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   -8.397871422849                   -0.90    5.2   29.1ms
  2   -8.400256360100       -2.62       -1.74    1.0   20.2ms
  3   -8.400406731886       -3.82       -2.96    1.5   84.1ms
  4   -8.400427867511       -4.67       -2.99    3.2   25.7ms
  5   -8.400427973109       -6.98       -3.08    1.0   20.3ms
  6   -8.400428147126       -6.76       -4.64    1.0   20.8ms
  7   -8.400428151843       -8.33       -4.49    2.2   23.4ms
  8   -8.400428152197       -9.45       -5.94    1.0   20.3ms
  9   -8.400428152209      -10.92       -6.01    2.5   24.2ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime 
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397873371344                   -0.90           5.2    1.33s
  2   -8.400384592063       -2.60       -1.79   0.80    2.2    489ms
  3   -8.400423044826       -4.42       -2.98   0.80    1.0    215ms
  4   -8.400428109568       -5.30       -3.46   0.80    2.5   22.7ms
  5   -8.400428149287       -7.40       -4.74   0.80    1.2   19.3ms
  6   -8.400428152190       -8.54       -5.66   0.80    2.8   23.0ms
  7   -8.400428152209      -10.73       -6.14   0.80    2.0   20.8ms

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   +0.851540917567                   -1.07    2.76s
  2   -1.693367225484        0.41       -0.68    146ms
  3   -4.371156220351        0.43       -0.41   46.9ms
  4   -5.987513648481        0.21       -0.50   46.8ms
  5   -7.560530922402        0.20       -0.71   47.2ms
  6   -7.915992780144       -0.45       -1.33   34.1ms
  7   -8.240117397411       -0.49       -1.57   96.1ms
  8   -8.301792812880       -1.21       -1.83   34.7ms
  9   -8.359339659398       -1.24       -2.28   35.3ms
 10   -8.373267170605       -1.86       -2.21   34.1ms
 11   -8.388053035236       -1.83       -2.87   35.0ms
 12   -8.391888896641       -2.42       -2.86   34.1ms
 13   -8.394357196809       -2.61       -2.91   34.0ms
 14   -8.395550409041       -2.92       -3.18   34.5ms
 15   -8.396736118751       -2.93       -3.15   63.5ms
 16   -8.398337276630       -2.80       -2.83   34.9ms
 17   -8.399151523072       -3.09       -3.09   35.0ms
 18   -8.400015173084       -3.06       -3.92   33.8ms
 19   -8.400224684349       -3.68       -3.07   34.5ms
 20   -8.400359054470       -3.87       -3.27   34.5ms
 21   -8.400388161030       -4.54       -4.00   44.0ms
 22   -8.400414513730       -4.58       -4.00   34.5ms
 23   -8.400421219811       -5.17       -4.36   34.5ms
 24   -8.400425888180       -5.33       -4.48   34.5ms
 25   -8.400426874589       -6.01       -4.71   34.7ms
 26   -8.400427779034       -6.04       -5.04   41.1ms
 27   -8.400427909159       -6.89       -5.24   34.1ms
 28   -8.400428066151       -6.80       -5.61   35.6ms
 29   -8.400428101593       -7.45       -5.41   34.5ms
 30   -8.400428134952       -7.48       -5.94   34.2ms
 31   -8.400428142422       -8.13       -5.78   42.0ms
 32   -8.400428149596       -8.14       -6.00   34.4ms
 33   -8.400428150611       -8.99       -6.14   34.8ms

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   -8.397844512611                   -0.90    5.2   29.5ms

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   -8.400427983814                   -1.78    8.59s
  2   -8.400428152209       -6.77       -4.03    3.26s
  3   -8.400428152209      -14.45       -7.84   81.0ms

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|  = 1.218622089202394e-6
|ρ_newton - ρ_scfv| = 1.445305486135598e-7
|ρ_newton - ρ_dm|   = 7.632944843918924e-7