bondscell_results$0c146256-a19d-43c7-a874-63c810ab2f08queued¤logsrunning¦outputbodyelementshamprefixHamiltonianelementsbasisPlaneWaveBasis discretization: architecture : CPU() num. mpi processes : 1 num. julia threads : 2 num. DFTK threads : 2 num. blas threads : 1 num. fft threads : 1 Ecut : 7.0 Ha fft_size : (150, 24, 24), 86400 total points kgrid : MonkhorstPack([1, 1, 1]) num. red. kpoints : 1 num. irred. kpoints : 1 Discretized Model(gga_x_pbe+gga_c_pbe, 3D): lattice (in Bohr) : [61.2271 , 0 , 0 ] [0 , 7.65339 , 0 ] [0 , 0 , 7.65339 ] unit cell volume : 3586.3 Bohr³ atoms : Al₃₂ atom potentials : ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") num. electrons : 96 spin polarization : none temperature : 0.001 Ha smearing : DFTK.Smearing.FermiDirac() terms : Kinetic() DFTK.AtomicLocal() DFTK.AtomicNonlocal() DFTK.Ewald(nothing) DFTK.PspCorrection() Hartree() Xc(gga_x_pbe, gga_c_pbe) DFTK.Entropy()text/plainblocksprefixDFTK.HamiltonianBlockelementsprefixDftHamiltonianBlockelementsbasisPlaneWaveBasis discretization: architecture : CPU() num. mpi processes : 1 num. julia threads : 2 num. DFTK threads : 2 num. blas threads : 1 num. fft threads : 1 Ecut : 7.0 Ha fft_size : (150, 24, 24), 86400 total points kgrid : MonkhorstPack([1, 1, 1]) num. red. kpoints : 1 num. irred. kpoints : 1 Discretized Model(gga_x_pbe+gga_c_pbe, 3D): lattice (in Bohr) : [61.2271 , 0 , 0 ] [0 , 7.65339 , 0 ] [0 , 0 , 7.65339 ] unit cell volume : 3586.3 Bohr³ atoms : Al₃₂ atom potentials : ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") num. electrons : 96 spin polarization : none temperature : 0.001 Ha smearing : DFTK.Smearing.FermiDirac() terms : Kinetic() DFTK.AtomicLocal() DFTK.AtomicNonlocal() DFTK.Ewald(nothing) DFTK.PspCorrection() Hartree() Xc(gga_x_pbe, gga_c_pbe) DFTK.Entropy()text/plainkpointBKPoint([ 0, 0, 0], spin = 1, num. G vectors = 3245)text/plainoperatorsprefixAnyelementsmoretypeArrayprefix_shortobjectid8a1ecc05e9fac956!application/vnd.pluto.tree+objectfourier_opprefix/FourierMultiplication{Float64, Vector{Float64}}elementsbasisPlaneWaveBasis discretization: architecture : CPU() num. mpi processes : 1 num. julia threads : 2 num. DFTK threads : 2 num. blas threads : 1 num. fft threads : 1 Ecut : 7.0 Ha fft_size : (150, 24, 24), 86400 total points kgrid : MonkhorstPack([1, 1, 1]) num. red. kpoints : 1 num. irred. kpoints : 1 Discretized Model(gga_x_pbe+gga_c_pbe, 3D): lattice (in Bohr) : [61.2271 , 0 , 0 ] [0 , 7.65339 , 0 ] [0 , 0 , 7.65339 ] unit cell volume : 3586.3 Bohr³ atoms : Al₃₂ atom potentials : ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") num. electrons : 96 spin polarization : none temperature : 0.001 Ha smearing : DFTK.Smearing.FermiDirac() terms : Kinetic() DFTK.AtomicLocal() DFTK.AtomicNonlocal() DFTK.Ewald(nothing) DFTK.PspCorrection() Hartree() Xc(gga_x_pbe, gga_c_pbe) DFTK.Entropy()text/plainkpointBKPoint([ 0, 0, 0], spin = 1, num. G vectors = 3245)text/plainmultiplierprefixFloat64elementsmoretypeArrayprefix_shortobjectid6c8829a6a85fc828!application/vnd.pluto.tree+objecttypestructprefix_shortFourierMultiplicationobjectid7e33d4c0de34028!application/vnd.pluto.tree+objectlocal_opprefix3RealSpaceMultiplication{Float64, Array{Float64, 3}}elementsbasisPlaneWaveBasis discretization: architecture : CPU() num. mpi processes : 1 num. julia threads : 2 num. DFTK threads : 2 num. blas threads : 1 num. fft threads : 1 Ecut : 7.0 Ha fft_size : (150, 24, 24), 86400 total points kgrid : MonkhorstPack([1, 1, 1]) num. red. kpoints : 1 num. irred. kpoints : 1 Discretized Model(gga_x_pbe+gga_c_pbe, 3D): lattice (in Bohr) : [61.2271 , 0 , 0 ] [0 , 7.65339 , 0 ] [0 , 0 , 7.65339 ] unit cell volume : 3586.3 Bohr³ atoms : Al₃₂ atom potentials : ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") num. electrons : 96 spin polarization : none temperature : 0.001 Ha smearing : DFTK.Smearing.FermiDirac() terms : Kinetic() DFTK.AtomicLocal() DFTK.AtomicNonlocal() DFTK.Ewald(nothing) DFTK.PspCorrection() Hartree() Xc(gga_x_pbe, gga_c_pbe) DFTK.Entropy()text/plainkpointBKPoint([ 0, 0, 0], spin = 1, num. G vectors = 3245)text/plainpotentialb150×24×24 Array{Float64, 3}: [:, :, 1] = -2.97231 -2.88677 -2.56524 -1.97507 … -1.97507 -2.56524 -2.88677 -2.80624 -2.70676 -2.37825 -1.78086 -1.78086 -2.37825 -2.70676 -2.28437 -2.15753 -1.78854 -1.20489 -1.20489 -1.78854 -2.15753 -1.28891 -1.18917 -0.928745 -0.649339 -0.649339 -0.928745 -1.18917 -0.554796 -0.526235 -0.460747 -0.38166 -0.38166 -0.460747 -0.526235 -0.286045 -0.275472 -0.244606 -0.203635 … -0.203635 -0.244606 -0.275472 -0.107015 -0.104806 -0.100639 -0.0978476 -0.0978476 -0.100639 -0.104806 ⋮ ⋱ -0.107015 -0.104806 -0.100639 -0.0978476 -0.0978476 -0.100639 -0.104806 -0.286045 -0.275472 -0.244606 -0.203635 … -0.203635 -0.244606 -0.275472 -0.554796 -0.526235 -0.460747 -0.38166 -0.38166 -0.460747 -0.526235 -1.28891 -1.18917 -0.928745 -0.649339 -0.649339 -0.928745 -1.18917 -2.28437 -2.15753 -1.78854 -1.20489 -1.20489 -1.78854 -2.15753 -2.80624 -2.70676 -2.37825 -1.78086 -1.78086 -2.37825 -2.70676 [:, :, 2] = -2.88677 -2.79045 -2.44658 -1.84824 … -1.84824 -2.44658 -2.79045 -2.70676 -2.59685 -2.25708 -1.65842 -1.65842 -2.25708 -2.59685 -2.15753 -2.02947 -1.66502 -1.1081 -1.1081 -1.66502 -2.02947 -1.18917 -1.09721 -0.857967 -0.611315 -0.611315 -0.857967 -1.09721 -0.526235 -0.500364 -0.442189 -0.366328 -0.366328 -0.442189 -0.500364 -0.275472 -0.265449 -0.234333 -0.195163 … -0.195163 -0.234333 -0.265449 -0.104806 -0.101796 -0.0981096 -0.0945414 -0.0945414 -0.0981096 -0.101796 ⋮ ⋱ -0.104806 -0.101796 -0.0981096 -0.0945414 -0.0945414 -0.0981096 -0.101796 -0.275472 -0.265449 -0.234333 -0.195163 … -0.195163 -0.234333 -0.265449 -0.526235 -0.500364 -0.442189 -0.366328 -0.366328 -0.442189 -0.500364 -1.18917 -1.09721 -0.857967 -0.611315 -0.611315 -0.857967 -1.09721 -2.15753 -2.02947 -1.66502 -1.1081 -1.1081 -1.66502 -2.02947 -2.70676 -2.59685 -2.25708 -1.65842 -1.65842 -2.25708 -2.59685 [:, :, 3] = -2.56524 -2.44658 -2.09526 -1.49155 … -1.49155 -2.09526 -2.44658 -2.37825 -2.25708 -1.90928 -1.30811 -1.30811 -1.90928 -2.25708 -1.78854 -1.66502 -1.31393 -0.861839 -0.861839 -1.31393 -1.66502 -0.928745 -0.857967 -0.688018 -0.524169 -0.524169 -0.688018 -0.857967 -0.460747 -0.442189 -0.39455 -0.322103 -0.322103 -0.39455 -0.442189 -0.244606 -0.234333 -0.206013 -0.173607 … -0.173607 -0.206013 -0.234333 -0.100639 -0.0981096 -0.0932905 -0.0860427 -0.0860427 -0.0932905 -0.0981096 ⋮ ⋱ -0.100639 -0.0981096 -0.0932905 -0.0860427 -0.0860427 -0.0932905 -0.0981096 -0.244606 -0.234333 -0.206013 -0.173607 … -0.173607 -0.206013 -0.234333 -0.460747 -0.442189 -0.39455 -0.322103 -0.322103 -0.39455 -0.442189 -0.928745 -0.857967 -0.688018 -0.524169 -0.524169 -0.688018 -0.857967 -1.78854 -1.66502 -1.31393 -0.861839 -0.861839 -1.31393 -1.66502 -2.37825 -2.25708 -1.90928 -1.30811 -1.30811 -1.90928 -2.25708 ;;; … [:, :, 22] = -1.97507 -1.84824 -1.49155 … -0.981374 -1.49155 -1.84824 -1.78086 -1.65842 -1.30811 -0.85554 -1.30811 -1.65842 -1.20489 -1.1081 -0.861839 -0.615953 -0.861839 -1.1081 -0.649339 -0.611315 -0.524169 -0.421282 -0.524169 -0.611315 -0.38166 -0.366328 -0.322103 -0.26032 -0.322103 -0.366328 -0.203635 -0.195163 -0.173607 … -0.145073 -0.173607 -0.195163 -0.0978476 -0.0945414 -0.0860427 -0.0783525 -0.0860427 -0.0945414 ⋮ ⋱ -0.0978476 -0.0945414 -0.0860427 -0.0783525 -0.0860427 -0.0945414 -0.203635 -0.195163 -0.173607 … -0.145073 -0.173607 -0.195163 -0.38166 -0.366328 -0.322103 -0.26032 -0.322103 -0.366328 -0.649339 -0.611315 -0.524169 -0.421282 -0.524169 -0.611315 -1.20489 -1.1081 -0.861839 -0.615953 -0.861839 -1.1081 -1.78086 -1.65842 -1.30811 -0.85554 -1.30811 -1.65842 [:, :, 23] = -2.56524 -2.44658 -2.09526 -1.49155 … -1.49155 -2.09526 -2.44658 -2.37825 -2.25708 -1.90928 -1.30811 -1.30811 -1.90928 -2.25708 -1.78854 -1.66502 -1.31393 -0.861839 -0.861839 -1.31393 -1.66502 -0.928745 -0.857967 -0.688018 -0.524169 -0.524169 -0.688018 -0.857967 -0.460747 -0.442189 -0.39455 -0.322103 -0.322103 -0.39455 -0.442189 -0.244606 -0.234333 -0.206013 -0.173607 … -0.173607 -0.206013 -0.234333 -0.100639 -0.0981096 -0.0932905 -0.0860427 -0.0860427 -0.0932905 -0.0981096 ⋮ ⋱ -0.100639 -0.0981096 -0.0932905 -0.0860427 -0.0860427 -0.0932905 -0.0981096 -0.244606 -0.234333 -0.206013 -0.173607 … -0.173607 -0.206013 -0.234333 -0.460747 -0.442189 -0.39455 -0.322103 -0.322103 -0.39455 -0.442189 -0.928745 -0.857967 -0.688018 -0.524169 -0.524169 -0.688018 -0.857967 -1.78854 -1.66502 -1.31393 -0.861839 -0.861839 -1.31393 -1.66502 -2.37825 -2.25708 -1.90928 -1.30811 -1.30811 -1.90928 -2.25708 [:, :, 24] = -2.88677 -2.79045 -2.44658 -1.84824 … -1.84824 -2.44658 -2.79045 -2.70676 -2.59685 -2.25708 -1.65842 -1.65842 -2.25708 -2.59685 -2.15753 -2.02947 -1.66502 -1.1081 -1.1081 -1.66502 -2.02947 -1.18917 -1.09721 -0.857967 -0.611315 -0.611315 -0.857967 -1.09721 -0.526235 -0.500364 -0.442189 -0.366328 -0.366328 -0.442189 -0.500364 -0.275472 -0.265449 -0.234333 -0.195163 … -0.195163 -0.234333 -0.265449 -0.104806 -0.101796 -0.0981096 -0.0945414 -0.0945414 -0.0981096 -0.101796 ⋮ ⋱ -0.104806 -0.101796 -0.0981096 -0.0945414 -0.0945414 -0.0981096 -0.101796 -0.275472 -0.265449 -0.234333 -0.195163 … -0.195163 -0.234333 -0.265449 -0.526235 -0.500364 -0.442189 -0.366328 -0.366328 -0.442189 -0.500364 -1.18917 -1.09721 -0.857967 -0.611315 -0.611315 -0.857967 -1.09721 -2.15753 -2.02947 -1.66502 -1.1081 -1.1081 -1.66502 -2.02947 -2.70676 -2.59685 -2.25708 -1.65842 -1.65842 -2.25708 -2.59685text/plaintypestructprefix_shortRealSpaceMultiplicationobjectid92ea692d035ddd12!application/vnd.pluto.tree+objectnonlocal_opprefix>NonlocalOperator{Float64, Matrix{ComplexF64}, Matrix{Float64}}elementsbasisPlaneWaveBasis discretization: architecture : CPU() num. mpi processes : 1 num. julia threads : 2 num. DFTK threads : 2 num. blas threads : 1 num. fft threads : 1 Ecut : 7.0 Ha fft_size : (150, 24, 24), 86400 total points kgrid : MonkhorstPack([1, 1, 1]) num. red. kpoints : 1 num. irred. kpoints : 1 Discretized Model(gga_x_pbe+gga_c_pbe, 3D): lattice (in Bohr) : [61.2271 , 0 , 0 ] [0 , 7.65339 , 0 ] [0 , 0 , 7.65339 ] unit cell volume : 3586.3 Bohr³ atoms : Al₃₂ atom potentials : ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") num. electrons : 96 spin polarization : none temperature : 0.001 Ha smearing : DFTK.Smearing.FermiDirac() terms : Kinetic() DFTK.AtomicLocal() DFTK.AtomicNonlocal() DFTK.Ewald(nothing) DFTK.PspCorrection() Hartree() Xc(gga_x_pbe, gga_c_pbe) DFTK.Entropy()text/plainkpointBKPoint([ 0, 0, 0], spin = 1, num. G vectors = 3245)text/plainP3245×576 Matrix{ComplexF64}: 0.015063+0.0im 0.0322237+0.0im 0.0-0.0im … 0.0+0.0im 0.0150591+0.0im 0.0321164+0.0im 0.0+0.0im -2.15613e-5-8.93099e-6im 0.0150472+0.0im 0.0317962+0.0im 0.0+0.0im -6.5896e-5-6.5896e-5im 0.0150267+0.0im 0.0312671+0.0im 0.0+0.0im -8.00108e-5-0.000193163im 0.0149968+0.0im 0.0305358+0.0im 0.0+0.0im -0.0-0.000370196im 0.0149563+0.0im 0.0296117+0.0im 0.0-0.0im … 0.000220202-0.000531615im 0.0149037+0.0im 0.0285066+0.0im 0.0-0.0im 0.000582149-0.000582149im ⋮ ⋱ ⋮ 0.0140722+0.0im 0.0174646+0.0im 0.0+0.00129703im 0.000418504+0.000418504im 0.0141576+0.0im 0.0182911+0.0im 0.0+0.00121419im … 0.000317701+0.000766997im 0.0142258+0.0im 0.0189834+0.0im 0.0+0.00114418im 0.0+0.00102764im 0.0142777+0.0im 0.0195321+0.0im 0.0+0.00108832im -0.000452609+0.00109269im 0.0143141+0.0im 0.0199295+0.0im 0.0+0.00104767im -0.000915218+0.000915218im 0.0143358+0.0im 0.0201701+0.0im 0.0+0.00102296im -0.00125795+0.000521059imtext/plainD576×576 Matrix{Float64}: 20.5067 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 2.91317 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 29.1502 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 3.32975 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 29.1502 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 ⋮ ⋱ ⋮ 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -11.082 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -2.55089 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -11.082 0.0 0.0 0.0 0.0 0.0 0.0 … 0.0 0.0 0.0 -2.55089text/plaintypestructprefix_shortNonlocalOperatorobjectidbd79b876e24222cd!application/vnd.pluto.tree+objectdivAgrad_opnothingtext/plainscratchprefix+@NamedTuple{ψ_reals::Array{ComplexF64, 3}}elementsmoretypeArrayprefix_shortobjectide32ad4d81c1b9661!application/vnd.pluto.tree+objecttypestructprefix_shortDftHamiltonianBlockobjectid5655a10cf6e9ef11!application/vnd.pluto.tree+objecttypeArrayprefix_shortobjectid886aaf0d6692c5e1!application/vnd.pluto.tree+objecttypestructprefix_shortHamiltonianobjectid675b4b69b2e6da7d!application/vnd.pluto.tree+objectbasisPlaneWaveBasis discretization: architecture : CPU() num. mpi processes : 1 num. julia threads : 2 num. DFTK threads : 2 num. blas threads : 1 num. fft threads : 1 Ecut : 7.0 Ha fft_size : (150, 24, 24), 86400 total points kgrid : MonkhorstPack([1, 1, 1]) num. red. kpoints : 1 num. irred. kpoints : 1 Discretized Model(gga_x_pbe+gga_c_pbe, 3D): lattice (in Bohr) : [61.2271 , 0 , 0 ] [0 , 7.65339 , 0 ] [0 , 0 , 7.65339 ] unit cell volume : 3586.3 Bohr³ atoms : Al₃₂ atom potentials : ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") ElementPsp(Al; psp="/home/runner/.julia/artifacts/897b9d17527039dd86ca882603c3af2ea0c0aada/Al.upf") num. electrons : 96 spin polarization : none temperature : 0.001 Ha smearing : DFTK.Smearing.FermiDirac() terms : Kinetic() DFTK.AtomicLocal() DFTK.AtomicNonlocal() DFTK.Ewald(nothing) DFTK.PspCorrection() Hartree() Xc(gga_x_pbe, gga_c_pbe) DFTK.Entropy()text/plainenergies]Energy breakdown (in Ha): Kinetic 29.1896150 AtomicLocal 4.1603003 AtomicNonlocal 11.9150162 Ewald -86.2650498 PspCorrection 3.4974064 Hartree 0.0938466 Xc -34.5689781 Entropy -0.0044987 total -71.982342052783text/plainconvergedtruetext/plainoccupation_threshold1.0e-6text/plainρa150×24×24×1 Array{Float64, 4}: [:, :, 1, 1] = 0.00142816 0.0033055 0.00897871 0.0173433 … 0.0173433 0.00897871 0.0033055 0.00401287 0.00594311 0.011533 0.0193528 0.0193528 0.011533 0.00594311 0.0121847 0.0139012 0.0185206 0.0243294 0.0243294 0.0185206 0.0139012 0.0225463 0.0235647 0.0261308 0.028981 0.028981 0.0261308 0.0235647 0.0282196 0.0285849 0.0294403 0.0302297 0.0302297 0.0294403 0.0285849 0.0275972 0.0277317 0.0280543 0.0283998 … 0.0283998 0.0280543 0.0277317 0.0243663 0.0245394 0.0250258 0.0257736 0.0257736 0.0250258 0.0245394 ⋮ ⋱ 0.0243663 0.0245394 0.0250258 0.0257736 0.0257736 0.0250258 0.0245394 0.0275972 0.0277317 0.0280543 0.0283998 … 0.0283998 0.0280543 0.0277317 0.0282196 0.0285849 0.0294403 0.0302297 0.0302297 0.0294403 0.0285849 0.0225463 0.0235647 0.0261308 0.028981 0.028981 0.0261308 0.0235647 0.0121847 0.0139012 0.0185206 0.0243294 0.0243294 0.0185206 0.0139012 0.00401287 0.00594311 0.011533 0.0193528 0.0193528 0.011533 0.00594311 [:, :, 2, 1] = 0.0033055 0.00515929 0.0106568 0.0185783 … 0.0185783 0.0106568 0.00515929 0.00594311 0.00780953 0.013144 0.0204739 0.0204739 0.013144 0.00780953 0.0139012 0.0155004 0.0197747 0.0250849 0.0250849 0.0197747 0.0155004 0.0235647 0.0244768 0.0267602 0.0292558 0.0292558 0.0267602 0.0244768 0.0285849 0.0288913 0.0295961 0.0302076 0.0302076 0.0295961 0.0288913 0.0277317 0.0278397 0.0280931 0.0283511 … 0.0283511 0.0280931 0.0278397 0.0245394 0.0246975 0.0251414 0.025824 0.025824 0.0251414 0.0246975 ⋮ ⋱ 0.0245394 0.0246975 0.0251414 0.025824 0.025824 0.0251414 0.0246975 0.0277317 0.0278397 0.0280931 0.0283511 … 0.0283511 0.0280931 0.0278397 0.0285849 0.0288913 0.0295961 0.0302076 0.0302076 0.0295961 0.0288913 0.0235647 0.0244768 0.0267602 0.0292558 0.0292558 0.0267602 0.0244768 0.0139012 0.0155004 0.0197747 0.0250849 0.0250849 0.0197747 0.0155004 0.00594311 0.00780953 0.013144 0.0204739 0.0204739 0.013144 0.00780953 [:, :, 3, 1] = 0.00897871 0.0106568 0.015432 0.021943 … 0.021943 0.015432 0.0106568 0.011533 0.013144 0.0176086 0.0234703 0.0234703 0.0176086 0.013144 0.0185206 0.0197747 0.0230644 0.0270063 0.0270063 0.0230644 0.0197747 0.0261308 0.0267602 0.0282994 0.0298783 0.0298783 0.0282994 0.0267602 0.0294403 0.0295961 0.0299181 0.0300841 0.0300841 0.0299181 0.0295961 0.0280543 0.0280931 0.0281669 0.0282014 … 0.0282014 0.0281669 0.0280931 0.0250258 0.0251414 0.0254659 0.0259695 0.0259695 0.0254659 0.0251414 ⋮ ⋱ 0.0250258 0.0251414 0.0254659 0.0259695 0.0259695 0.0254659 0.0251414 0.0280543 0.0280931 0.0281669 0.0282014 … 0.0282014 0.0281669 0.0280931 0.0294403 0.0295961 0.0299181 0.0300841 0.0300841 0.0299181 0.0295961 0.0261308 0.0267602 0.0282994 0.0298783 0.0298783 0.0282994 0.0267602 0.0185206 0.0197747 0.0230644 0.0270063 0.0270063 0.0230644 0.0197747 0.011533 0.013144 0.0176086 0.0234703 0.0234703 0.0176086 0.013144 ;;; … [:, :, 22, 1] = 0.0173433 0.0185783 0.021943 0.0262254 … 0.0262254 0.021943 0.0185783 0.0193528 0.0204739 0.0234703 0.0271626 0.0271626 0.0234703 0.0204739 0.0243294 0.0250849 0.0270063 0.0291512 0.0291512 0.0270063 0.0250849 0.028981 0.0292558 0.0298783 0.0303643 0.0303643 0.0298783 0.0292558 0.0302297 0.0302076 0.0300841 0.0297481 0.0297481 0.0300841 0.0302076 0.0283998 0.0283511 0.0282014 0.0279643 … 0.0279643 0.0282014 0.0283511 0.0257736 0.025824 0.0259695 0.0262151 0.0262151 0.0259695 0.025824 ⋮ ⋱ 0.0257736 0.025824 0.0259695 0.0262151 0.0262151 0.0259695 0.025824 0.0283998 0.0283511 0.0282014 0.0279643 … 0.0279643 0.0282014 0.0283511 0.0302297 0.0302076 0.0300841 0.0297481 0.0297481 0.0300841 0.0302076 0.028981 0.0292558 0.0298783 0.0303643 0.0303643 0.0298783 0.0292558 0.0243294 0.0250849 0.0270063 0.0291512 0.0291512 0.0270063 0.0250849 0.0193528 0.0204739 0.0234703 0.0271626 0.0271626 0.0234703 0.0204739 [:, :, 23, 1] = 0.00897871 0.0106568 0.015432 0.021943 … 0.021943 0.015432 0.0106568 0.011533 0.013144 0.0176086 0.0234703 0.0234703 0.0176086 0.013144 0.0185206 0.0197747 0.0230644 0.0270063 0.0270063 0.0230644 0.0197747 0.0261308 0.0267602 0.0282994 0.0298783 0.0298783 0.0282994 0.0267602 0.0294403 0.0295961 0.0299181 0.0300841 0.0300841 0.0299181 0.0295961 0.0280543 0.0280931 0.0281669 0.0282014 … 0.0282014 0.0281669 0.0280931 0.0250258 0.0251414 0.0254659 0.0259695 0.0259695 0.0254659 0.0251414 ⋮ ⋱ 0.0250258 0.0251414 0.0254659 0.0259695 0.0259695 0.0254659 0.0251414 0.0280543 0.0280931 0.0281669 0.0282014 … 0.0282014 0.0281669 0.0280931 0.0294403 0.0295961 0.0299181 0.0300841 0.0300841 0.0299181 0.0295961 0.0261308 0.0267602 0.0282994 0.0298783 0.0298783 0.0282994 0.0267602 0.0185206 0.0197747 0.0230644 0.0270063 0.0270063 0.0230644 0.0197747 0.011533 0.013144 0.0176086 0.0234703 0.0234703 0.0176086 0.013144 [:, :, 24, 1] = 0.0033055 0.00515929 0.0106568 0.0185783 … 0.0185783 0.0106568 0.00515929 0.00594311 0.00780953 0.013144 0.0204739 0.0204739 0.013144 0.00780953 0.0139012 0.0155004 0.0197747 0.0250849 0.0250849 0.0197747 0.0155004 0.0235647 0.0244768 0.0267602 0.0292558 0.0292558 0.0267602 0.0244768 0.0285849 0.0288913 0.0295961 0.0302076 0.0302076 0.0295961 0.0288913 0.0277317 0.0278397 0.0280931 0.0283511 … 0.0283511 0.0280931 0.0278397 0.0245394 0.0246975 0.0251414 0.025824 0.025824 0.0251414 0.0246975 ⋮ ⋱ 0.0245394 0.0246975 0.0251414 0.025824 0.025824 0.0251414 0.0246975 0.0277317 0.0278397 0.0280931 0.0283511 … 0.0283511 0.0280931 0.0278397 0.0285849 0.0288913 0.0295961 0.0302076 0.0302076 0.0295961 0.0288913 0.0235647 0.0244768 0.0267602 0.0292558 0.0292558 0.0267602 0.0244768 0.0139012 0.0155004 0.0197747 0.0250849 0.0250849 0.0197747 0.0155004 0.00594311 0.00780953 0.013144 0.0204739 0.0204739 0.013144 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-2.85538e-7-7.4739e-7im 2.10552e-9+1.29792e-9im -1.07127e-9-2.29921e-10im 1.87727e-6-1.8991e-7im -1.27856e-9-5.67525e-10im -3.64851e-10+3.49019e-10im -7.87116e-7-4.24616e-7im -9.3112e-10+7.23061e-10im 2.52113e-8+6.79505e-9im … 2.6215e-6+7.78621e-7im 1.87104e-8-6.7484e-9im -3.51654e-10+5.99946e-10im 1.89672e-5+2.14296e-5im ⋮ ⋱ 1.17433e-10-1.06053e-10im 3.68458e-11-1.98663e-11im 7.96099e-6+1.10203e-5im 1.45446e-10-3.23514e-11im 7.62553e-11+4.70936e-11im … 1.6117e-5-4.19804e-5im -4.24303e-12+5.52886e-11im 7.09011e-11-3.74461e-11im 1.07589e-5+1.62896e-5im 5.88251e-11-2.21428e-11im 1.88343e-9+5.86628e-10im -2.20279e-5-4.17436e-6im 2.3953e-9-7.26469e-10im 4.73655e-11-1.46363e-11im -6.11547e-5-2.29252e-6im 2.55278e-11+7.14418e-11im 1.44907e-9-1.12881e-9im 2.88866e-6-2.19166e-5imtext/plaintypeArrayprefix_shortobjectid215573f7d7f76ef8!application/vnd.pluto.tree+objectdiagonalizationprefix٩@NamedTuple{λ::Vector{Vector{Float64}}, X::Vector{Matrix{ComplexF64}}, 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0.000354231text/plainmore6.08797e-9text/plaintypeArrayprefix_shortobjectid6a5c5c0ded8cf063!application/vnd.pluto.tree+objecthistory_EtotprefixFloat64elements-71.9648text/plain-71.9781text/plain-71.9795text/plain-71.9786text/plain-71.982text/plain-71.9812text/plain-71.9821text/plain-71.9823text/plain -71.9823text/plainmore-71.9823text/plaintypeArrayprefix_shortobjectide700e0c066d5fa6a!application/vnd.pluto.tree+objectruntime_ns0x00000003402c9693text/plainalgorithm"SCF"text/plaintypeNamedTupleobjectidca1ce0b9e2461672mime!application/vnd.pluto.tree+objectrootassigneescfres_Al_smalllast_run_timestampAŤHUpersist_js_state·has_pluto_hook_features§cell_id$0c146256-a19d-43c7-a874-63c810ab2f08depends_on_disabled_cells§runtimeMpublished_object_keysdepends_on_skipped_cells§errored$82399cad-6e2e-4586-8a0c-40a9d420c463queued¤logsrunning¦outputbody%

