 -36.73596244634 + -2.27 -2.11 3.0 174ms\n",
" 17 -36.73736756507 -2.85 -2.21 3.0 157ms\n",
" 18 -36.74144784101 -2.39 -2.53 3.0 146ms\n",
" 19 -36.74150148859 -4.27 -2.50 3.0 147ms\n",
" 20 -36.74244708404 -3.02 -2.94 2.0 118ms\n",
" 21 -36.74247259177 -4.59 -3.09 2.0 136ms\n",
" 22 -36.74251298891 -4.39 -3.68 2.0 117ms\n",
" 23 -36.74251206447 + -6.03 -3.76 2.0 159ms\n",
" 24 -36.74251434697 -5.64 -4.10 1.0 99.0ms\n",
" 25 -36.74251450683 -6.80 -4.01 2.0 127ms\n",
" 26 -36.74251455560 -7.31 -4.33 2.0 115ms\n",
" 27 -36.74251474605 -6.72 -4.43 2.0 124ms\n",
" 28 -36.74251453687 + -6.68 -4.36 2.0 127ms\n",
" 29 -36.74251476405 -6.64 -4.97 2.0 117ms\n",
" 30 -36.74251476752 -8.46 -5.13 3.0 147ms\n",
" 31 -36.74251477028 -8.56 -5.20 3.0 178ms\n",
" 32 -36.74251477111 -9.08 -5.33 1.0 95.7ms\n",
" 33 -36.74251477251 -8.85 -5.66 2.0 117ms\n",
" 34 -36.74251477235 + -9.79 -5.55 3.0 148ms\n",
" 35 -36.74251477230 + -10.25 -5.56 3.0 139ms\n",
" 36 -36.74251477302 -9.14 -6.29 2.0 117ms\n",
" 37 -36.74251477290 + -9.91 -5.98 3.0 148ms\n",
" 38 -36.74251477303 -9.89 -6.59 3.0 146ms\n",
" 39 -36.74251477300 + -10.46 -6.15 3.0 176ms\n",
" 40 -36.74251477303 -10.45 -6.60 3.0 135ms\n",
" 41 -36.74251477304 -11.42 -6.98 2.0 117ms\n",
" 42 -36.74251477304 -12.58 -7.06 3.0 147ms\n",
" 43 -36.74251477304 -12.56 -7.12 1.0 98.3ms\n",
" 44 -36.74251477304 -12.89 -7.36 2.0 117ms\n",
" 45 -36.74251477304 -12.77 -7.77 1.0 98.4ms\n",
" 46 -36.74251477304 + -12.87 -7.39 3.0 155ms\n",
" 47 -36.74251477304 -13.30 -7.40 3.0 173ms\n",
" 48 -36.74251477304 + -Inf -7.55 3.0 177ms\n",
" 49 -36.74251477304 -13.15 -7.81 3.0 143ms\n",
" 50 -36.74251477304 -13.67 -8.21 2.0 114ms\n",
" 51 -36.74251477304 -14.15 -8.18 3.0 140ms\n",
" 52 -36.74251477304 + -14.15 -8.16 2.0 115ms\n",
" 53 -36.74251477304 + -Inf -8.22 3.0 123ms\n",
" 54 -36.74251477304 -14.15 -8.28 2.0 149ms\n",
" 55 -36.74251477304 + -Inf -8.90 1.0 99.3ms\n",
" 56 -36.74251477304 + -Inf -8.68 3.0 156ms\n",
" 57 -36.74251477304 + -Inf -8.67 2.0 119ms\n",
" 58 -36.74251477304 + -Inf -9.29 2.0 117ms\n",
" 59 -36.74251477304 + -13.85 -8.60 3.0 166ms\n",
" 60 -36.74251477304 -14.15 -9.27 4.0 167ms\n",
" 61 -36.74251477304 + -Inf -8.59 4.0 169ms\n",
" 62 -36.74251477304 -14.15 -9.10 3.0 180ms\n",
" 63 -36.74251477304 + -13.85 -9.24 2.0 129ms\n",
" 64 -36.74251477304 -14.15 -10.09 2.0 114ms\n",
" 65 -36.74251477304 -14.15 -9.70 4.0 176ms\n",
" 66 -36.74251477304 + -13.85 -10.20 3.0 146ms\n",
" 67 -36.74251477304 + -Inf -10.24 1.0 101ms\n",
" 68 -36.74251477304 + -Inf -10.57 1.0 163ms\n",
" 69 -36.74251477304 -14.15 -10.49 3.0 692ms\n",
" 70 -36.74251477304 + -14.15 -10.76 2.0 115ms\n",
" 71 -36.74251477304 + -Inf -10.80 2.0 115ms\n",
" 72 -36.74251477304 -13.85 -10.98 2.0 108ms\n",
