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   -8.397850840534                   -0.90    5.2   26.9ms
  2   -8.400238865929       -2.62       -1.74    1.0   19.1ms
  3   -8.400406299385       -3.78       -2.99    1.5   31.2ms
  4   -8.400427840485       -4.67       -2.99    3.0   24.0ms
  5   -8.400427964063       -6.91       -3.08    1.0   19.1ms
  6   -8.400428145362       -6.74       -4.59    1.0   19.1ms
  7   -8.400428151835       -8.19       -4.49    2.5   23.5ms
  8   -8.400428152181       -9.46       -5.10    1.0   19.3ms
  9   -8.400428152207      -10.58       -6.66    1.0   29.0ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397824602190                   -0.90           5.5   27.5ms
  2   -8.400385172332       -2.59       -1.79   0.80    2.0   19.7ms
  3   -8.400422888817       -4.42       -3.00   0.80    1.0   17.0ms
  4   -8.400428103020       -5.28       -3.42   0.80    2.2   20.7ms
  5   -8.400428149004       -7.34       -4.55   0.80    1.2   17.6ms
  6   -8.400428152179       -8.50       -5.76   0.80    2.5   20.8ms
  7   -8.400428152209      -10.52       -6.31   0.80    3.0   31.7ms

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/7krni/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/7krni/src/types.jl:120
n     Energy            log10(ΔE)   log10(Δρ)   Δtime
---   ---------------   ---------   ---------   ------
  1   +0.441868345458                   -1.09   57.9ms
  2   -1.835008751893        0.36       -0.68   32.4ms
  3   -4.825976714005        0.48       -0.40   43.5ms
  4   -6.411520373505        0.20       -0.49   50.8ms
  5   -7.752873286620        0.13       -0.79   43.6ms
  6   -8.122412381647       -0.43       -1.33   32.2ms
  7   -8.249151470917       -0.90       -1.57   32.2ms
  8   -8.306972391537       -1.24       -1.83   32.1ms
  9   -8.348288316082       -1.38       -2.15   32.0ms
 10   -8.370405984660       -1.66       -2.34   38.0ms
 11   -8.387410960860       -1.77       -2.28   32.3ms
 12   -8.394825059432       -2.13       -2.54   31.9ms
 13   -8.398661408926       -2.42       -2.96   32.3ms
 14   -8.399665611902       -3.00       -3.21   32.0ms
 15   -8.400157320591       -3.31       -3.12   32.3ms
 16   -8.400280750884       -3.91       -3.54   39.1ms
 17   -8.400376743234       -4.02       -3.40   32.3ms
 18   -8.400402667041       -4.59       -3.97   32.2ms
 19   -8.400418052491       -4.81       -3.97   32.2ms
 20   -8.400423579308       -5.26       -4.16   32.3ms
 21   -8.400425960583       -5.62       -4.51   32.1ms
 22   -8.400427143737       -5.93       -4.55   38.2ms
 23   -8.400427777539       -6.20       -4.73   32.2ms
 24   -8.400427985709       -6.68       -5.07   32.2ms
 25   -8.400428099345       -6.94       -5.10   32.1ms
 26   -8.400428131166       -7.50       -5.19   32.1ms
 27   -8.400428144888       -7.86       -5.67   32.0ms
 28   -8.400428149396       -8.35       -5.73   39.1ms
 29   -8.400428151122       -8.76       -6.26   32.3ms

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   -8.397798577809                   -0.90    4.8   26.0ms

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   -8.400427969307                   -1.78    576ms
  2   -8.400428152209       -6.74       -4.01    399ms
  3   -8.400428152209      -14.45       -7.80    118ms

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|  = 2.8917263727095035e-7
|ρ_newton - ρ_scfv| = 3.5798138102439843e-7
|ρ_newton - ρ_dm|   = 2.2752778931026034e-6