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   -8.397869930941                   -0.90    5.0   27.0ms
  2   -8.400242650011       -2.62       -1.74    1.0   19.6ms
  3   -8.400406177549       -3.79       -2.98    1.5   20.6ms
  4   -8.400427850036       -4.66       -2.98    3.2   25.3ms
  5   -8.400427941838       -7.04       -3.05    1.0   19.7ms
  6   -8.400428145821       -6.69       -4.51    1.0   19.8ms
  7   -8.400428151695       -8.23       -4.42    2.2   23.1ms
  8   -8.400428152179       -9.31       -5.16    1.0   19.8ms
  9   -8.400428152207      -10.56       -6.16    1.5   64.3ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397830272404                   -0.90           5.0   27.5ms
  2   -8.400383902420       -2.59       -1.78   0.80    2.2   20.8ms
  3   -8.400422943373       -4.41       -2.99   0.80    1.0   17.3ms
  4   -8.400428105784       -5.29       -3.45   0.80    2.8   21.6ms
  5   -8.400428149701       -7.36       -4.72   0.80    1.5   18.7ms
  6   -8.400428152188       -8.60       -5.68   0.80    2.5   21.1ms
  7   -8.400428152209      -10.67       -6.16   0.80    2.2   21.2ms

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   +0.866686241541                   -1.04   58.9ms
  2   -1.499489568951        0.37       -0.65   32.7ms
  3   -4.547845687275        0.48       -0.39   43.8ms
  4   -6.068610645817        0.18       -0.49   43.7ms
  5   -7.577347519852        0.18       -0.78   43.8ms
  6   -7.996827859288       -0.38       -1.29   32.5ms
  7   -8.201972864217       -0.69       -1.50   32.5ms
  8   -8.252972604863       -1.29       -1.75   32.6ms
  9   -8.296007314027       -1.37       -1.88   33.1ms
 10   -8.301859057674       -2.23       -1.98   45.6ms
 11   -8.317105693141       -1.82       -1.77   33.5ms
 12   -8.361203195060       -1.36       -1.70   45.2ms
 13   -8.380169696928       -1.72       -2.00   45.3ms
 14   -8.393851984409       -1.86       -2.56   33.5ms
 15   -8.394967150696       -2.95       -3.05   33.9ms
 16   -8.397669212604       -2.57       -2.97   33.6ms
 17   -8.398746961765       -2.97       -3.00   33.6ms
 18   -8.399588418315       -3.07       -3.23   33.3ms
 19   -8.400044259016       -3.34       -3.29   33.2ms
 20   -8.400262556747       -3.66       -3.52   33.4ms
 21   -8.400375133763       -3.95       -3.77   33.6ms
 22   -8.400412036900       -4.43       -3.84   32.8ms
 23   -8.400421315196       -5.03       -4.10   33.0ms
 24   -8.400425869884       -5.34       -4.23   42.6ms
 25   -8.400426988800       -5.95       -4.65   33.5ms
 26   -8.400427751839       -6.12       -4.75   33.3ms
 27   -8.400427997660       -6.61       -4.93   33.3ms
 28   -8.400428099384       -6.99       -5.00   33.3ms
 29   -8.400428126504       -7.57       -4.97   33.9ms
 30   -8.400428143226       -7.78       -5.33   33.4ms
 31   -8.400428146649       -8.47       -5.64   32.7ms
 32   -8.400428149910       -8.49       -5.63   32.5ms
 33   -8.400428151003       -8.96       -5.88   32.3ms
 34   -8.400428151836       -9.08       -6.16   32.4ms

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   -8.397802414684                   -0.90    5.2   34.6ms

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   -8.400427981975                   -1.79    605ms
  2   -8.400428152209       -6.77       -4.03    405ms
  3   -8.400428152209      -14.45       -7.84    101ms

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|  = 3.8517478841090336e-7
|ρ_newton - ρ_scfv| = 1.6844234814559423e-7
|ρ_newton - ρ_dm|   = 1.4441002405698352e-6