" 73 -36.74251477304 + -13.85 -11.19 2.0 138ms\n",
" 74 -36.74251477304 -13.85 -10.63 3.0 168ms\n",
" 75 -36.74251477304 + -Inf -11.35 3.0 165ms\n",
" 76 -36.74251477304 + -13.85 -11.02 3.0 161ms\n",
" 77 -36.74251477304 -13.85 -11.74 3.0 159ms\n",
" 78 -36.74251477304 + -14.15 -11.93 2.0 138ms\n",
" 79 -36.74251477304 + -Inf -12.23 2.0 121ms\n"
]
}
],
"cell_type": "code",
"source": [
"# Note: DFTK uses the self-adapting LdosMixing() by default, so to truly disable\n",
"# any preconditioning, we need to supply `mixing=SimpleMixing()` explicitly.\n",
"scfres_Al = self_consistent_field(basis_Al; tol=1e-12, mixing=SimpleMixing());"
],
"metadata": {},
"execution_count": 3
},
{
"cell_type": "markdown",
"source": [
"while when using the Kerker preconditioner it is much faster:"
],
"metadata": {}
},
{
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"n Energy log10(ΔE) log10(Δρ) Diag Δtime\n",
"--- --------------- --------- --------- ---- ------\n",
" 1 -36.73113129843 -0.88 11.0 412ms\n",
" 2 -36.73922462098 -2.09 -1.36 1.0 657ms\n",
" 3 -36.73897761884 + -3.61 -1.61 3.0 128ms\n",
" 4 -36.74219400837 -2.49 -2.16 2.0 112ms\n",
" 5 -36.74231044516 -3.93 -2.40 5.0 114ms\n",
" 6 -36.74244084393 -3.88 -2.45 2.0 103ms\n",
" 7 -36.74248499908 -4.36 -3.12 1.0 90.2ms\n",
" 8 -36.74251177551 -4.57 -3.22 3.0 151ms\n",
" 9 -36.74251295375 -5.93 -3.45 1.0 92.0ms\n",
" 10 -36.74251418318 -5.91 -3.76 2.0 101ms\n",
" 11 -36.74251471578 -6.27 -4.21 6.0 124ms\n",
" 12 -36.74251476845 -7.28 -4.66 2.0 129ms\n",
" 13 -36.74251476124 + -8.14 -4.66 2.0 107ms\n",
" 14 -36.74251477267 -7.94 -5.35 1.0 96.0ms\n",
" 15 -36.74251477297 -9.52 -5.53 3.0 140ms\n",
" 16 -36.74251477301 -10.40 -5.71 2.0 98.3ms\n",
" 17 -36.74251477292 + -10.04 -5.84 2.0 104ms\n",
" 18 -36.74251477304 -9.95 -6.58 1.0 114ms\n",
" 19 -36.74251477304 -12.10 -6.66 8.0 172ms\n",
" 20 -36.74251477304 + -11.98 -6.82 1.0 95.9ms\n",
" 21 -36.74251477304 -12.13 -7.08 2.0 131ms\n",
" 22 -36.74251477304 -12.28 -7.57 2.0 106ms\n",
" 23 -36.74251477304 + -14.15 -7.74 3.0 139ms\n",
" 24 -36.74251477304 -14.15 -7.94 1.0 96.1ms\n",
" 25 -36.74251477304 + -Inf -8.52 2.0 103ms\n",
" 26 -36.74251477304 + -13.85 -8.41 3.0 143ms\n",
" 27 -36.74251477304 -13.85 -8.65 1.0 112ms\n",
" 28 -36.74251477304 + -13.85 -9.20 2.0 105ms\n",
" 29 -36.74251477304 + -Inf -9.49 3.0 144ms\n",
" 30 -36.74251477304 -13.85 -9.91 1.0 96.2ms\n",
" 31 -36.74251477304 + -Inf -10.16 3.0 142ms\n",
" 32 -36.74251477304 + -Inf -10.56 1.0 96.0ms\n",
" 33 -36.74251477304 + -14.15 -10.86 2.0 111ms\n",
" 34 -36.74251477304 + -Inf -11.31 3.0 128ms\n",
" 35 -36.74251477304 + -Inf -11.36 2.0 131ms\n",
" 36 -36.74251477304 -14.15 -11.80 2.0 104ms\n",
" 37 -36.74251477304 + -14.15 -12.08 4.0 132ms\n"
