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 LocalNonlinearity(f)
term of DFTK, with f(ρ) = C ρ^α.
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])),
LocalNonlinearity(ρ -> C * ρ^α),
]
model = Model(lattice; n_electrons, terms, spin_polarization=:spinless); # 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 1.597855e+02 1.595717e+02 * time: 0.0005300045013427734 1 1.469986e+02 1.261816e+02 * time: 0.0022919178009033203 2 1.115577e+02 1.294368e+02 * time: 0.0041179656982421875 3 3.741183e+01 7.778237e+01 * time: 0.006308078765869141 4 1.132672e+01 2.650572e+01 * time: 0.008222103118896484 5 5.237033e+00 2.651336e+01 * time: 0.009653091430664062 6 3.525862e+00 5.232413e+00 * time: 0.010807037353515625 7 2.579229e+00 4.300211e+00 * time: 0.011939048767089844 8 1.857861e+00 5.166602e+00 * time: 0.013085126876831055 9 1.348603e+00 1.152814e+00 * time: 0.013997077941894531 10 1.266251e+00 8.369854e-01 * time: 0.014881134033203125 11 1.236860e+00 1.189976e+00 * time: 0.0157620906829834 12 1.180866e+00 1.116620e+00 * time: 0.016663074493408203 13 1.155944e+00 6.315887e-01 * time: 0.017580032348632812 14 1.146175e+00 1.323158e-01 * time: 0.01848292350769043 15 1.145246e+00 9.650417e-02 * time: 0.019381999969482422 16 1.144528e+00 5.546970e-02 * time: 0.020288944244384766 17 1.144217e+00 3.787587e-02 * time: 0.02118396759033203 18 1.144109e+00 4.298021e-02 * time: 0.02210402488708496 19 1.144089e+00 2.082608e-02 * time: 0.02300405502319336 20 1.144053e+00 1.406235e-02 * time: 0.023906946182250977 21 1.144046e+00 1.115889e-02 * time: 0.024810075759887695 22 1.144042e+00 7.881216e-03 * time: 0.02573394775390625 23 1.144039e+00 4.643694e-03 * time: 0.026634931564331055 24 1.144038e+00 2.360815e-03 * time: 0.027534008026123047 25 1.144037e+00 1.434339e-03 * time: 0.028425931930541992 26 1.144037e+00 1.088896e-03 * time: 0.029330015182495117 27 1.144037e+00 4.398326e-04 * time: 0.030247926712036133 28 1.144037e+00 3.776739e-04 * time: 0.03115105628967285 29 1.144037e+00 2.171108e-04 * time: 0.032058000564575195 30 1.144037e+00 1.694329e-04 * time: 0.03296613693237305 31 1.144037e+00 1.174734e-04 * time: 0.03389406204223633 32 1.144037e+00 6.650623e-05 * time: 0.03479599952697754 33 1.144037e+00 5.347064e-05 * time: 0.035427093505859375 34 1.144037e+00 4.466947e-05 * time: 0.036309003829956055 35 1.144037e+00 2.760755e-05 * time: 0.03720593452453613 36 1.144037e+00 2.058599e-05 * time: 0.03815007209777832 37 1.144037e+00 1.853899e-05 * time: 0.03906607627868652 38 1.144037e+00 7.690783e-06 * time: 0.0399630069732666 39 1.144037e+00 5.385035e-06 * time: 0.04087996482849121 40 1.144037e+00 7.139302e-06 * time: 0.041548967361450195 41 1.144037e+00 3.974504e-06 * time: 0.042472124099731445 42 1.144037e+00 2.237889e-06 * time: 0.04339313507080078 43 1.144037e+00 1.608997e-06 * time: 0.04430413246154785 44 1.144037e+00 6.633127e-07 * time: 0.045213937759399414 45 1.144037e+00 4.057094e-07 * time: 0.04614710807800293 46 1.144037e+00 2.924376e-07 * time: 0.04706096649169922 47 1.144037e+00 2.454457e-07 * time: 0.04796314239501953 48 1.144037e+00 1.680865e-07 * time: 0.04887104034423828
Energy breakdown (in Ha): Kinetic 0.2682057 ExternalFromReal 0.4707475 LocalNonlinearity 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))
8.096429383436293e-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)
3.048896039669727e-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))
0.00022347205815728655