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   -8.397860908587                   -0.90    5.2   27.6ms
  2   -8.400251696573       -2.62       -1.74    1.0   20.2ms
  3   -8.400408238057       -3.81       -2.97    1.8   33.6ms
  4   -8.400427857486       -4.71       -2.99    2.8   25.6ms
  5   -8.400427952377       -7.02       -3.06    1.0   20.6ms
  6   -8.400428145266       -6.71       -4.72    1.0   22.4ms
  7   -8.400428151844       -8.18       -4.47    3.2   27.2ms
  8   -8.400428152197       -9.45       -6.02    1.0   20.9ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397852946069                   -0.90           4.8   28.3ms
  2   -8.400381348873       -2.60       -1.77   0.80    2.0   20.4ms
  3   -8.400423113117       -4.38       -2.97   0.80    1.0   17.6ms
  4   -8.400428111592       -5.30       -3.46   0.80    2.2   21.8ms
  5   -8.400428149181       -7.42       -4.81   0.80    1.0   18.4ms
  6   -8.400428152189       -8.52       -5.50   0.80    2.8   22.1ms
  7   -8.400428152208      -10.72       -6.24   0.80    2.0   19.9ms

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   +0.873056137404                   -1.02   71.0ms
  2   -1.797410289569        0.43       -0.69   33.5ms
  3   -4.253168643873        0.39       -0.35   44.8ms
  4   -5.632712110226        0.14       -0.43   44.5ms
  5   -7.190959408273        0.19       -0.59   54.0ms
  6   -7.828504085345       -0.20       -0.92   46.3ms
  7   -8.075803389189       -0.61       -1.27   33.6ms
  8   -8.227688060627       -0.82       -1.81   32.9ms
  9   -8.308477027126       -1.09       -1.96   32.5ms
 10   -8.345647355831       -1.43       -2.33   41.0ms
 11   -8.371708316959       -1.58       -2.32   33.5ms
 12   -8.385243913635       -1.87       -2.12   32.9ms
 13   -8.393662790548       -2.07       -2.39   33.3ms
 14   -8.397392397064       -2.43       -2.79   33.2ms
 15   -8.399339312952       -2.71       -2.75   33.3ms
 16   -8.400032400232       -3.16       -2.95   41.5ms
 17   -8.400287517992       -3.59       -3.16   32.8ms
 18   -8.400372381923       -4.07       -3.52   32.9ms
 19   -8.400405946127       -4.47       -3.49   33.8ms
 20   -8.400419888455       -4.86       -4.09   33.4ms
 21   -8.400424683002       -5.32       -3.93   33.1ms
 22   -8.400426486623       -5.74       -4.44   40.3ms
 23   -8.400427372669       -6.05       -4.22   32.6ms
 24   -8.400427798212       -6.37       -5.31   32.9ms
 25   -8.400427981139       -6.74       -4.62   32.5ms
 26   -8.400428071575       -7.04       -5.08   32.4ms
 27   -8.400428120275       -7.31       -5.08   32.4ms
 28   -8.400428138551       -7.74       -5.31   41.6ms
 29   -8.400428146664       -8.09       -5.66   33.6ms
 30   -8.400428149551       -8.54       -5.51   33.3ms
 31   -8.400428151307       -8.76       -5.74   32.6ms
 32   -8.400428151780       -9.32       -6.01   33.2ms

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   -8.397880637314                   -0.90    5.0   36.6ms

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   -8.400427980396                   -1.79    615ms
  2   -8.400428152209       -6.76       -4.03    412ms
  3   -8.400428152209      -14.45       -7.83    119ms

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|  = 2.974858061180254e-6
|ρ_newton - ρ_scfv| = 4.993638387443451e-7
|ρ_newton - ρ_dm|   = 1.70376831078109e-6