]
}
],
"cell_type": "code",
"source": [
"scfres_Al = self_consistent_field(basis_Al; tol=1e-12, mixing=KerkerMixing());"
],
"metadata": {},
"execution_count": 4
},
{
"cell_type": "markdown",
"source": [
"Given this `scfres_Al` we construct functions representing\n",
"$\\varepsilon^\\dagger$ and $P^{-1}$:"
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "epsilon (generic function with 1 method)"
},
"metadata": {},
"execution_count": 5
}
],
"cell_type": "code",
"source": [
"# Function, which applies P^{-1} for the case of KerkerMixing\n",
"Pinv_Kerker(δρ) = DFTK.mix_density(KerkerMixing(), basis_Al, δρ)\n",
"\n",
"# Function which applies ε† = 1 - χ0 K\n",
"function epsilon(δρ)\n",
" δV = apply_kernel(basis_Al, δρ; ρ=scfres_Al.ρ)\n",
" χ0δV = apply_χ0(scfres_Al, δV)\n",
" δρ - χ0δV\n",
"end"
],
"metadata": {},
"execution_count": 5
},
{
"cell_type": "markdown",
"source": [
"With these functions available we can now compute the desired eigenvalues.\n",
"For simplicity we only consider the first few largest ones."
],
"metadata": {}
},
{
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"┌ Info: Arnoldi eigsolve finished after 1 iterations:\n",
"│ * 7 eigenvalues converged\n",
"│ * norm of residuals = (8.18e-06, 2.11e-08, 6.48e-09, 1.67e-05, 9.36e-06, 2.68e-05, 2.73e-05)\n",
"└ * number of operations = 15\n"
]
},
{
"output_type": "execute_result",
"data": {
"text/plain": "44.02448779307413"
},
"metadata": {},
"execution_count": 6
}
],
"cell_type": "code",
"source": [
"using KrylovKit\n",
"λ_Simple, X_Simple = eigsolve(epsilon, randn(size(scfres_Al.ρ)), 3, :LM;\n",
" tol=1e-3, eager=true, verbosity=2)\n",
"λ_Simple_max = maximum(real.(λ_Simple))"
],
"metadata": {},
"execution_count": 6
},
{
"cell_type": "markdown",
"source": [
"The smallest eigenvalue is a bit more tricky to obtain, so we will just assume"
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "0.952"
},
"metadata": {},
"execution_count": 7
}
],
"cell_type": "code",
"source": [
"λ_Simple_min = 0.952"
],
"metadata": {},
"execution_count": 7
},
{
"cell_type": "markdown",
"source": [
"This makes the condition number around 30:"
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "46.24420986667451"
},
"metadata": {},
"execution_count": 8
}
],
"cell_type": "code",
"source": [
"cond_Simple = λ_Simple_max / λ_Simple_min"
],
"metadata": {},
"execution_count": 8
},
{
"cell_type": "markdown",
"source": [
"This does not sound large compared to the condition numbers you might know\n",
"from linear systems.\n",
"\n",
"However, this is sufficient to cause a notable slowdown, which would be even more\n",
"pronounced if we did not use Anderson, since we also would need to drastically\n",
"reduce the damping (try it!)."
