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   -8.397902078475                   -0.90    5.2   27.9ms
  2   -8.400252313138       -2.63       -1.74    1.0   19.9ms
  3   -8.400406949396       -3.81       -2.98    1.5   20.7ms
  4   -8.400427864530       -4.68       -3.00    3.0   24.6ms
  5   -8.400427950102       -7.07       -3.05    1.0   19.7ms
  6   -8.400428145974       -6.71       -4.68    1.0   31.6ms
  7   -8.400428151816       -8.23       -4.47    2.8   24.9ms
  8   -8.400428152183       -9.43       -5.19    1.0   20.1ms
  9   -8.400428152207      -10.63       -6.29    1.2   20.6ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397887450077                   -0.90           5.5   28.4ms
  2   -8.400385295625       -2.60       -1.79   0.80    2.0   30.3ms
  3   -8.400422663555       -4.43       -3.01   0.80    1.0   17.4ms
  4   -8.400428118605       -5.26       -3.44   0.80    2.5   21.9ms
  5   -8.400428149791       -7.51       -4.85   0.80    1.2   17.8ms
  6   -8.400428152203       -8.62       -5.76   0.80    3.0   23.1ms
  7   -8.400428152209      -11.26       -6.35   0.80    1.8   19.3ms

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   +1.327526420825                   -1.06   58.6ms
  2   -1.205858674623        0.40       -0.67   42.4ms
  3   -4.266569519889        0.49       -0.35   44.9ms
  4   -5.832031466196        0.19       -0.44   44.5ms
  5   -7.412025995759        0.20       -0.66   44.6ms
  6   -7.858547694203       -0.35       -1.26   41.9ms
  7   -8.218725022551       -0.44       -1.51   33.2ms
  8   -8.315445717108       -1.01       -1.88   32.9ms
  9   -8.370729146880       -1.26       -2.23   33.0ms
 10   -8.383710138994       -1.89       -2.46   33.2ms
 11   -8.392941080978       -2.03       -2.58   33.2ms
 12   -8.397244605412       -2.37       -3.30   41.6ms
 13   -8.399182412687       -2.71       -3.00   33.9ms
 14   -8.399960931300       -3.11       -3.42   33.6ms
 15   -8.400170385799       -3.68       -3.52   32.9ms
 16   -8.400349865189       -3.75       -4.02   32.8ms
 17   -8.400384313982       -4.46       -3.78   32.8ms
 18   -8.400412018397       -4.56       -4.09   41.1ms
 19   -8.400419101520       -5.15       -4.19   33.1ms
 20   -8.400423949656       -5.31       -4.97   33.3ms
 21   -8.400425599965       -5.78       -4.64   32.9ms
 22   -8.400427192402       -5.80       -4.81   32.9ms
 23   -8.400427738293       -6.26       -4.89   32.7ms
 24   -8.400427983221       -6.61       -5.11   41.7ms
 25   -8.400428075070       -7.04       -5.20   33.1ms
 26   -8.400428121973       -7.33       -5.42   32.7ms
 27   -8.400428139799       -7.75       -5.42   32.7ms
 28   -8.400428146791       -8.16       -5.71   33.0ms
 29   -8.400428150322       -8.45       -6.13   33.0ms

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   -8.397861761834                   -0.90    5.0   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   -8.400427942446                   -1.78    555ms
  2   -8.400428152209       -6.68       -3.97    353ms
  3   -8.400428152209      -14.27       -7.73   92.2ms

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|  = 1.4171283963674658e-6
|ρ_newton - ρ_scfv| = 3.1610814103727794e-7
|ρ_newton - ρ_dm|   = 1.950184023618165e-6