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   -8.397828786723                   -0.90    5.2   27.7ms
  2   -8.400232933169       -2.62       -1.74    1.0   19.8ms
  3   -8.400405950093       -3.76       -2.98    1.5   20.8ms
  4   -8.400427852767       -4.66       -2.99    3.0   25.0ms
  5   -8.400427955744       -6.99       -3.06    1.0   19.6ms
  6   -8.400428145331       -6.72       -4.70    1.0   19.5ms
  7   -8.400428151767       -8.19       -4.43    3.0   34.8ms
  8   -8.400428152188       -9.37       -5.40    1.0   20.5ms
  9   -8.400428152207      -10.72       -6.34    1.8   22.0ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397796484264                   -0.90           4.8   27.1ms
  2   -8.400374913282       -2.59       -1.77   0.80    2.0   19.2ms
  3   -8.400422324153       -4.32       -2.98   0.80    1.0   16.5ms
  4   -8.400428108548       -5.24       -3.46   0.80    2.5   28.2ms
  5   -8.400428148562       -7.40       -5.05   0.80    1.2   17.3ms
  6   -8.400428152204       -8.44       -5.20   0.80    3.5   23.5ms
  7   -8.400428152209      -11.29       -6.28   0.80    1.0   16.9ms

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   +0.892185875672                   -1.03   57.3ms
  2   -1.669149508252        0.41       -0.66   33.0ms
  3   -4.591462868235        0.47       -0.35   51.4ms
  4   -6.174225043057        0.20       -0.44   43.5ms
  5   -7.614420947048        0.16       -0.68   43.6ms
  6   -8.068895202141       -0.34       -1.28   32.2ms
  7   -8.238559826318       -0.77       -1.68   32.2ms
  8   -8.300935025729       -1.20       -1.95   32.2ms
  9   -8.338741125448       -1.42       -2.10   38.7ms
 10   -8.353183744979       -1.84       -2.35   32.2ms
 11   -8.369810654194       -1.78       -2.23   32.2ms
 12   -8.384887836849       -1.82       -2.17   32.0ms
 13   -8.391707399890       -2.17       -2.58   32.2ms
 14   -8.397580652834       -2.23       -3.03   32.3ms
 15   -8.399268658227       -2.77       -2.78   39.3ms
 16   -8.399949531672       -3.17       -3.54   32.7ms
 17   -8.400226458519       -3.56       -3.35   32.4ms
 18   -8.400356156075       -3.89       -3.73   32.2ms
 19   -8.400391771839       -4.45       -3.48   32.4ms
 20   -8.400419429381       -4.56       -4.22   32.1ms
 21   -8.400423277126       -5.41       -3.97   37.8ms
 22   -8.400426319072       -5.52       -4.22   32.4ms
 23   -8.400427259287       -6.03       -4.38   32.5ms
 24   -8.400427766617       -6.29       -4.78   32.2ms
 25   -8.400427954221       -6.73       -4.78   32.2ms
 26   -8.400428082313       -6.89       -5.70   32.0ms
 27   -8.400428115891       -7.47       -5.19   31.9ms
 28   -8.400428139286       -7.63       -6.09   39.8ms

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   -8.397812518225                   -0.90    5.2   27.2ms

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   -8.400427977670                   -1.78    552ms
  2   -8.400428152209       -6.76       -4.03    365ms
  3   -8.400428152209      -14.45       -7.82   83.6ms

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|  = 1.2585501466496013e-6
|ρ_newton - ρ_scfv| = 2.397060241894607e-7
|ρ_newton - ρ_dm|   = 3.1254271045776822e-6