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   -8.397533475602                   -0.90    5.2   27.1ms
  2   -8.400190098924       -2.58       -1.73    1.0   18.7ms
  3   -8.400398784128       -3.68       -3.02    1.5   19.8ms
  4   -8.400427805933       -4.54       -2.95    3.5   25.2ms
  5   -8.400428056718       -6.60       -3.25    1.0   18.7ms
  6   -8.400428149451       -7.03       -5.04    1.0   67.4ms
  7   -8.400428152018       -8.59       -4.62    3.2   25.3ms
  8   -8.400428152188       -9.77       -5.06    1.8   21.3ms
  9   -8.400428152209      -10.69       -6.32    1.0   19.0ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime 
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397597610391                   -0.90           5.5    1.82s
  2   -8.400387568653       -2.55       -1.78   0.80    2.0    497ms
  3   -8.400423547171       -4.44       -2.98   0.80    1.0    232ms
  4   -8.400428107244       -5.34       -3.49   0.80    2.5   21.1ms
  5   -8.400428148316       -7.39       -4.91   0.80    1.0   17.2ms
  6   -8.400428152200       -8.41       -5.39   0.80    3.2   23.3ms
  7   -8.400428152208      -11.11       -6.20   0.80    1.2   17.9ms

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   +0.825542008740                   -1.06    3.39s
  2   -1.443459399722        0.36       -0.65    145ms
  3   -4.376234794314        0.47       -0.36   44.8ms
  4   -5.692409576195        0.12       -0.42   44.8ms
  5   -7.344055766914        0.22       -0.59   44.6ms
  6   -7.562390618382       -0.66       -1.32   72.7ms
  7   -7.954324430012       -0.41       -1.59   33.2ms
  8   -8.088628887579       -0.87       -1.69   32.9ms
  9   -8.257100741089       -0.77       -1.55   32.9ms
 10   -8.302040659293       -1.35       -1.85   33.1ms
 11   -8.350297898969       -1.32       -2.61   32.7ms
 12   -8.369543166194       -1.72       -2.41   57.9ms
 13   -8.382805613927       -1.88       -2.54   33.2ms
 14   -8.389152712879       -2.20       -2.74   33.1ms
 15   -8.393990214078       -2.32       -2.89   33.5ms
 16   -8.396730556740       -2.56       -2.77   33.3ms
 17   -8.397965307920       -2.91       -2.89   33.1ms
 18   -8.399080362913       -2.95       -3.48   39.6ms
 19   -8.399717658499       -3.20       -3.46   33.4ms
 20   -8.400051546710       -3.48       -3.57   33.7ms
 21   -8.400266049278       -3.67       -3.50   33.2ms
 22   -8.400359851361       -4.03       -3.49   33.0ms
 23   -8.400396655577       -4.43       -3.71   39.3ms
 24   -8.400412279475       -4.81       -4.12   33.3ms
 25   -8.400421256515       -5.05       -4.37   33.3ms
 26   -8.400424549145       -5.48       -4.46   33.2ms
 27   -8.400427145249       -5.59       -4.25   33.1ms
 28   -8.400427572955       -6.37       -4.74   39.2ms
 29   -8.400427963963       -6.41       -4.90   33.6ms
 30   -8.400428029436       -7.18       -4.95   33.2ms
 31   -8.400428102679       -7.14       -5.38   33.2ms
 32   -8.400428124385       -7.66       -5.28   33.1ms
 33   -8.400428143777       -7.71       -5.43   39.1ms
 34   -8.400428148447       -8.33       -5.73   33.2ms
 35   -8.400428150898       -8.61       -6.22   32.9ms

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   -8.397493151231                   -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   -8.400427980226                   -1.80    11.3s
  2   -8.400428152209       -6.76       -4.03    3.64s
  3   -8.400428152209      -14.75       -7.84   93.9ms

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|  = 5.514920433609086e-7
|ρ_newton - ρ_scfv| = 5.33750970933964e-7
|ρ_newton - ρ_dm|   = 1.945874739613956e-6