Eigenvalues of the dielectric matrix
We compute a few eigenvalues of the dielectric matrix ($q=0$, $ω=0$) iteratively.
using DFTK
using Plots
using KrylovKit
using Printf
# Calculation parameters
kgrid = [1, 1, 1]
Ecut = 5
# Silicon lattice
a = 10.26
lattice = a / 2 .* [[0 1 1.]; [1 0 1.]; [1 1 0.]]
Si = ElementPsp(:Si, psp=load_psp("hgh/lda/Si-q4"))
atoms = [Si, Si]
positions = [ones(3)/8, -ones(3)/8]
# Compute the dielectric operator without symmetries
model = model_LDA(lattice, atoms, positions, symmetries=false)
basis = PlaneWaveBasis(model; Ecut, kgrid)
scfres = self_consistent_field(basis, tol=1e-8);n Energy log10(ΔE) log10(Δρ) Diag Δtime
--- --------------- --------- --------- ---- ------
1 -7.234330793358 -0.50 8.0
2 -7.250259413926 -1.80 -1.40 1.0 12.9ms
3 -7.251165247710 -3.04 -2.15 1.0 11.7ms
4 -7.251298519288 -3.88 -2.22 2.0 14.2ms
5 -7.251336520537 -4.42 -2.87 1.0 12.0ms
6 -7.251337697847 -5.93 -3.12 2.0 14.0ms
7 -7.251338631755 -6.03 -3.52 1.0 11.5ms
8 -7.251338785910 -6.81 -4.08 1.0 12.1ms
9 -7.251338797752 -7.93 -4.55 2.0 16.4ms
10 -7.251338798605 -9.07 -4.99 2.0 15.2ms
11 -7.251338798676 -10.15 -5.36 2.0 14.6ms
12 -7.251338798700 -10.62 -5.68 1.0 13.1ms
13 -7.251338798704 -11.39 -6.20 2.0 128ms
14 -7.251338798704 -12.26 -6.59 2.0 15.3ms
15 -7.251338798705 -13.55 -6.93 1.0 12.4ms
16 -7.251338798705 -13.91 -7.36 2.0 16.2ms
17 -7.251338798705 -14.75 -7.77 1.0 14.1ms
18 -7.251338798705 -15.05 -8.15 2.0 15.2msApplying $ε^† ≔ (1- χ_0 K)$ …
function eps_fun(δρ)
δV = apply_kernel(basis, δρ; ρ=scfres.ρ)
χ0δV = apply_χ0(scfres, δV)
δρ - χ0δV
end;… eagerly diagonalizes the subspace matrix at each iteration
eigsolve(eps_fun, randn(size(scfres.ρ)), 5, :LM; eager=true, verbosity=3);[ Info: Arnoldi iteration step 1: normres = 0.032476931465942556
[ Info: Arnoldi iteration step 2: normres = 0.4869928040369925
[ Info: Arnoldi iteration step 3: normres = 0.3383774694574327
[ Info: Arnoldi iteration step 4: normres = 1.097204847224133
[ Info: Arnoldi iteration step 5: normres = 0.23177094180381658
[ Info: Arnoldi schursolve in iter 1, krylovdim = 5: 0 values converged, normres = (1.55e-01, 3.24e-02, 1.16e-01, 4.91e-02, 1.13e-01)
[ Info: Arnoldi iteration step 6: normres = 0.47514454185688004
[ Info: Arnoldi schursolve in iter 1, krylovdim = 6: 0 values converged, normres = (5.75e-02, 3.96e-02, 4.64e-01, 6.74e-02, 1.14e-02)
[ Info: Arnoldi iteration step 7: normres = 0.0992888159685404
[ Info: Arnoldi schursolve in iter 1, krylovdim = 7: 0 values converged, normres = (3.10e-03, 7.80e-03, 4.19e-02, 2.06e-02, 8.02e-02)
[ Info: Arnoldi iteration step 8: normres = 0.1135305672087734
[ Info: Arnoldi schursolve in iter 1, krylovdim = 8: 0 values converged, normres = (1.51e-04, 6.27e-04, 3.69e-03, 7.13e-03, 6.06e-02)
[ Info: Arnoldi iteration step 9: normres = 0.06430608489565225
[ Info: Arnoldi schursolve in iter 1, krylovdim = 9: 0 values converged, normres = (4.28e-06, 2.97e-05, 1.95e-04, 1.93e-03, 3.96e-02)
[ Info: Arnoldi iteration step 10: normres = 0.1023873199507804
[ Info: Arnoldi schursolve in iter 1, krylovdim = 10: 0 values converged, normres = (1.92e-07, 2.21e-06, 1.61e-05, 7.30e-04, 3.25e-02)
[ Info: Arnoldi iteration step 11: normres = 0.0748108237143604
[ Info: Arnoldi schursolve in iter 1, krylovdim = 11: 0 values converged, normres = (6.09e-09, 1.14e-07, 9.24e-07, 1.61e-04, 1.39e-02)
[ Info: Arnoldi iteration step 12: normres = 0.0842707841083441
[ Info: Arnoldi schursolve in iter 1, krylovdim = 12: 0 values converged, normres = (2.24e-10, 6.94e-09, 6.25e-08, 4.77e-05, 9.34e-03)
[ Info: Arnoldi iteration step 13: normres = 0.040252660925506493
[ Info: Arnoldi schursolve in iter 1, krylovdim = 13: 0 values converged, normres = (3.85e-12, 1.95e-10, 1.95e-09, 5.95e-06, 2.55e-03)
[ Info: Arnoldi iteration step 14: normres = 0.31204896047506425
[ Info: Arnoldi schursolve in iter 1, krylovdim = 14: 1 values converged, normres = (5.15e-13, 4.28e-11, 4.74e-10, 5.55e-06, 4.29e-03)
[ Info: Arnoldi iteration step 15: normres = 0.12283042108990469
