We use the DFTK and Optim packages in this example to find the minimal-energy bond length of the $H_2$ molecule. We setup $H_2$ in an LDA model just like in the Tutorial for silicon.
using DFTK
using Optim
using LinearAlgebra
using Printf
kgrid = [1, 1, 1] # k-point grid
Ecut = 5 # kinetic energy cutoff in Hartree
tol = 1e-8 # tolerance for the optimization routine
a = 10 # lattice constant in Bohr
lattice = a * I(3)
H = ElementPsp(:H, psp=load_psp("hgh/lda/h-q1"));
atoms = [H, H];
We define a Bloch wave and a density to be used as global variables so that we can transfer the solution from one iteration to another and therefore reduce the optimization time.
ψ = nothing
ρ = nothing
First, we create a function that computes the solution associated to the
position $x ∈ ℝ^6$ of the atoms in reduced coordinates
(cf. Reduced and cartesian coordinates for more
details on the coordinates system).
They are stored as a vector: x[1:3]
represents the position of the
first atom and x[4:6]
the position of the second.
We also update ψ
and ρ
for the next iteration.
function compute_scfres(x)
model = model_LDA(lattice, atoms, [x[1:3], x[4:6]])
basis = PlaneWaveBasis(model; Ecut, kgrid)
global ψ, ρ
if isnothing(ρ)
ρ = guess_density(basis)
end
scfres = self_consistent_field(basis; ψ, ρ, tol=tol / 10, callback=identity)
ψ = scfres.ψ
ρ = scfres.ρ
scfres
end;
Then, we create the function we want to optimize: fg!
is used to update the
value of the objective function F
, namely the energy, and its gradient G
,
here computed with the forces (which are, by definition, the negative gradient
of the energy).
function fg!(F, G, x)
scfres = compute_scfres(x)
if G != nothing
grad = compute_forces(scfres)
G .= -[grad[1]; grad[2]]
end
scfres.energies.total
end;
Now, we can optimize on the 6 parameters x = [x1, y1, z1, x2, y2, z2]
in
reduced coordinates, using LBFGS()
, the default minimization algorithm
in Optim. We start from x0
, which is a first guess for the coordinates. By
default, optimize
traces the output of the optimization algorithm during the
iterations. Once we have the minimizer xmin
, we compute the bond length in
Cartesian coordinates.
x0 = vcat(lattice \ [0., 0., 0.], lattice \ [1.4, 0., 0.])
xres = optimize(Optim.only_fg!(fg!), x0, LBFGS(),
Optim.Options(show_trace=true, f_tol=tol))
xmin = Optim.minimizer(xres)
dmin = norm(lattice*xmin[1:3] - lattice*xmin[4:6])
@printf "\nOptimal bond length for Ecut=%.2f: %.3f Bohr\n" Ecut dmin
Iter Function value Gradient norm 0 -1.061651e+00 6.219595e-01 * time: 5.1975250244140625e-5 1 -1.064076e+00 2.917716e-01 * time: 2.589946985244751 2 -1.066008e+00 4.814326e-02 * time: 3.065577983856201 3 -1.066049e+00 4.063030e-04 * time: 3.3044850826263428 4 -1.066049e+00 4.521245e-06 * time: 3.511681079864502 5 -1.066049e+00 1.667006e-07 * time: 3.7725119590759277 Optimal bond length for Ecut=5.00: 1.556 Bohr
We used here very rough parameters to generate the example and
setting Ecut
to 10 Ha yields a bond length of 1.523 Bohr,
which agrees with ABINIT.
!!! note "Degrees of freedom" We used here a very general setting where we optimized on the 6 variables representing the position of the 2 atoms and it can be easily extended to molecules with more atoms (such as $H_2O$). In the particular case of $H_2$, we could use only the internal degree of freedom which, in this case, is just the bond length.