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 * Diagonal(ones(3))
H = ElementPsp(:H, psp=load_psp("hgh/lda/h-q1"));
We define a blochwave 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 \in \mathbb{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)
atoms = [H => [x[1:3], x[4:6]]]
model = model_LDA(lattice, atoms)
basis = PlaneWaveBasis(model; Ecut, kgrid)
global ψ, ρ
if ρ === nothing
ρ = guess_density(basis)
end
scfres = self_consistent_field(basis; ψ=ψ, ρ=ρ,
tol=tol / 10, callback=info->nothing)
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][1]; grad[1][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 -1.061170e+00 6.234736e-01
* time: 6.198883056640625e-5
1 -1.065558e+00 4.367323e-02
* time: 1.5204880237579346
2 -1.065592e+00 8.306383e-04
* time: 1.8629989624023438
3 -1.065592e+00 5.377312e-06
* time: 2.1599440574645996
4 -1.065592e+00 1.721628e-06
* time: 2.286029100418091
5 -1.065592e+00 3.647087e-08
* time: 2.496251106262207
Optimal bond length for Ecut=5.00: 1.557 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.