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.oncvpsp3.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   -8.397792788160                   -0.90    5.5   27.9ms
  2   -8.400244155508       -2.61       -1.74    1.0   20.0ms
  3   -8.400406826641       -3.79       -2.96    1.5   20.4ms
  4   -8.400428741956       -4.66       -2.99    3.2   33.8ms
  5   -8.400428849800       -6.97       -3.09    1.0   19.5ms
  6   -8.400429019032       -6.77       -4.89    1.0   19.5ms
  7   -8.400429024083       -8.30       -4.48    3.2   26.2ms
  8   -8.400429024399       -9.50       -5.28    1.0   26.5ms
  9   -8.400429024418      -10.72       -6.53    1.2   20.3ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397858236010                   -0.90           5.2   32.9ms
  2   -8.400386738238       -2.60       -1.79   0.80    2.0   19.2ms
  3   -8.400423676335       -4.43       -3.02   0.80    1.0   16.0ms
  4   -8.400428977728       -5.28       -3.42   0.80    2.5   20.3ms
  5   -8.400429022180       -7.35       -4.40   0.80    1.2   16.7ms
  6   -8.400429024358       -8.66       -5.32   0.80    2.0   18.9ms
  7   -8.400429024419      -10.21       -6.21   0.80    1.8   24.6ms

Direct minimization

scfres_dm = direct_minimization(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   Δtime
---   ---------------   ---------   ---------   ------
  1   +1.487530978831                   -0.99   52.2ms
  2   -0.899226592009        0.38       -0.64   29.1ms
  3   -3.422398950364        0.40       -0.40   44.7ms
  4   -3.650528180984       -0.64       -0.48   40.2ms
  5   -5.741740944573        0.32       -0.47   44.4ms
  6   -6.247592878073       -0.30       -0.67   39.4ms
  7   -7.497558506948        0.10       -0.75   39.3ms
  8   -7.851986451043       -0.45       -1.26   45.4ms
  9   -8.043179728424       -0.72       -1.52   30.0ms
 10   -8.113910476602       -1.15       -1.75   29.5ms
 11   -8.190931774790       -1.11       -2.05   33.9ms
 12   -8.263397132377       -1.14       -1.97   29.7ms
 13   -8.272990384024       -2.02       -1.87   29.4ms
 14   -8.278327282764       -2.27       -2.10   29.1ms
 15   -8.299401395769       -1.68       -1.69   44.6ms
 16   -8.310726924435       -1.95       -1.90   39.7ms
 17   -8.341283528175       -1.51       -2.32   29.2ms
 18   -8.353516248182       -1.91       -2.60   34.5ms
 19   -8.370340665279       -1.77       -2.37   29.3ms
 20   -8.386115724579       -1.80       -2.60   29.4ms
 21   -8.394812949280       -2.06       -2.56   33.7ms
 22   -8.398944649780       -2.38       -2.68   29.4ms
 23   -8.399924689066       -3.01       -3.03   29.3ms
 24   -8.400152447377       -3.64       -3.49   29.1ms
 25   -8.400333546687       -3.74       -3.86   34.5ms
 26   -8.400378982171       -4.34       -3.77   29.2ms
 27   -8.400414561334       -4.45       -4.30   29.0ms
 28   -8.400421618345       -5.15       -4.07   29.3ms
 29   -8.400425941080       -5.36       -4.44   34.0ms
 30   -8.400427777407       -5.74       -4.65   29.3ms
 31   -8.400428518403       -6.13       -4.64   29.1ms
 32   -8.400428809793       -6.54       -4.88   29.2ms
 33   -8.400428928075       -6.93       -5.06   35.4ms
 34   -8.400429000826       -7.14       -5.30   29.7ms
 35   -8.400429012871       -7.92       -5.50   29.7ms
 36   -8.400429019942       -8.15       -5.51   34.4ms
 37   -8.400429021560       -8.79       -6.23   30.1ms

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   -8.397824312158                   -0.90    5.2   27.7ms

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   -8.400428845395                   -1.78    587ms
  2   -8.400429024420       -6.75       -4.02    374ms
  3   -8.400429024420      -14.75       -7.81    115ms

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|  = 1.3863623422489796e-6
|ρ_newton - ρ_scfv| = 3.616324255835933e-7
|ρ_newton - ρ_dm|   = 2.2201950055443067e-6