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   -8.397556467063                   -0.90    5.2   24.2ms
  2   -8.400200706131       -2.58       -1.72    1.0   18.4ms
  3   -8.400396329311       -3.71       -3.03    1.5   18.9ms
  4   -8.400427727662       -4.50       -2.93    3.2   22.6ms
  5   -8.400428044313       -6.50       -3.24    1.0   18.3ms
  6   -8.400428149262       -6.98       -4.76    1.0   18.4ms
  7   -8.400428151908       -8.58       -4.56    2.2   21.3ms
  8   -8.400428152173       -9.58       -5.05    1.0   18.5ms
  9   -8.400428152207      -10.46       -6.60    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   -8.397586539266                   -0.90           5.2    1.71s
  2   -8.400388203825       -2.55       -1.79   0.80    2.0    1.06s
  3   -8.400423805353       -4.45       -2.98   0.80    1.0    201ms
  4   -8.400428106049       -5.37       -3.46   0.80    2.2   19.2ms
  5   -8.400428150175       -7.36       -4.67   0.80    1.5   17.5ms
  6   -8.400428152174       -8.70       -5.70   0.80    2.5   19.8ms
  7   -8.400428152209      -10.46       -6.22   0.80    2.8   21.0ms

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   +0.902157582141                   -1.02    2.97s
  2   -1.979271000623        0.46       -0.64    152ms
  3   -4.746725829762        0.44       -0.40    132ms
  4   -6.540959124718        0.25       -0.42   48.3ms
  5   -7.712741556779        0.07       -0.90   47.2ms
  6   -8.049859317199       -0.47       -1.40   35.1ms
  7   -8.177389304149       -0.89       -1.84   35.1ms
  8   -8.257974473013       -1.09       -1.96   35.3ms
  9   -8.310384408654       -1.28       -2.07   35.2ms
 10   -8.350858515499       -1.39       -2.02   35.0ms
 11   -8.373838585096       -1.64       -2.01   77.1ms
 12   -8.387281844540       -1.87       -1.99   35.1ms
 13   -8.393501283430       -2.21       -2.53   35.0ms
 14   -8.395725727141       -2.65       -2.45   35.2ms
 15   -8.397144890372       -2.85       -3.12   35.2ms
 16   -8.398208646323       -2.97       -2.60   35.3ms
 17   -8.399170388075       -3.02       -3.14   35.2ms
 18   -8.399954374142       -3.11       -3.04   35.0ms
 19   -8.400230572955       -3.56       -3.35   44.1ms
 20   -8.400354549822       -3.91       -3.43   35.3ms
 21   -8.400390669773       -4.44       -3.27   35.2ms
 22   -8.400415345940       -4.61       -3.96   35.0ms
 23   -8.400421795599       -5.19       -3.68   36.1ms
 24   -8.400425785706       -5.40       -4.34   35.7ms
 25   -8.400427076567       -5.89       -4.09   35.1ms
 26   -8.400427687655       -6.21       -4.95   43.6ms
 27   -8.400427935837       -6.61       -4.53   35.1ms
 28   -8.400428050678       -6.94       -5.17   35.2ms
 29   -8.400428105936       -7.26       -4.92   35.1ms
 30   -8.400428130546       -7.61       -5.54   35.5ms
 31   -8.400428143463       -7.89       -5.26   35.6ms
 32   -8.400428148968       -8.26       -5.54   35.3ms
 33   -8.400428151010       -8.69       -5.83   44.0ms
 34   -8.400428151666       -9.18       -5.87   35.2ms
 35   -8.400428152002       -9.47       -6.37   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   -8.397496305861                   -0.90    4.8   24.0ms

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   -8.400427934186                   -1.77    8.75s
  2   -8.400428152209       -6.66       -3.97    3.32s
  3   -8.400428152209      -14.27       -7.73   88.1ms

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|  = 8.238354116595835e-7
|ρ_newton - ρ_scfv| = 1.8370271985249607e-7
|ρ_newton - ρ_dm|   = 1.752237202186734e-6