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   -8.397880059610                   -0.90    5.2   24.5ms
  2   -8.400244575049       -2.63       -1.74    1.0   17.6ms
  3   -8.400407389741       -3.79       -2.97    1.5   28.2ms
  4   -8.400427865074       -4.69       -3.00    2.8   22.3ms
  5   -8.400427969566       -6.98       -3.08    1.0   17.7ms
  6   -8.400428149762       -6.74       -4.40    1.0   17.7ms
  7   -8.400428155841       -8.22       -4.47    2.0   20.4ms
  8   -8.400428156236       -9.40       -5.02    1.0   18.0ms
  9   -8.400428156274      -10.42       -6.26    1.5   18.8ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397840930287                   -0.90           5.2   24.8ms
  2   -8.400386054131       -2.59       -1.79   0.80    2.0   17.8ms
  3   -8.400423050608       -4.43       -3.03   0.80    1.0   15.1ms
  4   -8.400428111756       -5.30       -3.42   0.80    2.8   19.3ms
  5   -8.400428154618       -7.37       -4.49   0.80    1.5   23.3ms
  6   -8.400428156245       -8.79       -5.65   0.80    2.0   17.8ms
  7   -8.400428156276      -10.50       -6.42   0.80    2.8   19.9ms

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/8dE7C/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/8dE7C/src/types.jl:120
n     Energy            log10(ΔE)   log10(Δρ)   Δtime
---   ---------------   ---------   ---------   ------
  1   +0.865547824673                   -1.01   50.7ms
  2   -1.288313265980        0.33       -0.66   28.7ms
  3   -4.097123720406        0.45       -0.34   38.1ms
  4   -5.360682000074        0.10       -0.42   37.9ms
  5   -7.367113135079        0.30       -0.53   44.3ms
  6   -7.962102776699       -0.23       -1.11   38.9ms
  7   -8.145764786062       -0.74       -1.27   28.4ms
  8   -8.264683713413       -0.92       -1.68   28.6ms
  9   -8.314504990127       -1.30       -1.84   28.3ms
 10   -8.344764904650       -1.52       -1.99   33.2ms
 11   -8.367016866406       -1.65       -2.09   28.9ms
 12   -8.382028788590       -1.82       -2.15   28.6ms
 13   -8.392178482550       -1.99       -2.41   28.7ms
 14   -8.396965448672       -2.32       -2.73   28.6ms
 15   -8.398962434081       -2.70       -2.72   28.5ms
 16   -8.399723479189       -3.12       -3.06   33.0ms
 17   -8.400137936897       -3.38       -3.20   28.1ms
 18   -8.400311930274       -3.76       -3.80   28.1ms
 19   -8.400387458276       -4.12       -3.70   28.3ms
 20   -8.400408624430       -4.67       -4.26   33.3ms
 21   -8.400421025257       -4.91       -3.93   28.5ms
 22   -8.400425118159       -5.39       -4.43   28.8ms
 23   -8.400427447175       -5.63       -4.34   32.3ms
 24   -8.400427802121       -6.45       -4.69   28.9ms
 25   -8.400428031797       -6.64       -5.37   29.0ms
 26   -8.400428087830       -7.25       -5.02   28.6ms
 27   -8.400428127487       -7.40       -5.36   32.9ms
 28   -8.400428140728       -7.88       -5.50   28.2ms
 29   -8.400428149684       -8.05       -5.56   28.3ms
 30   -8.400428152596       -8.54       -5.82   28.4ms
 31   -8.400428154766       -8.66       -5.97   32.6ms
 32   -8.400428155568       -9.10       -6.32   28.7ms

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   -8.397840924757                   -0.90    5.2   29.3ms

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   -8.400427982103                   -1.78    530ms
  2   -8.400428156277       -6.76       -4.03    355ms
  3   -8.400428156277      -14.45       -7.82    108ms

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|  = 7.264133849910721e-7
|ρ_newton - ρ_scfv| = 4.5754765066279303e-7
|ρ_newton - ρ_dm|   = 1.3915523196257542e-6