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   -8.397931706513                   -0.90    5.5   24.9ms
  2   -8.400242222110       -2.64       -1.74    1.0   17.2ms
  3   -8.400401457770       -3.80       -2.98    1.2   17.9ms
  4   -8.400427815698       -4.58       -2.94    3.2   22.7ms
  5   -8.400427943286       -6.89       -3.04    1.0   17.1ms
  6   -8.400428150111       -6.68       -4.79    1.0   17.2ms
  7   -8.400428155827       -8.24       -4.42    2.8   21.9ms
  8   -8.400428156262       -9.36       -5.96    1.0   17.7ms
  9   -8.400428156276      -10.85       -5.96    2.5   21.8ms
 10   -8.400428156277      -12.21       -6.63    1.0   18.1ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397804189276                   -0.90           5.2   31.2ms
  2   -8.400384024766       -2.59       -1.78   0.80    2.2   22.4ms
  3   -8.400423224439       -4.41       -3.01   0.80    1.0   19.3ms
  4   -8.400428108718       -5.31       -3.48   0.80    2.5   23.6ms
  5   -8.400428152170       -7.36       -4.71   0.80    1.2   20.1ms
  6   -8.400428156259       -8.39       -5.44   0.80    2.8   35.5ms
  7   -8.400428156276      -10.78       -6.92   0.80    1.8   17.9ms

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   +1.100854778973                   -1.06   51.0ms
  2   -1.645318362252        0.44       -0.68   28.6ms
  3   -4.550888789956        0.46       -0.39   38.4ms
  4   -6.195456648319        0.22       -0.49   38.8ms
  5   -7.601378730970        0.15       -0.76   38.6ms
  6   -8.023986560934       -0.37       -1.39   28.5ms
  7   -8.209256312844       -0.73       -1.65   28.3ms
  8   -8.281930897926       -1.14       -1.77   28.7ms
  9   -8.319365965735       -1.43       -2.36   28.1ms
 10   -8.339383707243       -1.70       -2.40   28.1ms
 11   -8.358547940460       -1.72       -2.16   28.0ms
 12   -8.374454821829       -1.80       -2.15   28.0ms
 13   -8.387923708235       -1.87       -2.35   28.0ms
 14   -8.395330133532       -2.13       -2.82   40.4ms
 15   -8.398010712697       -2.57       -2.92   28.3ms
 16   -8.399626832535       -2.79       -3.21   28.0ms
 17   -8.400052158648       -3.37       -3.61   28.2ms
 18   -8.400313372302       -3.58       -3.55   28.3ms
 19   -8.400369249438       -4.25       -3.80   28.2ms
 20   -8.400411605930       -4.37       -3.92   28.4ms
 21   -8.400418383925       -5.17       -4.17   28.5ms
 22   -8.400425009704       -5.18       -4.57   28.3ms
 23   -8.400426704576       -5.77       -4.56   28.3ms
 24   -8.400427635503       -6.03       -4.83   28.4ms
 25   -8.400427900928       -6.58       -4.74   28.3ms
 26   -8.400428038447       -6.86       -5.06   28.2ms
 27   -8.400428109675       -7.15       -5.09   34.3ms
 28   -8.400428132706       -7.64       -6.18   28.6ms

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   -8.397803133072                   -0.90    5.2   24.7ms

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   -8.400427976433                   -1.79    547ms
  2   -8.400428156277       -6.75       -4.02    369ms
  3   -8.400428156277      -14.45       -7.80    124ms

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|  = 1.444120027233851e-7
|ρ_newton - ρ_scfv| = 5.11477523540288e-7
|ρ_newton - ρ_dm|   = 7.241798945936816e-6