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   -8.397534019310                   -0.90    5.5   27.0ms
  2   -8.400214938849       -2.57       -1.72    1.0   18.1ms
  3   -8.400392867940       -3.75       -2.97    1.2   18.5ms
  4   -8.400427662012       -4.46       -2.90    3.2    107ms
  5   -8.400428106359       -6.35       -3.48    1.0   18.2ms
  6   -8.400428149230       -7.37       -4.47    1.0   18.3ms
  7   -8.400428152107       -8.54       -4.85    2.2   22.6ms
  8   -8.400428152196      -10.05       -5.36    1.0   18.3ms
  9   -8.400428152208      -10.92       -6.26    1.8   20.1ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime 
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397493653699                   -0.90           5.2    1.69s
  2   -8.400383742047       -2.54       -1.78   0.80    2.0    656ms
  3   -8.400423166357       -4.40       -2.99   0.80    1.0    336ms
  4   -8.400428102145       -5.31       -3.46   0.80    2.2   20.4ms
  5   -8.400428148027       -7.34       -4.79   0.80    1.2   17.1ms
  6   -8.400428152198       -8.38       -5.40   0.80    3.2   22.6ms
  7   -8.400428152208      -10.97       -6.81   0.80    1.2   17.4ms

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   +0.701853675602                   -1.08    3.29s
  2   -1.849017931266        0.41       -0.66    146ms
  3   -4.636944847902        0.45       -0.39    105ms
  4   -6.423057203887        0.25       -0.48   43.8ms
  5   -7.758110659317        0.13       -0.83   45.2ms
  6   -8.108895492792       -0.45       -1.33   32.7ms
  7   -8.206990099885       -1.01       -1.65   32.4ms
  8   -8.295576652992       -1.05       -1.90   32.2ms
  9   -8.325227911712       -1.53       -2.13   62.1ms
 10   -8.360866013725       -1.45       -2.30   32.6ms
 11   -8.379894611369       -1.72       -2.04   32.3ms
 12   -8.389826707140       -2.00       -2.31   32.3ms
 13   -8.395697484976       -2.23       -2.60   32.3ms
 14   -8.398279198017       -2.59       -3.18   32.1ms
 15   -8.399577423996       -2.89       -3.28   32.5ms
 16   -8.400080770561       -3.30       -3.54   39.7ms
 17   -8.400281178495       -3.70       -3.52   32.5ms
 18   -8.400371632670       -4.04       -3.90   32.5ms
 19   -8.400402123870       -4.52       -3.75   32.4ms
 20   -8.400416720110       -4.84       -4.13   32.8ms
 21   -8.400423409630       -5.17       -4.08   40.1ms
 22   -8.400425835304       -5.62       -4.43   32.6ms
 23   -8.400427203210       -5.86       -4.70   32.6ms
 24   -8.400427715321       -6.29       -4.90   32.8ms
 25   -8.400427970171       -6.59       -4.96   32.4ms
 26   -8.400428079586       -6.96       -5.10   38.2ms
 27   -8.400428119961       -7.39       -5.25   32.4ms
 28   -8.400428139712       -7.70       -5.36   32.3ms
 29   -8.400428147622       -8.10       -5.79   32.2ms
 30   -8.400428150336       -8.57       -5.85   32.1ms
 31   -8.400428151454       -8.95       -6.32   37.4ms

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   -8.397522006672                   -0.90    5.0   26.3ms

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   -8.400427983177                   -1.79    11.1s
  2   -8.400428152209       -6.77       -4.04    3.61s
  3   -8.400428152209      -14.45       -7.86   90.6ms

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|  = 2.946446237994852e-7
|ρ_newton - ρ_scfv| = 3.861534150920846e-7
|ρ_newton - ρ_dm|   = 1.9297730292991e-6