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   -8.397521039997                   -0.90    5.2   27.1ms
  2   -8.400194917171       -2.57       -1.72    1.0   18.5ms
  3   -8.400398667365       -3.69       -3.04    1.5   19.8ms
  4   -8.400427818253       -4.54       -2.97    3.2   24.3ms
  5   -8.400428073292       -6.59       -3.31    1.0   18.6ms
  6   -8.400428149701       -7.12       -5.13    1.0   18.7ms
  7   -8.400428152071       -8.63       -4.70    3.5   58.6ms
  8   -8.400428152194       -9.91       -5.14    1.8   21.1ms
  9   -8.400428152209      -10.82       -6.54    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   -8.397553352719                   -0.90           5.2    1.82s
  2   -8.400387981834       -2.55       -1.79   0.80    2.0    679ms
  3   -8.400423767489       -4.45       -2.98   0.80    1.0    326ms
  4   -8.400428113999       -5.36       -3.49   0.80    2.2   20.9ms
  5   -8.400428149412       -7.45       -4.86   0.80    1.2   17.5ms
  6   -8.400428152198       -8.55       -5.55   0.80    3.2   22.7ms
  7   -8.400428152209      -10.98       -6.06   0.80    1.5   18.3ms

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   +1.152334597589                   -1.10    3.38s
  2   -1.366769378124        0.40       -0.71    217ms
  3   -4.345440095705        0.47       -0.43   44.2ms
  4   -5.958875528274        0.21       -0.48   44.7ms
  5   -7.442598449178        0.17       -0.81   44.2ms
  6   -7.881444917747       -0.36       -1.26   33.0ms
  7   -8.134070278544       -0.60       -1.47   64.9ms
  8   -8.228074323837       -1.03       -1.75   33.0ms
  9   -8.269616702955       -1.38       -1.82   33.2ms
 10   -8.274254451957       -2.33       -1.85   32.7ms
 11   -8.339154336788       -1.19       -1.61   44.0ms
 12   -8.361822461346       -1.64       -1.80   51.9ms
 13   -8.387053734960       -1.60       -2.68   32.8ms
 14   -8.392887180505       -2.23       -2.85   32.7ms
 15   -8.397090026219       -2.38       -2.91   33.1ms
 16   -8.399118759992       -2.69       -2.88   32.7ms
 17   -8.399550320453       -3.36       -3.10   39.9ms
 18   -8.400135635829       -3.23       -3.46   32.9ms
 19   -8.400278504207       -3.85       -3.57   32.8ms
 20   -8.400367589997       -4.05       -4.02   32.8ms
 21   -8.400402284424       -4.46       -3.98   32.8ms
 22   -8.400416826575       -4.84       -3.98   40.7ms
 23   -8.400423047035       -5.21       -4.24   33.0ms
 24   -8.400425808341       -5.56       -5.25   33.4ms
 25   -8.400427227823       -5.85       -4.64   32.9ms
 26   -8.400427746547       -6.29       -4.72   32.8ms
 27   -8.400427991879       -6.61       -5.04   40.7ms
 28   -8.400428094242       -6.99       -5.42   33.1ms
 29   -8.400428126159       -7.50       -5.22   33.1ms
 30   -8.400428140362       -7.85       -5.46   33.2ms
 31   -8.400428146891       -8.19       -5.70   33.0ms
 32   -8.400428149809       -8.53       -5.79   32.8ms
 33   -8.400428151005       -8.92       -6.20   40.6ms

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   -8.397535836762                   -0.90    5.2   27.2ms

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   -8.400427980936                   -1.79    12.5s
  2   -8.400428152209       -6.77       -4.03    3.72s
  3   -8.400428152209      -14.45       -7.85   70.4ms

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|  = 4.536886537453675e-7
|ρ_newton - ρ_scfv| = 3.6220752953770615e-7
|ρ_newton - ρ_dm|   = 2.526715210847308e-6