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   -8.397492705270                   -0.90    5.2   27.5ms
  2   -8.400183003496       -2.57       -1.72    1.0   18.7ms
  3   -8.400397103459       -3.67       -3.04    1.5   19.9ms
  4   -8.400427796703       -4.51       -2.97    3.0   24.2ms
  5   -8.400428072605       -6.56       -3.32    1.0   19.1ms
  6   -8.400428149312       -7.12       -4.99    1.0   18.8ms
  7   -8.400428152062       -8.56       -4.69    3.2   25.1ms
  8   -8.400428152192       -9.88       -5.19    1.0   52.8ms
  9   -8.400428152208      -10.80       -6.67    1.0   19.1ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime 
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397541900527                   -0.90           5.0    1.90s
  2   -8.400386343134       -2.55       -1.78   0.80    2.0    527ms
  3   -8.400423699294       -4.43       -2.96   0.80    1.0    241ms
  4   -8.400428113232       -5.36       -3.47   0.80    2.5   20.9ms
  5   -8.400428149290       -7.44       -4.77   0.80    1.2   17.6ms
  6   -8.400428152190       -8.54       -5.50   0.80    2.5   21.6ms
  7   -8.400428152208      -10.73       -6.19   0.80    2.0   19.7ms

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   +0.903105440088                   -1.03    3.58s
  2   -1.704197207711        0.42       -0.64    150ms
  3   -4.259524850913        0.41       -0.35   44.4ms
  4   -5.513699787408        0.10       -0.41   44.1ms
  5   -7.282946395942        0.25       -0.51   79.4ms
  6   -7.902990153931       -0.21       -1.02   44.3ms
  7   -8.115170893175       -0.67       -1.26   32.9ms
  8   -8.257032680626       -0.85       -1.77   33.2ms
  9   -8.328810266029       -1.14       -1.82   33.0ms
 10   -8.365158009826       -1.44       -2.12   32.9ms
 11   -8.385723799705       -1.69       -2.39   57.6ms
 12   -8.392593543593       -2.16       -2.31   33.2ms
 13   -8.397930694856       -2.27       -2.78   33.0ms
 14   -8.398842588706       -3.04       -2.82   32.7ms
 15   -8.399906314447       -2.97       -3.00   32.5ms
 16   -8.400144841965       -3.62       -3.17   32.9ms
 17   -8.400313862540       -3.77       -3.76   40.6ms
 18   -8.400365535006       -4.29       -3.66   32.8ms
 19   -8.400409966585       -4.35       -4.02   32.9ms
 20   -8.400417670670       -5.11       -3.89   33.2ms
 21   -8.400424251378       -5.18       -4.62   32.8ms
 22   -8.400425730155       -5.83       -4.32   39.1ms
 23   -8.400427218010       -5.83       -4.95   32.8ms
 24   -8.400427668263       -6.35       -4.60   32.9ms
 25   -8.400427983346       -6.50       -5.11   32.7ms
 26   -8.400428080548       -7.01       -4.90   32.5ms
 27   -8.400428127963       -7.32       -5.73   38.5ms
 28   -8.400428136647       -8.06       -5.44   33.0ms
 29   -8.400428145819       -8.04       -5.96   33.0ms
 30   -8.400428148825       -8.52       -5.73   32.5ms
 31   -8.400428150689       -8.73       -6.24   32.6ms

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   -8.397558030985                   -0.90    5.2   26.7ms

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   -8.400427984907                   -1.79    11.8s
  2   -8.400428152209       -6.78       -4.03    3.87s
  3   -8.400428152209      -14.45       -7.86    1.27s

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|  = 5.015353288271713e-7
|ρ_newton - ρ_scfv| = 3.720706192050978e-7
|ρ_newton - ρ_dm|   = 1.547039825464894e-6