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   -8.397537016312                   -0.90    5.2   15.8ms
  2   -8.400212914427       -2.57       -1.72    1.0   11.2ms
  3   -8.400393708936       -3.74       -2.97    1.5   11.8ms
  4   -8.400427722869       -4.47       -2.91    3.2   14.3ms
  5   -8.400428105400       -6.42       -3.46    1.0   11.4ms
  6   -8.400428149382       -7.36       -4.63    1.0   11.4ms
  7   -8.400428152092       -8.57       -4.82    2.2   13.3ms
  8   -8.400428152181      -10.05       -5.07    1.0   11.7ms
  9   -8.400428152207      -10.58       -6.06    1.0   11.3ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime 
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397488664258                   -0.90           5.5    1.08s
  2   -8.400385998823       -2.54       -1.78   0.80    2.0    663ms
  3   -8.400423438876       -4.43       -2.96   0.80    1.0    132ms
  4   -8.400428099598       -5.33       -3.46   0.80    2.5   12.2ms
  5   -8.400428150114       -7.30       -4.72   0.80    1.2   10.5ms
  6   -8.400428152178       -8.69       -5.69   0.80    2.5   12.2ms
  7   -8.400428152209      -10.51       -5.93   0.80    2.5   12.3ms
  8   -8.400428152209      -12.65       -7.26   0.80    1.0   10.1ms

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   +0.802925759647                   -1.12    2.21s
  2   -1.630746337825        0.39       -0.68    159ms
  3   -4.237442741058        0.42       -0.43   28.3ms
  4   -5.588623354990        0.13       -0.54   28.0ms
  5   -7.230263725330        0.22       -0.65   29.0ms
  6   -7.836331878749       -0.22       -1.07   28.7ms
  7   -8.055938681926       -0.66       -1.29   20.8ms
  8   -8.208795089567       -0.82       -1.72   20.5ms
  9   -8.274637206606       -1.18       -1.76   54.3ms
 10   -8.330163409718       -1.26       -1.97   20.3ms
 11   -8.365901058848       -1.45       -2.03   20.6ms
 12   -8.385119660502       -1.72       -2.00   20.7ms
 13   -8.393754625766       -2.06       -2.78   20.1ms
 14   -8.397620800771       -2.41       -2.70   19.9ms
 15   -8.399335701315       -2.77       -3.15   19.9ms
 16   -8.399939125076       -3.22       -3.21   26.0ms
 17   -8.400251926749       -3.50       -3.28   20.1ms
 18   -8.400369332341       -3.93       -3.90   20.1ms
 19   -8.400403420125       -4.47       -3.86   19.9ms
 20   -8.400418140586       -4.83       -3.97   20.4ms
 21   -8.400422165410       -5.40       -4.06   20.4ms
 22   -8.400425430307       -5.49       -4.57   27.7ms
 23   -8.400426796627       -5.86       -4.17   20.8ms
 24   -8.400427602453       -6.09       -4.85   20.6ms
 25   -8.400427910435       -6.51       -4.58   20.7ms
 26   -8.400428066482       -6.81       -5.00   20.5ms
 27   -8.400428101590       -7.45       -4.99   27.1ms
 28   -8.400428135148       -7.47       -5.41   20.5ms
 29   -8.400428142263       -8.15       -5.49   20.6ms
 30   -8.400428149638       -8.13       -6.35   20.6ms

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   -8.397514397610                   -0.90    5.2   37.6ms

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   -8.400427984308                   -1.79    6.32s
  2   -8.400428152209       -6.77       -4.04    2.18s
  3   -8.400428152209      -14.45       -7.86   43.6ms

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|  = 9.17707223534843e-7
|ρ_newton - ρ_scfv| = 5.144617778324957e-8
|ρ_newton - ρ_dm|   = 2.47741057634323e-6