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   -8.397813632007                   -0.90    5.0   24.2ms
  2   -8.400241189226       -2.61       -1.73    1.0   25.8ms
  3   -8.400399136119       -3.80       -2.95    1.5   18.4ms
  4   -8.400427696881       -4.54       -2.89    3.2   22.8ms
  5   -8.400427910116       -6.67       -3.02    1.0   17.5ms
  6   -8.400428148559       -6.62       -5.06    1.0   17.6ms
  7   -8.400428155759       -8.14       -4.38    3.5   24.6ms
  8   -8.400428156273       -9.29       -5.44    2.0   20.5ms
  9   -8.400428156277      -11.45       -6.43    1.0   17.8ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397823224325                   -0.90           5.5   25.2ms
  2   -8.400375967824       -2.59       -1.79   0.80    2.0   23.2ms
  3   -8.400422330500       -4.33       -3.07   0.80    1.0   14.9ms
  4   -8.400428115139       -5.24       -3.42   0.80    2.5   19.0ms
  5   -8.400428153389       -7.42       -4.56   0.80    1.0   14.9ms
  6   -8.400428156238       -8.55       -5.67   0.80    2.2   18.4ms
  7   -8.400428156277      -10.42       -5.89   0.80    3.2   20.8ms
  8   -8.400428156277      -12.69       -7.05   0.80    1.0   15.1ms

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/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   +0.666741851717                   -1.06   49.5ms
  2   -2.039540939097        0.43       -0.65   27.8ms
  3   -4.635288465619        0.41       -0.38   43.5ms
  4   -6.418051145558        0.25       -0.48   37.5ms
  5   -7.702360377454        0.11       -0.82   37.4ms
  6   -8.054020480095       -0.45       -1.29   27.8ms
  7   -8.195212691035       -0.85       -1.74   27.7ms
  8   -8.268323700675       -1.14       -1.96   32.3ms
  9   -8.311121803617       -1.37       -2.33   27.8ms
 10   -8.341829687537       -1.51       -2.03   27.7ms
 11   -8.367458501220       -1.59       -2.09   27.7ms
 12   -8.385686452155       -1.74       -2.21   27.7ms
 13   -8.392813165442       -2.15       -2.56   32.4ms
 14   -8.397225566598       -2.36       -2.88   27.9ms
 15   -8.398695129057       -2.83       -2.61   27.8ms
 16   -8.399722873904       -2.99       -3.06   27.7ms
 17   -8.400105946813       -3.42       -2.82   31.5ms
 18   -8.400301814282       -3.71       -3.11   27.8ms
 19   -8.400363513504       -4.21       -3.59   27.7ms
 20   -8.400401685577       -4.42       -3.76   27.7ms
 21   -8.400416364354       -4.83       -3.98   31.3ms
 22   -8.400423217367       -5.16       -4.45   27.9ms
 23   -8.400426075587       -5.54       -4.21   27.8ms
 24   -8.400427323182       -5.90       -4.94   30.9ms
 25   -8.400427781015       -6.34       -4.82   28.4ms
 26   -8.400428019346       -6.62       -5.23   27.8ms
 27   -8.400428114018       -7.02       -5.01   27.8ms
 28   -8.400428135929       -7.66       -5.63   31.3ms
 29   -8.400428148960       -7.89       -5.70   27.7ms
 30   -8.400428151839       -8.54       -5.75   27.8ms
 31   -8.400428154351       -8.60       -6.03   27.8ms

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   -8.397864719616                   -0.90    4.8   23.9ms

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   -8.400427981609                   -1.79    524ms
  2   -8.400428156277       -6.76       -4.02    346ms
  3   -8.400428156277   +    -Inf       -7.82    111ms

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|  = 3.0632737107740273e-7
|ρ_newton - ρ_scfv| = 6.199899358817966e-8
|ρ_newton - ρ_dm|   = 3.662890961207304e-6