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   -8.397811793729                   -0.90    5.2   27.1ms
  2   -8.400237838323       -2.62       -1.74    1.0   19.4ms
  3   -8.400405312553       -3.78       -2.96    1.8   20.6ms
  4   -8.400427824130       -4.65       -2.97    3.0   24.1ms
  5   -8.400427948301       -6.91       -3.05    1.0   67.9ms
  6   -8.400428145490       -6.71       -4.67    1.0   19.2ms
  7   -8.400428151788       -8.20       -4.46    2.8   23.7ms
  8   -8.400428152194       -9.39       -5.72    1.0   19.4ms
  9   -8.400428152209      -10.82       -6.41    2.0   22.4ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime 
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397864551459                   -0.90           5.2    1.68s
  2   -8.400384461344       -2.60       -1.78   0.80    2.2    489ms
  3   -8.400423020092       -4.41       -3.00   0.80    1.0    215ms
  4   -8.400428105194       -5.29       -3.45   0.80    2.5   20.7ms
  5   -8.400428148814       -7.36       -4.73   0.80    1.2   17.4ms
  6   -8.400428152191       -8.47       -5.54   0.80    2.5   20.6ms
  7   -8.400428152208      -10.77       -6.51   0.80    2.0   19.1ms

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   +1.296835977040                   -1.06    3.38s
  2   -1.488896743330        0.44       -0.66    148ms
  3   -4.380695655419        0.46       -0.41   44.0ms
  4   -5.953162128670        0.20       -0.47   43.6ms
  5   -7.519029961086        0.19       -0.72   79.7ms
  6   -7.929262518392       -0.39       -1.29   32.6ms
  7   -8.218964207670       -0.54       -1.51   32.3ms
  8   -8.302727441045       -1.08       -1.68   32.4ms
  9   -8.357711442759       -1.26       -1.96   32.2ms
 10   -8.376406680480       -1.73       -2.34   32.3ms
 11   -8.388693554386       -1.91       -2.44   53.3ms
 12   -8.396543288987       -2.11       -2.55   32.3ms
 13   -8.398487023965       -2.71       -2.85   32.3ms
 14   -8.399841825416       -2.87       -3.12   32.7ms
 15   -8.400016839890       -3.76       -3.43   32.2ms
 16   -8.400289086349       -3.57       -3.51   38.1ms
 17   -8.400350066091       -4.21       -4.03   32.4ms
 18   -8.400400047888       -4.30       -4.75   32.1ms
 19   -8.400413130359       -4.88       -4.31   32.2ms
 20   -8.400423192917       -5.00       -4.15   32.4ms
 21   -8.400425882509       -5.57       -4.45   36.9ms
 22   -8.400427137003       -5.90       -4.90   32.4ms
 23   -8.400427596439       -6.34       -5.15   32.4ms
 24   -8.400427909618       -6.50       -5.21   32.3ms
 25   -8.400428017652       -6.97       -5.14   36.7ms
 26   -8.400428091836       -7.13       -5.24   32.8ms
 27   -8.400428126433       -7.46       -5.54   32.0ms
 28   -8.400428139212       -7.89       -5.97   32.3ms
 29   -8.400428147924       -8.06       -5.80   32.5ms
 30   -8.400428150045       -8.67       -5.75   36.8ms
 31   -8.400428151443       -8.85       -6.52   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   -8.397828591081                   -0.90    5.2   27.4ms

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   -8.400427986311                   -1.79    11.3s
  2   -8.400428152209       -6.78       -4.04    3.68s
  3   -8.400428152209      -14.45       -7.85   75.1ms

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|  = 4.1269006210954543e-7
|ρ_newton - ρ_scfv| = 3.81460313832426e-7
|ρ_newton - ρ_dm|   = 1.47840549987321e-6