This mode can be physically interprested 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 phaenomenon is called charge-sloshing.

mimetext/htmlrootassigneelast_run_timestampAŤ, ԰persist_js_state·has_pluto_hook_features§cell_id$82399cad-6e2e-4586-8a0c-40a9d420c463depends_on_disabled_cells§runtimepublished_object_keysdepends_on_skipped_cells§errored$1fff9a65-fc63-445c-a41d-17df98d56d3cqueued¤logsrunning¦outputbody
mimetext/htmlrootassigneelast_run_timestampAŤ;tpersist_js_state·has_pluto_hook_features§cell_id$1fff9a65-fc63-445c-a41d-17df98d56d3cdepends_on_disabled_cells§runtime^published_object_keysdepends_on_skipped_cells§errored$7c37fa06-2c88-4794-baab-81c408a89e6aqueued¤logsrunning¦outputbodymimetext/plainrootassigneelast_run_timestampAŤ1 8persist_js_state·has_pluto_hook_features§cell_id$7c37fa06-2c88-4794-baab-81c408a89e6adepends_on_disabled_cells§runtimee0published_object_keysdepends_on_skipped_cells§errored$a4cf819c-8568-49fc-bf0a-3ae209aa56bequeued¤logsrunning¦outputbodyJ

Select kinetic energy cutoff Ecut:

mimetext/htmlrootassigneelast_run_timestampAŤ2 հpersist_js_state·has_pluto_hook_features§cell_id$a4cf819c-8568-49fc-bf0a-3ae209aa56bedepends_on_disabled_cells§runtimeupublished_object_keysdepends_on_skipped_cells§errored$fae0a2bd-6e12-4fa7-8805-0948c81efe57queued¤logsrunning¦outputbody
mimetext/htmlrootassigneelast_run_timestampAŤ2'W.persist_js_state·has_pluto_hook_features§cell_id$fae0a2bd-6e12-4fa7-8805-0948c81efe57depends_on_disabled_cells§runtimeFGpublished_object_keysdepends_on_skipped_cells§errored$ecacc9c7-3b11-46ba-aebd-c57887074651queued¤logsrunning¦outputbody. mimeimage/svg+xmlrootassigneelast_run_timestampAŤ;tt!persist_js_state·has_pluto_hook_features§cell_id$ecacc9c7-3b11-46ba-aebd-c57887074651depends_on_disabled_cells§runtime3cpublished_object_keysdepends_on_skipped_cells§errored$bead2d43-47ad-48ff-949e-1521dd997e22queued¤logsrunning¦outputbodymimetext/plainrootassigneelast_run_timestampAŤ;agpersist_js_state·has_pluto_hook_features§cell_id$bead2d43-47ad-48ff-949e-1521dd997e22depends_on_disabled_cells§runtime?{published_object_keysdepends_on_skipped_cells§errored$00d596b2-7060-4578-bc1a-9841e0c68277queued¤logslinemsgEigensolver not convergedtext/plaincell_id$00d596b2-7060-4578-bc1a-9841e0c68277kwargsn_iterprefixInt64elements4text/plain4text/plain4text/plain48text/plaintypeArrayprefix_shortobjectid42ab7dbdd567bbbb!application/vnd.pluto.tree+objectidDFTK_62605705fileH/home/runner/.julia/packages/DFTK/xNYFf/src/scf/self_consistent_field.jlgroupself_consistent_fieldlevelWarnlinemsgEigensolver not convergedtext/plaincell_id$00d596b2-7060-4578-bc1a-9841e0c68277kwargsn_iterprefixInt64elements5text/plain5text/plain3text/plain48text/plaintypeArrayprefix_shortobjectid8e0c281b6fb14881!application/vnd.pluto.tree+objectidDFTK_62605705fileH/home/runner/.julia/packages/DFTK/xNYFf/src/scf/self_consistent_field.jlgroupself_consistent_fieldlevelWarnrunning¦outputbody* mimeimage/svg+xmlrootassigneelast_run_timestampAŤC?persist_js_state·has_pluto_hook_features§cell_id$00d596b2-7060-4578-bc1a-9841e0c68277depends_on_disabled_cells§runtime{apublished_object_keysdepends_on_skipped_cells§errored$e58421d4-8f8d-11ef-18ff-d316ae57fa3aqueued¤logsrunning¦outputbodyG

Dielectric eigenpairs

To understand the difference we now look at the eigenpairs of the Jacobian governing the convergence, that is

$$1 - α P^{-1} \varepsilon^\dagger \qquad \text{with} \qquad \varepsilon^\dagger = (1-\chi_0 K).$$

mimetext/htmlrootassigneelast_run_timestampAŤ, P'persist_js_state·has_pluto_hook_features§cell_id$e58421d4-8f8d-11ef-18ff-d316ae57fa3adepends_on_disabled_cells§runtime+published_object_keysdepends_on_skipped_cells§errored$af492cf0-8bdb-4cae-91c8-b189c969f8dfqueued¤logsrunning¦outputbody

Clearly the charge-sloshing mode is no longer dominating.

