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   -8.397811886237                   -0.90    5.5   26.9ms
  2   -8.400253533221       -2.61       -1.74    1.0   20.7ms
  3   -8.400397482785       -3.84       -2.91    1.2   19.6ms
  4   -8.400427779469       -4.52       -2.92    3.0   31.8ms
  5   -8.400427674702   +   -6.98       -2.84    1.0   19.1ms
  6   -8.400428142288       -6.33       -4.31    1.0   19.1ms
  7   -8.400428155176       -7.89       -4.24    2.2   22.5ms
  8   -8.400428156231       -8.98       -5.07    1.0   24.0ms
  9   -8.400428156273      -10.37       -6.24    1.2   19.7ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397799737108                   -0.90           5.2   30.8ms
  2   -8.400385248102       -2.59       -1.79   0.80    2.2   19.4ms
  3   -8.400422774312       -4.43       -3.05   0.80    1.0   16.4ms
  4   -8.400428110320       -5.27       -3.40   0.80    2.5   20.2ms
  5   -8.400428152968       -7.37       -4.45   0.80    1.0   16.5ms
  6   -8.400428156189       -8.49       -5.61   0.80    2.2   19.7ms
  7   -8.400428156277      -10.06       -6.13   0.80    3.2   28.1ms

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   +1.015142122654                   -1.09   58.4ms
  2   -1.496222509640        0.40       -0.69   32.8ms
  3   -4.291963730684        0.45       -0.42   47.9ms
  4   -5.669940207943        0.14       -0.53   44.1ms
  5   -7.338045367684        0.22       -0.68   44.5ms
  6   -7.497436872008       -0.80       -1.41   36.4ms
  7   -8.021504091459       -0.28       -1.55   32.8ms
  8   -8.122686219232       -0.99       -1.79   32.6ms
  9   -8.215991592012       -1.03       -1.68   32.6ms
 10   -8.237914579347       -1.66       -1.89   36.1ms
 11   -8.255931323429       -1.74       -1.83   32.7ms
 12   -8.293781212957       -1.42       -1.64   44.4ms
 13   -8.310198098145       -1.78       -1.63   47.8ms
 14   -8.352927358479       -1.37       -1.85   44.3ms
 15   -8.385001005696       -1.49       -2.22   33.3ms
 16   -8.391529762152       -2.19       -2.30   33.7ms
 17   -8.396870965654       -2.27       -2.91   36.8ms
 18   -8.398407530843       -2.81       -3.08   33.1ms
 19   -8.399541194369       -2.95       -2.99   33.5ms
 20   -8.400091474958       -3.26       -3.37   32.9ms
 21   -8.400248621688       -3.80       -3.47   36.7ms
 22   -8.400358543643       -3.96       -3.74   33.0ms
 23   -8.400395850582       -4.43       -3.60   33.2ms
 24   -8.400417587224       -4.66       -3.92   36.1ms
 25   -8.400423759293       -5.21       -4.16   33.2ms
 26   -8.400426225927       -5.61       -4.31   32.7ms
 27   -8.400427319521       -5.96       -4.62   33.0ms
 28   -8.400427745308       -6.37       -5.02   36.2ms
 29   -8.400427997733       -6.60       -4.83   33.3ms
 30   -8.400428091321       -7.03       -5.27   32.8ms
 31   -8.400428131338       -7.40       -5.16   32.9ms
 32   -8.400428146928       -7.81       -5.76   36.3ms
 33   -8.400428152179       -8.28       -5.51   33.4ms
 34   -8.400428154666       -8.60       -6.50   32.9ms

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   -8.397823981485                   -0.90    5.0   25.6ms

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   -8.400427995172                   -1.79    575ms
  2   -8.400428156277       -6.79       -4.04    380ms
  3   -8.400428156277      -14.45       -7.87    103ms

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|  = 9.079478852921349e-7
|ρ_newton - ρ_scfv| = 9.047653528040841e-8
|ρ_newton - ρ_dm|   = 2.061922894339901e-6