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   -8.397893623604                   -0.90    5.2   27.0ms
  2   -8.400241063616       -2.63       -1.74    1.0   19.4ms
  3   -8.400403411189       -3.79       -2.97    1.5   20.5ms
  4   -8.400427813025       -4.61       -2.95    3.2   25.2ms
  5   -8.400427945534       -6.88       -3.06    1.0   68.6ms
  6   -8.400428145969       -6.70       -4.81    1.0   19.7ms
  7   -8.400428151760       -8.24       -4.42    3.2   25.5ms
  8   -8.400428152189       -9.37       -5.33    1.0   19.7ms
  9   -8.400428152207      -10.76       -6.16    1.0   19.6ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime 
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397830086846                   -0.90           5.2    1.71s
  2   -8.400378716699       -2.59       -1.78   0.80    2.2    504ms
  3   -8.400422820048       -4.36       -3.01   0.80    1.0    216ms
  4   -8.400428098144       -5.28       -3.45   0.80    2.5   21.0ms
  5   -8.400428148663       -7.30       -4.61   0.80    1.5   18.2ms
  6   -8.400428152178       -8.45       -5.71   0.80    2.0   19.9ms
  7   -8.400428152209      -10.52       -6.31   0.80    3.0   22.0ms

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   +0.676711003861                   -1.14    3.42s
  2   -1.674584149586        0.37       -0.64    149ms
  3   -4.398523426463        0.44       -0.39   43.9ms
  4   -5.779794607133        0.14       -0.43   44.1ms
  5   -7.382924313603        0.20       -0.62   78.8ms
  6   -7.908752217167       -0.28       -1.02   47.3ms
  7   -8.088043251853       -0.75       -1.38   32.6ms
  8   -8.193777142342       -0.98       -1.78   32.6ms
  9   -8.264169493573       -1.15       -1.81   33.1ms
 10   -8.322153408959       -1.24       -1.80   56.0ms
 11   -8.359285921040       -1.43       -2.13   33.6ms
 12   -8.381528551771       -1.65       -2.06   33.8ms
 13   -8.392393201711       -1.96       -2.24   33.0ms
 14   -8.397729601180       -2.27       -2.50   33.4ms
 15   -8.399587507666       -2.73       -2.60   33.1ms
 16   -8.400162781466       -3.24       -2.84   41.0ms
 17   -8.400343668241       -3.74       -2.93   33.3ms
 18   -8.400405513536       -4.21       -3.22   33.1ms
 19   -8.400415601890       -5.00       -3.41   33.0ms
 20   -8.400423960312       -5.08       -3.74   38.6ms
 21   -8.400425663148       -5.77       -3.86   32.9ms
 22   -8.400427291341       -5.79       -4.25   33.2ms
 23   -8.400427692196       -6.40       -4.23   33.3ms
 24   -8.400427980587       -6.54       -4.86   33.0ms
 25   -8.400428056878       -7.12       -4.61   40.6ms
 26   -8.400428120768       -7.19       -5.31   33.2ms
 27   -8.400428134585       -7.86       -4.93   33.1ms
 28   -8.400428146810       -7.91       -5.72   32.9ms
 29   -8.400428149313       -8.60       -5.28   39.0ms
 30   -8.400428151308       -8.70       -5.96   33.1ms
 31   -8.400428151764       -9.34       -5.82   33.3ms
 32   -8.400428152060       -9.53       -6.54   33.4ms

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   -8.397851628949                   -0.90    5.2   49.7ms

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   -8.400427980482                   -1.78    11.7s
  2   -8.400428152209       -6.77       -4.02    3.69s
  3   -8.400428152209      -14.75       -7.83   91.5ms

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|  = 2.0152472189209338e-6
|ρ_newton - ρ_scfv| = 1.4784777299239654e-7
|ρ_newton - ρ_dm|   = 1.1344652303167526e-6