Overview of largest eigenvalues:

Since the smallest eigenvalue is around $1$ in both cases, this implies that the conditioning improves noticably 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.

mimetext/htmlrootassigneelast_run_timestampAŤpersist_js_state·has_pluto_hook_features§cell_id$af492cf0-8bdb-4cae-91c8-b189c969f8dfdepends_on_disabled_cells§runtime쎵published_object_keysdepends_on_skipped_cells§errored$f2f201cf-0256-4363-b3c1-17104054e594queued¤logsrunning¦outputbody;

Dielectric eigenpairs

mimetext/htmlrootassigneelast_run_timestampAŤ+Tpersist_js_state·has_pluto_hook_features§cell_id$f2f201cf-0256-4363-b3c1-17104054e594depends_on_disabled_cells§runtimebŵpublished_object_keysdepends_on_skipped_cells§errored$a513d7cd-2431-4275-8965-286510668621queued¤logsrunning¦outputbodyP

We can also e.g. plot the largest eigenmode.

mimetext/htmlrootassigneelast_run_timestampAŤ, persist_js_state·has_pluto_hook_features§cell_id$a513d7cd-2431-4275-8965-286510668621depends_on_disabled_cells§runtime뜵published_object_keysdepends_on_skipped_cells§errored$a6c74c32-9329-4d11-8b05-9effa23a108aqueued¤logslinemsgiArnoldi schursolve in iter 1, krylovdim = 3: 0 values converged, normres = (3.00e+00, 1.22e+01, 8.58e-02)text/plaincell_id$a6c74c32-9329-4d11-8b05-9effa23a108akwargsidKrylovKit_21da2b25fileD/home/runner/.julia/packages/KrylovKit/Y0zG7/src/eigsolve/arnoldi.jlgrouparnoldilevelInfolinemsgiArnoldi schursolve in iter 1, krylovdim = 4: 0 values converged, normres = (9.47e-02, 1.68e+00, 4.95e+00)text/plaincell_id$a6c74c32-9329-4d11-8b05-9effa23a108akwargsidKrylovKit_21da2b25fileD/home/runner/.julia/packages/KrylovKit/Y0zG7/src/eigsolve/arnoldi.jlgrouparnoldilevelInfolinemsgiArnoldi schursolve in iter 1, krylovdim = 5: 0 values converged, normres = (3.44e-03, 2.66e-01, 4.55e+00)text/plaincell_id$a6c74c32-9329-4d11-8b05-9effa23a108akwargsidKrylovKit_21da2b25fileD/home/runner/.julia/packages/KrylovKit/Y0zG7/src/eigsolve/arnoldi.jlgrouparnoldilevelInfolinemsgiArnoldi schursolve in iter 1, krylovdim = 6: 1 values converged, normres = (7.21e-05, 2.64e-02, 2.42e+00)text/plaincell_id$a6c74c32-9329-4d11-8b05-9effa23a108akwargsidKrylovKit_21da2b25fileD/home/runner/.julia/packages/KrylovKit/Y0zG7/src/eigsolve/arnoldi.jlgrouparnoldilevelInfolinemsgiArnoldi schursolve in iter 1, krylovdim = 7: 1 values converged, normres = (1.84e-06, 2.78e-03, 8.97e-01)text/plaincell_id$a6c74c32-9329-4d11-8b05-9effa23a108akwargsidKrylovKit_21da2b25fileD/home/runner/.julia/packages/KrylovKit/Y0zG7/src/eigsolve/arnoldi.jlgrouparnoldilevelInfolinemsgiArnoldi schursolve in iter 1, krylovdim = 8: 1 values converged, normres = (2.18e-06, 8.55e+01, 5.33e-02)text/plaincell_id$a6c74c32-9329-4d11-8b05-9effa23a108akwargsidKrylovKit_21da2b25fileD/home/runner/.julia/packages/KrylovKit/Y0zG7/src/eigsolve/arnoldi.jlgrouparnoldilevelInfolinemsgiArnoldi schursolve in iter 1, krylovdim = 9: 1 values converged, normres = (2.18e-04, 1.61e+00, 1.31e-04)text/plaincell_id$a6c74c32-9329-4d11-8b05-9effa23a108akwargsidKrylovKit_21da2b25fileD/home/runner/.julia/packages/KrylovKit/Y0zG7/src/eigsolve/arnoldi.jlgrouparnoldilevelInfolinemsgjArnoldi schursolve in iter 1, krylovdim = 10: 0 values converged, normres = (3.54e-03, 1.31e-02, 3.41e-06)text/plaincell_id$a6c74c32-9329-4d11-8b05-9effa23a108akwargsidKrylovKit_21da2b25fileD/home/runner/.julia/packages/KrylovKit/Y0zG7/src/eigsolve/arnoldi.jlgrouparnoldilevelInfolinemsgjArnoldi schursolve in iter 1, krylovdim = 11: 0 values converged, normres = (1.71e-03, 9.87e-04, 2.97e-06)text/plaincell_id$a6c74c32-9329-4d11-8b05-9effa23a108akwargsidKrylovKit_21da2b25fileD/home/runner/.julia/packages/KrylovKit/Y0zG7/src/eigsolve/arnoldi.jlgrouparnoldilevelInfolinemsgپArnoldi eigsolve finished after 1 iterations: * 3 eigenvalues converged * norm of residuals = (6.945869607507194e-5, 2.63461549594983e-5, 1.663739390057295e-5) * number of operations = 12text/plaincell_id$a6c74c32-9329-4d11-8b05-9effa23a108akwargsidKrylovKit_626ee90bfileD/home/runner/.julia/packages/KrylovKit/Y0zG7/src/eigsolve/arnoldi.jlgrouparnoldilevelInforunning¦outputbodyelementsprefixComplexF64elements179.774+0.0imtext/plain179.774+0.0imtext/plain49.2165+0.0imtext/plaintypeArrayprefix_shortobjectidfbcd8f78d9b5a4ce!application/vnd.pluto.tree+objectprefixArray{ComplexF64, 4}elements150×24×24×1 Array{ComplexF64, 4}: [:, :, 1, 1] = 2.76184e-5+0.0im 7.24501e-5+0.0im … 0.000219807+0.0im 7.24498e-5+0.0im -0.000176596+0.0im -0.000169872+0.0im -0.000112887+0.0im -0.000169871+0.0im -0.000607259+0.0im -0.00059865+0.0im -0.000543117+0.0im -0.000598649+0.0im -0.00116222+0.0im -0.00112798+0.0im -0.00101485+0.0im -0.00112798+0.0im -0.00151854+0.0im -0.00146816+0.0im -0.00131752+0.0im -0.00146816+0.0im -0.00156945+0.0im -0.00152279+0.0im … -0.00138378+0.0im -0.00152279+0.0im -0.00147147+0.0im -0.00143815+0.0im -0.00133602+0.0im -0.00143815+0.0im ⋮ ⋱ 0.00329207+0.0im 0.00323848+0.0im 0.00307797+0.0im 0.00323848+0.0im 0.00346097+0.0im 0.00338765+0.0im … 0.00317494+0.0im 0.00338765+0.0im 0.00333813+0.0im 0.00327038+0.0im 0.0030723+0.0im 0.00327038+0.0im 0.00257824+0.0im 0.00256267+0.0im 0.00250205+0.0im 0.00256267+0.0im 0.00137147+0.0im 0.00142659+0.0im 0.00155849+0.0im 0.00142658+0.0im 0.000412164+0.0im 0.000492107+0.0im 0.00070852+0.0im 0.000492106+0.0im [:, :, 2, 1] = 7.24498e-5+0.0im 0.000113597+0.0im … 0.000248857+0.0im 0.000113596+0.0im -0.000169871+0.0im -0.000162013+0.0im -0.000105186+0.0im -0.000162013+0.0im -0.00059865+0.0im -0.00058865+0.0im -0.00053185+0.0im -0.000588649+0.0im -0.00112798+0.0im -0.00109426+0.0im -0.000983849+0.0im -0.00109426+0.0im -0.00146816+0.0im -0.00141926+0.0im -0.0012734+0.0im -0.00141926+0.0im -0.00152279+0.0im -0.00147715+0.0im … -0.00134149+0.0im -0.00147715+0.0im -0.00143815+0.0im -0.00140484+0.0im -0.00130321+0.0im -0.00140484+0.0im ⋮ ⋱ 0.00323848+0.0im 0.00318381+0.0im 0.00302084+0.0im 0.00318381+0.0im 0.00338765+0.0im 0.00331402+0.0im … 0.00310111+0.0im 0.00331402+0.0im 0.00327038+0.0im 0.0032013+0.0im 0.00300068+0.0im 0.0032013+0.0im 0.00256267+0.0im 0.0025416+0.0im 0.00246779+0.0im 0.0025416+0.0im 0.00142658+0.0im 0.00147166+0.0im 0.00157819+0.0im 0.00147166+0.0im 0.000492106+0.0im 0.000562579+0.0im 0.000753616+0.0im 0.000562578+0.0im [:, :, 3, 1] = 0.000219807+0.0im 0.000248858+0.0im … 0.000346307+0.0im 0.000248857+0.0im -0.000112888+0.0im -0.000105186+0.0im -5.64325e-5+0.0im -0.000105187+0.0im -0.000543119+0.0im -0.000531851+0.0im -0.000477534+0.0im -0.000531851+0.0im -0.00101485+0.0im -0.000983851+0.0im -0.000884041+0.0im -0.00098385+0.0im -0.00131752+0.0im -0.00127341+0.0im -0.00114238+0.0im -0.00127341+0.0im -0.00138378+0.0im -0.00134149+0.0im … -0.00121643+0.0im -0.00134149+0.0im -0.00133602+0.0im -0.00130321+0.0im -0.00120435+0.0im -0.00130321+0.0im ⋮ ⋱ 0.00307797+0.0im 0.00302084+0.0im 0.0028526+0.0im 0.00302084+0.0im 0.00317494+0.0im 0.00310111+0.0im … 0.00288955+0.0im 0.00310111+0.0im 0.0030723+0.0im 0.00300068+0.0im 0.00279598+0.0im 0.00300068+0.0im 0.00250205+0.0im 0.0024678+0.0im 0.00236256+0.0im 0.00246779+0.0im 0.00155849+0.0im 0.0015782+0.0im 0.00162036+0.0im 0.00157819+0.0im 0.000708521+0.0im 0.000753619+0.0im 0.000876567+0.0im 0.000753617+0.0im ;;; … [:, :, 22, 1] = 0.000455907+0.0im 0.000465658+0.0im … 0.000505362+0.0im 0.00046566+0.0im 3.69089e-5+0.0im 3.8624e-5+0.0im 6.30329e-5+0.0im 3.86259e-5+0.0im -0.000400249+0.0im -0.000392041+0.0im -0.000352347+0.0im -0.000392039+0.0im -0.00081353+0.0im -0.000788939+0.0im -0.00071016+0.0im -0.000788937+0.0im -0.00107648+0.0im -0.00104053+0.0im -0.00093408+0.0im -0.00104053+0.0im -0.00116104+0.0im -0.00112472+0.0im … -0.00101803+0.0im -0.00112472+0.0im -0.00116513+0.0im -0.00113426+0.0im -0.00104282+0.0im -0.00113426+0.0im ⋮ ⋱ 0.00282315+0.0im 0.00276359+0.0im 0.00259145+0.0im 0.00276359+0.0im 0.00285034+0.0im 0.00277789+0.0im … 0.00257288+0.0im 0.00277789+0.0im 0.00276433+0.0im 0.00269206+0.0im 0.00248909+0.0im 0.00269206+0.0im 0.00236713+0.0im 0.00231962+0.0im 0.00218326+0.0im 0.00231962+0.0im 0.00168047+0.0im 0.00167109+0.0im 0.00163913+0.0im 0.00167109+0.0im 0.000978902+0.0im 0.000991955+0.0im 0.00102864+0.0im 0.000991955+0.0im [:, :, 23, 1] = 0.000219808+0.0im 0.000248857+0.0im … 0.000346309+0.0im 0.000248858+0.0im -0.000112887+0.0im -0.000105186+0.0im -5.64301e-5+0.0im -0.000105184+0.0im -0.000543117+0.0im -0.000531851+0.0im -0.000477532+0.0im -0.000531849+0.0im -0.00101485+0.0im -0.000983849+0.0im -0.000884039+0.0im -0.000983848+0.0im -0.00131752+0.0im -0.0012734+0.0im -0.00114238+0.0im -0.0012734+0.0im -0.00138378+0.0im -0.00134149+0.0im … -0.00121642+0.0im -0.00134149+0.0im -0.00133602+0.0im -0.00130321+0.0im -0.00120435+0.0im -0.00130321+0.0im ⋮ ⋱ 0.00307797+0.0im 0.00302084+0.0im 0.0028526+0.0im 0.00302084+0.0im 0.00317494+0.0im 0.00310111+0.0im … 0.00288955+0.0im 0.00310111+0.0im 0.0030723+0.0im 0.00300068+0.0im 0.00279598+0.0im 0.00300068+0.0im 0.00250205+0.0im 0.0024678+0.0im 0.00236256+0.0im 0.0024678+0.0im 0.00155849+0.0im 0.0015782+0.0im 0.00162037+0.0im 0.00157819+0.0im 0.000708522+0.0im 0.000753618+0.0im 0.000876569+0.0im 0.000753619+0.0im [:, :, 24, 1] = 7.24502e-5+0.0im 0.000113597+0.0im … 0.000248858+0.0im 0.000113597+0.0im -0.000169871+0.0im -0.000162013+0.0im -0.000105185+0.0im -0.000162011+0.0im -0.000598649+0.0im -0.000588649+0.0im -0.000531849+0.0im -0.000588648+0.0im -0.00112798+0.0im -0.00109426+0.0im -0.000983848+0.0im -0.00109426+0.0im -0.00146816+0.0im -0.00141926+0.0im -0.0012734+0.0im -0.00141926+0.0im -0.00152279+0.0im -0.00147715+0.0im … -0.00134149+0.0im -0.00147715+0.0im -0.00143815+0.0im -0.00140484+0.0im -0.00130321+0.0im -0.00140484+0.0im ⋮ ⋱ 0.00323848+0.0im 0.00318382+0.0im 0.00302084+0.0im 0.00318382+0.0im 0.00338765+0.0im 0.00331402+0.0im … 0.00310111+0.0im 0.00331402+0.0im 0.00327038+0.0im 0.0032013+0.0im 0.00300068+0.0im 0.0032013+0.0im 0.00256267+0.0im 0.0025416+0.0im 0.00246779+0.0im 0.0025416+0.0im 0.00142659+0.0im 0.00147166+0.0im 0.00157819+0.0im 0.00147166+0.0im 0.000492106+0.0im 0.000562579+0.0im 0.000753618+0.0im 0.000562579+0.0imtext/plain150×24×24×1 Array{ComplexF64, 4}: [:, :, 1, 1] = -0.000176197+0.0im -0.000462208+0.0im … -0.0014023+0.0im -0.000462208+0.0im -0.000790791+0.0im -0.00107214+0.0im -0.00195491+0.0im -0.00107214+0.0im -0.00257003+0.0im -0.00277641+0.0im -0.00337942+0.0im -0.00277641+0.0im -0.00476698+0.0im -0.00482322+0.0im -0.00497911+0.0im -0.00482322+0.0im -0.00612894+0.0im -0.00606567+0.0im -0.00589099+0.0im -0.00606567+0.0im -0.00637003+0.0im -0.00627698+0.0im … -0.00601834+0.0im -0.00627698+0.0im -0.00612595+0.0im -0.00605547+0.0im -0.00585171+0.0im -0.00605547+0.0im ⋮ ⋱ -0.00548889+0.0im -0.00543003+0.0im -0.0052614+0.0im -0.00543003+0.0im -0.00569728+0.0im -0.00562027+0.0im … -0.00540867+0.0im -0.00562027+0.0im -0.00547942+0.0im -0.00543195+0.0im -0.0053039+0.0im -0.00543195+0.0im -0.00426674+0.0im -0.00432965+0.0im -0.00450877+0.0im -0.00432965+0.0im -0.0023054+0.0im -0.00250556+0.0im -0.00309835+0.0im -0.00250556+0.0im -0.000712052+0.0im -0.000983613+0.0im -0.00184505+0.0im -0.000983613+0.0im [:, :, 2, 1] = -0.000462208+0.0im -0.000724712+0.0im … -0.00158763+0.0im -0.000724712+0.0im -0.00107214+0.0im -0.0013262+0.0im -0.00212582+0.0im -0.0013262+0.0im -0.00277641+0.0im -0.00295444+0.0im -0.00347877+0.0im -0.00295444+0.0im -0.00482322+0.0im -0.00485992+0.0im -0.00496436+0.0im -0.00485992+0.0im -0.00606567+0.0im -0.00599339+0.0im -0.00579554+0.0im -0.00599339+0.0im -0.00627698+0.0im -0.00617971+0.0im … -0.00591001+0.0im -0.00617971+0.0im -0.00605547+0.0im -0.0059815+0.0im -0.00576812+0.0im -0.0059815+0.0im ⋮ ⋱ -0.00543003+0.0im -0.00536782+0.0im -0.00518983+0.0im -0.00536782+0.0im -0.00562027+0.0im -0.00553895+0.0im … -0.00531586+0.0im -0.00553895+0.0im -0.00543195+0.0im -0.00537544+0.0im -0.00522393+0.0im -0.00537544+0.0im -0.00432965+0.0im -0.00437367+0.0im -0.00450274+0.0im -0.00437367+0.0im -0.00250556+0.0im -0.0026789+0.0im -0.00319657+0.0im -0.0026789+0.0im -0.000983613+0.0im -0.00122929+0.0im -0.00201097+0.0im -0.00122929+0.0im [:, :, 3, 1] = -0.0014023+0.0im -0.00158763+0.0im … -0.00220934+0.0im -0.00158763+0.0im -0.00195491+0.0im -0.00212582+0.0im -0.00267849+0.0im -0.00212582+0.0im -0.00337942+0.0im -0.00347877+0.0im -0.00378574+0.0im -0.00347877+0.0im -0.00497911+0.0im -0.00496436+0.0im -0.00493333+0.0im -0.00496436+0.0im -0.00589099+0.0im -0.00579554+0.0im -0.00553808+0.0im -0.00579554+0.0im -0.00601834+0.0im -0.00591001+0.0im … -0.00561157+0.0im -0.00591001+0.0im -0.00585171+0.0im -0.00576812+0.0im -0.00552894+0.0im -0.00576812+0.0im ⋮ ⋱ -0.0052614+0.0im -0.00518983+0.0im -0.00498637+0.0im -0.00518983+0.0im -0.00540867+0.0im -0.00531586+0.0im … -0.00506245+0.0im -0.00531586+0.0im -0.0053039+0.0im -0.00522393+0.0im -0.00501138+0.0im -0.00522393+0.0im -0.00450877+0.0im -0.00450274+0.0im -0.00449914+0.0im -0.00450274+0.0im -0.00309835+0.0im -0.00319658+0.0im -0.00350517+0.0im -0.00319657+0.0im -0.00184505+0.0im -0.00201097+0.0im -0.00255372+0.0im -0.00201097+0.0im ;;; … [:, :, 22, 1] = -0.00290854+0.0im -0.00297075+0.0im … -0.00322404+0.0im -0.00297075+0.0im -0.00330326+0.0im -0.00335113+0.0im -0.00354682+0.0im -0.00335113+0.0im -0.00422283+0.0im -0.00421793+0.0im -0.0042378+0.0im -0.00421793+0.0im -0.00516841+0.0im -0.00509048+0.0im -0.00489242+0.0im -0.00509048+0.0im -0.0056408+0.0im -0.0055177+0.0im -0.00518912+0.0im -0.0055177+0.0im -0.00565683+0.0im -0.00553432+0.0im … -0.00519987+0.0im -0.00553432+0.0im -0.00555551+0.0im -0.00545797+0.0im -0.0051829+0.0im -0.00545797+0.0im ⋮ ⋱ -0.00502212+0.0im -0.00493668+0.0im -0.00469686+0.0im -0.00493668+0.0im -0.00512036+0.0im -0.00501239+0.0im … -0.00471963+0.0im -0.00501239+0.0im -0.00512714+0.0im -0.00501851+0.0im -0.00473132+0.0im -0.00501851+0.0im -0.00474304+0.0im -0.00467475+0.0im -0.00450546+0.0im -0.00467475+0.0im -0.00394456+0.0im -0.00394201+0.0im -0.00397146+0.0im -0.00394201+0.0im -0.00317728+0.0im -0.00322364+0.0im -0.00341768+0.0im -0.00322364+0.0im [:, :, 23, 1] = -0.0014023+0.0im -0.00158763+0.0im … -0.00220934+0.0im -0.00158763+0.0im -0.00195491+0.0im -0.00212582+0.0im -0.0026785+0.0im -0.00212582+0.0im -0.00337942+0.0im -0.00347877+0.0im -0.00378574+0.0im -0.00347877+0.0im -0.00497911+0.0im -0.00496436+0.0im -0.00493334+0.0im -0.00496436+0.0im -0.00589099+0.0im -0.00579554+0.0im -0.00553809+0.0im -0.00579554+0.0im -0.00601834+0.0im -0.00591001+0.0im … -0.00561157+0.0im -0.00591001+0.0im -0.00585171+0.0im -0.00576812+0.0im -0.00552894+0.0im -0.00576812+0.0im ⋮ ⋱ -0.0052614+0.0im -0.00518984+0.0im -0.00498637+0.0im -0.00518984+0.0im -0.00540867+0.0im -0.00531586+0.0im … -0.00506245+0.0im -0.00531586+0.0im -0.00530391+0.0im -0.00522393+0.0im -0.00501138+0.0im -0.00522393+0.0im -0.00450877+0.0im -0.00450274+0.0im -0.00449914+0.0im -0.00450274+0.0im -0.00309835+0.0im -0.00319658+0.0im -0.00350517+0.0im -0.00319658+0.0im -0.00184505+0.0im -0.00201097+0.0im -0.00255372+0.0im -0.00201097+0.0im [:, :, 24, 1] = -0.000462208+0.0im -0.000724712+0.0im … -0.00158763+0.0im -0.000724712+0.0im -0.00107214+0.0im -0.0013262+0.0im -0.00212582+0.0im -0.0013262+0.0im -0.00277641+0.0im -0.00295444+0.0im -0.00347877+0.0im -0.00295444+0.0im -0.00482322+0.0im -0.00485992+0.0im -0.00496436+0.0im -0.00485992+0.0im -0.00606567+0.0im -0.00599339+0.0im -0.00579554+0.0im -0.00599339+0.0im -0.00627698+0.0im -0.00617971+0.0im … -0.00591001+0.0im -0.00617971+0.0im -0.00605547+0.0im -0.0059815+0.0im -0.00576812+0.0im -0.0059815+0.0im ⋮ ⋱ -0.00543003+0.0im -0.00536782+0.0im -0.00518984+0.0im -0.00536782+0.0im -0.00562027+0.0im -0.00553895+0.0im … -0.00531586+0.0im -0.00553895+0.0im -0.00543195+0.0im -0.00537545+0.0im -0.00522393+0.0im -0.00537545+0.0im -0.00432965+0.0im -0.00437367+0.0im -0.00450274+0.0im -0.00437367+0.0im -0.00250556+0.0im -0.0026789+0.0im -0.00319658+0.0im -0.0026789+0.0im -0.000983613+0.0im -0.00122929+0.0im -0.00201097+0.0im -0.00122929+0.0imtext/plain150×24×24×1 Array{ComplexF64, 4}: [:, :, 1, 1] = 3.36231e-5+0.0im 0.000230132+0.0im … 0.000921248+0.0im 0.000230132+0.0im 0.000394378+0.0im 0.00055878+0.0im 0.00114778+0.0im 0.000558779+0.0im 0.00127335+0.0im 0.00138232+0.0im 0.00177775+0.0im 0.00138232+0.0im 0.00220764+0.0im 0.00226003+0.0im 0.00246024+0.0im 0.00226003+0.0im 0.0029433+0.0im 0.00295176+0.0im 0.00300461+0.0im 0.00295176+0.0im 0.00342244+0.0im 0.0034066+0.0im … 0.00337378+0.0im 0.0034066+0.0im 0.00368568+0.0im 0.00366353+0.0im 0.00359844+0.0im 0.00366353+0.0im ⋮ ⋱ 0.00524346+0.0im 0.00518727+0.0im 0.00501971+0.0im 0.00518727+0.0im 0.00506768+0.0im 0.00500663+0.0im … 0.00484161+0.0im 0.00500663+0.0im 0.0045129+0.0im 0.00447528+0.0im 0.00439367+0.0im 0.00447528+0.0im 0.00339863+0.0im 0.00342472+0.0im 0.00354176+0.0im 0.00342472+0.0im 0.00189029+0.0im 0.00200324+0.0im 0.00239485+0.0im 0.00200325+0.0im 0.000570341+0.0im 0.000750811+0.0im 0.00137202+0.0im 0.000750812+0.0im [:, :, 2, 1] = 0.000230132+0.0im 0.000389785+0.0im … 0.000982176+0.0im 0.000389785+0.0im 0.00055878+0.0im 0.000691258+0.0im 0.00119437+0.0im 0.000691258+0.0im 0.00138232+0.0im 0.00146779+0.0im 0.00179997+0.0im 0.00146778+0.0im 0.00226003+0.0im 0.00229684+0.0im 0.00245514+0.0im 0.00229684+0.0im 0.00295176+0.0im 0.00295023+0.0im 0.00297611+0.0im 0.00295023+0.0im 0.0034066+0.0im 0.00338414+0.0im … 0.00333329+0.0im 0.00338414+0.0im 0.00366353+0.0im 0.00363681+0.0im 0.00355906+0.0im 0.00363681+0.0im ⋮ ⋱ 0.00518727+0.0im 0.00512671+0.0im 0.00494734+0.0im 0.00512671+0.0im 0.00500663+0.0im 0.00493975+0.0im … 0.00475923+0.0im 0.00493975+0.0im 0.00447528+0.0im 0.00442845+0.0im 0.00432245+0.0im 0.00442845+0.0im 0.00342472+0.0im 0.0034346+0.0im 0.00350886+0.0im 0.0034346+0.0im 0.00200325+0.0im 0.00209005+0.0im 0.00241248+0.0im 0.00209005+0.0im 0.000750812+0.0im 0.000896598+0.0im 0.00142552+0.0im 0.000896599+0.0im [:, :, 3, 1] = 0.000921248+0.0im 0.000982175+0.0im … 0.00130523+0.0im 0.000982176+0.0im 0.00114778+0.0im 0.00119436+0.0im 0.00146226+0.0im 0.00119437+0.0im 0.00177775+0.0im 0.00179997+0.0im 0.00195912+0.0im 0.00179997+0.0im 0.00246024+0.0im 0.00245514+0.0im 0.00249919+0.0im 0.00245514+0.0im 0.00300461+0.0im 0.00297611+0.0im 0.00292862+0.0im 0.00297611+0.0im 0.00337378+0.0im 0.00333329+0.0im … 0.0032335+0.0im 0.00333329+0.0im 0.00359844+0.0im 0.00355906+0.0im 0.00344682+0.0im 0.00355906+0.0im ⋮ ⋱ 0.00501971+0.0im 0.00494734+0.0im 0.00473663+0.0im 0.00494734+0.0im 0.00484161+0.0im 0.00475923+0.0im … 0.00453763+0.0im 0.00475923+0.0im 0.00439367+0.0im 0.00432245+0.0im 0.00415142+0.0im 0.00432245+0.0im 0.00354176+0.0im 0.00350886+0.0im 0.00346862+0.0im 0.00350886+0.0im 0.00239485+0.0im 0.00241247+0.0im 0.00254884+0.0im 0.00241247+0.0im 0.00137202+0.0im 0.00142552+0.0im 0.00170444+0.0im 0.00142552+0.0im ;;; … [:, :, 22, 1] = 0.00211911+0.0im 0.00205801+0.0im … 0.00203551+0.0im 0.00205801+0.0im 0.00219048+0.0im 0.00212988+0.0im 0.00209341+0.0im 0.00212987+0.0im 0.00249667+0.0im 0.00243892+0.0im 0.002372+0.0im 0.00243892+0.0im 0.00285178+0.0im 0.00279261+0.0im 0.00268522+0.0im 0.00279261+0.0im 0.00315008+0.0im 0.00308567+0.0im 0.00293894+0.0im 0.00308567+0.0im 0.00336575+0.0im 0.00330035+0.0im … 0.003133+0.0im 0.00330035+0.0im 0.00351457+0.0im 0.00345672+0.0im 0.00329531+0.0im 0.00345672+0.0im ⋮ ⋱ 0.00476969+0.0im 0.00468071+0.0im 0.00442746+0.0im 0.00468071+0.0im 0.00462789+0.0im 0.00452497+0.0im … 0.00424941+0.0im 0.00452497+0.0im 0.00433041+0.0im 0.00422807+0.0im 0.00397268+0.0im 0.00422807+0.0im 0.0037891+0.0im 0.00370296+0.0im 0.00351614+0.0im 0.00370296+0.0im 0.00307181+0.0im 0.00300436+0.0im 0.00290414+0.0im 0.00300436+0.0im 0.00243069+0.0im 0.0023708+0.0im 0.00233122+0.0im 0.0023708+0.0im [:, :, 23, 1] = 0.000921247+0.0im 0.000982176+0.0im … 0.00130523+0.0im 0.000982175+0.0im 0.00114778+0.0im 0.00119437+0.0im 0.00146226+0.0im 0.00119436+0.0im 0.00177775+0.0im 0.00179997+0.0im 0.00195912+0.0im 0.00179996+0.0im 0.00246024+0.0im 0.00245514+0.0im 0.00249919+0.0im 0.00245514+0.0im 0.00300461+0.0im 0.00297611+0.0im 0.00292862+0.0im 0.00297611+0.0im 0.00337378+0.0im 0.00333329+0.0im … 0.00323349+0.0im 0.00333329+0.0im 0.00359844+0.0im 0.00355906+0.0im 0.00344682+0.0im 0.00355906+0.0im ⋮ ⋱ 0.0050197+0.0im 0.00494734+0.0im 0.00473663+0.0im 0.00494734+0.0im 0.00484161+0.0im 0.00475923+0.0im … 0.00453763+0.0im 0.00475923+0.0im 0.00439367+0.0im 0.00432245+0.0im 0.00415142+0.0im 0.00432245+0.0im 0.00354176+0.0im 0.00350886+0.0im 0.00346862+0.0im 0.00350886+0.0im 0.00239485+0.0im 0.00241247+0.0im 0.00254884+0.0im 0.00241247+0.0im 0.00137202+0.0im 0.00142552+0.0im 0.00170444+0.0im 0.00142552+0.0im [:, :, 24, 1] = 0.000230132+0.0im 0.000389785+0.0im … 0.000982175+0.0im 0.000389785+0.0im 0.000558779+0.0im 0.000691258+0.0im 0.00119436+0.0im 0.000691257+0.0im 0.00138232+0.0im 0.00146778+0.0im 0.00179996+0.0im 0.00146778+0.0im 0.00226003+0.0im 0.00229684+0.0im 0.00245514+0.0im 0.00229684+0.0im 0.00295176+0.0im 0.00295023+0.0im 0.00297611+0.0im 0.00295023+0.0im 0.0034066+0.0im 0.00338414+0.0im … 0.00333329+0.0im 0.00338414+0.0im 0.00366353+0.0im 0.00363681+0.0im 0.00355906+0.0im 0.00363681+0.0im ⋮ ⋱ 0.00518727+0.0im 0.00512671+0.0im 0.00494734+0.0im 0.00512671+0.0im 0.00500663+0.0im 0.00493975+0.0im … 0.00475923+0.0im 0.00493975+0.0im 0.00447528+0.0im 0.00442844+0.0im 0.00432245+0.0im 0.00442845+0.0im 0.00342472+0.0im 0.0034346+0.0im 0.00350886+0.0im 0.0034346+0.0im 0.00200324+0.0im 0.00209005+0.0im 0.00241247+0.0im 0.00209005+0.0im 0.000750811+0.0im 0.000896598+0.0im 0.00142552+0.0im 0.000896598+0.0imtext/plaintypeArrayprefix_shortobjectidda6019a84fec7705!application/vnd.pluto.tree+objectConvergenceInfo: 3 converged values after 1 iterations and 12 applications of the linear map; norms of residuals are given by (6.945869607507194e-5, 2.63461549594983e-5, 1.663739390057295e-5). text/plaintypeTupleobjectidd5d7ac52f4a0ddddmime!application/vnd.pluto.tree+objectrootassigneelast_run_timestampAŤn¡persist_js_state·has_pluto_hook_features§cell_id$a6c74c32-9329-4d11-8b05-9effa23a108adepends_on_disabled_cells§runtime"1published_object_keysdepends_on_skipped_cells§errored$5f8776fb-bea2-4b30-a733-8899392e0aa2queued¤logsrunning¦outputbody

