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   -8.397870395058                   -0.90    5.2   39.0ms
  2   -8.400248242784       -2.62       -1.74    1.0   19.7ms
  3   -8.400405279453       -3.80       -2.96    1.5   20.8ms
  4   -8.400427836862       -4.65       -2.97    3.0   25.0ms
  5   -8.400427970370       -6.87       -3.08    1.0   19.3ms
  6   -8.400428147250       -6.75       -4.78    1.0   19.6ms
  7   -8.400428151852       -8.34       -4.49    2.5   23.8ms
  8   -8.400428152191       -9.47       -5.27    1.0   30.1ms
  9   -8.400428152208      -10.77       -6.30    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   -8.397804622323                   -0.90           5.0   27.4ms
  2   -8.400383114103       -2.59       -1.79   0.80    2.0   19.7ms
  3   -8.400422382548       -4.41       -3.02   0.80    1.0   16.8ms
  4   -8.400428111447       -5.24       -3.44   0.80    2.5   20.7ms
  5   -8.400428149690       -7.42       -4.60   0.80    1.0   25.9ms
  6   -8.400428152187       -8.60       -5.76   0.80    2.5   21.3ms
  7   -8.400428152209      -10.66       -6.17   0.80    2.8   21.5ms

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   +0.798586406526                   -1.05   58.3ms
  2   -1.567143741012        0.37       -0.63   33.9ms
  3   -4.485441310758        0.47       -0.38   52.9ms
  4   -6.057696227312        0.20       -0.46   44.2ms
  5   -7.577622778738        0.18       -0.74   44.2ms
  6   -7.997350920034       -0.38       -1.42   32.4ms
  7   -8.200447472971       -0.69       -1.64   32.6ms
  8   -8.271715967352       -1.15       -1.92   40.6ms
  9   -8.326518448416       -1.26       -2.10   33.3ms
 10   -8.355414863002       -1.54       -2.08   32.8ms
 11   -8.381111968009       -1.59       -2.21   33.0ms
 12   -8.390387067738       -2.03       -2.26   32.8ms
 13   -8.395406596581       -2.30       -2.79   32.7ms
 14   -8.397799615367       -2.62       -3.28   32.7ms
 15   -8.399243782916       -2.84       -3.46   40.7ms
 16   -8.399832940692       -3.23       -3.51   32.8ms
 17   -8.400146742923       -3.50       -3.75   32.9ms
 18   -8.400281836888       -3.87       -3.72   32.7ms
 19   -8.400366921395       -4.07       -3.75   32.7ms
 20   -8.400402814670       -4.44       -3.73   32.7ms
 21   -8.400416439117       -4.87       -4.72   41.8ms
 22   -8.400424036416       -5.12       -4.07   33.0ms
 23   -8.400426489422       -5.61       -4.36   33.4ms
 24   -8.400427471914       -6.01       -4.93   33.0ms
 25   -8.400427821779       -6.46       -5.32   32.8ms
 26   -8.400428010392       -6.72       -5.28   32.9ms
 27   -8.400428074518       -7.19       -5.23   40.8ms
 28   -8.400428124209       -7.30       -5.32   32.8ms
 29   -8.400428134242       -8.00       -5.85   33.1ms
 30   -8.400428146138       -7.92       -5.53   32.9ms
 31   -8.400428148911       -8.56       -5.86   32.8ms
 32   -8.400428150974       -8.69       -6.24   32.5ms

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   -8.397847327823                   -0.90    5.2   27.3ms

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   -8.400427981782                   -1.79    595ms
  2   -8.400428152209       -6.77       -4.03    390ms
  3   -8.400428152209      -14.45       -7.83    108ms

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|  = 5.277287891702114e-7
|ρ_newton - ρ_scfv| = 2.0495469659677585e-7
|ρ_newton - ρ_dm|   = 1.821428428830081e-6