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   -8.397547376235                   -0.90    5.2   26.3ms
  2   -8.400189504963       -2.58       -1.72    1.0   18.2ms
  3   -8.400397228022       -3.68       -3.03    1.5    112ms
  4   -8.400427786427       -4.51       -2.95    3.2   25.2ms
  5   -8.400428074624       -6.54       -3.32    1.0   18.8ms
  6   -8.400428149511       -7.13       -4.98    1.0   18.9ms
  7   -8.400428152046       -8.60       -4.66    3.2   25.0ms
  8   -8.400428152187       -9.85       -5.13    1.0   19.2ms
  9   -8.400428152208      -10.68       -6.61    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   -8.397526947266                   -0.90           5.2    1.89s
  2   -8.400384400915       -2.54       -1.79   0.80    2.0    711ms
  3   -8.400423439247       -4.41       -3.00   0.80    1.0    356ms
  4   -8.400428119668       -5.33       -3.50   0.80    2.5   20.9ms
  5   -8.400428149401       -7.53       -4.95   0.80    1.0   16.8ms
  6   -8.400428152204       -8.55       -5.81   0.80    3.2   23.0ms
  7   -8.400428152209      -11.30       -6.32   0.80    2.0   19.2ms

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   +0.768383939682                   -1.06    3.76s
  2   -1.985291885426        0.44       -0.69    176ms
  3   -4.718546065378        0.44       -0.41    118ms
  4   -6.373119270069        0.22       -0.52   44.5ms
  5   -7.808987855985        0.16       -0.76   43.9ms
  6   -8.117275244651       -0.51       -1.28   32.5ms
  7   -8.264101779478       -0.83       -1.69   32.5ms
  8   -8.325377116755       -1.21       -1.86   32.6ms
  9   -8.359491226300       -1.47       -2.18   66.8ms
 10   -8.375148720432       -1.81       -2.33   32.9ms
 11   -8.384879426870       -2.01       -2.44   33.0ms
 12   -8.389786609289       -2.31       -2.79   32.9ms
 13   -8.392298936012       -2.60       -2.85   32.9ms
 14   -8.393337480888       -2.98       -3.08   32.6ms
 15   -8.393768405962       -3.37       -3.70   32.8ms
 16   -8.394170182345       -3.40       -3.55   41.7ms
 17   -8.394715575373       -3.26       -3.28   32.7ms
 18   -8.395868328655       -2.94       -2.73   32.7ms
 19   -8.397288786214       -2.85       -2.97   32.8ms
 20   -8.399159784168       -2.73       -3.04   32.7ms
 21   -8.399958263242       -3.10       -3.76   39.4ms
 22   -8.400211856210       -3.60       -3.44   33.0ms
 23   -8.400358470380       -3.83       -3.95   32.7ms
 24   -8.400381830985       -4.63       -3.92   32.7ms
 25   -8.400411391341       -4.53       -4.34   32.8ms
 26   -8.400418444481       -5.15       -4.16   39.4ms
 27   -8.400424615619       -5.21       -4.97   33.1ms
 28   -8.400425971987       -5.87       -4.30   33.1ms
 29   -8.400427575178       -5.80       -4.69   32.6ms
 30   -8.400427868719       -6.53       -4.93   44.1ms
 31   -8.400428043817       -6.76       -5.17   40.0ms
 32   -8.400428103566       -7.22       -5.21   33.0ms
 33   -8.400428135701       -7.49       -5.77   33.0ms
 34   -8.400428141056       -8.27       -5.44   32.7ms
 35   -8.400428149142       -8.09       -5.85   32.8ms
 36   -8.400428150157       -8.99       -5.74   39.9ms
 37   -8.400428151748       -8.80       -5.87   32.8ms
 38   -8.400428151943       -9.71       -6.55   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   -8.397515962947                   -0.90    5.2   26.7ms

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   -8.400427980481                   -1.79    11.7s
  2   -8.400428152209       -6.77       -4.03    3.76s
  3   -8.400428152209      -14.75       -7.85   75.1ms

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|  = 5.77362856843328e-7
|ρ_newton - ρ_scfv| = 2.3368383343805158e-7
|ρ_newton - ρ_dm|   = 6.246168601287243e-7