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   -8.397843450839                   -0.90    5.2   27.3ms
  2   -8.400236257349       -2.62       -1.74    1.0   78.8ms
  3   -8.400406924908       -3.77       -2.97    1.5   20.8ms
  4   -8.400427865047       -4.68       -3.01    3.2   25.4ms
  5   -8.400427969456       -6.98       -3.08    1.0   19.9ms
  6   -8.400428146645       -6.75       -4.61    1.0   19.9ms
  7   -8.400428151839       -8.28       -4.50    2.2   23.6ms
  8   -8.400428152173       -9.48       -5.05    1.0   19.9ms
  9   -8.400428152206      -10.48       -6.13    1.0   49.5ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime 
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397855275854                   -0.90           5.2    1.69s
  2   -8.400386008693       -2.60       -1.79   0.80    2.0    482ms
  3   -8.400423470890       -4.43       -2.98   0.80    1.0    216ms
  4   -8.400428092439       -5.34       -3.43   0.80    2.2   96.9ms
  5   -8.400428149826       -7.24       -4.53   0.80    1.2   18.1ms
  6   -8.400428152183       -8.63       -5.85   0.80    2.2   20.2ms
  7   -8.400428152209      -10.59       -6.11   0.80    3.2   22.6ms

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   +0.921715324439                   -1.05    3.33s
  2   -1.016226088281        0.29       -0.66    204ms
  3   -4.311415988979        0.52       -0.34   43.5ms
  4   -5.454148747734        0.06       -0.46   43.1ms
  5   -7.454053876390        0.30       -0.56   43.2ms
  6   -7.599500262128       -0.84       -1.36   32.1ms
  7   -8.142760096497       -0.26       -1.56   58.5ms
  8   -8.242616770017       -1.00       -1.84   32.1ms
  9   -8.307469131809       -1.19       -2.20   31.9ms
 10   -8.341032440902       -1.47       -2.05   32.1ms
 11   -8.365908391247       -1.60       -2.26   31.9ms
 12   -8.382859004430       -1.77       -2.18   31.8ms
 13   -8.392276611469       -2.03       -2.40   37.7ms
 14   -8.397821152092       -2.26       -2.60   32.0ms
 15   -8.399240425245       -2.85       -2.85   31.9ms
 16   -8.399900520381       -3.18       -3.47   31.9ms
 17   -8.400189380011       -3.54       -3.30   31.9ms
 18   -8.400339579900       -3.82       -3.62   36.4ms
 19   -8.400394007897       -4.26       -3.56   31.9ms
 20   -8.400414886268       -4.68       -3.75   32.0ms
 21   -8.400423366806       -5.07       -4.06   32.8ms
 22   -8.400425658970       -5.64       -4.33   36.5ms
 23   -8.400427232825       -5.80       -4.50   32.0ms
 24   -8.400427675101       -6.35       -4.77   32.4ms
 25   -8.400427995260       -6.49       -5.22   32.0ms
 26   -8.400428080274       -7.07       -5.11   36.2ms
 27   -8.400428116392       -7.44       -5.47   31.8ms
 28   -8.400428134236       -7.75       -5.26   31.7ms
 29   -8.400428143617       -8.03       -5.67   32.0ms
 30   -8.400428149178       -8.25       -5.76   32.0ms
 31   -8.400428150962       -8.75       -6.10   36.7ms

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   -8.397887522445                   -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   -8.400427987828                   -1.78    7.85s
  2   -8.400428152209       -6.78       -4.04    3.07s
  3   -8.400428152209      -14.45       -7.85   83.6ms

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|  = 1.8949611561832562e-6
|ρ_newton - ρ_scfv| = 1.0492500454414279e-7
|ρ_newton - ρ_dm|   = 1.7146795383827009e-6