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   -8.397845319135                   -0.90    5.2   26.6ms
  2   -8.400251875023       -2.62       -1.74    1.0   18.9ms
  3   -8.400407437159       -3.81       -2.97    1.5   19.7ms
  4   -8.400427836984       -4.69       -2.98    3.0   23.8ms
  5   -8.400427938122       -7.00       -3.04    1.0   19.1ms
  6   -8.400428145555       -6.68       -4.77    1.0   19.1ms
  7   -8.400428151778       -8.21       -4.43    3.2   24.6ms
  8   -8.400428152184       -9.39       -5.26    1.0   19.0ms
  9   -8.400428152206      -10.64       -6.43    1.0   19.2ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397850737087                   -0.90           5.0   25.8ms
  2   -8.400386509718       -2.60       -1.79   0.80    2.0   21.2ms
  3   -8.400423091101       -4.44       -3.04   0.80    1.0   20.4ms
  4   -8.400428109497       -5.30       -3.45   0.80    2.2   24.8ms
  5   -8.400428148506       -7.41       -4.81   0.80    1.0   21.3ms
  6   -8.400428152193       -8.43       -5.61   0.80    3.0   26.7ms
  7   -8.400428152209      -10.81       -6.24   0.80    2.2   24.2ms

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/7krni/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/7krni/src/types.jl:120
n     Energy            log10(ΔE)   log10(Δρ)   Δtime
---   ---------------   ---------   ---------   ------
  1   +0.808642976091                   -1.00   72.1ms
  2   -1.735241709001        0.41       -0.63   49.0ms
  3   -4.468315432957        0.44       -0.36   44.5ms
  4   -6.067644117166        0.20       -0.39   44.5ms
  5   -7.576717567012        0.18       -0.69   44.7ms
  6   -7.956103653720       -0.42       -1.30   32.7ms
  7   -8.179165787509       -0.65       -1.56   33.0ms
  8   -8.263346094757       -1.07       -1.85   33.4ms
  9   -8.316888978739       -1.27       -2.11   32.6ms
 10   -8.348294917335       -1.50       -1.90   32.7ms
 11   -8.376247433220       -1.55       -2.27   32.5ms
 12   -8.387639251527       -1.94       -2.46   32.8ms
 13   -8.395454959556       -2.11       -3.03   32.5ms
 14   -8.397896891478       -2.61       -3.07   32.4ms
 15   -8.399447821643       -2.81       -3.22   32.6ms
 16   -8.399972687530       -3.28       -3.17   32.6ms
 17   -8.400249265716       -3.56       -3.74   39.1ms
 18   -8.400362113945       -3.95       -3.49   33.1ms
 19   -8.400398395550       -4.44       -3.62   32.8ms
 20   -8.400420143724       -4.66       -3.96   32.8ms
 21   -8.400424413403       -5.37       -4.62   32.6ms
 22   -8.400426653721       -5.65       -4.81   32.6ms
 23   -8.400427464677       -6.09       -5.20   32.6ms
 24   -8.400427901493       -6.36       -5.05   32.6ms
 25   -8.400428037740       -6.87       -5.19   32.5ms
 26   -8.400428116875       -7.10       -5.13   33.2ms
 27   -8.400428132364       -7.81       -5.58   32.9ms
 28   -8.400428144658       -7.91       -5.64   32.9ms
 29   -8.400428148735       -8.39       -5.80   32.4ms
 30   -8.400428150579       -8.73       -6.50   38.5ms

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   -8.397777615146                   -0.90    5.0   25.5ms

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   -8.400427974948                   -1.79    568ms
  2   -8.400428152209       -6.75       -4.03    400ms
  3   -8.400428152209      -14.45       -7.81    139ms

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|  = 1.7391182297066575e-6
|ρ_newton - ρ_scfv| = 2.410341469367132e-7
|ρ_newton - ρ_dm|   = 1.5256117482491983e-6