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   -8.397558050258                   -0.90    5.0   26.3ms
  2   -8.400194544067       -2.58       -1.72    1.0   18.0ms
  3   -8.400394663199       -3.70       -2.96    1.5   18.9ms
  4   -8.400427711684       -4.48       -2.90    3.2   23.8ms
  5   -8.400428084549       -6.43       -3.34    1.0   17.9ms
  6   -8.400428148949       -7.19       -4.80    1.0   68.6ms
  7   -8.400428152089       -8.50       -4.77    3.0   24.0ms
  8   -8.400428152199       -9.96       -5.35    1.0   18.4ms
  9   -8.400428152208      -11.03       -6.71    1.0   18.5ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime 
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397533914939                   -0.90           5.2    1.86s
  2   -8.400387700819       -2.54       -1.79   0.80    2.0    667ms
  3   -8.400423547059       -4.45       -3.02   0.80    1.0    229ms
  4   -8.400428125112       -5.34       -3.53   0.80    2.5   20.5ms
  5   -8.400428149951       -7.60       -5.19   0.80    1.2   17.1ms
  6   -8.400428152205       -8.65       -5.56   0.80    3.8   24.0ms
  7   -8.400428152209      -11.41       -6.25   0.80    1.0   17.0ms

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   +1.286243257903                   -1.08    3.31s
  2   -1.728439362783        0.48       -0.63    143ms
  3   -4.246626904820        0.40       -0.34   43.8ms
  4   -5.950161462482        0.23       -0.43   43.8ms
  5   -7.570249317767        0.21       -0.68   43.6ms
  6   -7.864228531463       -0.53       -1.41    200ms
  7   -8.187542297452       -0.49       -1.56   40.0ms
  8   -8.264659310151       -1.11       -1.91   39.9ms
  9   -8.328929000062       -1.19       -1.90   39.8ms
 10   -8.354561067888       -1.59       -2.15   40.1ms
 11   -8.371859193693       -1.76       -2.27   39.3ms
 12   -8.385145931872       -1.88       -2.19    1.10s
 13   -8.392350409612       -2.14       -2.79   33.3ms
 14   -8.396300812007       -2.40       -3.04   32.9ms
 15   -8.398694214127       -2.62       -2.97   32.8ms
 16   -8.399464410377       -3.11       -3.17   32.8ms
 17   -8.399989724449       -3.28       -3.44   32.8ms
 18   -8.400234059400       -3.61       -3.68   32.9ms
 19   -8.400339566754       -3.98       -3.95   34.1ms
 20   -8.400389467186       -4.30       -4.11   36.1ms
 21   -8.400411953276       -4.65       -4.34   34.8ms
 22   -8.400419749581       -5.11       -4.23   36.2ms
 23   -8.400424689147       -5.31       -4.55   36.0ms
 24   -8.400426498166       -5.74       -4.84   34.4ms
 25   -8.400427495334       -6.00       -4.77   34.5ms
 26   -8.400427907321       -6.39       -5.00   36.6ms
 27   -8.400428030920       -6.91       -5.20   40.3ms
 28   -8.400428111860       -7.09       -5.51   38.5ms
 29   -8.400428132306       -7.69       -5.29   33.7ms
 30   -8.400428143313       -7.96       -5.90   34.3ms
 31   -8.400428147242       -8.41       -5.72   33.3ms
 32   -8.400428150275       -8.52       -6.15   35.2ms

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   -8.397589109252                   -0.90    5.2   27.4ms

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   -8.400427981270                   -1.79    11.2s
  2   -8.400428152209       -6.77       -4.03    3.62s
  3   -8.400428152209      -14.75       -7.85   90.1ms

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|  = 8.131439750951786e-7
|ρ_newton - ρ_scfv| = 3.427322959759338e-7
|ρ_newton - ρ_dm|   = 8.295665561022536e-7