],
"metadata": {}
},
{
"cell_type": "markdown",
"source": [
" Having computed the eigenvalues of the dielectric matrix\n",
"we can now also look at the eigenmodes, which are responsible for\n",
"the bad convergence behaviour. The largest eigenmode for example:"
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "Plot{Plots.GRBackend() n=1}",
"image/png": 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]
},
"metadata": {},
"execution_count": 9
}
],
"cell_type": "code",
"source": [
"using Statistics\n",
"using Plots\n",
"mode_xy = mean(real.(X_Simple[1]), dims=3)[:, :, 1, 1] # Average along z axis\n",
"heatmap(mode_xy', c=:RdBu_11, aspect_ratio=1, grid=false,\n",
" legend=false, clim=(-0.006, 0.006))"
],
"metadata": {},
"execution_count": 9
},
{
"cell_type": "markdown",
"source": [
"This mode can be physically interpreted as the reason why this SCF converges\n",
"slowly. For example in this case it displays a displacement of electron\n",
"density from the centre to the extremal parts of the unit cell. This\n",
"phenomenon is called charge-sloshing."
],
"metadata": {}
},
{
"cell_type": "markdown",
"source": [
"We repeat the exercise for the Kerker-preconditioned dielectric operator:"
],
"metadata": {}
},
{
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"┌ Info: Arnoldi eigsolve finished after 1 iterations:\n",
"│ * 3 eigenvalues converged\n",
"│ * norm of residuals = (2.12e-04, 4.21e-04, 2.33e-04)\n",
"└ * number of operations = 12\n"
]
},
{
"output_type": "execute_result",
"data": {
"text/plain": "Plot{Plots.GRBackend() n=1}",
"image/png": 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},
"metadata": {},
"execution_count": 10
}
],
"cell_type": "code",
"source": [
"λ_Kerker, X_Kerker = eigsolve(Pinv_Kerker ∘ epsilon,\n",
" randn(size(scfres_Al.ρ)), 3, :LM;\n",
" tol=1e-3, eager=true, verbosity=2)\n",
"\n",
"mode_xy = mean(real.(X_Kerker[1]), dims=3)[:, :, 1, 1] # Average along z axis\n",
"heatmap(mode_xy', c=:RdBu_11, aspect_ratio=1, grid=false,\n",
" legend=false, clim=(-0.006, 0.006))"
],
"metadata": {},
"execution_count": 10
},
{
"cell_type": "markdown",
"source": [
"Clearly the charge-sloshing mode is no longer dominating.\n",
"\n",
"The largest eigenvalue is now"
],
"metadata": {}
},
{
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": "4.7236110498662836"
},
"metadata": {},
"execution_count": 11
}
],
"cell_type": "code",
"source": [
"maximum(real.(λ_Kerker))"
],
"metadata": {},
"execution_count": 11
},
{
"cell_type": "markdown",
"source": [
"Since the smallest eigenvalue in this case remains of similar size (it is now\n",
"around 0.8), this implies that the conditioning improves noticeably when\n",
"`KerkerMixing` is used.\n",
"\n",
"**Note:** Since LdosMixing requires solving a linear system at each\n",
"application of $P^{-1}$, determining the eigenvalues of\n",
"$P^{-1} \\varepsilon^\\dagger$ is slightly more expensive and thus not shown. The\n",
"results are similar to `KerkerMixing`, however."
],
"metadata": {}
},
{
"cell_type": "markdown",
"source": [
"We could repeat the exercise for an insulating system (e.g. a Helium chain).\n",
"In this case you would notice that the condition number without mixing\n",
"is actually smaller than the condition number with Kerker mixing. In other\n",
"words employing Kerker mixing makes the convergence *worse*. A closer\n",
"investigation of the eigenvalues shows that Kerker mixing reduces the\n",
"smallest eigenvalue of the dielectric operator this time, while keeping\n",
"the largest value unchanged. Overall the conditioning thus workens."
],
"metadata": {}
},
{
"cell_type": "markdown",
"source": [
"**Takeaways:**\n",
"- For metals the conditioning of the dielectric matrix increases steeply with system size.\n",
"- The Kerker preconditioner tames this and makes SCFs on large metallic\n",
" systems feasible by keeping the condition number of order 1.\n",
"- For insulating systems the best approach is to not use any mixing.\n",
"- **The ideal mixing** strongly depends on the dielectric properties of\n",
" system which is studied (metal versus insulator versus semiconductor)."
],
"metadata": {}
}
],
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