Expensive notebook

Some computations in this notebook take extended time. To see the "real" effects. Try increasing the n_repeat_aluminium and n_repeat_helium and Ecut as high as you can afford. Running the simulations take minutes. This is normal.

mimetext/htmlrootassigneelast_run_timestampAŤ,9persist_js_state·has_pluto_hook_features§cell_id$5f8776fb-bea2-4b30-a733-8899392e0aa2depends_on_disabled_cells§runtime63published_object_keysdepends_on_skipped_cells§errored$7fb6f4bf-3ccb-44be-8a69-735cb004e48dqueued¤logsrunning¦outputbodym

We repeat the exercise for the Kerker-preconditioned dielectric operator:

mimetext/htmlrootassigneelast_run_timestampAŤ, persist_js_state·has_pluto_hook_features§cell_id$7fb6f4bf-3ccb-44be-8a69-735cb004e48ddepends_on_disabled_cells§runtimepublished_object_keysdepends_on_skipped_cells§errored$13446c99-ae13-47ee-9c29-f6acd2a9fff4queued¤logsrunning¦outputbodymimetext/plainrootassigneelast_run_timestampAŤ5tpersist_js_state·has_pluto_hook_features§cell_id$13446c99-ae13-47ee-9c29-f6acd2a9fff4depends_on_disabled_cells§runtimeGTݵpublished_object_keysdepends_on_skipped_cells§errored$e297eedc-d45e-466c-834e-e59cf2814de3queued¤logsrunning¦outputbodyȑb mimeimage/svg+xmlrootassigneelast_run_timestampAŤ[lpersist_js_state·has_pluto_hook_features§cell_id$e297eedc-d45e-466c-834e-e59cf2814de3depends_on_disabled_cells§runtimeEpublished_object_keysdepends_on_skipped_cells§errored$a5cb6eac-57a8-45fe-8436-7ab7ea8d8f09queued¤logsrunning¦outputbodyȚ mimeimage/svg+xmlrootassigneelast_run_timestampAŤoUQpersist_js_state·has_pluto_hook_features§cell_id$a5cb6eac-57a8-45fe-8436-7ab7ea8d8f09depends_on_disabled_cells§runtimeW֩published_object_keysdepends_on_skipped_cells§errored$a70d053e-4257-4458-86bf-aef38557493bqueued¤logsrunning¦outputbody٪

