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   -8.397866551447                   -0.90    5.2   27.9ms
  2   -8.400234848488       -2.63       -1.74    1.0   91.9ms
  3   -8.400406766425       -3.76       -2.99    1.5   20.6ms
  4   -8.400427882565       -4.68       -2.99    3.2   25.5ms
  5   -8.400427959397       -7.11       -3.06    1.0   19.1ms
  6   -8.400428146929       -6.73       -4.67    1.0   19.1ms
  7   -8.400428151847       -8.31       -4.49    2.5   23.2ms
  8   -8.400428152194       -9.46       -5.40    1.0   20.0ms
  9   -8.400428152208      -10.86       -6.33    1.5   20.8ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime 
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397819612817                   -0.90           5.0    1.75s
  2   -8.400386966863       -2.59       -1.79   0.80    2.2    543ms
  3   -8.400423446565       -4.44       -3.00   0.80    1.0    239ms
  4   -8.400428106556       -5.33       -3.47   0.80    2.2   21.3ms
  5   -8.400428149157       -7.37       -4.67   0.80    1.2   18.0ms
  6   -8.400428152186       -8.52       -5.76   0.80    2.8   22.3ms
  7   -8.400428152209      -10.64       -6.08   0.80    2.5    115ms

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   +1.097597716533                   -1.06    3.42s
  2   -1.707213938995        0.45       -0.67    151ms
  3   -4.420403483846        0.43       -0.41   46.2ms
  4   -5.998866724531        0.20       -0.48    117ms
  5   -7.449420901549        0.16       -0.69   46.1ms
  6   -7.813226714640       -0.44       -1.32   33.9ms
  7   -8.046532874119       -0.63       -1.62   33.9ms
  8   -8.167489053390       -0.92       -1.99   33.8ms
  9   -8.268963574219       -0.99       -2.00   67.1ms
 10   -8.299199620088       -1.52       -2.28   33.9ms
 11   -8.320982476191       -1.66       -2.12   34.0ms
 12   -8.328827142036       -2.11       -2.32   33.7ms
 13   -8.338142183589       -2.03       -2.66   33.7ms
 14   -8.346394871575       -2.08       -2.54   33.8ms
 15   -8.358809908561       -1.91       -2.09   33.8ms
 16   -8.369575955230       -1.97       -2.24   42.5ms
 17   -8.387646271244       -1.74       -2.28   33.7ms
 18   -8.393603751344       -2.22       -2.39   33.7ms
 19   -8.398619555914       -2.30       -2.78   33.5ms
 20   -8.399334823518       -3.15       -3.10   40.0ms
 21   -8.400128847045       -3.10       -3.19   33.8ms
 22   -8.400230110784       -3.99       -3.48   33.9ms
 23   -8.400362972382       -3.88       -3.85   33.7ms
 24   -8.400383886788       -4.68       -3.78   33.6ms
 25   -8.400410173516       -4.58       -4.09   40.2ms
 26   -8.400417449503       -5.14       -3.91   33.9ms
 27   -8.400424675669       -5.14       -4.12   33.9ms
 28   -8.400426027067       -5.87       -4.45   34.0ms
 29   -8.400427562592       -5.81       -4.56   33.7ms
 30   -8.400427715703       -6.81       -4.92   40.1ms
 31   -8.400428001792       -6.54       -5.13   33.8ms
 32   -8.400428057136       -7.26       -5.27   34.0ms
 33   -8.400428115009       -7.24       -5.48   34.3ms
 34   -8.400428132924       -7.75       -5.80   33.7ms
 35   -8.400428146534       -7.87       -6.15   40.3ms

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   -8.397836963643                   -0.90    5.2   28.3ms

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   -8.400427987240                   -1.79    11.7s
  2   -8.400428152209       -6.78       -4.03    3.82s
  3   -8.400428152209      -14.75       -7.85   97.5ms

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|  = 8.858849988741973e-7
|ρ_newton - ρ_scfv| = 1.187990723241967e-7
|ρ_newton - ρ_dm|   = 4.662489102106395e-6