In this example we will use DFTK to solve the Gross-Pitaevskii equation, and use this opportunity to explore a few internals.
The Gross-Pitaevskii equation (GPE)
is a simple non-linear equation used to model bosonic systems
in a mean-field approach. Denoting by ψ
the effective one-particle bosonic
wave function, the time-independent GPE reads in atomic units:
$$
H ψ = \left(-\frac12 Δ + V + 2 C |ψ|^2\right) ψ = μ ψ \qquad \|ψ\|_{L^2} = 1
$$
where C
provides the strength of the boson-boson coupling.
It's in particular a favorite model of applied mathematicians because it
has a structure simpler than but similar to that of DFT, and displays
interesting behavior (especially in higher dimensions with magnetic fields, see
Gross-Pitaevskii equation with external magnetic field).
We wish to model this equation in 1D using DFTK. First 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]];
which is special cased in DFTK to support 1D models.
For the potential term V
we just pick a harmonic
potential. The real-space grid is in [0,1)
in fractional coordinates( see
Lattices and lattice vectors),
therefore:
pot(x) = (x - a/2)^2;
We setup each energy term in sequence: kinetic, potential and nonlinear term.
For the non-linearity we use the PowerNonlinearity(C, α)
term of DFTK.
This object introduces an energy term C ∫ ρ(r)^α dr
to the total energy functional, thus a potential term α C ρ^{α-1}
.
In our case we thus need the parameters
C = 1.0
α = 2;
... and with this build the model
using DFTK
using LinearAlgebra
n_electrons = 1 # Increase this for fun
terms = [Kinetic(),
ExternalFromReal(r -> pot(r[1])),
PowerNonlinearity(C, α),
]
model = Model(lattice; n_electrons=n_electrons, terms=terms,
spin_polarization=:spinless); # use "spinless electrons"
We discretize using a moderate Ecut (For 1D values up to 5000
are completely fine)
and run a direct minimization algorithm:
basis = PlaneWaveBasis(model, Ecut=500, kgrid=(1, 1, 1))
scfres = direct_minimization(basis, tol=1e-8) # This is a constrained preconditioned LBFGS
scfres.energies
Iter Function value Gradient norm 0 1.569950e+02 1.475165e+02 * time: 0.0006210803985595703 1 1.462999e+02 1.280535e+02 * time: 0.0018641948699951172 2 1.099572e+02 1.388987e+02 * time: 0.003020048141479492 3 3.461111e+01 7.750219e+01 * time: 0.00440216064453125 4 9.792206e+00 2.028901e+01 * time: 0.005523204803466797 5 4.643768e+00 1.457883e+01 * time: 0.006398200988769531 6 3.692633e+00 1.198520e+01 * time: 0.007093191146850586 7 2.907833e+00 4.781847e+00 * time: 0.007870197296142578 8 2.415187e+00 3.373766e+00 * time: 0.008682012557983398 9 2.325554e+00 7.641153e+00 * time: 0.009289026260375977 10 1.875915e+00 9.739840e+00 * time: 0.010096073150634766 11 1.503594e+00 4.792395e+00 * time: 0.010890007019042969 12 1.312164e+00 1.918003e+00 * time: 0.011541128158569336 13 1.197278e+00 5.809415e-01 * time: 0.01223611831665039 14 1.165462e+00 4.538327e-01 * time: 0.012847185134887695 15 1.152961e+00 3.375983e-01 * time: 0.013442039489746094 16 1.146802e+00 1.153249e-01 * time: 0.01402902603149414 17 1.146042e+00 2.007782e-01 * time: 0.014696121215820312 18 1.144605e+00 5.419894e-02 * time: 0.015351057052612305 19 1.144371e+00 6.250033e-02 * time: 0.016062021255493164 20 1.144189e+00 3.115222e-02 * time: 0.016746997833251953 21 1.144114e+00 3.187875e-02 * time: 0.017208099365234375 22 1.144084e+00 2.420941e-02 * time: 0.01781010627746582 23 1.144048e+00 1.928183e-02 * time: 0.018447160720825195 24 1.144041e+00 8.211231e-03 * time: 0.019070148468017578 25 1.144038e+00 2.602573e-03 * time: 0.019697189331054688 