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   -8.397617985862                   -0.90    5.2   24.4ms
  2   -8.400224584391       -2.58       -1.72    1.0   18.2ms
  3   -8.400395978336       -3.77       -2.97    1.2   18.5ms
  4   -8.400427706393       -4.50       -2.91    3.2   22.7ms
  5   -8.400428064297       -6.45       -3.28    1.0   18.1ms
  6   -8.400428149447       -7.07       -4.76    1.0   18.1ms
  7   -8.400428151931       -8.60       -4.58    2.2    206ms
  8   -8.400428152168       -9.63       -5.00    1.0   19.0ms
  9   -8.400428152207      -10.41       -6.28    1.0   18.7ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime 
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397614301244                   -0.90           5.0    1.56s
  2   -8.400388986182       -2.56       -1.78   0.80    2.2    1.06s
  3   -8.400424306799       -4.45       -2.95   0.80    1.0    200ms
  4   -8.400428114256       -5.42       -3.51   0.80    2.2   19.0ms
  5   -8.400428149127       -7.46       -4.83   0.80    1.0   16.4ms
  6   -8.400428152198       -8.51       -5.39   0.80    3.0    1.29s
  7   -8.400428152208      -10.99       -6.25   0.80    1.5   18.5ms

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   +1.292024224759                   -1.02    2.87s
  2   -2.011443612502        0.52       -0.64    148ms
  3   -4.330354200346        0.37       -0.39   47.2ms
  4   -6.006496484021        0.22       -0.44   47.1ms
  5   -7.568458307716        0.19       -0.68   47.0ms
  6   -7.805069211644       -0.63       -1.39   34.8ms
  7   -8.162258318299       -0.45       -1.60   34.9ms
  8   -8.232626233547       -1.15       -1.92   34.9ms
  9   -8.295549657961       -1.20       -1.83   34.8ms
 10   -8.297342190346       -2.75       -1.81   34.9ms
 11   -8.357222412224       -1.22       -1.72   48.9ms
 12   -8.376654866557       -1.71       -1.66   47.1ms
 13   -8.385057117082       -2.08       -2.82   34.9ms
 14   -8.394060982148       -2.05       -3.17   34.8ms
 15   -8.396851526467       -2.55       -2.75   35.3ms
 16   -8.397829842387       -3.01       -2.88   34.8ms
 17   -8.399178718736       -2.87       -3.38   34.8ms
 18   -8.399723670784       -3.26       -3.03   34.8ms
 19   -8.400150678565       -3.37       -3.42    125ms
 20   -8.400265859392       -3.94       -3.59   35.5ms
 21   -8.400384605939       -3.93       -3.79   35.1ms
 22   -8.400407181614       -4.65       -4.16   35.0ms
 23   -8.400421080241       -4.86       -4.26   35.1ms
 24   -8.400425627337       -5.34       -4.60   34.8ms
 25   -8.400427178062       -5.81       -4.72   34.8ms
 26   -8.400427660574       -6.32       -5.30   34.9ms
 27   -8.400427948216       -6.54       -5.14   34.9ms
 28   -8.400428047506       -7.00       -5.37   34.9ms
 29   -8.400428112826       -7.18       -5.46   34.9ms
 30   -8.400428136098       -7.63       -5.78   34.7ms
 31   -8.400428146529       -7.98       -5.62   34.7ms
 32   -8.400428149767       -8.49       -5.98   35.0ms
 33   -8.400428151144       -8.86       -6.63   34.8ms

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   -8.397630558151                   -0.90    5.0   24.0ms

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   -8.400427982787                   -1.79    8.54s
  2   -8.400428152209       -6.77       -4.03    3.29s
  3   -8.400428152209      -14.45       -7.85   71.7ms

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|  = 8.072359069419976e-7
|ρ_newton - ρ_scfv| = 4.4899376058431184e-7
|ρ_newton - ρ_dm|   = 1.1385530124080065e-6