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:
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 1.803271e+02 1.687193e+02 * time: 0.0005929470062255859 1 1.601613e+02 1.408108e+02 * time: 0.0019040107727050781 2 1.260739e+02 1.559715e+02 * time: 0.0030488967895507812 3 5.005600e+01 1.034580e+02 * time: 0.004617929458618164 4 2.562358e+01 7.426721e+01 * time: 0.005960941314697266 5 7.775187e+00 4.265292e+00 * time: 0.007016897201538086 6 5.938181e+00 9.590303e+00 * time: 0.007807016372680664 7 5.303142e+00 2.436536e+01 * time: 0.00855398178100586 8 3.146115e+00 1.014774e+01 * time: 0.009292840957641602 9 2.180215e+00 2.588234e+00 * time: 0.0101318359375 10 1.714000e+00 2.342090e+00 * time: 0.010967016220092773 11 1.385824e+00 1.603570e+00 * time: 0.011739969253540039 12 1.262433e+00 1.010181e+00 * time: 0.012336015701293945 13 1.193454e+00 8.628967e-01 * time: 0.012928962707519531 14 1.168538e+00 9.568073e-01 * time: 0.013535022735595703 15 1.151669e+00 6.331508e-01 * time: 0.014189958572387695 16 1.148250e+00 5.083292e-01 * time: 0.014854907989501953 17 1.146282e+00 1.382512e-01 * time: 0.015578031539916992 18 1.146192e+00 1.878342e-01 * time: 0.01607799530029297 19 1.145136e+00 1.445251e-01 * time: 0.016679048538208008 20 1.144409e+00 1.156383e-01 * time: 0.01729106903076172 21 1.144244e+00 3.846202e-02 * time: 0.017920970916748047 22 1.144143e+00 2.496072e-02 * time: 0.01853489875793457 23 1.144075e+00 1.052256e-02 * time: 0.01916193962097168 24 1.144048e+00 9.322276e-03 * time: 0.019819974899291992 25 1.144041e+00 5.145441e-03 * time: 0.020519018173217773 26 1.144039e+00 2.955804e-03 * time: 0.021149873733520508 27 1.144038e+00 2.301091e-03 * time: 0.02176189422607422 28 1.144037e+00 4.981745e-03 * time: 0.022357940673828125 29 1.144037e+00 3.642429e-03 * time: 0.022958993911743164 30 1.144037e+00 1.534142e-03 * time: 0.0235898494720459 31 1.144037e+00 6.220448e-04 * time: 0.024198055267333984 32 1.144037e+00 3.882335e-04 * time: 0.024888992309570312 33 1.144037e+00 3.317794e-04 * time: 0.025586843490600586 34 1.144037e+00 1.604869e-04 * time: 0.026239871978759766 35 1.144037e+00 1.243124e-04 * time: 0.0268399715423584 36 1.144037e+00 1.524377e-04 * time: 0.02746295928955078 37 1.144037e+00 6.338290e-05 * time: 0.02789592742919922 38 1.144037e+00 3.728940e-05 * time: 0.0285189151763916 39 1.144037e+00 3.866383e-05 * time: 0.029130935668945312 40 1.144037e+00 1.915725e-05 * time: 0.029855966567993164 41 1.144037e+00 1.303086e-05 * time: 0.030536890029907227 42 1.144037e+00 4.742708e-06 * time: 0.03122091293334961 43 1.144037e+00 3.254390e-06 * time: 0.03182697296142578 44 1.144037e+00 2.711997e-06 * time: 0.03243088722229004 45 1.144037e+00 2.139196e-06 * time: 0.03302288055419922 46 1.144037e+00 1.146952e-06 * time: 0.03362083435058594 47 1.144037e+00 1.153891e-06 * time: 0.034214019775390625 48 1.144037e+00 8.805688e-07 * time: 0.03488302230834961 49 1.144037e+00 7.859604e-07 * time: 0.03557300567626953 50 1.144037e+00 6.630402e-07 * time: 0.03619790077209473 51 1.144037e+00 2.651445e-07 * time: 0.036778926849365234 52 1.144037e+00 1.375474e-07 * time: 0.03736686706542969
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))
1.1045183068376976e-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)
2.0076092799647823e-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))
0.00022341788149503677