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   -8.397871040479                   -0.90    4.8   25.7ms
  2   -8.400249627308       -2.62       -1.73    1.0   18.9ms
  3   -8.400403760156       -3.81       -2.95    1.5   19.6ms
  4   -8.400427784361       -4.62       -2.92    3.2   33.4ms
  5   -8.400427978138       -6.71       -3.07    1.0   19.3ms
  6   -8.400428150548       -6.76       -4.93    1.0   19.5ms
  7   -8.400428155958       -8.27       -4.50    3.5   25.8ms
  8   -8.400428156266       -9.51       -5.86    1.0   19.6ms
  9   -8.400428156277      -10.95       -6.25    2.0   22.4ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397784177301                   -0.90           4.5   25.5ms
  2   -8.400379242924       -2.59       -1.77   0.80    2.2   19.3ms
  3   -8.400423258062       -4.36       -2.96   0.80    1.0   16.4ms
  4   -8.400428112097       -5.31       -3.48   0.80    2.2   26.0ms
  5   -8.400428152548       -7.39       -5.00   0.80    1.2   17.2ms
  6   -8.400428156273       -8.43       -5.35   0.80    3.8   23.4ms
  7   -8.400428156276      -11.44       -6.82   0.80    1.0   16.9ms

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/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   +0.811579736385                   -1.05   57.7ms
  2   -2.175644224432        0.48       -0.70   33.1ms
  3   -4.651360318919        0.39       -0.40   44.6ms
  4   -6.331133238248        0.23       -0.50   49.7ms
  5   -7.761514758414        0.16       -0.76   44.8ms
  6   -8.089835691822       -0.48       -1.28   33.2ms
  7   -8.246842081362       -0.80       -1.71   32.8ms
  8   -8.274081976884       -1.56       -1.96   32.5ms
  9   -8.293563757163       -1.71       -1.82   32.8ms
 10   -8.354497393615       -1.22       -1.66   49.7ms
 11   -8.375125649007       -1.69       -1.90   44.8ms
 12   -8.389642704499       -1.84       -2.66   33.3ms
 13   -8.394791362619       -2.29       -2.82   33.3ms
 14   -8.398777967727       -2.40       -2.96   38.1ms
 15   -8.399642347400       -3.06       -3.12   33.4ms
 16   -8.400048043555       -3.39       -3.33   33.0ms
 17   -8.400263254949       -3.67       -3.65   33.0ms
 18   -8.400337650572       -4.13       -3.65   36.9ms
 19   -8.400396809478       -4.23       -3.91   33.3ms
 20   -8.400412494755       -4.80       -3.88   33.1ms
 21   -8.400423803973       -4.95       -4.08   32.9ms
 22   -8.400426241651       -5.61       -4.13   36.3ms
 23   -8.400427313622       -5.97       -4.56   33.1ms
 24   -8.400427713565       -6.40       -4.70   32.7ms
 25   -8.400427964199       -6.60       -4.88   36.6ms
 26   -8.400428079449       -6.94       -4.90   33.3ms
 27   -8.400428128607       -7.31       -5.10   32.9ms
 28   -8.400428144307       -7.80       -5.27   33.0ms
 29   -8.400428150495       -8.21       -5.56   36.5ms
 30   -8.400428153936       -8.46       -5.83   33.3ms
 31   -8.400428155081       -8.94       -5.90   33.2ms
 32   -8.400428155760       -9.17       -6.41   37.2ms

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   -8.397842791157                   -0.90    5.0   27.4ms

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   -8.400427994786                   -1.79    601ms
  2   -8.400428156277       -6.79       -4.05    405ms
  3   -8.400428156277      -14.75       -7.86    105ms

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|  = 1.664401693530924e-7
|ρ_newton - ρ_scfv| = 2.8974785243296963e-7
|ρ_newton - ρ_dm|   = 1.3665981130416105e-6