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   -8.397836327215                   -0.90    5.0   27.3ms
  2   -8.400254094267       -2.62       -1.74    1.0   19.6ms
  3   -8.400401015308       -3.83       -2.93    1.2   19.7ms
  4   -8.400427776864       -4.57       -2.92    3.2   25.4ms
  5   -8.400427864782       -7.06       -2.95    1.0   31.6ms
  6   -8.400428144530       -6.55       -4.71    1.0   19.7ms
  7   -8.400428151598       -8.15       -4.35    3.2   25.6ms
  8   -8.400428152189       -9.23       -5.60    1.0   20.1ms
  9   -8.400428152209      -10.71       -5.94    2.0   23.0ms
 10   -8.400428152209      -13.08       -6.12    1.0   20.2ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397851979858                   -0.90           5.2   28.3ms
  2   -8.400384309608       -2.60       -1.79   0.80    2.0   20.1ms
  3   -8.400422296693       -4.42       -3.02   0.80    1.0   17.1ms
  4   -8.400428104806       -5.24       -3.40   0.80    2.5   21.1ms
  5   -8.400428148532       -7.36       -4.54   0.80    1.0   16.8ms
  6   -8.400428152180       -8.44       -5.93   0.80    2.5   21.1ms
  7   -8.400428152209      -10.54       -5.98   0.80    3.2   32.1ms
  8   -8.400428152209      -12.98       -7.20   0.80    1.0   17.6ms

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   +0.627633972220                   -1.13   58.6ms
  2   -1.609282063153        0.35       -0.67   33.7ms
  3   -4.838616526367        0.51       -0.40   44.6ms
  4   -6.469094884107        0.21       -0.52   52.7ms
  5   -7.745535535309        0.11       -0.84   44.0ms
  6   -8.135542340726       -0.41       -1.24   32.6ms
  7   -8.254683404827       -0.92       -1.72   32.7ms
  8   -8.333650530402       -1.10       -1.84   33.3ms
  9   -8.356100846995       -1.65       -2.28   40.6ms
 10   -8.378145442172       -1.66       -2.51   32.8ms
 11   -8.389295518810       -1.95       -2.16   33.3ms
 12   -8.396159524440       -2.16       -2.43   34.2ms
 13   -8.398556642290       -2.62       -2.74   34.8ms
 14   -8.399632648904       -2.97       -3.07   40.4ms
 15   -8.400041729243       -3.39       -3.01   33.1ms
 16   -8.400266951883       -3.65       -3.46   32.6ms
 17   -8.400354827046       -4.06       -3.43   32.9ms
 18   -8.400394549276       -4.40       -3.65   32.8ms
 19   -8.400413498843       -4.72       -3.87   33.1ms
 20   -8.400421341827       -5.11       -3.90   40.5ms
 21   -8.400425595635       -5.37       -4.22   33.4ms
 22   -8.400426783197       -5.93       -4.41   33.0ms
 23   -8.400427483135       -6.15       -5.26   32.8ms
 24   -8.400427820929       -6.47       -4.61   32.9ms
 25   -8.400428005442       -6.73       -5.16   32.5ms
 26   -8.400428086362       -7.09       -4.93   40.9ms
 27   -8.400428124561       -7.42       -5.62   32.8ms
 28   -8.400428134936       -7.98       -5.32   32.9ms
 29   -8.400428145992       -7.96       -5.71   32.7ms
 30   -8.400428148720       -8.56       -5.69   33.4ms
 31   -8.400428150713       -8.70       -6.05   40.8ms

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   -8.397873013063                   -0.90    5.0   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   -8.400427981527                   -1.78    607ms
  2   -8.400428152209       -6.77       -4.03    414ms
  3   -8.400428152209      -14.45       -7.83    107ms

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|  = 6.500430119291558e-7
|ρ_newton - ρ_scfv| = 5.81697582032615e-8
|ρ_newton - ρ_dm|   = 8.70689564998361e-7