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   -8.397608061713                   -0.90    5.2   27.4ms
  2   -8.400208110714       -2.59       -1.73    1.0   18.7ms
  3   -8.400398251586       -3.72       -3.01    1.5   19.6ms
  4   -8.400427756905       -4.53       -2.93    3.2   85.1ms
  5   -8.400428028286       -6.57       -3.20    1.0   19.1ms
  6   -8.400428149332       -6.92       -5.01    1.0   18.9ms
  7   -8.400428151976       -8.58       -4.58    3.2   25.0ms
  8   -8.400428152181       -9.69       -5.01    1.8   21.3ms
  9   -8.400428152208      -10.57       -6.25    1.0   19.1ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime 
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397512401267                   -0.90           5.2    1.74s
  2   -8.400386677427       -2.54       -1.78   0.80    2.2    683ms
  3   -8.400424136693       -4.43       -3.01   0.80    1.0    243ms
  4   -8.400428113725       -5.40       -3.57   0.80    2.2   20.8ms
  5   -8.400428148973       -7.45       -4.69   0.80    1.5   17.7ms
  6   -8.400428152185       -8.49       -5.27   0.80    2.5   21.0ms
  7   -8.400428152207      -10.64       -6.12   0.80    1.2   17.4ms

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   +1.104023082474                   -1.05    3.48s
  2   -1.137950282150        0.35       -0.66    152ms
  3   -4.212266229424        0.49       -0.36   44.5ms
  4   -5.440145125008        0.09       -0.47   44.6ms
  5   -7.312688061762        0.27       -0.53   88.3ms
  6   -7.920009719918       -0.22       -1.15   44.7ms
  7   -8.110428839432       -0.72       -1.31   32.9ms
  8   -8.260577643586       -0.82       -1.76   32.6ms
  9   -8.331566223218       -1.15       -1.88   32.7ms
 10   -8.368706450728       -1.43       -2.03   57.1ms
 11   -8.387136499057       -1.73       -2.32   32.9ms
 12   -8.393037978036       -2.23       -2.44   32.8ms
 13   -8.397591217131       -2.34       -2.76   32.8ms
 14   -8.399081512512       -2.83       -2.97   32.7ms
 15   -8.399879670975       -3.10       -3.14   39.4ms
 16   -8.400157667920       -3.56       -3.52   32.6ms
 17   -8.400318081015       -3.79       -3.70   32.7ms
 18   -8.400372177441       -4.27       -4.02   32.7ms
 19   -8.400407096196       -4.46       -3.95   32.7ms
 20   -8.400417623987       -4.98       -4.28   38.5ms
 21   -8.400423779408       -5.21       -4.16   33.0ms
 22   -8.400425908056       -5.67       -4.62   32.9ms
 23   -8.400427381343       -5.83       -4.47   32.6ms
 24   -8.400427857298       -6.32       -5.27   32.6ms
 25   -8.400428020962       -6.79       -4.97   38.9ms
 26   -8.400428095637       -7.13       -5.64   32.7ms
 27   -8.400428123764       -7.55       -5.20   33.0ms
 28   -8.400428139845       -7.79       -5.55   33.5ms
 29   -8.400428146165       -8.20       -5.82   33.0ms
 30   -8.400428149850       -8.43       -5.89   39.3ms
 31   -8.400428150940       -8.96       -5.96   32.7ms
 32   -8.400428151738       -9.10       -6.19   33.1ms

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   -8.397805832321                   -0.90    4.8   30.3ms

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   -8.400427943651                   -1.77    12.6s
  2   -8.400428152209       -6.68       -3.98    3.65s
  3   -8.400428152209      -14.75       -7.73   86.9ms

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|  = 6.74943963455187e-7
|ρ_newton - ρ_scfv| = 6.117630487432075e-7
|ρ_newton - ρ_dm|   = 1.5887233912905272e-6