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.oncvpsp3.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   -8.397836896156                   -0.90    5.2   36.0ms
  2   -8.400252748251       -2.62       -1.74    1.0   24.9ms
  3   -8.400400135389       -3.83       -2.94    1.2   27.2ms
  4   -8.400428595710       -4.55       -2.91    3.2   24.7ms
  5   -8.400428745668       -6.82       -2.98    1.0   18.5ms
  6   -8.400429017443       -6.57       -4.71    1.0   18.6ms
  7   -8.400429023933       -8.19       -4.40    3.0   30.6ms
  8   -8.400429024391       -9.34       -5.11    1.0   19.4ms
  9   -8.400429024418      -10.57       -6.27    1.0   18.9ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397773014239                   -0.90           5.2   26.5ms
  2   -8.400382928736       -2.58       -1.79   0.80    2.2   18.9ms
  3   -8.400423178463       -4.40       -3.01   0.80    1.0   15.4ms
  4   -8.400428967159       -5.24       -3.38   0.80    2.8   20.2ms
  5   -8.400429021308       -7.27       -4.44   0.80    1.2   16.5ms
  6   -8.400429024370       -8.51       -5.58   0.80    2.2   25.2ms
  7   -8.400429024420      -10.31       -6.03   0.80    2.8   20.8ms

Direct minimization

scfres_dm = direct_minimization(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   Δtime
---   ---------------   ---------   ---------   ------
  1   +0.934356000086                   -1.08   52.0ms
  2   -1.829151898816        0.44       -0.65   34.2ms
  3   -3.691492741717        0.27       -0.37   39.2ms
  4   -4.597929108750       -0.04       -0.42   39.1ms
  5   -6.341671948801        0.24       -0.43   43.4ms
  6   -7.252268521930       -0.04       -0.56   38.9ms
  7   -7.995470135500       -0.13       -0.90   44.1ms
  8   -8.162544937083       -0.78       -1.44   29.0ms
  9   -8.228101894488       -1.18       -1.77   28.8ms
 10   -8.274853831208       -1.33       -2.03   28.7ms
 11   -8.300492431911       -1.59       -2.17   33.7ms
 12   -8.326197509491       -1.59       -2.07   29.4ms
 13   -8.353735859756       -1.56       -1.95   29.0ms
 14   -8.378155204297       -1.61       -1.94   33.1ms
 15   -8.390554005365       -1.91       -2.67   29.2ms
 16   -8.396026008886       -2.26       -2.44   28.7ms
 17   -8.398297519344       -2.64       -3.09   28.6ms
 18   -8.399528339438       -2.91       -2.85   34.0ms
 19   -8.400111466114       -3.23       -3.26   29.2ms
 20   -8.400289393872       -3.75       -3.43   33.7ms
 21   -8.400371604633       -4.09       -3.74   29.5ms
 22   -8.400393272690       -4.66       -3.74   33.2ms
 23   -8.400418724903       -4.59       -4.18   28.9ms
 24   -8.400423927409       -5.28       -4.45   28.8ms
 25   -8.400426877138       -5.53       -4.22   28.7ms
 26   -8.400427808527       -6.03       -4.66   33.9ms
 27   -8.400428589852       -6.11       -4.53   28.8ms
 28   -8.400428792591       -6.69       -4.78   28.6ms
 29   -8.400428937052       -6.84       -4.89   33.1ms
 30   -8.400428968561       -7.50       -5.22   29.2ms
 31   -8.400428995853       -7.56       -5.17   29.1ms
 32   -8.400429007153       -7.95       -5.33   29.0ms
 33   -8.400429017351       -7.99       -5.50   33.7ms
 34   -8.400429021261       -8.41       -5.63   29.1ms
 35   -8.400429023092       -8.74       -5.95   28.7ms
 36   -8.400429023858       -9.12       -5.92   28.7ms
 37   -8.400429024187       -9.48       -6.18   33.3ms

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   -8.397850966313                   -0.90    5.2   30.4ms

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   -8.400428849711                   -1.78    529ms
  2   -8.400429024420       -6.76       -4.02    348ms
  3   -8.400429024420      -14.45       -7.82   94.2ms

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|  = 1.3082743108085614e-6
|ρ_newton - ρ_scfv| = 2.5126585093097167e-7
|ρ_newton - ρ_dm|   = 1.507832152650422e-6