With these functions available we can now compute the desired eigenvalues. For simplicity we only consider the first few largest ones.

mimetext/htmlrootassigneelast_run_timestampAŤ, jfpersist_js_state·has_pluto_hook_features§cell_id$a70d053e-4257-4458-86bf-aef38557493bdepends_on_disabled_cells§runtime ٵpublished_object_keysdepends_on_skipped_cells§errored$7434967b-b988-447a-b346-cac2b4c11776queued¤logsrunning¦outputbodyMFlexibleSystem(Al₈, periodic = TTT): bounding_box : [ 8.1 0 0; 0 4.05 0; 0 0 4.05]u"Å" Atom(Al, [ 0, 0, 0]u"Å") Atom(Al, [ 0, 2.025, 2.025]u"Å") Atom(Al, [ 2.025, 0, 2.025]u"Å") Atom(Al, [ 2.025, 2.025, 0]u"Å") Atom(Al, [ 4.05, 0, 0]u"Å") Atom(Al, [ 4.05, 2.025, 2.025]u"Å") Atom(Al, [ 6.075, 0, 2.025]u"Å") Atom(Al, [ 6.075, 2.025, 0]u"Å") .-------------------. /| | * | | |Al Al | | | | | .--Al--------Al-----. |/ Al Al / Al--------Al--------* mimetext/plainrootassigneelast_run_timestampAŤ3persist_js_state·has_pluto_hook_features§cell_id$7434967b-b988-447a-b346-cac2b4c11776depends_on_disabled_cells§runtimeΧzpublished_object_keysdepends_on_skipped_cells§errored$0574ad92-2554-4469-9776-4a42caad7256queued¤logsrunning¦outputbody6

