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   -8.397863751843                   -0.90    5.2   26.2ms
  2   -8.400238836002       -2.62       -1.74    1.0   24.9ms
  3   -8.400405164893       -3.78       -2.98    1.5   19.8ms
  4   -8.400427857909       -4.64       -2.99    3.0   23.9ms
  5   -8.400427921932       -7.19       -3.02    1.0   18.9ms
  6   -8.400428144057       -6.65       -4.55    1.0   23.8ms
  7   -8.400428151597       -8.12       -4.36    2.8   23.5ms
  8   -8.400428152173       -9.24       -5.10    1.0   19.3ms
  9   -8.400428152207      -10.47       -6.24    1.2   20.0ms

Potential-based SCF

scfres_scfv = DFTK.scf_potential_mixing(basis; tol);

n     Energy            log10(ΔE)   log10(Δρ)   α      Diag   Δtime
---   ---------------   ---------   ---------   ----   ----   ------
  1   -8.397796881798                   -0.90           5.5   28.2ms
  2   -8.400384894315       -2.59       -1.79   0.80    2.2   19.1ms
  3   -8.400422730030       -4.42       -3.01   0.80    1.0   16.1ms
  4   -8.400428106113       -5.27       -3.43   0.80    2.2   24.0ms
  5   -8.400428149660       -7.36       -4.62   0.80    1.2   16.8ms
  6   -8.400428152185       -8.60       -5.70   0.80    2.5   20.0ms
  7   -8.400428152209      -10.62       -6.00   0.80    2.5   20.1ms
  8   -8.400428152209      -12.59       -7.10   0.80    1.0   16.3ms

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   +1.051648144054                   -1.09   61.7ms
  2   -1.859552167927        0.46       -0.71   32.7ms
  3   -4.552250590873        0.43       -0.47   44.6ms
  4   -6.172299567834        0.21       -0.54   47.6ms
  5   -7.604966352285        0.16       -0.83   44.2ms
  6   -7.968828670400       -0.44       -1.34   32.4ms
  7   -8.195406774940       -0.64       -1.55   36.3ms
  8   -8.257493620890       -1.21       -1.68   32.9ms
  9   -8.328531831544       -1.15       -1.90   33.0ms
 10   -8.359099405008       -1.51       -2.11   36.5ms
 11   -8.384647230904       -1.59       -2.53   32.5ms
 12   -8.393349879024       -2.06       -2.59   32.6ms
 13   -8.397091360335       -2.43       -3.00   32.5ms
 14   -8.398964668611       -2.73       -3.29   35.9ms
 15   -8.399782260021       -3.09       -3.41   32.4ms
 16   -8.400059388742       -3.56       -3.20   32.4ms
 17   -8.400276814040       -3.66       -3.33   32.4ms
 18   -8.400353446940       -4.12       -3.92   36.0ms
 19   -8.400396969208       -4.36       -3.80   32.5ms
 20   -8.400418357490       -4.67       -4.21   32.9ms
 21   -8.400424594455       -5.21       -4.10   32.4ms
 22   -8.400426796142       -5.66       -4.37   36.2ms
 23   -8.400427691398       -6.05       -4.52   32.8ms
 24   -8.400427947513       -6.59       -5.35   32.4ms
 25   -8.400428070850       -6.91       -5.17   36.3ms
 26   -8.400428115764       -7.35       -5.50   32.8ms
 27   -8.400428136957       -7.67       -5.74   32.9ms
 28   -8.400428146309       -8.03       -5.82   32.9ms
 29   -8.400428149795       -8.46       -5.82   36.2ms
 30   -8.400428151043       -8.90       -6.22   32.7ms

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   -8.397910971377                   -0.90    5.0   25.4ms

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   -8.400427991194                   -1.79    575ms
  2   -8.400428152209       -6.79       -4.05    382ms
  3   -8.400428152209   +    -Inf       -7.86    120ms

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|  = 1.3828366765974826e-6
|ρ_newton - ρ_scfv| = 1.951945286376085e-7
|ρ_newton - ρ_dm|   = 1.691272756887701e-6