Comparison of DFT solvers

We compare four different approaches for solving the DFT minimisation problem, namely a density-based SCF, a potential-based SCF, direct minimisation and Newton.

First we setup our problem

using AtomsBuilder
using DFTK
using LinearAlgebra
using PseudoPotentialData

pseudopotentials = PseudoFamily("dojo.nc.sr.pbesol.v0_4_1.standard.upf")
model = model_DFT(bulk(:Si); functionals=PBEsol(), pseudopotentials)
basis = PlaneWaveBasis(model; Ecut=5, kgrid=[3, 3, 3])

# Convergence we desire in the density
tol = 1e-6

1.0e-6

Density-based self-consistent field

scfres_scf = self_consistent_field(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   Diag   Δtime
---   ---------------   ---------   ---------   ----   ------
  1   -8.397784993516                   -0.90    5.0   27.7ms
  2   -8.400242366402       -2.61       -1.74    1.0   20.4ms
  3   -8.400399386720       -3.80       -2.94    1.5   21.3ms
  4   -8.400427774461       -4.55       -2.92    3.0   37.7ms
  5   -8.400427937432       -6.79       -3.02    1.0   20.4ms
  6   -8.400428144721       -6.68       -4.90    1.0   20.3ms
  7   -8.400428151718       -8.16       -4.39    3.5   27.2ms
  8   -8.400428152194       -9.32       -5.82    1.2   21.1ms
  9   -8.400428152209      -10.83       -6.37    2.0   23.3ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397798252234                   -0.90           5.2   28.7ms
  2   -8.400376774580       -2.59       -1.77   0.80    2.2   20.5ms
  3   -8.400422723782       -4.34       -2.96   0.80    1.0   17.3ms
  4   -8.400428105275       -5.27       -3.51   0.80    2.5   21.2ms
  5   -8.400428148047       -7.37       -5.20   0.80    1.2   18.3ms
  6   -8.400428152206       -8.38       -5.22   0.80    3.8   25.0ms
  7   -8.400428152209      -11.60       -6.34   0.80    1.0   17.6ms

Direct minimization

scfres_dm = direct_minimization(basis; tol);

┌ Warning: x_tol is deprecated. Use x_abstol or x_reltol instead. The provided value (-1) will be used as x_abstol.
└ @ Optim ~/.julia/packages/Optim/7krni/src/types.jl:110
┌ Warning: f_tol is deprecated. Use f_abstol or f_reltol instead. The provided value (-1) will be used as f_reltol.
└ @ Optim ~/.julia/packages/Optim/7krni/src/types.jl:120
n     Energy            log10(ΔE)   log10(Δρ)   Δtime
---   ---------------   ---------   ---------   ------
  1   +1.138759181862                   -1.10   67.7ms
  2   -1.617531458491        0.44       -0.63   33.6ms
  3   -4.250502228829        0.42       -0.32   45.5ms
  4   -5.671731198349        0.15       -0.41   45.3ms
  5   -7.448415030139        0.25       -0.52   51.8ms
  6   -7.556091592187       -0.97       -1.23   33.5ms
  7   -8.175130719880       -0.21       -1.33   33.7ms
  8   -8.266315813733       -1.04       -1.78   33.8ms
  9   -8.322491472162       -1.25       -1.97   34.0ms
 10   -8.355173290880       -1.49       -2.09   41.6ms
 11   -8.376266636642       -1.68       -2.43   33.7ms
 12   -8.388260235780       -1.92       -2.80   33.7ms
 13   -8.394491894338       -2.21       -2.53   33.6ms
 14   -8.398096572867       -2.44       -2.82   33.8ms
 15   -8.399448283173       -2.87       -3.25   33.7ms
 16   -8.400018972910       -3.24       -3.38   41.3ms
 17   -8.400206770763       -3.73       -3.55   34.1ms
 18   -8.400342601191       -3.87       -3.44   33.6ms
 19   -8.400383585910       -4.39       -3.81   33.9ms
 20   -8.400410661684       -4.57       -3.93   34.0ms
 21   -8.400418806240       -5.09       -4.10   33.7ms
 22   -8.400423908374       -5.29       -4.41   41.4ms
 23   -8.400425850109       -5.71       -4.73   33.9ms
 24   -8.400427075096       -5.91       -4.61   33.5ms
 25   -8.400427678392       -6.22       -5.31   33.8ms
 26   -8.400427958292       -6.55       -4.92   33.7ms
 27   -8.400428052572       -7.03       -5.25   33.7ms
 28   -8.400428113606       -7.21       -5.24   39.5ms
 29   -8.400428132372       -7.73       -5.42   33.8ms
 30   -8.400428145814       -7.87       -5.55   33.6ms
 31   -8.400428149396       -8.45       -5.87   33.7ms
 32   -8.400428151065       -8.78       -6.17   33.7ms

Newton algorithm

Start not too far from the solution to ensure convergence: We run first a very crude SCF to get close and then switch to Newton.

scfres_start = self_consistent_field(basis; tol=0.5);

n     Energy            log10(ΔE)   log10(Δρ)   Diag   Δtime
---   ---------------   ---------   ---------   ----   ------
  1   -8.397891778556                   -0.90    5.2   28.6ms

Remove the virtual orbitals (which Newton cannot treat yet)

ψ = DFTK.select_occupied_orbitals(basis, scfres_start.ψ, scfres_start.occupation).ψ
scfres_newton = newton(basis, ψ; tol);

n     Energy            log10(ΔE)   log10(Δρ)   Δtime
---   ---------------   ---------   ---------   ------
  1   -8.400427989879                   -1.79    616ms
  2   -8.400428152209       -6.79       -4.04    400ms
  3   -8.400428152209      -14.45       -7.86    103ms

Comparison of results

println("|ρ_newton - ρ_scf|  = ", norm(scfres_newton.ρ - scfres_scf.ρ))
println("|ρ_newton - ρ_scfv| = ", norm(scfres_newton.ρ - scfres_scfv.ρ))
println("|ρ_newton - ρ_dm|   = ", norm(scfres_newton.ρ - scfres_dm.ρ))

|ρ_newton - ρ_scf|  = 5.265036059276644e-7
|ρ_newton - ρ_scfv| = 3.1391134817814887e-7
|ρ_newton - ρ_dm|   = 2.571024482390791e-6