# Custom potential

We solve the 1D Gross-Pitaevskii equation with a custom potential. This is similar to Gross-Pitaevskii equation in one dimension and we show how to define local potentials attached to atoms, which allows for instance to compute forces.

```
using DFTK
using LinearAlgebra
```

First, we define a new element which represents a nucleus generating a custom potential

```
struct ElementCustomPotential <: DFTK.Element
pot_real::Function # Real potential
pot_fourier::Function # Fourier potential
end
```

We need to extend two methods to access the real and Fourier forms of the potential during the computations performed by DFTK

```
function DFTK.local_potential_fourier(el::ElementCustomPotential, q::Real)
return el.pot_fourier(q)
end
function DFTK.local_potential_real(el::ElementCustomPotential, r::Real)
return el.pot_real(r)
end
```

We set up the lattice. For a 1D case we supply two zero lattice vectors

```
a = 10
lattice = a .* [[1 0 0.]; [0 0 0]; [0 0 0]];
```

In this example, we want to generate two Gaussian potentials generated by two nuclei localized at positions $x_1$ and $x_2$, that are expressed in $[0,1)$ in fractional coordinates. $|x_1 - x_2|$ should be different from $0.5$ to break symmetry and get nonzero forces.

```
x1 = 0.2
x2 = 0.8;
```

We define the width of the Gaussian potential generated by one nucleus

`L = 0.5;`

We set the potential in its real and Fourier forms

```
pot_real(x) = exp(-(x/L)^2)
pot_fourier(q::T) where {T <: Real} = exp(- (q*L)^2 / 4);
```

And finally we build the elements and set their positions in the `atoms`

array. Note that in this example `pot_real`

is not required as all applications of local potentials are done in the Fourier space.

```
nucleus = ElementCustomPotential(pot_real, pot_fourier)
atoms = [nucleus => [x1*[1,0,0], x2*[1,0,0]]];
```

Setup the Gross-Pitaevskii model

```
C = 1.0
α = 2;
n_electrons = 1 # Increase this for fun
terms = [Kinetic(),
AtomicLocal(),
PowerNonlinearity(C, α),
]
model = Model(lattice; atoms=atoms, n_electrons=n_electrons, terms=terms,
spin_polarization=:spinless); # use "spinless electrons"
```

We discretize using a moderate Ecut and run a SCF algorithm to compute forces afterwards. As there is no ionic charge associated to `nucleus`

we have to specify a starting density and we choose to start from a zero density.

```
Ecut = 500
basis = PlaneWaveBasis(model, Ecut, kgrid=(1, 1, 1))
ρ = zeros(complex(eltype(basis)), basis.fft_size)
scfres = self_consistent_field(basis, tol=1e-8, ρ=from_fourier(basis, ρ))
scfres.energies
```

Energy breakdown: Kinetic 0.0304518 AtomicLocal 0.0972469 PowerNonlinearity 0.1149347 total 0.242633376339

Computing the forces can then be done as usual:

`hcat(compute_forces(scfres)...)`

2×1 Matrix{StaticArrays.SVector{3, Float64}}: [-0.03869064065391005, 0.0, 0.0] [0.038693672636929605, 0.0, 0.0]

Extract the converged total local potential

`tot_local_pot = DFTK.total_local_potential(scfres.ham)[:, 1, 1]; # use only dimension 1`

Extract other quantities before plotting them

```
ρ = real(scfres.ρ.real)[:, 1, 1] # converged density
ψ_fourier = scfres.ψ[1][:, 1]; # first kpoint, all G components, first eigenvector
ψ = G_to_r(basis, basis.kpoints[1], ψ_fourier)[:, 1, 1]
ψ /= (ψ[div(end, 2)] / abs(ψ[div(end, 2)]));
using Plots
x = a * vec(first.(DFTK.r_vectors(basis)))
p = plot(x, real.(ψ), label="real(ψ)")
plot!(p, x, imag.(ψ), label="imag(ψ)")
plot!(p, x, ρ, label="ρ")
plot!(p, x, tot_local_pot, label="tot local pot")
```