# Custom potential

We solve the 1D Gross-Pitaevskii equation with a custom potential. This is similar to Gross-Pitaevskii equation in 1D example and we show how to define local potentials attached to atoms, which allows for instance to compute forces. The custom potential is actually already defined as `ElementGaussian`

in DFTK, and could be used as is.

```
using DFTK
using LinearAlgebra
```

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

```
struct CustomPotential <: DFTK.Element
α # Prefactor
L # Width of the Gaussian nucleus
end
```

Some default values

`CustomPotential() = CustomPotential(1.0, 0.5);`

We extend the two methods providing access to the real and Fourier representation of the potential to DFTK.

```
function DFTK.local_potential_real(el::CustomPotential, r::Real)
-el.α / (√(2π) * el.L) * exp(- (r / el.L)^2 / 2)
end
function DFTK.local_potential_fourier(el::CustomPotential, q::Real)
# = ∫ V(r) exp(-ix⋅q) dx
-el.α * exp(- (q * el.L)^2 / 2)
end
```

DFTK already implements `CustomPotential`

in form of the `DFTK.ElementGaussian`

, so this explicit re-implementation is only provided for demonstration purposes.

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
positions = [[x1, 0, 0], [x2, 0, 0]]
gauss = CustomPotential()
atoms = [gauss, gauss];
```

We setup a Gross-Pitaevskii model

```
C = 1.0
α = 2;
n_electrons = 1 # Increase this for fun
terms = [Kinetic(),
AtomicLocal(),
LocalNonlinearity(ρ -> C * ρ^α)]
model = Model(lattice, atoms, positions; n_electrons, 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 `gauss`

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

```
basis = PlaneWaveBasis(model; Ecut=500, kgrid=(1, 1, 1))
ρ = zeros(eltype(basis), basis.fft_size..., 1)
scfres = self_consistent_field(basis; tol=1e-5, ρ)
scfres.energies
```

```
Energy breakdown (in Ha):
Kinetic 0.0380294
AtomicLocal -0.3163466
LocalNonlinearity 0.1212608
total -0.157056406908
```

Computing the forces can then be done as usual:

`compute_forces(scfres)`

```
2-element Vector{StaticArraysCore.SVector{3, Float64}}:
[-0.0556886918408491, 0.0, 0.0]
[0.05568135132239964, 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 = scfres.ρ[:, 1, 1, 1] # converged density, first spin component
ψ_fourier = scfres.ψ[1][:, 1] # first k-point, all G components, first eigenvector
ψ = ifft(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")
```