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   -8.397869408670                   -0.90    5.5   26.7ms
  2   -8.400252184274       -2.62       -1.74    1.0   19.0ms
  3   -8.400407569531       -3.81       -2.97    1.5   26.6ms
  4   -8.400427888941       -4.69       -3.00    3.0   24.1ms
  5   -8.400427947965       -7.23       -3.04    1.0   19.3ms
  6   -8.400428145803       -6.70       -4.60    1.0   19.4ms
  7   -8.400428151794       -8.22       -4.46    2.5   28.5ms
  8   -8.400428152181       -9.41       -5.09    1.0   19.6ms
  9   -8.400428152207      -10.57       -6.29    1.0   19.8ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397846568465                   -0.90           5.0   26.2ms
  2   -8.400382305693       -2.60       -1.79   0.80    2.2   19.5ms
  3   -8.400422496929       -4.40       -3.01   0.80    1.0   16.4ms
  4   -8.400428101172       -5.25       -3.41   0.80    2.5   20.3ms
  5   -8.400428147512       -7.33       -4.67   0.80    1.0   21.2ms
  6   -8.400428152187       -8.33       -5.34   0.80    2.8   21.2ms
  7   -8.400428152208      -10.68       -6.84   0.80    2.0   18.9ms

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   +0.500756078601                   -1.08   61.9ms
  2   -1.802990078003        0.36       -0.64   32.9ms
  3   -4.704024560266        0.46       -0.33   44.1ms
  4   -6.194045821553        0.17       -0.44   47.8ms
  5   -7.682048059060        0.17       -0.66   44.4ms
  6   -7.975201065137       -0.53       -1.37   32.6ms
  7   -8.226856529765       -0.60       -1.64   37.5ms
  8   -8.295949022353       -1.16       -1.97   32.9ms
  9   -8.345950122910       -1.30       -2.15   32.6ms
 10   -8.372843766151       -1.57       -2.41   33.2ms
 11   -8.388384810173       -1.81       -2.64   36.8ms
 12   -8.393329010000       -2.31       -2.48   33.0ms
 13   -8.396662502572       -2.48       -2.84   32.7ms
 14   -8.398049741498       -2.86       -3.01   32.7ms
 15   -8.399148424016       -2.96       -3.06   37.3ms
 16   -8.399789308482       -3.19       -2.91   32.9ms
 17   -8.400132025574       -3.47       -3.38   33.3ms
 18   -8.400305685271       -3.76       -3.47   37.3ms
 19   -8.400381097777       -4.12       -3.67   32.9ms
 20   -8.400410901053       -4.53       -3.83   33.0ms
 21   -8.400418343507       -5.13       -4.21   33.8ms
 22   -8.400424156710       -5.24       -4.27   36.4ms
 23   -8.400426365891       -5.66       -4.55   32.8ms
 24   -8.400427208784       -6.07       -4.85   33.0ms
 25   -8.400427691450       -6.32       -5.02   33.1ms
 26   -8.400427933554       -6.62       -4.99   36.6ms
 27   -8.400428061100       -6.89       -5.18   32.6ms
 28   -8.400428112637       -7.29       -5.45   32.9ms
 29   -8.400428133615       -7.68       -5.59   37.3ms
 30   -8.400428143658       -8.00       -5.93   32.3ms
 31   -8.400428148464       -8.32       -5.82   32.1ms
 32   -8.400428150896       -8.61       -6.33   33.2ms

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   -8.397845481102                   -0.90    5.0   25.5ms

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   -8.400427981114                   -1.79    575ms
  2   -8.400428152209       -6.77       -4.03    389ms
  3   -8.400428152209      -14.45       -7.83    119ms

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|  = 4.4857527455555035e-7
|ρ_newton - ρ_scfv| = 3.489946670649073e-7
|ρ_newton - ρ_dm|   = 3.290745223565668e-6