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   -8.397844207748                   -0.90    5.2   27.8ms
  2   -8.400236234605       -2.62       -1.74    1.0   18.8ms
  3   -8.400406823827       -3.77       -2.97    1.5   19.9ms
  4   -8.400427870063       -4.68       -3.00    3.0   23.3ms
  5   -8.400427968691       -7.01       -3.08    1.0   18.6ms
  6   -8.400428145869       -6.75       -4.51    1.0   18.4ms
  7   -8.400428151814       -8.23       -4.48    2.2   72.5ms
  8   -8.400428152172       -9.45       -5.04    1.0   18.9ms
  9   -8.400428152207      -10.45       -6.42    1.5   19.7ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime 
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397843323003                   -0.90           5.0    1.74s
  2   -8.400382511707       -2.60       -1.80   0.80    2.5    592ms
  3   -8.400422133296       -4.40       -3.01   0.80    1.0    219ms
  4   -8.400428097608       -5.22       -3.39   0.80    2.5   21.2ms
  5   -8.400428148457       -7.29       -4.43   0.80    1.5   18.3ms
  6   -8.400428152159       -8.43       -5.13   0.80    2.5   20.7ms
  7   -8.400428152207      -10.32       -6.54   0.80    1.5   18.5ms

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   +0.889846505268                   -1.03    3.46s
  2   -1.506281979837        0.38       -0.67    151ms
  3   -4.364949466781        0.46       -0.39   44.1ms
  4   -5.831104741969        0.17       -0.49   43.9ms
  5   -7.439552899907        0.21       -0.69   44.0ms
  6   -7.623919327222       -0.73       -1.41   63.2ms
  7   -8.103466488537       -0.32       -1.58   32.4ms
  8   -8.183383653682       -1.10       -1.68   32.4ms
  9   -8.204395629238       -1.68       -1.74   32.4ms
 10   -8.279037095057       -1.13       -1.67   43.3ms
 11   -8.311593170501       -1.49       -1.83   65.5ms
 12   -8.340953712539       -1.53       -2.29   32.5ms
 13   -8.358990386328       -1.74       -2.63   32.5ms
 14   -8.371774249563       -1.89       -2.45   32.9ms
 15   -8.386978529973       -1.82       -2.65   32.4ms
 16   -8.394863398386       -2.10       -2.65   32.9ms
 17   -8.397546947171       -2.57       -2.85   39.3ms
 18   -8.399316117527       -2.75       -3.16   32.3ms
 19   -8.399885858717       -3.24       -3.69   32.9ms
 20   -8.400207409301       -3.49       -3.63   33.1ms
 21   -8.400346710612       -3.86       -3.75   38.0ms
 22   -8.400393740813       -4.33       -3.90   32.8ms
 23   -8.400412967768       -4.72       -3.83   32.3ms
 24   -8.400421393005       -5.07       -4.40   32.5ms
 25   -8.400424697451       -5.48       -4.11   32.4ms
 26   -8.400426995990       -5.64       -4.39   39.6ms
 27   -8.400427639626       -6.19       -4.71   32.7ms
 28   -8.400427945708       -6.51       -5.04   32.4ms
 29   -8.400428071063       -6.90       -4.97   32.4ms
 30   -8.400428111831       -7.39       -5.93   32.4ms
 31   -8.400428134929       -7.64       -5.35   38.3ms
 32   -8.400428142954       -8.10       -5.65   32.8ms
 33   -8.400428148719       -8.24       -5.55   32.6ms
 34   -8.400428150578       -8.73       -6.06   33.1ms

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   -8.397823692873                   -0.90    5.0   47.2ms

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   -8.400427974014                   -1.78    11.5s
  2   -8.400428152209       -6.75       -4.03    3.65s
  3   -8.400428152209      -14.75       -7.80   89.5ms

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|  = 2.268097354622436e-7
|ρ_newton - ρ_scfv| = 7.866198218055216e-7
|ρ_newton - ρ_dm|   = 9.339721700357298e-7