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   -8.397835367120                   -0.89    5.0   27.0ms
  2   -8.400261736631       -2.62       -1.73    1.0   88.3ms
  3   -8.400401250669       -3.86       -2.88    1.2   20.2ms
  4   -8.400427754327       -4.58       -2.91    3.2   25.4ms
  5   -8.400427885269       -6.88       -2.97    1.0   19.8ms
  6   -8.400428144088       -6.59       -4.66    1.0   19.9ms
  7   -8.400428151667       -8.12       -4.37    3.0   25.7ms
  8   -8.400428152184       -9.29       -5.32    1.0   51.3ms
  9   -8.400428152205      -10.68       -5.88    1.5   21.5ms
 10   -8.400428152209      -11.45       -5.98    2.0   24.2ms
 11   -8.400428152209      -12.46       -6.85    1.0   20.7ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime 
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397872626871                   -0.90           5.2    1.85s
  2   -8.400384914352       -2.60       -1.78   0.80    2.0    524ms
  3   -8.400422691503       -4.42       -3.00   0.80    1.0    228ms
  4   -8.400428104593       -5.27       -3.41   0.80    2.2   21.0ms
  5   -8.400428148663       -7.36       -4.57   0.80    1.0   17.3ms
  6   -8.400428152179       -8.45       -5.74   0.80    2.8   21.8ms
  7   -8.400428152209      -10.53       -6.12   0.80    2.8   22.3ms

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   +1.018398868414                   -1.05    3.73s
  2   -1.628498611167        0.42       -0.69    161ms
  3   -4.444848762327        0.45       -0.37   45.2ms
  4   -6.117351721311        0.22       -0.50   45.6ms
  5   -7.598689816813        0.17       -0.76   80.7ms
  6   -8.059468101271       -0.34       -1.20   33.4ms
  7   -8.249154659435       -0.72       -1.50   33.3ms
  8   -8.322020221663       -1.14       -1.76   33.6ms
  9   -8.366455367347       -1.35       -2.09   33.5ms
 10   -8.380860911353       -1.84       -2.33   54.9ms
 11   -8.391157430699       -1.99       -2.33   33.4ms
 12   -8.395488231763       -2.36       -2.56   33.4ms
 13   -8.398123700406       -2.58       -3.15   33.8ms
 14   -8.399364218982       -2.91       -3.01   33.5ms
 15   -8.399895445802       -3.27       -3.42   33.5ms
 16   -8.400188276385       -3.53       -3.15   39.5ms
 17   -8.400320927741       -3.88       -3.94   33.7ms
 18   -8.400385963111       -4.19       -3.56   33.4ms
 19   -8.400410183550       -4.62       -3.80   33.2ms
 20   -8.400419558876       -5.03       -4.12   39.2ms
 21   -8.400424550286       -5.30       -4.48   34.2ms
 22   -8.400426073289       -5.82       -4.55   33.5ms
 23   -8.400427410717       -5.87       -4.56   33.3ms
 24   -8.400427751617       -6.47       -5.07   38.9ms
 25   -8.400427987597       -6.63       -5.04   33.6ms
 26   -8.400428068159       -7.09       -5.15   33.4ms
 27   -8.400428116181       -7.32       -5.54   33.2ms
 28   -8.400428136412       -7.69       -5.24   39.0ms
 29   -8.400428144687       -8.08       -5.44   42.0ms
 30   -8.400428148879       -8.38       -5.97   33.8ms
 31   -8.400428150457       -8.80       -5.93   33.3ms
 32   -8.400428151542       -8.96       -6.23   33.4ms

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   -8.397810849394                   -0.90    4.8   27.1ms

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   -8.400427946144                   -1.78    9.84s
  2   -8.400428152209       -6.69       -3.98    2.63s
  3   -8.400428152209      -14.27       -7.74   94.7ms

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|  = 1.8642623093202785e-7
|ρ_newton - ρ_scfv| = 2.3506703116563346e-7
|ρ_newton - ρ_dm|   = 6.269172317753076e-7