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   -8.397803491316                   -0.90    5.0   25.4ms
  2   -8.400247842464       -2.61       -1.74    1.0   25.0ms
  3   -8.400400158381       -3.82       -2.93    1.2   19.5ms
  4   -8.400427820361       -4.56       -2.93    3.5   24.3ms
  5   -8.400427843918       -7.63       -2.93    1.0   18.9ms
  6   -8.400428143517       -6.52       -4.85    1.0   23.6ms
  7   -8.400428151659       -8.09       -4.35    3.2   25.0ms
  8   -8.400428152188       -9.28       -5.32    1.0   19.1ms
  9   -8.400428152206      -10.73       -6.24    1.2   19.6ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397783895886                   -0.90           4.8   25.6ms
  2   -8.400375079640       -2.59       -1.77   0.80    2.0   18.7ms
  3   -8.400422579348       -4.32       -2.98   0.80    1.0   16.2ms
  4   -8.400428097197       -5.26       -3.47   0.80    2.2   19.6ms
  5   -8.400428146339       -7.31       -5.07   0.80    1.2   21.6ms
  6   -8.400428152201       -8.23       -5.11   0.80    3.5   23.0ms
  7   -8.400428152208      -11.10       -6.09   0.80    1.0   16.6ms

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/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   +0.770751616314                   -1.08   57.8ms
  2   -1.974847677590        0.44       -0.68   37.3ms
  3   -4.549990948652        0.41       -0.41   45.9ms
  4   -6.296444665171        0.24       -0.52   43.8ms
  5   -7.678488592807        0.14       -0.82   47.6ms
  6   -8.120425408081       -0.35       -1.23   32.4ms
  7   -8.301331615857       -0.74       -1.64   32.3ms
  8   -8.348200075642       -1.33       -1.99   36.7ms
  9   -8.379034791541       -1.51       -2.28   32.8ms
 10   -8.387102024781       -2.09       -2.30   32.8ms
 11   -8.394604125935       -2.12       -2.60   32.6ms
 12   -8.396683147532       -2.68       -3.02   36.0ms
 13   -8.398741335371       -2.69       -3.25   32.4ms
 14   -8.399477915917       -3.13       -3.11   32.3ms
 15   -8.400015511962       -3.27       -3.33   32.3ms
 16   -8.400231743397       -3.67       -3.72   35.6ms
 17   -8.400352002041       -3.92       -3.86   36.4ms
 18   -8.400398112254       -4.34       -4.24   33.1ms
 19   -8.400413354026       -4.82       -4.46   32.4ms
 20   -8.400421459434       -5.09       -4.35   35.8ms
 21   -8.400425237909       -5.42       -4.87   32.4ms
 22   -8.400426784577       -5.81       -4.78   32.4ms
 23   -8.400427509456       -6.14       -5.03   32.7ms
 24   -8.400427915859       -6.39       -5.37   36.0ms
 25   -8.400428042437       -6.90       -5.06   32.5ms
 26   -8.400428113421       -7.15       -5.25   32.3ms
 27   -8.400428125904       -7.90       -5.91   32.4ms
 28   -8.400428144267       -7.74       -5.66   35.7ms
 29   -8.400428147806       -8.45       -6.07   32.2ms

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   -8.397814918866                   -0.90    5.2   26.0ms

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   -8.400427982521                   -1.78    601ms
  2   -8.400428152209       -6.77       -4.03    380ms
  3   -8.400428152209      -14.45       -7.84    102ms

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|  = 2.241678877515628e-6
|ρ_newton - ρ_scfv| = 3.821740981839904e-7
|ρ_newton - ρ_dm|   = 2.1309095011256974e-6