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   -8.397797399189                   -0.90    4.8   31.7ms
  2   -8.400226111919       -2.61       -1.74    1.0   18.7ms
  3   -8.400397474637       -3.77       -2.94    1.5   19.5ms
  4   -8.400427739110       -4.52       -2.89    3.2   24.2ms
  5   -8.400427751393       -7.91       -2.88    1.0   23.6ms
  6   -8.400428140520       -6.41       -4.96    1.0   18.9ms
  7   -8.400428151354       -7.97       -4.25    3.5   25.8ms
  8   -8.400428152202       -9.07       -5.56    2.0   26.1ms
  9   -8.400428152208      -11.22       -5.74    1.5   20.6ms
 10   -8.400428152209      -12.06       -6.24    1.0   19.9ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397786612086                   -0.90           5.2   30.9ms
  2   -8.400383130068       -2.59       -1.78   0.80    2.2   19.5ms
  3   -8.400422583889       -4.40       -3.00   0.80    1.0   16.1ms
  4   -8.400428099923       -5.26       -3.42   0.80    2.5   19.8ms
  5   -8.400428148628       -7.31       -4.56   0.80    1.2   16.8ms
  6   -8.400428152188       -8.45       -5.67   0.80    2.8   25.1ms
  7   -8.400428152209      -10.67       -6.14   0.80    2.5   20.7ms

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/7krni/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/7krni/src/types.jl:120
n     Energy            log10(ΔE)   log10(Δρ)   Δtime
---   ---------------   ---------   ---------   ------
  1   +1.152313660900                   -1.11   58.3ms
  2   -1.538376776795        0.43       -0.65   37.1ms
  3   -4.229463550170        0.43       -0.39   44.0ms
  4   -5.842754278622        0.21       -0.47   43.7ms
  5   -7.405089622628        0.19       -0.72   47.3ms
  6   -7.773866451747       -0.43       -1.47   32.5ms
  7   -8.034363701784       -0.58       -1.75   32.6ms
  8   -8.128249719866       -1.03       -1.87   32.4ms
  9   -8.226399601375       -1.01       -1.99   36.0ms
 10   -8.280857042385       -1.26       -1.62   32.4ms
 11   -8.339071407013       -1.23       -1.98   32.4ms
 12   -8.375261719150       -1.44       -2.05   36.0ms
 13   -8.388546571953       -1.88       -2.37   32.5ms
 14   -8.395153561923       -2.18       -2.68   32.4ms
 15   -8.397704624541       -2.59       -2.70   32.4ms
 16   -8.399107922830       -2.85       -3.22   36.0ms
 17   -8.399875283332       -3.12       -3.37   32.6ms
 18   -8.400172688652       -3.53       -3.40   32.6ms
 19   -8.400311637132       -3.86       -3.65   32.5ms
 20   -8.400381514244       -4.16       -3.92   35.8ms
 21   -8.400407027710       -4.59       -4.00   32.5ms
 22   -8.400419155366       -4.92       -4.50   32.4ms
 23   -8.400424555019       -5.27       -4.21   35.7ms
 24   -8.400426892911       -5.63       -4.39   32.7ms
 25   -8.400427708263       -6.09       -4.68   32.5ms
 26   -8.400427984959       -6.56       -4.92   32.4ms
 27   -8.400428096255       -6.95       -5.10   35.7ms
 28   -8.400428126402       -7.52       -5.15   32.4ms
 29   -8.400428142748       -7.79       -5.85   32.4ms
 30   -8.400428147664       -8.31       -5.81   32.6ms
 31   -8.400428150315       -8.58       -6.57   35.9ms

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   -8.397818424611                   -0.90    5.2   25.9ms

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   -8.400427976288                   -1.79    590ms
  2   -8.400428152209       -6.75       -4.03    379ms
  3   -8.400428152209      -14.45       -7.81    118ms

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|  = 5.446089927868523e-7
|ρ_newton - ρ_scfv| = 2.736491459183635e-7
|ρ_newton - ρ_dm|   = 2.6106906301697153e-6