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   -8.397783097643                   -0.90    5.2   26.3ms
  2   -8.400232125236       -2.61       -1.74    1.0   18.8ms
  3   -8.400404610549       -3.76       -2.97    1.5   26.2ms
  4   -8.400427818704       -4.63       -2.97    3.0   23.8ms
  5   -8.400427902253       -7.08       -3.02    1.0   18.8ms
  6   -8.400428144293       -6.62       -4.53    1.0   22.9ms
  7   -8.400428151620       -8.14       -4.37    3.0   24.2ms
  8   -8.400428152188       -9.25       -5.48    1.0   19.2ms
  9   -8.400428152208      -10.71       -6.10    1.8   21.3ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397794151797                   -0.90           5.0   25.8ms
  2   -8.400378653892       -2.59       -1.78   0.80    2.2   19.1ms
  3   -8.400422261617       -4.36       -2.98   0.80    1.0   16.1ms
  4   -8.400428094630       -5.23       -3.40   0.80    2.5   24.3ms
  5   -8.400428148335       -7.27       -4.59   0.80    1.2   17.3ms
  6   -8.400428152173       -8.42       -5.38   0.80    2.5   19.8ms
  7   -8.400428152207      -10.47       -6.29   0.80    1.8   18.3ms

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/7krni/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/7krni/src/types.jl:120
n     Energy            log10(ΔE)   log10(Δρ)   Δtime
---   ---------------   ---------   ---------   ------
  1   +1.127154337369                   -1.05   62.1ms
  2   -1.733596683750        0.46       -0.63   32.8ms
  3   -4.643059596334        0.46       -0.38   43.8ms
  4   -6.329362109434        0.23       -0.47   47.4ms
  5   -7.622975725904        0.11       -0.80   43.9ms
  6   -8.007867463797       -0.41       -1.39   32.6ms
  7   -8.191576771524       -0.74       -1.64   36.0ms
  8   -8.275875311660       -1.07       -1.87   32.5ms
  9   -8.313777596077       -1.42       -2.18   32.5ms
 10   -8.354013752573       -1.40       -2.33   32.8ms
 11   -8.378643970930       -1.61       -2.14   36.1ms
 12   -8.390911315511       -1.91       -2.34   32.8ms
 13   -8.396611497525       -2.24       -2.86   32.5ms
 14   -8.398705974326       -2.68       -2.76   35.9ms
 15   -8.399733245477       -2.99       -2.97   32.5ms
 16   -8.400123106356       -3.41       -3.55   32.4ms
 17   -8.400292001743       -3.77       -3.31   32.3ms
 18   -8.400376254990       -4.07       -4.01   35.6ms
 19   -8.400408188799       -4.50       -3.56   32.3ms
 20   -8.400420687697       -4.90       -4.24   32.4ms
 21   -8.400425103118       -5.36       -4.12   35.7ms
 22   -8.400426779087       -5.78       -4.60   32.5ms
 23   -8.400427539916       -6.12       -4.66   32.4ms
 24   -8.400427860922       -6.49       -4.87   32.3ms
 25   -8.400428043198       -6.74       -4.99   35.7ms
 26   -8.400428096394       -7.27       -5.22   32.3ms
 27   -8.400428131427       -7.46       -5.38   32.6ms
 28   -8.400428142596       -7.95       -5.43   32.3ms
 29   -8.400428148705       -8.21       -5.95   35.8ms
 30   -8.400428150419       -8.77       -5.76   32.7ms
 31   -8.400428151528       -8.95       -6.57   32.3ms

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   -8.397846609589                   -0.90    5.2   26.1ms

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   -8.400427986823                   -1.79    590ms
  2   -8.400428152209       -6.78       -4.04    396ms
  3   -8.400428152209      -14.75       -7.85    103ms

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|  = 1.5169881095948585e-6
|ρ_newton - ρ_scfv| = 5.470216222855008e-7
|ρ_newton - ρ_dm|   = 1.7241801397733728e-6