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   -8.397646196048                   -0.90    5.2   24.3ms
  2   -8.400230471764       -2.59       -1.72    1.0   18.0ms
  3   -8.400396267974       -3.78       -2.97    1.0   17.9ms
  4   -8.400427701261       -4.50       -2.91    3.0   22.2ms
  5   -8.400428058299       -6.45       -3.26    1.0   81.2ms
  6   -8.400428148881       -7.04       -4.81    1.0   18.1ms
  7   -8.400428151910       -8.52       -4.55    2.8   21.9ms
  8   -8.400428152174       -9.58       -5.00    1.2   19.4ms
  9   -8.400428152208      -10.48       -6.57    1.0   18.4ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime 
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397600897681                   -0.90           5.2    1.61s
  2   -8.400387990954       -2.55       -1.78   0.80    2.0    1.04s
  3   -8.400423917409       -4.44       -2.95   0.80    1.0    195ms
  4   -8.400428114454       -5.38       -3.51   0.80    2.2   19.1ms
  5   -8.400428149719       -7.45       -4.83   0.80    1.2   16.9ms
  6   -8.400428152199       -8.61       -5.44   0.80    3.0   20.3ms
  7   -8.400428152209      -11.02       -6.25   0.80    1.5   17.2ms

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   +1.089786769601                   -1.01    3.21s
  2   -1.434472443980        0.40       -0.61    148ms
  3   -3.988566399645        0.41       -0.35   47.6ms
  4   -5.425211978186        0.16       -0.39   46.9ms
  5   -7.223129113687        0.25       -0.58   46.8ms
  6   -7.846388837815       -0.21       -1.08   83.3ms
  7   -8.061205871455       -0.67       -1.21   35.0ms
  8   -8.220554323183       -0.80       -1.64   34.8ms
  9   -8.284277443722       -1.20       -1.86   34.8ms
 10   -8.336654019924       -1.28       -1.97   34.6ms
 11   -8.363565859038       -1.57       -2.08   35.1ms
 12   -8.385182385322       -1.67       -2.24   59.7ms
 13   -8.393145708141       -2.10       -2.74   35.1ms
 14   -8.397233260440       -2.39       -2.91   35.0ms
 15   -8.398694489326       -2.84       -3.02   35.0ms
 16   -8.399496298273       -3.10       -3.34   35.0ms
 17   -8.399916967296       -3.38       -3.13   34.9ms
 18   -8.400201448867       -3.55       -3.37   42.2ms
 19   -8.400350190603       -3.83       -3.93   35.0ms
 20   -8.400397705073       -4.32       -3.98   35.0ms
 21   -8.400417006184       -4.71       -4.41   34.7ms
 22   -8.400424343849       -5.13       -4.17   34.7ms
 23   -8.400426702520       -5.63       -4.51   39.9ms
 24   -8.400427557130       -6.07       -4.45   35.1ms
 25   -8.400427948114       -6.41       -5.35   35.2ms
 26   -8.400428044986       -7.01       -4.80   35.2ms
 27   -8.400428122804       -7.11       -5.56   37.4ms
 28   -8.400428139869       -7.77       -5.16   40.2ms
 29   -8.400428148325       -8.07       -5.94   35.1ms
 30   -8.400428150183       -8.73       -5.82   35.0ms
 31   -8.400428151585       -8.85       -6.77   35.0ms

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   -8.397604435974                   -0.90    5.2   48.2ms

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   -8.400427976858                   -1.79    8.80s
  2   -8.400428152209       -6.76       -4.02    3.29s
  3   -8.400428152209      -14.75       -7.83    104ms

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|  = 3.092543598911769e-7
|ρ_newton - ρ_scfv| = 4.075536574504995e-7
|ρ_newton - ρ_dm|   = 8.006918754912265e-7