Modelling helium

mimetext/htmlrootassigneelast_run_timestampAŤ, 0 persist_js_state·has_pluto_hook_features§cell_id$0574ad92-2554-4469-9776-4a42caad7256depends_on_disabled_cells§runtimetYpublished_object_keysdepends_on_skipped_cells§errored$114d7d12-e317-4f1e-86bf-d75d8ab3cc23queued¤logslinemsgiArnoldi schursolve in iter 1, krylovdim = 3: 0 values converged, normres = (3.80e-01, 4.27e-02, 1.35e-01)text/plaincell_id$114d7d12-e317-4f1e-86bf-d75d8ab3cc23kwargsidKrylovKit_21da2b25fileD/home/runner/.julia/packages/KrylovKit/Y0zG7/src/eigsolve/arnoldi.jlgrouparnoldilevelInfolinemsgiArnoldi schursolve in iter 1, krylovdim = 4: 0 values converged, normres = (1.21e+00, 5.10e-01, 1.38e-02)text/plaincell_id$114d7d12-e317-4f1e-86bf-d75d8ab3cc23kwargsidKrylovKit_21da2b25fileD/home/runner/.julia/packages/KrylovKit/Y0zG7/src/eigsolve/arnoldi.jlgrouparnoldilevelInfolinemsgiArnoldi schursolve in iter 1, krylovdim = 5: 0 values converged, normres = (3.16e-01, 1.52e-01, 5.33e-01)text/plaincell_id$114d7d12-e317-4f1e-86bf-d75d8ab3cc23kwargsidKrylovKit_21da2b25fileD/home/runner/.julia/packages/KrylovKit/Y0zG7/src/eigsolve/arnoldi.jlgrouparnoldilevelInfolinemsgiArnoldi schursolve in iter 1, krylovdim = 6: 0 values converged, normres = (5.98e-02, 9.45e-02, 2.76e-01)text/plaincell_id$114d7d12-e317-4f1e-86bf-d75d8ab3cc23kwargsidKrylovKit_21da2b25fileD/home/runner/.julia/packages/KrylovKit/Y0zG7/src/eigsolve/arnoldi.jlgrouparnoldilevelInfolinemsgiArnoldi schursolve in iter 1, krylovdim = 7: 0 values converged, normres = (2.89e-02, 1.21e-01, 3.15e-01)text/plaincell_id$114d7d12-e317-4f1e-86bf-d75d8ab3cc23kwargsidKrylovKit_21da2b25fileD/home/runner/.julia/packages/KrylovKit/Y0zG7/src/eigsolve/arnoldi.jlgrouparnoldilevelInfolinemsgiArnoldi schursolve in iter 1, krylovdim = 8: 0 values converged, normres = (3.36e-03, 4.16e-02, 9.95e-02)text/plaincell_id$114d7d12-e317-4f1e-86bf-d75d8ab3cc23kwargsidKrylovKit_21da2b25fileD/home/runner/.julia/packages/KrylovKit/Y0zG7/src/eigsolve/arnoldi.jlgrouparnoldilevelInfolinemsgiArnoldi schursolve in iter 1, krylovdim = 9: 1 values converged, normres = (6.20e-04, 1.41e-02, 3.89e-02)text/plaincell_id$114d7d12-e317-4f1e-86bf-d75d8ab3cc23kwargsidKrylovKit_21da2b25fileD/home/runner/.julia/packages/KrylovKit/Y0zG7/src/eigsolve/arnoldi.jlgrouparnoldilevelInfolinemsgjArnoldi schursolve in iter 1, krylovdim = 10: 1 values converged, normres = (9.89e-05, 4.53e-03, 1.55e-02)text/plaincell_id$114d7d12-e317-4f1e-86bf-d75d8ab3cc23kwargsidKrylovKit_21da2b25fileD/home/runner/.julia/packages/KrylovKit/Y0zG7/src/eigsolve/arnoldi.jlgrouparnoldilevelInfolinemsgjArnoldi schursolve in iter 1, krylovdim = 11: 1 values converged, normres = (3.04e-05, 2.86e-03, 1.23e-02)text/plaincell_id$114d7d12-e317-4f1e-86bf-d75d8ab3cc23kwargsidKrylovKit_21da2b25fileD/home/runner/.julia/packages/KrylovKit/Y0zG7/src/eigsolve/arnoldi.jlgrouparnoldilevelInfolinemsgjArnoldi schursolve in iter 1, krylovdim = 12: 1 values converged, normres = (6.97e-05, 9.12e-01, 2.07e-01)text/plaincell_id$114d7d12-e317-4f1e-86bf-d75d8ab3cc23kwargsidKrylovKit_21da2b25fileD/home/runner/.julia/packages/KrylovKit/Y0zG7/src/eigsolve/arnoldi.jlgrouparnoldilevelInfolinemsgjArnoldi schursolve in iter 1, krylovdim = 13: 1 values converged, normres = (2.84e-04, 2.22e-01, 7.86e-03)text/plaincell_id$114d7d12-e317-4f1e-86bf-d75d8ab3cc23kwargsidKrylovKit_21da2b25fileD/home/runner/.julia/packages/KrylovKit/Y0zG7/src/eigsolve/arnoldi.jlgrouparnoldilevelInfolinemsgjArnoldi schursolve in iter 1, krylovdim = 14: 1 values converged, normres = (8.89e-04, 3.70e-02, 2.99e-04)text/plaincell_id$114d7d12-e317-4f1e-86bf-d75d8ab3cc23kwargsidKrylovKit_21da2b25fileD/home/runner/.julia/packages/KrylovKit/Y0zG7/src/eigsolve/arnoldi.jlgrouparnoldilevelInfolinemsgjArnoldi schursolve in iter 1, krylovdim = 15: 1 values converged, normres = (9.56e-04, 4.75e-03, 6.66e-05)text/plaincell_id$114d7d12-e317-4f1e-86bf-d75d8ab3cc23kwargsidKrylovKit_21da2b25fileD/home/runner/.julia/packages/KrylovKit/Y0zG7/src/eigsolve/arnoldi.jlgrouparnoldilevelInfolinemsgArnoldi eigsolve finished after 1 iterations: * 4 eigenvalues converged * norm of residuals = (0.00010962060099514753, 0.0008696529910375382, 2.220453924156357e-5, 0.0001727166939958009) * number of operations = 16text/plaincell_id$114d7d12-e317-4f1e-86bf-d75d8ab3cc23kwargsidKrylovKit_626ee90bfileD/home/runner/.julia/packages/KrylovKit/Y0zG7/src/eigsolve/arnoldi.jlgrouparnoldilevelInforunning¦outputbodyelementsprefixComplexF64elements4.32902+0.0imtext/plain4.32881+0.0imtext/plain3.13704+0.0imtext/plain2.91958+0.0imtext/plaintypeArrayprefix_shortobjectid6aa837396aae6cb5!application/vnd.pluto.tree+objectprefixArray{ComplexF64, 4}elements150×24×24×1 Array{ComplexF64, 4}: [:, :, 1, 1] = -1.44013e-5+0.0im -1.29199e-5+0.0im … -7.15191e-6+0.0im -1.28827e-5+0.0im -3.80687e-5+0.0im 0.000243078+0.0im 0.00088305+0.0im 0.000243068+0.0im 0.00358534+0.0im 0.00375276+0.0im 0.00407467+0.0im 0.00375275+0.0im 0.00916906+0.0im 0.00895652+0.0im 0.00831345+0.0im 0.0089566+0.0im 0.012423+0.0im 0.0119498+0.0im 0.0106436+0.0im 0.0119499+0.0im 0.0120136+0.0im 0.011541+0.0im … 0.0102362+0.0im 0.011541+0.0im 0.00983531+0.0im 0.00949549+0.0im 0.00853815+0.0im 0.0094955+0.0im ⋮ ⋱ -0.00984153+0.0im -0.00950199+0.0im -0.00854513+0.0im -0.00950203+0.0im -0.0120058+0.0im -0.0115338+0.0im … -0.0102304+0.0im -0.0115338+0.0im -0.0124046+0.0im -0.0119317+0.0im -0.0106259+0.0im -0.0119317+0.0im -0.00915829+0.0im -0.00894405+0.0im -0.00829651+0.0im -0.00894405+0.0im -0.00359677+0.0im -0.00376049+0.0im -0.00407192+0.0im -0.00376049+0.0im 1.18758e-5+0.0im -0.00026563+0.0im -0.000893423+0.0im -0.000265628+0.0im [:, :, 2, 1] = -1.29159e-5+0.0im -1.16175e-5+0.0im … -6.49463e-6+0.0im -1.15736e-5+0.0im 0.000243099+0.0im 0.000476369+0.0im 0.00100066+0.0im 0.000476404+0.0im 0.00375274+0.0im 0.00386504+0.0im 0.00405354+0.0im 0.00386508+0.0im 0.00895656+0.0im 0.00871186+0.0im 0.0079917+0.0im 0.0087119+0.0im 0.0119499+0.0im 0.0114647+0.0im 0.0101307+0.0im 0.0114647+0.0im 0.011541+0.0im 0.0110612+0.0im … 0.00973962+0.0im 0.0110613+0.0im 0.00949548+0.0im 0.00914563+0.0im 0.00816165+0.0im 0.00914564+0.0im ⋮ ⋱ -0.00950202+0.0im -0.00915236+0.0im -0.00816874+0.0im -0.0091524+0.0im -0.0115338+0.0im -0.0110546+0.0im … -0.00973445+0.0im -0.0110547+0.0im -0.0119317+0.0im -0.0114468+0.0im -0.0101139+0.0im -0.0114468+0.0im -0.00894403+0.0im -0.00869811+0.0im -0.00797463+0.0im -0.00869809+0.0im -0.0037605+0.0im -0.00386979+0.0im -0.0040494+0.0im -0.00386975+0.0im -0.00026564+0.0im -0.000495829+0.0im -0.00100938+0.0im -0.0004958+0.0im [:, :, 3, 1] = -7.1482e-6+0.0im -6.52595e-6+0.0im … -3.47071e-6+0.0im -6.49558e-6+0.0im 0.000883016+0.0im 0.00100057+0.0im 0.00124444+0.0im 0.00100059+0.0im 0.00407457+0.0im 0.00405346+0.0im 0.00391878+0.0im 0.00405346+0.0im 0.00831336+0.0im 0.00799159+0.0im 0.00708679+0.0im 0.00799161+0.0im 0.0106435+0.0im 0.0101307+0.0im 0.00873456+0.0im 0.0101307+0.0im 0.0102362+0.0im 0.0097396+0.0im … 0.00837947+0.0im 0.0097396+0.0im 0.0085382+0.0im 0.00816162+0.0im 0.00710859+0.0im 0.00816166+0.0im ⋮ ⋱ -0.00854509+0.0im -0.00816869+0.0im -0.00711575+0.0im -0.00816873+0.0im -0.0102304+0.0im -0.00973444+0.0im … -0.00837573+0.0im -0.00973444+0.0im -0.0106259+0.0im -0.0101139+0.0im -0.00871973+0.0im -0.0101139+0.0im -0.00829653+0.0im -0.00797464+0.0im -0.00706953+0.0im -0.0079746+0.0im -0.00407195+0.0im -0.00404946+0.0im -0.00391051+0.0im -0.00404943+0.0im -0.000893454+0.0im -0.00100942+0.0im -0.00124721+0.0im -0.00100939+0.0im ;;; … [:, :, 22, 1] = 2.96999e-6+0.0im 2.55843e-6+0.0im … 2.31689e-6+0.0im 2.53576e-6+0.0im 0.00147607+0.0im 0.00146758+0.0im 0.00140569+0.0im 0.0014676+0.0im 0.00420383+0.0im 0.00403873+0.0im 0.00355519+0.0im 0.00403873+0.0im 0.00727072+0.0im 0.00686975+0.0im 0.00577631+0.0im 0.00686978+0.0im 0.00880592+0.0im 0.00826943+0.0im 0.00682331+0.0im 0.00826943+0.0im 0.00841911+0.0im 0.00790567+0.0im … 0.00651027+0.0im 0.00790563+0.0im 0.00718363+0.0im 0.00677012+0.0im 0.00562565+0.0im 0.00677011+0.0im ⋮ ⋱ -0.00719008+0.0im -0.00677663+0.0im -0.00563215+0.0im -0.00677664+0.0im -0.00841468+0.0im -0.00790187+0.0im … -0.00650812+0.0im -0.00790185+0.0im -0.00878953+0.0im -0.00825426+0.0im -0.00681133+0.0im -0.00825425+0.0im -0.00724935+0.0im -0.00684957+0.0im -0.0057593+0.0im -0.00684953+0.0im -0.00418747+0.0im -0.0040232+0.0im -0.00354126+0.0im -0.00402318+0.0im -0.00146731+0.0im -0.00145945+0.0im -0.00139822+0.0im -0.00145946+0.0im [:, :, 23, 1] = -7.22097e-6+0.0im -6.51652e-6+0.0im … -3.50623e-6+0.0im -6.51759e-6+0.0im 0.00088301+0.0im 0.00100056+0.0im 0.00124444+0.0im 0.00100054+0.0im 0.00407459+0.0im 0.00405347+0.0im 0.00391874+0.0im 0.00405348+0.0im 0.00831337+0.0im 0.00799163+0.0im 0.00708679+0.0im 0.00799165+0.0im 0.0106435+0.0im 0.0101307+0.0im 0.00873454+0.0im 0.0101307+0.0im 0.0102362+0.0im 0.00973957+0.0im … 0.00837939+0.0im 0.00973957+0.0im 0.0085382+0.0im 0.00816164+0.0im 0.00710852+0.0im 0.00816165+0.0im ⋮ ⋱ -0.00854506+0.0im -0.00816869+0.0im -0.00711565+0.0im -0.00816866+0.0im -0.0102304+0.0im -0.00973438+0.0im … -0.00837572+0.0im -0.00973439+0.0im -0.0106259+0.0im -0.0101139+0.0im -0.00871967+0.0im -0.0101138+0.0im -0.00829652+0.0im -0.00797462+0.0im -0.00706952+0.0im -0.00797466+0.0im -0.00407196+0.0im -0.00404945+0.0im -0.00391059+0.0im -0.00404946+0.0im -0.000893442+0.0im -0.00100939+0.0im -0.00124727+0.0im -0.00100943+0.0im [:, :, 24, 1] = -1.29246e-5+0.0im -1.16254e-5+0.0im … -6.4947e-6+0.0im -1.16093e-5+0.0im 0.000243025+0.0im 0.000476356+0.0im 0.0010006+0.0im 0.000476375+0.0im 0.00375275+0.0im 0.00386506+0.0im 0.0040535+0.0im 0.00386506+0.0im 0.00895656+0.0im 0.00871184+0.0im 0.0079917+0.0im 0.00871192+0.0im 0.0119499+0.0im 0.0114647+0.0im 0.0101308+0.0im 0.0114647+0.0im 0.011541+0.0im 0.0110613+0.0im … 0.00973959+0.0im 0.0110613+0.0im 0.0094955+0.0im 0.00914564+0.0im 0.00816163+0.0im 0.0091456+0.0im ⋮ ⋱ -0.00950201+0.0im -0.00915234+0.0im -0.00816867+0.0im -0.00915233+0.0im -0.0115338+0.0im -0.0110546+0.0im … -0.00973441+0.0im -0.0110546+0.0im -0.0119317+0.0im -0.0114468+0.0im -0.0101139+0.0im -0.0114468+0.0im -0.00894404+0.0im -0.00869807+0.0im -0.00797461+0.0im -0.00869809+0.0im -0.00376051+0.0im -0.00386976+0.0im -0.00404946+0.0im -0.00386977+0.0im -0.00026563+0.0im -0.000495811+0.0im -0.0010094+0.0im -0.00049581+0.0imtext/plainW150×24×24×1 Array{ComplexF64, 4}: [:, :, 1, 1] = 0.00434079+0.0im 0.00389403+0.0im … 0.00217316+0.0im 0.00389432+0.0im 0.00394836+0.0im 0.00340409+0.0im 0.00157895+0.0im 0.00340401+0.0im 0.00173489+0.0im 0.00118211+0.0im -0.000385544+0.0im 0.00118208+0.0im -0.00159628+0.0im -0.00185773+0.0im -0.00251275+0.0im -0.00185707+0.0im -0.00273333+0.0im -0.00270956+0.0im -0.0026192+0.0im -0.00270881+0.0im -0.00114057+0.0im -0.00105251+0.0im … -0.000842383+0.0im -0.00105254+0.0im 0.000978028+0.0im 0.0010168+0.0im 0.00106805+0.0im 0.00101689+0.0im ⋮ ⋱ 0.000951215+0.0im 0.000990995+0.0im 0.00104455+0.0im 0.000990665+0.0im -0.0011732+0.0im -0.0010836+0.0im … -0.000870088+0.0im -0.00108384+0.0im -0.00276651+0.0im -0.00274083+0.0im -0.0026483+0.0im -0.00274099+0.0im -0.00162038+0.0im -0.00188126+0.0im -0.00253539+0.0im -0.00188122+0.0im 0.00172542+0.0im 0.00117237+0.0im -0.000396882+0.0im 0.00117236+0.0im 0.00394894+0.0im 0.00340361+0.0im 0.00157662+0.0im 0.00340363+0.0im [:, :, 2, 1] = 0.00389406+0.0im 0.00350012+0.0im … 0.00196863+0.0im 0.00350047+0.0im 0.00340426+0.0im 0.00293285+0.0im 0.00133722+0.0im 0.00293313+0.0im 0.001182+0.0im 0.000721623+0.0im -0.000592887+0.0im 0.000721965+0.0im -0.00185741+0.0im -0.00205239+0.0im -0.00253578+0.0im -0.00205204+0.0im -0.00270911+0.0im -0.00266121+0.0im -0.0025115+0.0im -0.00266094+0.0im -0.00105233+0.0im -0.000964851+0.0im … -0.000753146+0.0im -0.000964757+0.0im 0.00101671+0.0im 0.00104959+0.0im 0.00108749+0.0im 0.00104966+0.0im ⋮ ⋱ 0.000990701+0.0im 0.00102448+0.0im 0.00106427+0.0im 0.00102417+0.0im -0.00108386+0.0im -0.000994782+0.0im … -0.000780347+0.0im -0.000994933+0.0im -0.00274114+0.0im -0.00269195+0.0im -0.00253909+0.0im -0.00269189+0.0im -0.00188108+0.0im -0.00207548+0.0im -0.0025578+0.0im -0.00207528+0.0im 0.00117223+0.0im 0.000711308+0.0im -0.000603901+0.0im 0.00071162+0.0im 0.00340354+0.0im 0.00293169+0.0im 0.00133412+0.0im 0.00293193+0.0im [:, :, 3, 1] = 0.00217319+0.0im 0.00196839+0.0im … 0.0010579+0.0im 0.00196863+0.0im 0.00157868+0.0im 0.00133656+0.0im 0.000429534+0.0im 0.00133672+0.0im -0.000386411+0.0im -0.000593542+0.0im -0.0012204+0.0im -0.000593518+0.0im -0.00251342+0.0im -0.00253662+0.0im -0.00257848+0.0im -0.00253644+0.0im -0.00261972+0.0im -0.00251165+0.0im -0.00220967+0.0im -0.00251162+0.0im -0.000842491+0.0im -0.000753352+0.0im … -0.000534201+0.0im -0.000753315+0.0im 0.00106846+0.0im 0.00108721+0.0im 0.00109717+0.0im 0.00108757+0.0im ⋮ ⋱ 0.00104484+0.0im 0.00106466+0.0im 0.00107631+0.0im 0.0010644+0.0im -0.00087015+0.0im -0.000780241+0.0im … -0.000557539+0.0im -0.000780217+0.0im -0.00264821+0.0im -0.00253916+0.0im -0.00223379+0.0im -0.00253913+0.0im -0.0025356+0.0im -0.00255786+0.0im -0.00259783+0.0im -0.00255759+0.0im -0.000397094+0.0im -0.000604408+0.0im -0.00123107+0.0im -0.000604172+0.0im 0.00157638+0.0im 0.00133381+0.0im 0.000426314+0.0im 0.0013341+0.0im ;;; … [:, :, 22, 1] = -0.000878934+0.0im -0.000760403+0.0im … -0.000685899+0.0im -0.000760583+0.0im -0.00131708+0.0im -0.00121373+0.0im -0.00111764+0.0im -0.0012136+0.0im -0.00245114+0.0im -0.00233146+0.0im -0.00207602+0.0im -0.00233147+0.0im -0.00320534+0.0im -0.00302452+0.0im -0.0025436+0.0im -0.00302429+0.0im -0.00244507+0.0im -0.00226614+0.0im -0.00178544+0.0im -0.00226618+0.0im -0.000647687+0.0im -0.000551673+0.0im … -0.000308416+0.0im -0.000551946+0.0im 0.000992943+0.0im 0.00100133+0.0im 0.000995318+0.0im 0.0010012+0.0im ⋮ ⋱ 