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 magnetism).
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:
Ecut = 500
basis = PlaneWaveBasis(model, Ecut, 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 1.620254e+02 1.447057e+02 * time: 0.0006029605865478516 1 1.387679e+02 1.079574e+02 * time: 0.001878976821899414 2 1.005616e+02 1.217431e+02 * time: 0.0030519962310791016 3 2.097994e+01 4.182211e+01 * time: 0.004261016845703125 4 1.149958e+01 2.748740e+01 * time: 0.005278825759887695 5 9.125888e+00 9.082666e+00 * time: 0.006073951721191406 6 7.043560e+00 1.917138e+01 * time: 0.0068929195404052734 7 3.988754e+00 5.945317e+00 * time: 0.007701873779296875 8 2.726113e+00 3.024515e+00 * time: 0.008566856384277344 9 1.579391e+00 2.833376e+00 * time: 0.009402036666870117 10 1.413821e+00 2.784967e+00 * time: 0.010223865509033203 11 1.316640e+00 2.903979e+00 * time: 0.010879993438720703 12 1.225098e+00 1.191196e+00 * time: 0.011534929275512695 13 1.183430e+00 5.483881e-01 * time: 0.01220703125 14 1.157298e+00 2.956812e-01 * time: 0.012887954711914062 15 1.148557e+00 2.021497e-01 * time: 0.013510942459106445 16 1.146062e+00 1.049308e-01 * time: 0.014078855514526367 17 1.144636e+00 1.088491e-01 * time: 0.014629840850830078 18 1.144163e+00 3.123473e-02 * time: 0.015193939208984375 19 1.144122e+00 2.765210e-02 * time: 0.01575303077697754 20 1.144109e+00 4.863846e-02 * time: 0.016216039657592773 21 1.144064e+00 1.773289e-02 * time: 0.01669001579284668 22 1.144058e+00 1.421626e-02 * time: 0.017282962799072266 23 1.144051e+00 1.498466e-02 * time: 0.017679929733276367 24 1.144044e+00 9.733992e-03 * time: 0.018204927444458008 25 1.144038e+00 3.650658e-03 * time: 0.018730878829956055 26 1.144037e+00 3.983450e-03 * time: 0.019284963607788086 27 1.144037e+00 3.426817e-03 * time: 0.019841909408569336 28 1.144037e+00 1.854724e-03 * time: 0.020424842834472656 29 1.144037e+00 1.374631e-03 * time: 0.020973920822143555 30 1.144037e+00 5.714051e-04 * time: 0.021512985229492188 31 1.144037e+00 3.041005e-04 * time: 0.02206897735595703 32 1.144037e+00 3.243686e-04 * time: 0.022650957107543945 33 1.144037e+00 3.128633e-04 * time: 0.023231029510498047 34 1.144037e+00 2.327409e-04 * time: 0.023794889450073242 35 1.144037e+00 1.546077e-04 * time: 0.02422499656677246 36 1.144037e+00 5.593509e-05 * time: 0.02480602264404297 37 1.144037e+00 2.952055e-05 * time: 0.025367021560668945 38 1.144037e+00 1.658163e-05 * time: 0.02593088150024414 39 1.144037e+00 9.627992e-06 * time: 0.026556968688964844 40 1.144037e+00 7.030475e-06 * time: 0.027151823043823242 41 1.144037e+00 5.072021e-06 * time: 0.02775287628173828 42 1.144037e+00 3.266220e-06 * time: 0.02842998504638672 43 1.144037e+00 1.813561e-06 * time: 0.029094934463500977 44 1.144037e+00 1.296418e-06 * time: 0.029744863510131836 45 1.144037e+00 1.452421e-06 * time: 0.030373811721801758 46 1.144037e+00 1.276097e-06 * time: 0.03082895278930664 47 1.144037e+00 5.119396e-07 * time: 0.03144383430480957 48 1.144037e+00 3.767626e-07 * time: 0.032013893127441406 49 1.144037e+00 1.981879e-07 * time: 0.032629966735839844 50 1.144037e+00 1.486966e-07 * time: 0.03319883346557617
Energy breakdown: 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 kpoint, 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))
1.3777528460094525e-15
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 kpoints. Here, we just have one kpoint:
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 kpoint, 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)
2.5509757446571617e-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))
0.0002234290790088227