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   -8.397851962696                   -0.90    5.5   27.3ms
  2   -8.400249413406       -2.62       -1.74    1.0   18.1ms
  3   -8.400405454081       -3.81       -2.97    1.5   19.2ms
  4   -8.400427869000       -4.65       -2.97    3.2   23.8ms
  5   -8.400427950075       -7.09       -3.03    1.0   18.2ms
  6   -8.400428145736       -6.71       -4.74    1.0   18.4ms
  7   -8.400428151824       -8.22       -4.47    2.8   23.3ms
  8   -8.400428152193       -9.43       -5.44    1.0   18.5ms
  9   -8.400428152208      -10.82       -6.07    1.8   20.6ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime 
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397814467753                   -0.90           5.0    1.70s
  2   -8.400384887702       -2.59       -1.79   0.80    2.0    500ms
  3   -8.400422719182       -4.42       -3.05   0.80    1.0    218ms
  4   -8.400428110073       -5.27       -3.43   0.80    2.5   20.9ms
  5   -8.400428148986       -7.41       -4.51   0.80    1.2   17.3ms
  6   -8.400428152174       -8.50       -5.37   0.80    2.2   20.2ms
  7   -8.400428152208      -10.47       -6.61   0.80    2.0   19.8ms

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/gmigl/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/gmigl/src/types.jl:120
n     Energy            log10(ΔE)   log10(Δρ)   Δtime 
---   ---------------   ---------   ---------   ------
  1   +0.808857559312                   -1.13    3.39s
  2   -1.848929498169        0.42       -0.65    151ms
  3   -4.989621271578        0.50       -0.36   43.5ms
  4   -6.622388116710        0.21       -0.51    168ms
  5   -7.825305257357        0.08       -0.83   43.7ms
  6   -8.135342528174       -0.51       -1.27   32.5ms
  7   -8.238179142487       -0.99       -1.72   32.8ms
  8   -8.303041387494       -1.19       -1.99   32.2ms
  9   -8.323745433133       -1.68       -2.22   32.3ms
 10   -8.348920508014       -1.60       -2.40   32.3ms
 11   -8.369628501606       -1.68       -2.13   32.5ms
 12   -8.384919945107       -1.82       -2.03   32.0ms
 13   -8.392329821416       -2.13       -2.66   77.7ms
 14   -8.397135701883       -2.32       -2.76   32.5ms
 15   -8.398996916706       -2.73       -2.90   32.5ms
 16   -8.399902168165       -3.04       -3.26   32.4ms
 17   -8.400193754581       -3.54       -3.30   32.5ms
 18   -8.400337855633       -3.84       -3.52   32.3ms
 19   -8.400393607760       -4.25       -3.82   32.1ms
 20   -8.400414918268       -4.67       -3.81   32.1ms
 21   -8.400421853245       -5.16       -3.96   38.9ms
 22   -8.400425373892       -5.45       -4.58   32.2ms
 23   -8.400426902177       -5.82       -4.70   32.3ms
 24   -8.400427533393       -6.20       -4.74   32.7ms
 25   -8.400427894648       -6.44       -4.89   32.3ms
 26   -8.400428043998       -6.83       -5.04   32.0ms
 27   -8.400428109913       -7.18       -5.33   32.2ms
 28   -8.400428132398       -7.65       -5.31   39.7ms
 29   -8.400428144216       -7.93       -5.83   32.4ms
 30   -8.400428147945       -8.43       -5.86   32.4ms
 31   -8.400428150415       -8.61       -6.21   32.2ms

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   -8.397819497038                   -0.90    5.2   50.4ms

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   -8.400427965688                   -1.78    11.4s
  2   -8.400428152209       -6.73       -4.01    3.67s
  3   -8.400428152209      -14.45       -7.78   90.0ms

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|  = 3.6306445100299206e-7
|ρ_newton - ρ_scfv| = 2.2144158228218754e-7
|ρ_newton - ρ_dm|   = 1.5351451977452087e-6