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   -8.397801087242                   -0.90    5.0   26.8ms
  2   -8.400224467530       -2.62       -1.74    1.0   19.2ms
  3   -8.400404064260       -3.75       -2.98    1.5   20.3ms
  4   -8.400427856890       -4.62       -2.98    3.2   25.8ms
  5   -8.400427945768       -7.05       -3.06    1.0   29.0ms
  6   -8.400428145960       -6.70       -4.60    1.0   19.5ms
  7   -8.400428151747       -8.24       -4.43    2.5   23.4ms
  8   -8.400428152163       -9.38       -4.97    1.0   19.4ms
  9   -8.400428152207      -10.35       -6.38    1.0   19.3ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397824725821                   -0.90           5.2   27.5ms
  2   -8.400386017017       -2.59       -1.79   0.80    2.0   19.6ms
  3   -8.400423136524       -4.43       -2.99   0.80    1.0   16.7ms
  4   -8.400428112254       -5.30       -3.46   0.80    2.8   21.4ms
  5   -8.400428150826       -7.41       -4.55   0.80    1.5   18.2ms
  6   -8.400428152187       -8.87       -6.20   0.80    2.0   19.3ms

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   +0.865595046922                   -1.06   64.5ms
  2   -1.997999666647        0.46       -0.68   32.6ms
  3   -4.636607285953        0.42       -0.44   43.7ms
  4   -6.373948030885        0.24       -0.57   43.7ms
  5   -7.699573113249        0.12       -0.83   43.4ms
  6   -8.063858902257       -0.44       -1.26   38.8ms
  7   -8.230025841563       -0.78       -1.53   32.3ms
  8   -8.298568922229       -1.16       -1.88   32.3ms
  9   -8.338241603154       -1.40       -2.21   33.0ms
 10   -8.359060102912       -1.68       -2.38   32.2ms
 11   -8.377527875036       -1.73       -2.37   32.1ms
 12   -8.388036236656       -1.98       -2.39   38.7ms
 13   -8.392624872114       -2.34       -2.75   32.6ms
 14   -8.396130105406       -2.46       -2.81   32.4ms
 15   -8.398077576684       -2.71       -2.84   32.6ms
 16   -8.399164770762       -2.96       -2.97   32.1ms
 17   -8.399845822868       -3.17       -3.53   32.1ms
 18   -8.400113502639       -3.57       -3.47   38.9ms
 19   -8.400310964347       -3.70       -3.61   32.7ms
 20   -8.400375314637       -4.19       -4.11   32.8ms
 21   -8.400410183363       -4.46       -3.85   32.7ms
 22   -8.400416588566       -5.19       -4.18   32.2ms
 23   -8.400424088653       -5.12       -4.64   32.4ms
 24   -8.400426106147       -5.70       -4.56   38.5ms
 25   -8.400427386736       -5.89       -5.27   32.2ms
 26   -8.400427714006       -6.49       -5.19   32.3ms
 27   -8.400428019180       -6.52       -5.24   32.4ms
 28   -8.400428079654       -7.22       -5.21   32.2ms
 29   -8.400428126907       -7.33       -5.89   32.3ms
 30   -8.400428137434       -7.98       -5.64   38.9ms
 31   -8.400428146349       -8.05       -6.26   32.8ms

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   -8.397859840876                   -0.90    5.0   26.6ms

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   -8.400427990966                   -1.79    593ms
  2   -8.400428152209       -6.79       -4.04    401ms
  3   -8.400428152209      -14.75       -7.86    100ms

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|  = 4.090741530237962e-7
|ρ_newton - ρ_scfv| = 4.120697582631537e-6
|ρ_newton - ρ_dm|   = 2.635327211071386e-6