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   -8.397538845565                   -0.90    5.2   27.1ms
  2   -8.400194756857       -2.58       -1.72    1.0   18.7ms
  3   -8.400395266894       -3.70       -3.02    1.5   19.8ms
  4   -8.400427836430       -4.49       -2.98    3.2   93.3ms
  5   -8.400428087441       -6.60       -3.36    1.0   18.8ms
  6   -8.400428149986       -7.20       -5.07    1.0   18.7ms
  7   -8.400428152073       -8.68       -4.72    3.2   24.6ms
  8   -8.400428152187       -9.94       -5.08    1.5   20.2ms
  9   -8.400428152208      -10.67       -6.79    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   -8.397533237549                   -0.90           5.2    1.82s
  2   -8.400385704566       -2.54       -1.80   0.80    2.0    703ms
  3   -8.400423160694       -4.43       -3.02   0.80    1.0    258ms
  4   -8.400428114330       -5.31       -3.52   0.80    2.2   20.8ms
  5   -8.400428148638       -7.46       -5.06   0.80    1.2   17.6ms
  6   -8.400428152206       -8.45       -5.35   0.80    3.5   24.0ms
  7   -8.400428152209      -11.55       -6.56   0.80    1.0   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   +0.832296113287                   -1.05    3.63s
  2   -1.459921759952        0.36       -0.65    156ms
  3   -4.557706194883        0.49       -0.36   44.0ms
  4   -6.082620736756        0.18       -0.47   43.8ms
  5   -7.650391623259        0.20       -0.67   91.5ms
  6   -8.093207847246       -0.35       -1.25   32.6ms
  7   -8.256505027431       -0.79       -1.56   32.7ms
  8   -8.305920782550       -1.31       -1.85   32.5ms
  9   -8.344917725077       -1.41       -2.16   32.6ms
 10   -8.365991629860       -1.68       -2.37   56.9ms
 11   -8.383229413726       -1.76       -2.36   32.9ms
 12   -8.390696160542       -2.13       -2.38   32.7ms
 13   -8.394922017078       -2.37       -2.93   32.8ms
 14   -8.397524494449       -2.58       -2.92   32.8ms
 15   -8.398900778159       -2.86       -3.02   32.7ms
 16   -8.399680206844       -3.11       -3.01   40.0ms
 17   -8.400098726912       -3.38       -3.65   32.8ms
 18   -8.400270074798       -3.77       -3.54   33.0ms
 19   -8.400360291649       -4.04       -3.85   32.4ms
 20   -8.400398139441       -4.42       -4.27   32.6ms
 21   -8.400411617883       -4.87       -4.16   39.4ms
 22   -8.400421671378       -5.00       -4.35   32.9ms
 23   -8.400424839369       -5.50       -4.37   33.0ms
 24   -8.400426681772       -5.73       -4.66   33.2ms
 25   -8.400427422021       -6.13       -5.00   32.7ms
 26   -8.400427881622       -6.34       -5.14   38.7ms
 27   -8.400428018266       -6.86       -5.51   33.2ms
 28   -8.400428088998       -7.15       -5.38   32.8ms
 29   -8.400428125927       -7.43       -5.93   32.4ms
 30   -8.400428141621       -7.80       -5.52   32.8ms
 31   -8.400428148013       -8.19       -6.00   38.5ms
 32   -8.400428150256       -8.65       -6.09   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   -8.397541562927                   -0.90    5.2   26.6ms

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   -8.400427982316                   -1.79    11.7s
  2   -8.400428152209       -6.77       -4.04    3.76s
  3   -8.400428152209      -14.45       -7.84   91.1ms

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|  = 4.379856343796421e-7
|ρ_newton - ρ_scfv| = 2.1745553699176832e-7
|ρ_newton - ρ_dm|   = 1.8865964914831804e-6