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   -8.397859715427                   -0.90    5.2   30.9ms
  2   -8.400244549470       -2.62       -1.74    1.0   21.6ms
  3   -8.400404884656       -3.79       -2.97    1.5   22.1ms
  4   -8.400427855685       -4.64       -2.97    3.0   27.6ms
  5   -8.400427918944       -7.20       -3.01    1.0   36.3ms
  6   -8.400428144658       -6.65       -4.64    1.0   21.3ms
  7   -8.400428151720       -8.15       -4.41    2.8   27.0ms
  8   -8.400428152174       -9.34       -5.04    1.0   22.6ms
  9   -8.400428152207      -10.48       -6.33    1.0   21.7ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397888199463                   -0.90           5.5   44.8ms
  2   -8.400382702182       -2.60       -1.77   0.80    2.0   21.6ms
  3   -8.400423437013       -4.39       -2.97   0.80    1.0   18.6ms
  4   -8.400428109114       -5.33       -3.48   0.80    2.2   22.4ms
  5   -8.400428148565       -7.40       -4.99   0.80    1.2   18.3ms
  6   -8.400428152204       -8.44       -5.29   0.80    3.2   25.2ms
  7   -8.400428152208      -11.30       -6.57   0.80    1.0   18.5ms

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   +0.803836481554                   -1.08   74.0ms
  2   -1.757237103634        0.41       -0.65   34.6ms
  3   -4.480260876681        0.44       -0.37   46.3ms
  4   -5.940225066462        0.16       -0.45   46.8ms
  5   -7.471375716783        0.19       -0.70   53.5ms
  6   -7.506094127091       -1.46       -1.41   34.7ms
  7   -7.922888276770       -0.38       -1.41   36.2ms
  8   -8.213982608467       -0.54       -1.12   46.8ms
  9   -8.253643871422       -1.40       -1.82   35.1ms
 10   -8.339385978615       -1.07       -2.43   44.5ms
 11   -8.356267519297       -1.77       -2.32   36.0ms
 12   -8.379601246680       -1.63       -2.56   34.9ms
 13   -8.387822795578       -2.09       -2.37   34.9ms
 14   -8.395591318888       -2.11       -2.77   34.4ms
 15   -8.398154523478       -2.59       -2.87   34.7ms
 16   -8.399536862771       -2.86       -3.19   41.2ms
 17   -8.399957911674       -3.38       -3.04   34.5ms
 18   -8.400237500102       -3.55       -3.53   36.2ms
 19   -8.400331902592       -4.03       -3.27   34.3ms
 20   -8.400388376074       -4.25       -4.00   33.9ms
 21   -8.400404942190       -4.78       -3.65   34.5ms
 22   -8.400418462281       -4.87       -4.42   43.5ms
 23   -8.400423740155       -5.28       -3.98   34.2ms
 24   -8.400426699525       -5.53       -4.54   34.0ms
 25   -8.400427557860       -6.07       -4.37   34.6ms
 26   -8.400427903517       -6.46       -5.02   34.4ms
 27   -8.400428044534       -6.85       -4.80   34.4ms
 28   -8.400428112974       -7.16       -5.14   42.5ms
 29   -8.400428138255       -7.60       -5.11   34.7ms
 30   -8.400428145610       -8.13       -5.47   36.5ms
 31   -8.400428149265       -8.44       -5.66   34.4ms
 32   -8.400428150568       -8.88       -5.71   34.7ms
 33   -8.400428151538       -9.01       -6.00   34.3ms

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   -8.397839294351                   -0.90    5.0   29.4ms

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   -8.400427985826                   -1.79    612ms
  2   -8.400428152209       -6.78       -4.04    406ms
  3   -8.400428152209      -14.75       -7.84    124ms

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|  = 7.034236832991148e-7
|ρ_newton - ρ_scfv| = 3.473726560179103e-7
|ρ_newton - ρ_dm|   = 2.2438275793953397e-6