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   -8.397846879200                   -0.90    5.5   27.9ms
  2   -8.400252852220       -2.62       -1.74    1.0   19.5ms
  3   -8.400398155760       -3.84       -2.94    1.2   19.8ms
  4   -8.400427774730       -4.53       -2.91    3.2   34.3ms
  5   -8.400427903971       -6.89       -2.99    1.0   19.2ms
  6   -8.400428149098       -6.61       -5.00    1.0   19.0ms
  7   -8.400428155753       -8.18       -4.36    3.5   26.5ms
  8   -8.400428156237       -9.32       -5.05    1.2   19.9ms
  9   -8.400428156274      -10.43       -5.94    1.0   25.4ms
 10   -8.400428156276      -11.56       -5.94    2.0   22.2ms
 11   -8.400428156277      -12.61       -6.81    1.0   19.4ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397845642657                   -0.90           5.5   33.2ms
  2   -8.400383915525       -2.60       -1.79   0.80    2.0   18.8ms
  3   -8.400422652342       -4.41       -3.01   0.80    1.2   16.1ms
  4   -8.400428104402       -5.26       -3.43   0.80    2.5   19.8ms
  5   -8.400428154154       -7.30       -4.61   0.80    1.2   16.5ms
  6   -8.400428156249       -8.68       -6.09   0.80    2.5   19.9ms

Direct minimization

scfres_dm = direct_minimization(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   Δtime
---   ---------------   ---------   ---------   ------
  1   +1.359177533015                   -1.09   58.0ms
  2   -1.293394058816        0.42       -0.67   29.0ms
  3   -3.665976947168        0.38       -0.41   39.7ms
  4   -4.457951506555       -0.10       -0.51   45.9ms
  5   -6.494527136139        0.31       -0.57   39.1ms
  6   -7.246752334556       -0.12       -0.87   39.0ms
  7   -7.498629649210       -0.60       -1.23   33.9ms
  8   -7.914327629359       -0.38       -1.47   28.9ms
  9   -7.985405714064       -1.15       -1.46   28.8ms
 10   -8.119665574510       -0.87       -1.29   39.1ms
 11   -8.182217915813       -1.20       -1.51   56.2ms
 12   -8.208404125764       -1.58       -1.92   39.2ms
 13   -8.278121600492       -1.16       -1.78   39.1ms
 14   -8.303636209860       -1.59       -1.85   54.5ms
 15   -8.364906661510       -1.21       -2.09   29.0ms
 16   -8.385118848588       -1.69       -2.60   28.8ms
 17   -8.393654699717       -2.07       -2.33   28.9ms
 18   -8.396192966943       -2.60       -2.45   33.8ms
 19   -8.398937507512       -2.56       -2.91   28.9ms
 20   -8.399782161125       -3.07       -2.74   28.7ms
 21   -8.400145925577       -3.44       -3.09   28.8ms
 22   -8.400205597100       -4.22       -3.15   33.9ms
 23   -8.400337597211       -3.88       -3.35   29.0ms
 24   -8.400390860627       -4.27       -3.65   28.8ms
 25   -8.400411869696       -4.68       -4.23   28.8ms
 26   -8.400420706640       -5.05       -4.06   34.2ms
 27   -8.400425234126       -5.34       -4.78   29.0ms
 28   -8.400426515553       -5.89       -4.22   28.8ms
 29   -8.400427606322       -5.96       -4.84   29.0ms
 30   -8.400427856860       -6.60       -4.67   29.3ms
 31   -8.400427982382       -6.90       -5.18   34.1ms
 32   -8.400428066899       -7.07       -5.14   29.0ms
 33   -8.400428117767       -7.29       -5.54   28.8ms
 34   -8.400428147546       -7.53       -5.54   29.3ms
 35   -8.400428153076       -8.26       -5.58   38.2ms
 36   -8.400428155063       -8.70       -5.92   29.0ms
 37   -8.400428155503       -9.36       -5.93   29.0ms
 38   -8.400428155975       -9.33       -6.11   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   -8.397881970538                   -0.90    5.0   32.4ms

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   -8.400427946340                   -1.77    542ms
  2   -8.400428156277       -6.68       -3.97    368ms
  3   -8.400428156277      -14.27       -7.73    110ms

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|  = 2.2078565037565425e-7
|ρ_newton - ρ_scfv| = 3.945019700244033e-6
|ρ_newton - ρ_dm|   = 9.051430392926838e-7