[ Info: Arnoldi schursolve in iter 1, krylovdim = 15: 1 values converged, normres = (5.92e-14, 4.02e-11, 1.15e-01, 1.98e-02, 5.56e-05)
[ Info: Arnoldi iteration step 16: normres = 0.5896326612609463
[ Info: Arnoldi schursolve in iter 1, krylovdim = 16: 1 values converged, normres = (1.67e-14, 2.17e-11, 6.28e-02, 2.05e-03, 9.93e-06)
[ Info: Arnoldi iteration step 17: normres = 0.07349273946635694
[ Info: Arnoldi schursolve in iter 1, krylovdim = 17: 1 values converged, normres = (1.15e-15, 3.39e-10, 5.86e-02, 6.64e-04, 2.97e-02)
[ Info: Arnoldi iteration step 18: normres = 0.017954678406131772
[ Info: Arnoldi schursolve in iter 1, krylovdim = 18: 1 values converged, normres = (8.59e-18, 4.75e-04, 5.14e-04, 3.04e-04, 2.34e-04)
[ Info: Arnoldi iteration step 19: normres = 0.23499177350070719
[ Info: Arnoldi schursolve in iter 1, krylovdim = 19: 1 values converged, normres = (8.99e-19, 1.06e-09, 1.23e-04, 8.01e-09, 7.52e-05)
[ Info: Arnoldi iteration step 20: normres = 0.027066669896302836
[ Info: Arnoldi schursolve in iter 1, krylovdim = 20: 1 values converged, normres = (1.12e-20, 1.68e-09, 2.67e-06, 1.67e-06, 7.93e-07)
[ Info: Arnoldi iteration step 21: normres = 0.028139393528501137
[ Info: Arnoldi schursolve in iter 1, krylovdim = 21: 1 values converged, normres = (1.30e-22, 2.89e-08, 3.98e-08, 1.25e-08, 3.54e-08)
[ Info: Arnoldi iteration step 22: normres = 0.031532861508480205
[ Info: Arnoldi schursolve in iter 1, krylovdim = 22: 1 values converged, normres = (1.71e-24, 8.34e-10, 6.20e-10, 5.21e-10, 7.01e-10)
[ Info: Arnoldi iteration step 23: normres = 0.25016471438313154
[ Info: Arnoldi schursolve in iter 1, krylovdim = 23: 1 values converged, normres = (1.79e-25, 1.41e-10, 1.06e-10, 1.03e-10, 1.27e-10)
[ Info: Arnoldi iteration step 24: normres = 0.09088823480646846
[ Info: Arnoldi schursolve in iter 1, krylovdim = 24: 1 values converged, normres = (1.55e-26, 8.03e-11, 6.21e-11, 6.65e-10, 9.35e-10)
[ Info: Arnoldi iteration step 25: normres = 0.027435071533682177
[ Info: Arnoldi schursolve in iter 1, krylovdim = 25: 1 values converged, normres = (1.71e-28, 1.58e-12, 1.19e-12, 6.10e-09, 1.10e-08)
[ Info: Arnoldi iteration step 26: normres = 0.11166486845599458
[ Info: Arnoldi schursolve in iter 1, krylovdim = 26: 3 values converged, normres = (8.48e-30, 1.31e-13, 9.87e-14, 1.50e-04, 4.74e-05)
[ Info: Arnoldi iteration step 27: normres = 0.034376465853517585
[ Info: Arnoldi schursolve in iter 1, krylovdim = 27: 3 values converged, normres = (1.22e-31, 3.07e-15, 2.30e-15, 3.02e-09, 5.57e-09)
[ Info: Arnoldi iteration step 28: normres = 0.1179941022344992
[ Info: Arnoldi schursolve in iter 1, krylovdim = 28: 3 values converged, normres = (6.34e-33, 2.64e-16, 1.98e-16, 3.86e-07, 4.02e-08)
[ Info: Arnoldi iteration step 29: normres = 0.16874130257600892
[ Info: Arnoldi schursolve in iter 1, krylovdim = 29: 3 values converged, normres = (4.73e-34, 3.30e-17, 2.48e-17, 3.92e-09, 1.09e-08)
[ Info: Arnoldi iteration step 30: normres = 0.02927110513154585
[ Info: Arnoldi schursolve in iter 1, krylovdim = 30: 3 values converged, normres = (6.56e-36, 8.05e-19, 6.05e-19, 1.26e-09, 8.88e-10)
[ Info: Arnoldi schursolve in iter 2, krylovdim = 19: 3 values converged, normres = (6.62e-36, 8.05e-19, 6.05e-19, 1.26e-09, 8.88e-10)
[ Info: Arnoldi iteration step 20: normres = 0.05368525788249321
[ Info: Arnoldi schursolve in iter 2, krylovdim = 20: 3 values converged, normres = (1.46e-37, 2.86e-20, 2.15e-20, 5.92e-11, 1.16e-11)
[ Info: Arnoldi iteration step 21: normres = 0.05302623811279329
[ Info: Arnoldi schursolve in iter 2, krylovdim = 21: 3 values converged, normres = (3.32e-39, 1.07e-21, 8.03e-22, 4.84e-13, 2.45e-12)
[ Info: Arnoldi iteration step 22: normres = 0.012959246321441948
┌ Info: Arnoldi eigsolve finished after 2 iterations:
│ * 6 eigenvalues converged
│ * norm of residuals = (1.7798038901264966e-41, 9.167590957616759e-24, 1.0224183446019647e-24, 1.4151196953919328e-14, 1.4151196953919328e-14, 5.1741145982640815e-15)
└ * number of operations = 33