26 1.144037e+00 1.869669e-03 * time: 0.020338058471679688 27 1.144037e+00 1.072517e-03 * time: 0.020920991897583008 28 1.144037e+00 6.269688e-04 * time: 0.021499156951904297 29 1.144037e+00 3.584544e-04 * time: 0.022119998931884766 30 1.144037e+00 3.676293e-04 * time: 0.022752046585083008 31 1.144037e+00 5.429743e-04 * time: 0.02339315414428711 32 1.144037e+00 1.205961e-04 * time: 0.02405405044555664 33 1.144037e+00 6.787509e-05 * time: 0.024689197540283203 34 1.144037e+00 7.920861e-05 * time: 0.025269031524658203 35 1.144037e+00 4.074150e-05 * time: 0.025886058807373047 36 1.144037e+00 3.299029e-05 * time: 0.02650308609008789 37 1.144037e+00 1.246370e-05 * time: 0.027100086212158203 38 1.144037e+00 1.310536e-05 * time: 0.027744054794311523 39 1.144037e+00 9.293200e-06 * time: 0.028406143188476562 40 1.144037e+00 3.987355e-06 * time: 0.029052019119262695 41 1.144037e+00 3.673421e-06 * time: 0.029731035232543945 42 1.144037e+00 2.799374e-06 * time: 0.030381202697753906 43 1.144037e+00 2.361704e-06 * time: 0.031016111373901367 44 1.144037e+00 1.226682e-06 * time: 0.031699180603027344 45 1.144037e+00 6.938885e-07 * time: 0.03235316276550293 46 1.144037e+00 3.865066e-07 * time: 0.032981157302856445 47 1.144037e+00 2.257562e-07 * time: 0.03363919258117676 48 1.144037e+00 1.805372e-07 * time: 0.034243106842041016
Energy breakdown (in Ha): Kinetic 0.2682057 ExternalFromReal 0.4707475 PowerNonlinearity 0.4050836 total 1.144036852755
We use the opportunity to explore some of DFTK internals.
Extract the converged density and the obtained wave function:
ρ = real(scfres.ρ)[:, 1, 1, 1] # converged density, first spin component
ψ_fourier = scfres.ψ[1][:, 1]; # first k-point, all G components, first eigenvector
Transform the wave function to real space and fix the phase:
ψ = G_to_r(basis, basis.kpoints[1], ψ_fourier)[:, 1, 1]
ψ /= (ψ[div(end, 2)] / abs(ψ[div(end, 2)]));
Check whether ψ
is normalised:
x = a * vec(first.(DFTK.r_vectors(basis)))
N = length(x)
dx = a / N # real-space grid spacing
@assert sum(abs2.(ψ)) * dx ≈ 1.0
The density is simply built from ψ:
norm(scfres.ρ - abs2.(ψ))
7.490189186705262e-16
We summarize the ground state in a nice plot:
using Plots
p = plot(x, real.(ψ), label="real(ψ)")
plot!(p, x, imag.(ψ), label="imag(ψ)")
plot!(p, x, ρ, label="ρ")
The energy_hamiltonian
function can be used to get the energy and
effective Hamiltonian (derivative of the energy with respect to the density matrix)
of a particular state (ψ, occupation).
The density ρ associated to this state is precomputed
and passed to the routine as an optimization.
E, ham = energy_hamiltonian(basis, scfres.ψ, scfres.occupation; ρ=scfres.ρ)
@assert E.total == scfres.energies.total
Now the Hamiltonian contains all the blocks corresponding to k-points. Here, we just have one k-point:
H = ham.blocks[1];
H
can be used as a linear operator (efficiently using FFTs), or converted to a dense matrix:
ψ11 = scfres.ψ[1][:, 1] # first k-point, first eigenvector
Hmat = Array(H) # This is now just a plain Julia matrix,
# which we can compute and store in this simple 1D example
@assert norm(Hmat * ψ11 - H * ψ11) < 1e-10
Let's check that ψ11 is indeed an eigenstate:
norm(H * ψ11 - dot(ψ11, H * ψ11) * ψ11)
3.5111470189989075e-7
Build a finite-differences version of the GPE operator H
, as a sanity check:
A = Array(Tridiagonal(-ones(N - 1), 2ones(N), -ones(N - 1)))
A[1, end] = A[end, 1] = -1
K = A / dx^2 / 2
V = Diagonal(pot.(x) + C .* α .* (ρ.^(α-1)))
H_findiff = K + V;
maximum(abs.(H_findiff*ψ - (dot(ψ, H_findiff*ψ) / dot(ψ, ψ)) * ψ))
0.00022344684174090268