0.000973238+0.0im 0.000982952+0.0im 0.000979394+0.0im 0.000982912+0.0im -0.000670222+0.0im -0.000573226+0.0im … -0.000326155+0.0im -0.000573099+0.0im -0.00246807+0.0im -0.0022881+0.0im -0.00180378+0.0im -0.00228798+0.0im -0.00322476+0.0im -0.00304298+0.0im -0.00255893+0.0im -0.00304262+0.0im -0.00246247+0.0im -0.00234265+0.0im -0.00208592+0.0im -0.0023425+0.0im -0.00132092+0.0im -0.00121772+0.0im -0.00112175+0.0im -0.00121783+0.0im [:, :, 23, 1] = 0.00217261+0.0im 0.00196846+0.0im … 0.00105762+0.0im 0.00196845+0.0im 0.00157862+0.0im 0.00133643+0.0im 0.000429533+0.0im 0.00133627+0.0im -0.000386209+0.0im -0.000593417+0.0im -0.00122071+0.0im -0.000593398+0.0im -0.00251337+0.0im -0.00253629+0.0im -0.00257847+0.0im -0.00253619+0.0im -0.00261975+0.0im -0.00251204+0.0im -0.00220982+0.0im -0.00251167+0.0im -0.000842289+0.0im -0.000753541+0.0im … -0.00053482+0.0im -0.000753569+0.0im 0.00106841+0.0im 0.00108739+0.0im 0.00109664+0.0im 0.00108745+0.0im ⋮ ⋱ 0.00104512+0.0im 0.00106469+0.0im 0.00107709+0.0im 0.00106496+0.0im -0.000870073+0.0im -0.000779765+0.0im … -0.000557512+0.0im -0.000779814+0.0im -0.00264815+0.0im -0.00253884+0.0im -0.0022333+0.0im -0.00253866+0.0im -0.0025355+0.0im -0.00255774+0.0im -0.00259776+0.0im -0.00255803+0.0im -0.000397188+0.0im -0.000604323+0.0im -0.00123165+0.0im -0.000604386+0.0im 0.00157647+0.0im 0.0013341+0.0im 0.000425806+0.0im 0.00133378+0.0im [:, :, 24, 1] = 0.00389399+0.0im 0.00350005+0.0im … 0.00196863+0.0im 0.00350018+0.0im 0.00340367+0.0im 0.00293275+0.0im 0.00133677+0.0im 0.0029329+0.0im 0.00118207+0.0im 0.00072182+0.0im -0.000593191+0.0im 0.000721814+0.0im -0.00185742+0.0im -0.00205256+0.0im -0.00253573+0.0im -0.00205192+0.0im -0.00270925+0.0im -0.00266144+0.0im -0.00251147+0.0im -0.00266097+0.0im -0.0010524+0.0im -0.00096468+0.0im … -0.000753438+0.0im -0.000964686+0.0im 0.00101684+0.0im 0.00104964+0.0im 0.00108729+0.0im 0.0010493+0.0im ⋮ ⋱ 0.00099083+0.0im 0.00102465+0.0im 0.00106483+0.0im 0.00102469+0.0im -0.00108357+0.0im -0.000994707+0.0im … -0.000779986+0.0im -0.000994567+0.0im -0.0027409+0.0im -0.00269172+0.0im -0.0025388+0.0im -0.00269176+0.0im -0.00188118+0.0im -0.00207515+0.0im -0.00255761+0.0im -0.0020753+0.0im 0.00117219+0.0im 0.000711559+0.0im -0.000604398+0.0im 0.000711472+0.0im 0.00340361+0.0im 0.00293184+0.0im 0.00133402+0.0im 0.00293184+0.0imtext/plain150×24×24×1 Array{ComplexF64, 4}: [:, :, 1, 1] = -0.00801585+0.0im -0.0076628+0.0im … -0.00629646+0.0im -0.00766281+0.0im -0.00703061+0.0im -0.00679209+0.0im -0.00575091+0.0im -0.00679208+0.0im -0.0053839+0.0im -0.00527694+0.0im -0.00467874+0.0im -0.00527694+0.0im -0.00399789+0.0im -0.0039488+0.0im -0.00362121+0.0im -0.00394883+0.0im -0.002578+0.0im -0.00255754+0.0im -0.00240388+0.0im -0.00255757+0.0im -0.00101574+0.0im -0.0010288+0.0im … -0.00103477+0.0im -0.0010288+0.0im 0.000294698+0.0im 0.000264627+0.0im 0.000165193+0.0im 0.000264624+0.0im ⋮ ⋱ 0.00328619+0.0im 0.00316586+0.0im 0.00278963+0.0im 0.00316588+0.0im 0.00258366+0.0im 0.00243885+0.0im … 0.00204534+0.0im 0.00243886+0.0im 0.00111335+0.0im 0.000996367+0.0im 0.000746218+0.0im 0.000996373+0.0im -0.00126715+0.0im -0.00128568+0.0im -0.00118353+0.0im -0.00128569+0.0im -0.00428107+0.0im -0.00413462+0.0im -0.00348396+0.0im -0.00413462+0.0im -0.00699779+0.0im -0.00668475+0.0im -0.00548481+0.0im -0.00668475+0.0im [:, :, 2, 1] = -0.0076628+0.0im -0.007415+0.0im … -0.00631708+0.0im -0.00741501+0.0im -0.00679209+0.0im -0.00663736+0.0im -0.0058123+0.0im -0.00663737+0.0im -0.00527693+0.0im -0.00522179+0.0im -0.00476049+0.0im -0.00522181+0.0im -0.00394881+0.0im -0.00393098+0.0im -0.00368831+0.0im -0.00393099+0.0im -0.00255756+0.0im -0.00255914+0.0im -0.00246579+0.0im -0.00255915+0.0im -0.0010288+0.0im -0.0010581+0.0im … -0.00110848+0.0im -0.0010581+0.0im 0.000264631+0.0im 0.000223299+0.0im 9.28274e-5+0.0im 0.000223297+0.0im ⋮ ⋱ 0.00316588+0.0im 0.0030344+0.0im 0.00262832+0.0im 0.00303441+0.0im 0.00243887+0.0im 0.00228004+0.0im … 0.00184937+0.0im 0.00228005+0.0im 0.000996379+0.0im 0.000859494+0.0im 0.000556419+0.0im 0.000859492+0.0im -0.00128569+0.0im -0.00133865+0.0im -0.00132695+0.0im -0.00133866+0.0im -0.00413461+0.0im -0.00405072+0.0im -0.00356139+0.0im -0.00405074+0.0im -0.00668474+0.0im -0.00646644+0.0im -0.00550809+0.0im -0.00646645+0.0im [:, :, 3, 1] = -0.00629646+0.0im -0.00631707+0.0im … -0.00592281+0.0im -0.00631708+0.0im -0.0057509+0.0im -0.00581227+0.0im -0.00556081+0.0im -0.00581228+0.0im -0.0046787+0.0im -0.00476047+0.0im -0.00467131+0.0im -0.00476047+0.0im -0.00362118+0.0im -0.00368828+0.0im -0.00368129+0.0im -0.00368829+0.0im -0.00240386+0.0im -0.00246579+0.0im -0.00253938+0.0im -0.00246579+0.0im -0.00103477+0.0im -0.00110847+0.0im … -0.00127974+0.0im -0.00110847+0.0im 0.000165177+0.0im 9.28382e-5+0.0im -0.000121775+0.0im 9.28242e-5+0.0im ⋮ ⋱ 0.00278962+0.0im 0.0026283+0.0im 0.00214406+0.0im 0.00262831+0.0im 0.00204534+0.0im 0.00184937+0.0im … 0.00132088+0.0im 0.00184937+0.0im 0.000746214+0.0im 0.000556422+0.0im 0.000111244+0.0im 0.00055642+0.0im -0.00118352+0.0im -0.00132695+0.0im -0.00155813+0.0im -0.00132696+0.0im -0.00348395+0.0im -0.00356137+0.0im -0.00349875+0.0im -0.00356138+0.0im -0.0054848+0.0im -0.00550808+0.0im -0.00518258+0.0im -0.00550809+0.0im ;;; … [:, :, 22, 1] = -0.00377106+0.0im -0.00409383+0.0im … -0.00453891+0.0im -0.00409382+0.0im -0.00365389+0.0im -0.00396327+0.0im -0.00441091+0.0im -0.00396328+0.0im -0.00328303+0.0im -0.00353026+0.0im -0.00391766+0.0im -0.00353026+0.0im -0.00276857+0.0im -0.00294626+0.0im -0.00325909+0.0im -0.00294627+0.0im -0.00196013+0.0im -0.00210435+0.0im -0.00240986+0.0im -0.00210434+0.0im -0.000943093+0.0im -0.00107856+0.0im … -0.00141759+0.0im -0.00107855+0.0im 3.05189e-5+0.0im -8.70356e-5+0.0im -0.000420798+0.0im -8.70305e-5+0.0im ⋮ ⋱ 0.00219568+0.0im 0.00199294+0.0im 0.00140492+0.0im 0.00199294+0.0im 0.00153031+0.0im 0.00128553+0.0im … 0.000629986+0.0im 0.00128552+0.0im 0.000558279+0.0im 0.000300674+0.0im -0.000330574+0.0im 0.00030067+0.0im -0.000744963+0.0im -0.0009991+0.0im -0.00154102+0.0im -0.000999114+0.0im -0.00215139+0.0im -0.00241635+0.0im -0.00287745+0.0im -0.00241636+0.0im -0.00327619+0.0im -0.0035726+0.0im -0.00400401+0.0im -0.00357259+0.0im [:, :, 23, 1] = -0.00629644+0.0im -0.00631707+0.0im … -0.0059228+0.0im -0.00631707+0.0im -0.00575089+0.0im -0.00581227+0.0im -0.00556081+0.0im -0.00581226+0.0im -0.00467871+0.0im -0.00476047+0.0im -0.0046713+0.0im -0.00476047+0.0im -0.00362118+0.0im -0.00368829+0.0im -0.00368129+0.0im 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-0.0024658+0.0im -0.00255914+0.0im -0.0010288+0.0im -0.0010581+0.0im … -0.00110847+0.0im -0.0010581+0.0im 0.000264625+0.0im 0.000223297+0.0im 9.2835e-5+0.0im 0.000223311+0.0im ⋮ ⋱ 0.00316587+0.0im 0.0030344+0.0im 0.00262829+0.0im 0.00303439+0.0im 0.00243885+0.0im 0.00228004+0.0im … 0.00184936+0.0im 0.00228004+0.0im 0.00099637+0.0im 0.000859485+0.0im 0.000556407+0.0im 0.000859486+0.0im -0.00128569+0.0im -0.00133867+0.0im -0.00132696+0.0im -0.00133866+0.0im -0.00413461+0.0im -0.00405073+0.0im -0.00356137+0.0im -0.00405073+0.0im -0.00668475+0.0im -0.00646645+0.0im -0.00550809+0.0im -0.00646645+0.0imtext/plain150×24×24×1 Array{ComplexF64, 4}: [:, :, 1, 1] = 0.0181635+0.0im 0.0169575+0.0im … 0.0129133+0.0im 0.0169576+0.0im 0.0158131+0.0im 0.0145068+0.0im 0.0104535+0.0im 0.0145067+0.0im 0.00732084+0.0im 0.0063569+0.0im 0.0036053+0.0im 0.00635689+0.0im -0.00352755+0.0im -0.00371665+0.0im -0.00419946+0.0im -0.00371643+0.0im -0.00985477+0.0im -0.00945119+0.0im -0.00831458+0.0im -0.00945093+0.0im -0.0101699+0.0im -0.00966486+0.0im … -0.00824079+0.0im -0.00966488+0.0im -0.00808383+0.0im -0.00774279+0.0im -0.00672883+0.0im -0.00774276+0.0im ⋮ ⋱ -0.00217208+0.0im -0.00202636+0.0im -0.00161562+0.0im -0.00202647+0.0im -0.00235644+0.0im -0.00215029+0.0im … -0.00160838+0.0im -0.00215037+0.0im -0.00135482+0.0im -0.00125011+0.0im -0.00100556+0.0im -0.00125017+0.0im 0.00285424+0.0im 0.002574+0.0im 0.00172512+0.0im 0.00257402+0.0im 0.00972596+0.0im 0.00896493+0.0im 0.00660397+0.0im 0.00896493+0.0im 0.01565+0.0im 0.0146096+0.0im 0.0111504+0.0im 0.0146096+0.0im [:, :, 2, 1] = 0.0169576+0.0im 0.0158574+0.0im … 0.0121611+0.0im 0.0158575+0.0im 0.0145068+0.0im 0.0133379+0.0im 0.00969397+0.0im 0.013338+0.0im 0.00635686+0.0im 0.00553586+0.0im 0.00317303+0.0im 0.00553598+0.0im -0.00371654+0.0im -0.00381272+0.0im -0.00405561+0.0im -0.0038126+0.0im -0.00945103+0.0im -0.00900933+0.0im -0.00777816+0.0im -0.00900924+0.0im -0.0096648+0.0im -0.00914333+0.0im … -0.00767713+0.0im -0.0091433+0.0im -0.00774283+0.0im -0.0073852+0.0im -0.00632682+0.0im -0.00738518+0.0im ⋮ ⋱ -0.00202646+0.0im -0.00186865+0.0im -0.0014231+0.0im -0.00186875+0.0im -0.00215038+0.0im -0.00192691+0.0im … -0.0013372+0.0im -0.00192696+0.0im -0.00125022+0.0im -0.00110872+0.0im -0.000766582+0.0im -0.0011087+0.0im 0.00257406+0.0im 0.00236901+0.0im 0.00172198+0.0im 0.00236908+0.0im 0.00896488+0.0im 0.00830823+0.0im 0.00624631+0.0im 0.00830833+0.0im 0.0146096+0.0im 0.0136714+0.0im 0.0105374+0.0im 0.0136715+0.0im [:, :, 3, 1] = 0.0129134+0.0im 0.0121611+0.0im … 0.00955345+0.0im 0.0121611+0.0im 0.0104534+0.0im 0.00969375+0.0im 0.00724405+0.0im 0.0096938+0.0im 0.00360501+0.0im 0.00317281+0.0im 0.00186523+0.0im 0.00317282+0.0im -0.00419969+0.0im -0.00405589+0.0im -0.00367863+0.0im -0.00405584+0.0im -0.00831475+0.0im -0.0077782+0.0im -0.00631095+0.0im -0.0077782+0.0im -0.00824083+0.0im -0.0076772+0.0im … -0.00610715+0.0im -0.00767719+0.0im -0.00672869+0.0im -0.00632691+0.0im -0.00515282+0.0im -0.00632679+0.0im ⋮ ⋱ -0.00161552+0.0im -0.00142297+0.0im -0.000879435+0.0im -0.00142306+0.0im -0.0016084+0.0im -0.00133717+0.0im … -0.000617336+0.0im -0.00133716+0.0im -0.00100552+0.0im -0.000766606+0.0im -0.000166606+0.0im -0.000766597+0.0im 0.00172505+0.0im 0.00172196+0.0im 0.00161613+0.0im 0.00172205+0.0im 0.00660389+0.0im 0.00624613+0.0im 0.00502961+0.0im 0.00624621+0.0im 0.0111503+0.0im 0.0105373+0.0im 0.00839748+0.0im 0.0105374+0.0im ;;; … [:, :, 22, 1] = 0.0063719+0.0im 0.00617051+0.0im … 0.00521415+0.0im 0.00617045+0.0im 0.00439723+0.0im 0.00422306+0.0im 0.0034538+0.0im 0.0042231+0.0im -7.37332e-5+0.0im -8.51984e-6+0.0im 3.51875e-5+0.0im -8.52436e-6+0.0im -0.00463297+0.0im -0.0042059+0.0im -0.00309345+0.0im -0.00420583+0.0im -0.00663939+0.0im -0.00599746+0.0im -0.00426974+0.0im -0.00599747+0.0im -0.00619199+0.0im -0.00557863+0.0im … -0.00389094+0.0im -0.00557873+0.0im -0.00518496+0.0im -0.00472103+0.0im -0.00339377+0.0im -0.00472107+0.0im ⋮ ⋱ -0.00107371+0.0im -0.000825829+0.0im -0.000133953+0.0im -0.000825843+0.0im -0.000940792+0.0im -0.000602903+0.0im … 0.000294757+0.0im -0.000602859+0.0im -0.000748233+0.0im -0.000386609+0.0im 0.000538751+0.0im -0.000386569+0.0im 0.000495775+0.0im 0.000747562+0.0im 0.00132741+0.0im 0.000747682+0.0im 0.00306483+0.0im 0.0031153+0.0im 0.00304455+0.0im 0.00311535+0.0im 0.00559391+0.0im 0.00547898+0.0im 0.00481153+0.0im 0.00547895+0.0im [:, :, 23, 1] = 0.0129132+0.0im 0.0121611+0.0im … 0.00955336+0.0im 0.0121611+0.0im 0.0104534+0.0im 0.00969371+0.0im 0.00724406+0.0im 0.00969365+0.0im 0.00360508+0.0im 0.00317286+0.0im 0.00186513+0.0im 0.00317286+0.0im -0.00419967+0.0im -0.00405577+0.0im -0.00367863+0.0im -0.00405574+0.0im -0.00831475+0.0im -0.00777833+0.0im -0.006311+0.0im -0.00777821+0.0im -0.00824075+0.0im -0.00767726+0.0im … -0.00610736+0.0im -0.00767727+0.0im -0.00672871+0.0im -0.00632685+0.0im -0.005153+0.0im -0.00632683+0.0im ⋮ ⋱ -0.00161543+0.0im -0.00142296+0.0im -0.000879173+0.0im -0.00142287+0.0im -0.00160837+0.0im -0.001337+0.0im … -0.000617328+0.0im -0.00133702+0.0im -0.0010055+0.0im -0.000766498+0.0im -0.000166437+0.0im -0.000766436+0.0im 0.00172509+0.0im 0.001722+0.0im 0.00161616+0.0im 0.0017219+0.0im 0.00660386+0.0im 0.00624616+0.0im 0.00502942+0.0im 0.00624614+0.0im 0.0111503+0.0im 0.0105374+0.0im 0.00839731+0.0im 0.0105373+0.0im [:, :, 24, 1] = 0.0169575+0.0im 0.0158574+0.0im … 0.0121611+0.0im 0.0158574+0.0im 0.0145066+0.0im 0.0133379+0.0im 0.00969382+0.0im 0.013338+0.0im 0.00635689+0.0im 0.00553593+0.0im 0.00317293+0.0im 0.00553593+0.0im -0.00371654+0.0im -0.00381278+0.0im -0.00405559+0.0im -0.00381256+0.0im -0.00945108+0.0im -0.0090094+0.0im -0.00777814+0.0im -0.00900925+0.0im -0.00966482+0.0im -0.00914327+0.0im … -0.00767723+0.0im -0.00914327+0.0im -0.00774278+0.0im -0.00738518+0.0im -0.00632689+0.0im -0.0073853+0.0im ⋮ ⋱ -0.00202641+0.0im -0.00186859+0.0im -0.00142291+0.0im -0.00186857+0.0im -0.00215028+0.0im -0.00192688+0.0im … -0.00133708+0.0im -0.00192683+0.0im -0.00125014+0.0im -0.00110864+0.0im -0.000766482+0.0im -0.00110865+0.0im 0.00257403+0.0im 0.00236912+0.0im 0.00172204+0.0im 0.00236907+0.0im 0.00896487+0.0im 0.00830831+0.0im 0.00624614+0.0im 0.00830828+0.0im 0.0146096+0.0im 0.0136715+0.0im 0.0105373+0.0im 0.0136715+0.0imtext/plaintypeArrayprefix_shortobjectid487d54ea2658c3b7!application/vnd.pluto.tree+objectConvergenceInfo: 4 converged values after 1 iterations and 16 applications of the linear map; norms of residuals are given by (0.00010962060099514753, 0.0008696529910375382, 2.220453924156357e-5, 0.0001727166939958009). text/plaintypeTupleobjectidc975eabbbd22c1e9mime!application/vnd.pluto.tree+objectrootassigneelast_run_timestampAŤpersist_js_state·has_pluto_hook_features§cell_id$114d7d12-e317-4f1e-86bf-d75d8ab3cc23depends_on_disabled_cells§runtime&qepublished_object_keysdepends_on_skipped_cells§errored$5d127331-8884-45b3-ae28-80ce3d5531f5queued¤logsrunning¦outputbodymimetext/plainrootassigneelast_run_timestampAŤ1~Zpersist_js_state·has_pluto_hook_features§cell_id$5d127331-8884-45b3-ae28-80ce3d5531f5depends_on_disabled_cells§runtimetpublished_object_keysdepends_on_skipped_cells§errored$387b07a5-bd5d-4b45-a495-b26477e8f704queued¤logsrunning¦outputbodyC

