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   -8.397803592136                   -0.90    5.2   28.0ms
  2   -8.400251125963       -2.61       -1.74    1.0   19.7ms
  3   -8.400397569857       -3.83       -2.91    1.2   20.0ms
  4   -8.400427754119       -4.52       -2.93    3.2   25.1ms
  5   -8.400427835278       -7.09       -2.94    1.0   19.3ms
  6   -8.400428143205       -6.51       -4.61    1.0   30.7ms
  7   -8.400428151566       -8.08       -4.34    2.8   25.2ms
  8   -8.400428152180       -9.21       -5.27    1.0   20.0ms
  9   -8.400428152207      -10.57       -6.13    1.5   21.2ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397813490122                   -0.90           5.2   29.6ms
  2   -8.400383888019       -2.59       -1.79   0.80    2.2   20.3ms
  3   -8.400422725128       -4.41       -3.00   0.80    1.0   27.6ms
  4   -8.400428098999       -5.27       -3.38   0.80    2.5   20.9ms
  5   -8.400428149463       -7.30       -4.38   0.80    1.2   17.8ms
  6   -8.400428152156       -8.57       -5.61   0.80    2.0   19.9ms
  7   -8.400428152209      -10.27       -6.02   0.80    2.8   22.3ms

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   +0.640306604471                   -1.00   58.8ms
  2   -1.490725562186        0.33       -0.63   41.6ms
  3   -4.372689316092        0.46       -0.43   44.2ms
  4   -5.784497741939        0.15       -0.50   44.1ms
  5   -7.348509937362        0.19       -0.76   44.0ms
  6   -7.565199462645       -0.66       -1.41   32.6ms
  7   -8.088982227880       -0.28       -1.57   40.4ms
  8   -8.245085145918       -0.81       -1.81   32.8ms
  9   -8.333261502237       -1.05       -1.90   32.7ms
 10   -8.367624898186       -1.46       -2.01   32.8ms
 11   -8.383827277995       -1.79       -2.28   33.0ms
 12   -8.391258636473       -2.13       -2.51   33.4ms
 13   -8.396308193574       -2.30       -2.49   40.9ms
 14   -8.398290670932       -2.70       -2.95   33.1ms
 15   -8.399417774233       -2.95       -3.06   33.1ms
 16   -8.399899494929       -3.32       -3.36   32.9ms
 17   -8.400199800701       -3.52       -3.34   33.3ms
 18   -8.400324692081       -3.90       -3.44   32.9ms
 19   -8.400386317578       -4.21       -3.55   39.7ms
 20   -8.400410267434       -4.62       -4.34   32.8ms
 21   -8.400422720362       -4.90       -3.93   32.9ms
 22   -8.400425731252       -5.52       -4.41   32.8ms
 23   -8.400427439841       -5.77       -4.41   32.7ms
 24   -8.400427805317       -6.44       -4.55   32.8ms
 25   -8.400428041200       -6.63       -4.72   40.5ms
 26   -8.400428105111       -7.19       -5.71   32.8ms
 27   -8.400428132420       -7.56       -5.43   32.8ms
 28   -8.400428143635       -7.95       -6.02   32.6ms

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   -8.397835124598                   -0.90    5.2   27.5ms

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   -8.400427999165                   -1.80    609ms
  2   -8.400428152209       -6.82       -4.06    399ms
  3   -8.400428152209      -14.45       -7.89    108ms

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|  = 1.1514951237542666e-6
|ρ_newton - ρ_scfv| = 1.1191537872309611e-7
|ρ_newton - ρ_dm|   = 2.8733858467448253e-6