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   -8.397787161399                   -0.90    5.0   32.1ms
  2   -8.400241368650       -2.61       -1.74    1.0   22.8ms
  3   -8.400400355915       -3.80       -2.92    1.5   24.6ms
  4   -8.400427743621       -4.56       -2.91    3.0   28.9ms
  5   -8.400427869051       -6.90       -2.97    1.0   22.8ms
  6   -8.400428143667       -6.56       -4.76    1.0   24.0ms
  7   -8.400428151617       -8.10       -4.34    3.5   69.5ms
  8   -8.400428152181       -9.25       -5.21    1.0   24.4ms
  9   -8.400428152205      -10.62       -6.06    1.0   22.6ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime 
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397861436397                   -0.90           5.2    1.39s
  2   -8.400384526419       -2.60       -1.79   0.80    2.0    511ms
  3   -8.400423037896       -4.41       -3.02   0.80    1.0    321ms
  4   -8.400428106904       -5.30       -3.45   0.80    2.5   24.9ms
  5   -8.400428148576       -7.38       -4.61   0.80    1.2   21.8ms
  6   -8.400428152184       -8.44       -6.02   0.80    2.8   26.7ms

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   +0.937800392760                   -1.04    2.73s
  2   -2.055427954915        0.48       -0.69    228ms
  3   -4.666134892812        0.42       -0.40   49.6ms
  4   -6.299557764255        0.21       -0.47   48.4ms
  5   -7.681961336102        0.14       -0.72   48.5ms
  6   -8.053222586822       -0.43       -1.25   38.1ms
  7   -8.226020547837       -0.76       -1.63   36.7ms
  8   -8.276482700629       -1.30       -2.00   72.8ms
  9   -8.305318472471       -1.54       -2.37   37.1ms
 10   -8.326123774754       -1.68       -2.05   35.7ms
 11   -8.346879701161       -1.68       -2.26   37.8ms
 12   -8.366848794681       -1.70       -2.31   35.3ms
 13   -8.383620986798       -1.78       -2.56   37.4ms
 14   -8.393623711048       -2.00       -2.51   44.5ms
 15   -8.398027377503       -2.36       -2.80   36.4ms
 16   -8.399197497578       -2.93       -2.94   37.4ms
 17   -8.399958811797       -3.12       -3.12   35.4ms
 18   -8.400243089919       -3.55       -3.89   37.2ms
 19   -8.400349358486       -3.97       -3.81   42.5ms
 20   -8.400406242717       -4.25       -4.16   37.7ms
 21   -8.400415186693       -5.05       -3.95   39.8ms
 22   -8.400423374588       -5.09       -4.89   36.1ms
 23   -8.400425736259       -5.63       -4.21   42.1ms
 24   -8.400426994418       -5.90       -4.60   36.4ms
 25   -8.400427763343       -6.11       -4.79   35.6ms
 26   -8.400428010610       -6.61       -4.98   36.1ms
 27   -8.400428088930       -7.11       -5.00   36.6ms
 28   -8.400428129324       -7.39       -5.31   43.1ms
 29   -8.400428143652       -7.84       -5.21   36.5ms
 30   -8.400428147461       -8.42       -5.58   36.1ms
 31   -8.400428150595       -8.50       -5.86   35.7ms
 32   -8.400428151487       -9.05       -5.78   36.5ms
 33   -8.400428151867       -9.42       -6.46   42.9ms

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   -8.397883873880                   -0.90    5.2   33.7ms

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   -8.400427988460                   -1.79    10.4s
  2   -8.400428152209       -6.79       -4.04    3.39s
  3   -8.400428152209      -14.75       -7.85   98.1ms

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|  = 2.5210269646082138e-6
|ρ_newton - ρ_scfv| = 4.922273956104773e-6
|ρ_newton - ρ_dm|   = 1.931993493322587e-6