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 1.755505e+02 1.599729e+02 * time: 0.0004899501800537109 1 1.610994e+02 1.377143e+02 * time: 0.0016510486602783203 2 1.217723e+02 1.526572e+02 * time: 0.003050088882446289 3 4.913712e+01 1.010788e+02 * time: 0.004614114761352539 4 2.652192e+01 7.768343e+01 * time: 0.00592494010925293 5 7.521172e+00 4.704053e+00 * time: 0.007122039794921875 6 6.532983e+00 5.301687e+00 * time: 0.007838964462280273 7 5.552310e+00 3.404753e+01 * time: 0.008712053298950195 8 4.458500e+00 1.650916e+01 * time: 0.009562969207763672 9 3.300532e+00 1.607191e+01 * time: 0.010463953018188477 10 2.070245e+00 1.019517e+01 * time: 0.011322021484375 11 1.767372e+00 4.951547e+00 * time: 0.012186050415039062 12 1.550104e+00 2.946036e+00 * time: 0.012905120849609375 13 1.410975e+00 2.125734e+00 * time: 0.013620138168334961 14 1.375322e+00 2.328545e+00 * time: 0.014403104782104492 15 1.307995e+00 2.575039e+00 * time: 0.015725135803222656 16 1.211577e+00 2.053909e+00 * time: 0.016564130783081055 17 1.205535e+00 8.903566e-01 * time: 0.017141103744506836 18 1.173329e+00 7.974770e-01 * time: 0.01771402359008789 19 1.156801e+00 3.910967e-01 * time: 0.018522024154663086 20 1.146309e+00 1.858864e-01 * time: 0.019355058670043945 21 1.144562e+00 1.562937e-01 * time: 0.020115137100219727 22 1.144410e+00 1.520720e-01 * time: 0.02083301544189453 23 1.144228e+00 8.848874e-02 * time: 0.02154397964477539 24 1.144131e+00 4.414995e-02 * time: 0.022284984588623047 25 1.144099e+00 2.488609e-02 * time: 0.022999048233032227 26 1.144078e+00 1.697392e-02 * time: 0.023713111877441406 27 1.144052e+00 1.950068e-02 * time: 0.024424076080322266 28 1.144043e+00 2.142163e-02 * time: 0.02513599395751953 29 1.144039e+00 1.240663e-02 * time: 0.025846004486083984 30 1.144038e+00 6.180939e-03 * time: 0.026591062545776367 31 1.144037e+00 3.052281e-03 * time: 0.027302980422973633 32 1.144037e+00 7.743455e-04 * time: 0.028010129928588867 33 1.144037e+00 4.706603e-04 * time: 0.028717994689941406 34 1.144037e+00 4.072362e-04 * time: 0.029428958892822266 35 1.144037e+00 3.423622e-04 * time: 0.030187129974365234 36 1.144037e+00 3.063016e-04 * time: 0.030903100967407227 37 1.144037e+00 2.882213e-04 * time: 0.03162407875061035 38 1.144037e+00 1.797990e-04 * time: 0.03228902816772461 39 1.144037e+00 9.520380e-05 * time: 0.0330049991607666 40 1.144037e+00 5.032534e-05 * time: 0.0337069034576416 41 1.144037e+00 3.164985e-05 * time: 0.03456306457519531 42 1.144037e+00 2.292945e-05 * time: 0.03520798683166504 43 1.144037e+00 1.698935e-05 * time: 0.035932064056396484 44 1.144037e+00 6.683249e-06 * time: 0.03664398193359375 45 1.144037e+00 6.134734e-06 * time: 0.03734993934631348 46 1.144037e+00 3.275605e-06 * time: 0.03808093070983887 47 1.144037e+00 1.721814e-06 * time: 0.03879690170288086 48 1.144037e+00 1.706302e-06 * time: 0.03946495056152344 49 1.144037e+00 1.878129e-06 * time: 0.04014396667480469 50 1.144037e+00 1.561906e-06 * time: 0.04085111618041992 51 1.144037e+00 8.420301e-07 * time: 0.041558027267456055 52 1.144037e+00 8.571096e-07 * time: 0.04229402542114258 53 1.144037e+00 6.736076e-07 * time: 0.042997121810913086 54 1.144037e+00 2.660108e-07 * time: 0.04355502128601074 55 1.144037e+00 2.104323e-07 * time: 0.04423403739929199 56 1.144037e+00 1.231348e-07 * time: 0.044934988021850586
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.real)[:, 1, 1] # converged density
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.real - abs2.(ψ))
9.385285437428026e-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 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)
1.677075901831831e-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))
0.00022342696110591457