Based on this discretisation we run the SCF

$$ \rho_{n+1} = \rho_n + \texttt{damping} * P^{-1} \Big[ \rho \Big( V(\rho_n) \Big) - \rho_n \Big].$$

using three different preconditioners, namely

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Modelling Aluminium

We want to understand SCF convergence. For simplicity we select a system consisting of 4 layers of aluminium (a prottypical metal).

mimetext/htmlrootassigneelast_run_timestampAŤ2,ᨰpersist_js_state·has_pluto_hook_features§cell_id$c50ed373-4462-462b-924a-a57b1665e348depends_on_disabled_cells§runtime/published_object_keysdepends_on_skipped_cells§errored$eee2759b-5055-4143-820e-0f14b3d01caequeued¤logsrunning¦outputbodykFlexibleSystem(He₂₅, periodic = TTT): bounding_box : [ 65 0 0; 0 2.6 0; 0 0 2.6]u"Å" .---------------------------------------------------------------------------------------------. *| | |.---------------------------------------------------------------------------------------------. He--He--He-He--He--He--He-He--He--He--He-He--He--He--He-He--He--He--He-He--He--He--He-He--He--* mimetext/plainrootassigneelast_run_timestampAŤ;*|persist_js_state·has_pluto_hook_features§cell_id$eee2759b-5055-4143-820e-0f14b3d01caedepends_on_disabled_cells§runtime%>Ƶpublished_object_keysdepends_on_skipped_cells§errored±cell_dependencies$0c146256-a19d-43c7-a874-63c810ab2f08precedence_heuristic cell_id$0c146256-a19d-43c7-a874-63c810ab2f08downstream_cells_mapscfres_Al_small$a78e34b8-0ab9-4400-a437-ccb9648535d3$a6c74c32-9329-4d11-8b05-9effa23a108a$114d7d12-e317-4f1e-86bf-d75d8ab3cc23upstream_cells_mapPlaneWaveBasismodel_PBEbulkKerkerMixingself_consistent_field*@artifact_stridentityattach_psp$82399cad-6e2e-4586-8a0c-40a9d420c463precedence_heuristic cell_id$82399cad-6e2e-4586-8a0c-40a9d420c463downstream_cells_mapupstream_cells_map@md_strgetindex$1fff9a65-fc63-445c-a41d-17df98d56d3cprecedence_heuristic cell_id$1fff9a65-fc63-445c-a41d-17df98d56d3cdownstream_cells_mapn_repeat_helium$eee2759b-5055-4143-820e-0f14b3d01cae$00d596b2-7060-4578-bc1a-9841e0c68277upstream_cells_map@md_strCore:Base.get@bindSliderBasePlutoRunnerPlutoRunner.create_bondCore.applicablegetindex$7c37fa06-2c88-4794-baab-81c408a89e6aprecedence_heuristiccell_id$7c37fa06-2c88-4794-baab-81c408a89e6adownstream_cells_mapUnitfulAtomicStatisticsUnitfulAtomsBuilderLazyArtifactsLinearAlgebraPlutoUI$a4cf819c-8568-49fc-bf0a-3ae209aa56bePlotsDFTK$a78e34b8-0ab9-4400-a437-ccb9648535d3KrylovKitupstream_cells_map$a4cf819c-8568-49fc-bf0a-3ae209aa56beprecedence_heuristic cell_id$a4cf819c-8568-49fc-bf0a-3ae209aa56bedownstream_cells_mapEcut$13446c99-ae13-47ee-9c29-f6acd2a9fff4$bead2d43-47ad-48ff-949e-1521dd997e22upstream_cells_map@md_strCorePlutoUI$7c37fa06-2c88-4794-baab-81c408a89e6aBase.get@bindBasePlutoRunnerPlutoRunner.create_bondCore.applicablePlutoUI.Selectgetindex$fae0a2bd-6e12-4fa7-8805-0948c81efe57precedence_heuristic cell_id$fae0a2bd-6e12-4fa7-8805-0948c81efe57downstream_cells_mapn_repeat_aluminium$c50ed373-4462-462b-924a-a57b1665e348$7434967b-b988-447a-b346-cac2b4c11776$ecacc9c7-3b11-46ba-aebd-c57887074651upstream_cells_map@md_strCore:Base.get@bindSliderBasePlutoRunnerPlutoRunner.create_bondCore.applicablegetindex$ecacc9c7-3b11-46ba-aebd-c57887074651precedence_heuristic cell_id$ecacc9c7-3b11-46ba-aebd-c57887074651downstream_cells_mappupstream_cells_mapstringKerkerMixingplotSimpleMixingplot!n_repeat_aluminium$fae0a2bd-6e12-4fa7-8805-0948c81efe57identitybasis_Al$13446c99-ae13-47ee-9c29-f6acd2a9fff4self_consistent_fieldLdosMixing$bead2d43-47ad-48ff-949e-1521dd997e22precedence_heuristic cell_id$bead2d43-47ad-48ff-949e-1521dd997e22downstream_cells_mapmodel_Hebasis_He$00d596b2-7060-4578-bc1a-9841e0c68277upstream_cells_mapPlaneWaveBasismodel_PBEEcut$a4cf819c-8568-49fc-bf0a-3ae209aa56besystem_He$eee2759b-5055-4143-820e-0f14b3d01cae$00d596b2-7060-4578-bc1a-9841e0c68277precedence_heuristic 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cell_id$eee2759b-5055-4143-820e-0f14b3d01caedownstream_cells_maphe_singlebounding_boxsystem_He$bead2d43-47ad-48ff-949e-1521dd997e22upstream_cells_mapperiodic_system@u_strzeros*=>@artifact_strn_repeat_helium$1fff9a65-fc63-445c-a41d-17df98d56d3cattach_pspcell_execution_order$7c37fa06-2c88-4794-baab-81c408a89e6a$f2f201cf-0256-4363-b3c1-17104054e594$b0bb59d7-cb8d-4a8f-88c1-88e8fe70d791$5d127331-8884-45b3-ae28-80ce3d5531f5$a4cf819c-8568-49fc-bf0a-3ae209aa56be$5f8776fb-bea2-4b30-a733-8899392e0aa2$fae0a2bd-6e12-4fa7-8805-0948c81efe57$c50ed373-4462-462b-924a-a57b1665e348$7434967b-b988-447a-b346-cac2b4c11776$13446c99-ae13-47ee-9c29-f6acd2a9fff4$387b07a5-bd5d-4b45-a495-b26477e8f704$ecacc9c7-3b11-46ba-aebd-c57887074651$0574ad92-2554-4469-9776-4a42caad7256$1fff9a65-fc63-445c-a41d-17df98d56d3c$eee2759b-5055-4143-820e-0f14b3d01cae$bead2d43-47ad-48ff-949e-1521dd997e22$00d596b2-7060-4578-bc1a-9841e0c68277$e58421d4-8f8d-11ef-18ff-d316ae57fa3a$0c146256-a19d-43c7-a874-63c810ab2f08$a78e34b8-0ab9-4400-a437-ccb9648535d3$a70d053e-4257-4458-86bf-aef38557493b$a6c74c32-9329-4d11-8b05-9effa23a108a$a513d7cd-2431-4275-8965-286510668621$a5cb6eac-57a8-45fe-8436-7ab7ea8d8f09$82399cad-6e2e-4586-8a0c-40a9d420c463$7fb6f4bf-3ccb-44be-8a69-735cb004e48d$114d7d12-e317-4f1e-86bf-d75d8ab3cc23$e297eedc-d45e-466c-834e-e59cf2814de3$af492cf0-8bdb-4cae-91c8-b189c969f8dflast_hot_reload_timeshortpath5_dielectric_eigenpairs.jlprocess_statusreadypath`/home/runner/work/demo-molssi-workshop-dftk/demo-molssi-workshop-dftk/5_dielectric_eigenpairs.jlpluto_versionv0.19.47last_save_timeAŤ+HBcell_order$f2f201cf-0256-4363-b3c1-17104054e594$7c37fa06-2c88-4794-baab-81c408a89e6a$b0bb59d7-cb8d-4a8f-88c1-88e8fe70d791$5d127331-8884-45b3-ae28-80ce3d5531f5$a4cf819c-8568-49fc-bf0a-3ae209aa56be$5f8776fb-bea2-4b30-a733-8899392e0aa2$c50ed373-4462-462b-924a-a57b1665e348$fae0a2bd-6e12-4fa7-8805-0948c81efe57$7434967b-b988-447a-b346-cac2b4c11776$13446c99-ae13-47ee-9c29-f6acd2a9fff4$387b07a5-bd5d-4b45-a495-b26477e8f704$ecacc9c7-3b11-46ba-aebd-c57887074651$0574ad92-2554-4469-9776-4a42caad7256$1fff9a65-fc63-445c-a41d-17df98d56d3c$eee2759b-5055-4143-820e-0f14b3d01cae$bead2d43-47ad-48ff-949e-1521dd997e22$00d596b2-7060-4578-bc1a-9841e0c68277$e58421d4-8f8d-11ef-18ff-d316ae57fa3a$0c146256-a19d-43c7-a874-63c810ab2f08$a78e34b8-0ab9-4400-a437-ccb9648535d3$a70d053e-4257-4458-86bf-aef38557493b$a6c74c32-9329-4d11-8b05-9effa23a108a$a513d7cd-2431-4275-8965-286510668621$a5cb6eac-57a8-45fe-8436-7ab7ea8d8f09$82399cad-6e2e-4586-8a0c-40a9d420c463$7fb6f4bf-3ccb-44be-8a69-735cb004e48d$114d7d12-e317-4f1e-86bf-d75d8ab3cc23$e297eedc-d45e-466c-834e-e59cf2814de3$af492cf0-8bdb-4cae-91c8-b189c969f8dfpublished_objectsnbpkginstall_time_nsFinstantiatedòinstalled_versionsUnitful1.21.0Statistics1.11.1AtomsBuilder0.1.0UnitfulAtomic1.0.0LazyArtifactsstdlibLinearAlgebrastdlibPlutoUI0.7.60DFTK0.6.20Plots1.40.8KrylovKit0.8.1terminal_outputsUnitfulAtomic@ Instantiating... === Resolving... ===  No Changes to `/tmp/jl_Q7Sbii/Project.toml`  No Changes to `/tmp/jl_Q7Sbii/Manifest.toml` Precompiling... ===  Activating project at `/tmp/jl_Q7Sbii`Statistics@ Instantiating... === Resolving... ===  No Changes to `/tmp/jl_Q7Sbii/Project.toml`  No Changes to `/tmp/jl_Q7Sbii/Manifest.toml` Precompiling... ===  Activating project at `/tmp/jl_Q7Sbii`PlutoUI@ Instantiating... === Resolving... ===  No Changes to `/tmp/jl_Q7Sbii/Project.toml`  No Changes to `/tmp/jl_Q7Sbii/Manifest.toml` Precompiling... ===  Activating project at `/tmp/jl_Q7Sbii`DFTK@ Instantiating... === Resolving... ===  No Changes to `/tmp/jl_Q7Sbii/Project.toml`  No Changes to `/tmp/jl_Q7Sbii/Manifest.toml` Precompiling... ===  Activating project at `/tmp/jl_Q7Sbii`KrylovKit@ Instantiating... === Resolving... ===  No Changes to `/tmp/jl_Q7Sbii/Project.toml`  No Changes to `/tmp/jl_Q7Sbii/Manifest.toml` Precompiling... ===  Activating project at `/tmp/jl_Q7Sbii`AtomsBuilder@ Instantiating... === Resolving... ===  No Changes to `/tmp/jl_Q7Sbii/Project.toml`  No Changes to `/tmp/jl_Q7Sbii/Manifest.toml` Precompiling... ===  Activating project at `/tmp/jl_Q7Sbii`LazyArtifacts@ Instantiating... === Resolving... ===  No Changes to `/tmp/jl_Q7Sbii/Project.toml`  No Changes to `/tmp/jl_Q7Sbii/Manifest.toml` Precompiling... ===  Activating project at `/tmp/jl_Q7Sbii`LinearAlgebra@ Instantiating... === Resolving... ===  No Changes to `/tmp/jl_Q7Sbii/Project.toml`  No Changes to `/tmp/jl_Q7Sbii/Manifest.toml` Precompiling... ===  Activating project at `/tmp/jl_Q7Sbii`Plots@ Instantiating... === Resolving... ===  No Changes to `/tmp/jl_Q7Sbii/Project.toml`  No Changes to `/tmp/jl_Q7Sbii/Manifest.toml` Precompiling... ===  Activating project at `/tmp/jl_Q7Sbii`Unitful@ Instantiating... === Resolving... ===  No Changes to `/tmp/jl_Q7Sbii/Project.toml`  No Changes to `/tmp/jl_Q7Sbii/Manifest.toml` Precompiling... ===  Activating project at `/tmp/jl_Q7Sbii`nbpkg_sync@ Instantiating... === Resolving... ===  No Changes to `/tmp/jl_Q7Sbii/Project.toml`  No Changes to `/tmp/jl_Q7Sbii/Manifest.toml` Precompiling... ===  Activating project at `/tmp/jl_Q7Sbii`enabled÷restart_recommended_msgrestart_required_msgbusy_packageswaiting_for_permission,waiting_for_permission_but_probably_disabled«cell_inputs$0c146256-a19d-43c7-a874-63c810ab2f08cell_id$0c146256-a19d-43c7-a874-63c810ab2f08codemscfres_Al_small = let system_Al = attach_psp(bulk(:Al; cubic=true) * (8, 1, 1); Al=artifact"pd_nc_sr_pbe_standard_0.4.1_upf/Al.upf") model_Al = model_PBE(system_Al; temperature=1e-3, symmetries=false) basis_Al = PlaneWaveBasis(model_Al; Ecut=7, kgrid=(1, 1, 1)) self_consistent_field(basis_Al; mixing=KerkerMixing(), tol=1e-8, callback=identity) endmetadatashow_logsèdisabled®skip_as_script«code_folded$82399cad-6e2e-4586-8a0c-40a9d420c463cell_id$82399cad-6e2e-4586-8a0c-40a9d420c463codemd"This mode can be physically interprested 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 phaenomenon is called charge-sloshing."metadatashow_logsèdisabled®skip_as_script«code_folded$1fff9a65-fc63-445c-a41d-17df98d56d3ccell_id$1fff9a65-fc63-445c-a41d-17df98d56d3ccodeemd""" - `n_repeat_helium` = $(@bind n_repeat_helium Slider(25:5:40; show_value=true, default=25)) """metadatashow_logsèdisabled®skip_as_script«code_folded$7c37fa06-2c88-4794-baab-81c408a89e6acell_id$7c37fa06-2c88-4794-baab-81c408a89e6acodeٶbegin using PlutoUI using AtomsBuilder using DFTK using LazyArtifacts using Plots using KrylovKit using Statistics using LinearAlgebra using Unitful using UnitfulAtomic endmetadatashow_logsèdisabled®skip_as_script«code_folded$a4cf819c-8568-49fc-bf0a-3ae209aa56becell_id$a4cf819c-8568-49fc-bf0a-3ae209aa56becodenmd"""Select kinetic energy cutoff `Ecut`: - `Ecut = ` $(@bind Ecut PlutoUI.Select([7, 15, 25]; default=7)) """metadatashow_logsèdisabled®skip_as_script«code_folded$fae0a2bd-6e12-4fa7-8805-0948c81efe57cell_id$fae0a2bd-6e12-4fa7-8805-0948c81efe57codejmd""" - `n_repeat_aluminium = ` $(@bind n_repeat_aluminium Slider(2:2:10; show_value=true, default=4)) """metadatashow_logsèdisabled®skip_as_script«code_folded$ecacc9c7-3b11-46ba-aebd-c57887074651cell_id$ecacc9c7-3b11-46ba-aebd-c57887074651code6begin p = plot(yaxis=:log, title="Aluminium with $n_repeat_aluminium layers") for mixtype in [SimpleMixing, KerkerMixing, LdosMixing] scfres_Al = self_consistent_field(basis_Al; damping=1.0, # Use no damping callback=identity, # Disable output mixing=mixtype(), tol=1e-8, maxiter=20) plot!(p, scfres_Al.history_Δρ, label=string(mixtype), lw=2, mark=:x) end p endmetadatashow_logsèdisabled®skip_as_script«code_folded$bead2d43-47ad-48ff-949e-1521dd997e22cell_id$bead2d43-47ad-48ff-949e-1521dd997e22codeًbegin model_He = model_PBE(system_He; temperature=1e-3, symmetries=false) basis_He = PlaneWaveBasis(model_He; Ecut, kgrid=(1, 2, 2)) end;metadatashow_logsèdisabled®skip_as_script«code_folded$00d596b2-7060-4578-bc1a-9841e0c68277cell_id$00d596b2-7060-4578-bc1a-9841e0c68277code0begin q = plot(yaxis=:log, title="Helium with $n_repeat_helium layers") for mixtype in [SimpleMixing, KerkerMixing, LdosMixing] scfres_He = self_consistent_field(basis_He; damping=1.0, # Use no damping callback=identity, # Disable output mixing=mixtype(), tol=1e-8, maxiter=20) plot!(q, scfres_He.history_Δρ, label=string(mixtype), lw=2, mark=:x) end q endmetadatashow_logsèdisabled®skip_as_script«code_folded$e58421d4-8f8d-11ef-18ff-d316ae57fa3acell_id$e58421d4-8f8d-11ef-18ff-d316ae57fa3acodemd""" ## Dielectric eigenpairs To understand the difference we now look at the eigenpairs of the Jacobian governing the convergence, that is ```math 1 - α P^{-1} \varepsilon^\dagger \qquad \text{with} \qquad \varepsilon^\dagger = (1-\chi_0 K). ``` """metadatashow_logsèdisabled®skip_as_script«code_folded$af492cf0-8bdb-4cae-91c8-b189c969f8dfcell_id$af492cf0-8bdb-4cae-91c8-b189c969f8dfcodeOmd""" Clearly the charge-sloshing mode is no longer dominating. Overview of largest eigenvalues: - `SimpleMixing`: $(maximum(real.(λ_Simple))) - `KerkerMixing`: $(maximum(real.(λ_Kerker))) Since the smallest eigenvalue is around $1$ in both cases, this implies that the conditioning improves noticably 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. """metadatashow_logsèdisabled®skip_as_script«code_folded$f2f201cf-0256-4363-b3c1-17104054e594cell_id$f2f201cf-0256-4363-b3c1-17104054e594code!md""" # Dielectric eigenpairs """metadatashow_logsèdisabled®skip_as_script«code_folded$a513d7cd-2431-4275-8965-286510668621cell_id$a513d7cd-2431-4275-8965-286510668621code0md"We can also e.g. plot the largest eigenmode."metadatashow_logsèdisabled®skip_as_script«code_folded$a6c74c32-9329-4d11-8b05-9effa23a108acell_id$a6c74c32-9329-4d11-8b05-9effa23a108acodeّλ_Simple, X_Simple = eigsolve(epsilon, randn(size(scfres_Al_small.ρ)), 3, :LM; tol=1e-3, eager=true, verbosity=2)metadatashow_logsèdisabled®skip_as_script«code_folded$5f8776fb-bea2-4b30-a733-8899392e0aa2cell_id$5f8776fb-bea2-4b30-a733-8899392e0aa2codemd""" !!! note "Expensive notebook" Some computations in this notebook take extended time. To see the "real" effects. Try increasing the `n_repeat_aluminium` and `n_repeat_helium` and `Ecut` as high as you can afford. Running the simulations take minutes. This is normal. """metadatashow_logsèdisabled®skip_as_script«code_folded$7fb6f4bf-3ccb-44be-8a69-735cb004e48dcell_id$7fb6f4bf-3ccb-44be-8a69-735cb004e48dcodeMmd"We repeat the exercise for the Kerker-preconditioned dielectric operator:"metadatashow_logsèdisabled®skip_as_script«code_folded$13446c99-ae13-47ee-9c29-f6acd2a9fff4cell_id$13446c99-ae13-47ee-9c29-f6acd2a9fff4codeًbegin model_Al = model_PBE(system_Al; temperature=1e-3, symmetries=false) basis_Al = PlaneWaveBasis(model_Al; Ecut, kgrid=(1, 2, 2)) end;metadatashow_logsèdisabled®skip_as_script«code_folded$e297eedc-d45e-466c-834e-e59cf2814de3cell_id$e297eedc-d45e-466c-834e-e59cf2814de3codeٹlet mode_xy = mean(real.(X_Kerker[1]), dims=3)[:, :, 1, 1] # Average along z axis heatmap(mode_xy', c=:RdBu_11, aspect_ratio=1, grid=false, legend=false, clim=(-0.006, 0.006)) endmetadatashow_logsèdisabled®skip_as_script«code_folded$a5cb6eac-57a8-45fe-8436-7ab7ea8d8f09cell_id$a5cb6eac-57a8-45fe-8436-7ab7ea8d8f09codeٹlet mode_xy = mean(real.(X_Simple[1]), dims=3)[:, :, 1, 1] # Average along z axis heatmap(mode_xy', c=:RdBu_11, aspect_ratio=1, grid=false, legend=false, clim=(-0.006, 0.006)) endmetadatashow_logsèdisabled®skip_as_script«code_folded$a70d053e-4257-4458-86bf-aef38557493bcell_id$a70d053e-4257-4458-86bf-aef38557493bcodeِmd""" With these functions available we can now compute the desired eigenvalues. For simplicity we only consider the first few largest ones. """metadatashow_logsèdisabled®skip_as_script«code_folded$7434967b-b988-447a-b346-cac2b4c11776cell_id$7434967b-b988-447a-b346-cac2b4c11776codebegin n_supercell = floor(Int, n_repeat_aluminium/2) # Cubic aluminium has 2 layers per cell system_Al = attach_psp(bulk(:Al; cubic=true) * (n_supercell, 1, 1); Al=artifact"pd_nc_sr_pbe_standard_0.4.1_upf/Al.upf") endmetadatashow_logsèdisabled®skip_as_script«code_folded$0574ad92-2554-4469-9776-4a42caad7256cell_id$0574ad92-2554-4469-9776-4a42caad7256codemd""" ## Modelling helium """metadatashow_logsèdisabled®skip_as_script«code_folded$114d7d12-e317-4f1e-86bf-d75d8ab3cc23cell_id$114d7d12-e317-4f1e-86bf-d75d8ab3cc23codeٕλ_Kerker, X_Kerker = eigsolve(Pinv_Kerker ∘ epsilon, randn(size(scfres_Al_small.ρ)), 3, :LM; tol=1e-3, eager=true, verbosity=2)metadatashow_logsèdisabled®skip_as_script«code_folded$5d127331-8884-45b3-ae28-80ce3d5531f5cell_id$5d127331-8884-45b3-ae28-80ce3d5531f5code# Dump the Artifacts.toml file, which tells Julia where to get the pseudopotentials from begin content = """ ["pd_nc_sr_pbe_standard_0.4.1_upf"] git-tree-sha1 = "897b9d17527039dd86ca882603c3af2ea0c0aada" lazy = true [["pd_nc_sr_pbe_standard_0.4.1_upf".download]] sha256 = "1ee2aec5d651012557ebcfcd33d7a3192e4789b1c49af1bd846a6f7c6a032f84" url = "https://github.com/JuliaMolSim/PseudoLibrary/raw/main/artifacts/pd_nc_sr_pbe_standard_0.4.1_upf.tar.gz" """ write("Artifacts.toml", content) end;metadatashow_logsèdisabled®skip_as_script«code_folded$387b07a5-bd5d-4b45-a495-b26477e8f704cell_id$387b07a5-bd5d-4b45-a495-b26477e8f704codeVmd"""Based on this discretisation we run the SCF ```math \rho_{n+1} = \rho_n + \texttt{damping} * P^{-1} \Big[ \rho \Big( V(\rho_n) \Big) - \rho_n \Big]. ``` using three different preconditioners, namely - `SimpleMixing`, i.e. no preconditioner ($P = I$) - `KerkerMixing` a standard preconditioner for metals - Our proposed `LdosMixing`. """metadatashow_logsèdisabled®skip_as_script«code_folded$a78e34b8-0ab9-4400-a437-ccb9648535d3cell_id$a78e34b8-0ab9-4400-a437-ccb9648535d3codebegin # Function, which applies P^{-1} for the case of KerkerMixing Pinv_Kerker(δρ) = DFTK.mix_density(KerkerMixing(), scfres_Al_small.basis, δρ) # Function which applies ε† = 1 - χ0 K function epsilon(δρ) δV = apply_kernel(scfres_Al_small.basis, δρ; ρ=scfres_Al_small.ρ) χ0δV = apply_χ0(scfres_Al_small, δV) δρ - χ0δV end end;metadatashow_logsèdisabled®skip_as_script«code_folded$b0bb59d7-cb8d-4a8f-88c1-88e8fe70d791cell_id$b0bb59d7-cb8d-4a8f-88c1-88e8fe70d791codesetup_threading(n_blas=1)metadatashow_logsèdisabled®skip_as_script«code_folded$c50ed373-4462-462b-924a-a57b1665e348cell_id$c50ed373-4462-462b-924a-a57b1665e348codeټmd""" ## Modelling Aluminium We want to understand SCF convergence. For simplicity we select a system consisting of $n_repeat_aluminium layers of **aluminium** (a prottypical metal). """metadatashow_logsèdisabled®skip_as_script«code_folded$eee2759b-5055-4143-820e-0f14b3d01caecell_id$eee2759b-5055-4143-820e-0f14b3d01caecode,begin bounding_box = 2.6 * [[1, 0, 0], [0, 1, 0], [0, 0, 1]]u"Å" he_single = periodic_system([:He => zeros(3)u"Å", ], bounding_box) system_He = attach_psp(he_single * (n_repeat_helium, 1, 1); He=artifact"pd_nc_sr_pbe_standard_0.4.1_upf/He.upf") endmetadatashow_logsèdisabled®skip_as_script«code_folded«notebook_id$be56bf0a-8fd2-11ef-1a5d-f32ed357cb73in_temp_dir¨metadata