# Geometry optimization

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
is_converged = ScfConvergenceForce(tol / 10)
scfres = self_consistent_field(basis; ψ, ρ, is_converged, 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 !isnothing(G)
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 6.219313e-01
* time: 4.7206878662109375e-5
1 -1.064073e+00 2.919808e-01
* time: 0.9077260494232178
2 -1.066008e+00 4.821016e-02
* time: 1.581482172012329
3 -1.066049e+00 4.314813e-04
* time: 2.0173211097717285
4 -1.066049e+00 6.954595e-09
* time: 2.294365167617798
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.

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.