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 1.659236e+02 1.273413e+02 * time: 0.0005431175231933594 1 1.629640e+02 1.076857e+02 * time: 0.0017230510711669922 2 1.189965e+02 1.233141e+02 * time: 0.0031621456146240234 3 6.382899e+01 8.841720e+01 * time: 0.0046901702880859375 4 5.285382e+01 8.996111e+01 * time: 0.006206035614013672 5 3.672449e+01 6.707361e+01 * time: 0.007630109786987305 6 9.657292e+00 2.334245e+01 * time: 0.009122133255004883 7 5.714108e+00 5.499301e+00 * time: 0.010296106338500977 8 5.460833e+00 6.627074e+00 * time: 0.011226177215576172 9 4.599566e+00 4.976298e+00 * time: 0.012139081954956055 10 3.679981e+00 3.796006e+00 * time: 0.013181209564208984 11 3.566766e+00 9.160392e+00 * time: 0.013978004455566406 12 2.671424e+00 4.426654e+00 * time: 0.01498723030090332 13 1.940267e+00 1.816700e+00 * time: 0.015896081924438477 14 1.709309e+00 2.622563e+00 * time: 0.016662120819091797 15 1.482670e+00 2.091593e+00 * time: 0.01723504066467285 16 1.291693e+00 1.296827e+00 * time: 0.01830005645751953 17 1.200778e+00 7.728255e-01 * time: 0.019262075424194336 18 1.156592e+00 5.778878e-01 * time: 0.02006220817565918 19 1.152121e+00 4.035080e-01 * time: 0.02081918716430664 20 1.149436e+00 2.725012e-01 * time: 0.021610021591186523 21 1.146264e+00 1.498671e-01 * time: 0.022431135177612305 22 1.144894e+00 6.952439e-02 * time: 0.02320718765258789 23 1.144380e+00 4.165815e-02 * time: 0.02394700050354004 24 1.144285e+00 5.653299e-02 * time: 0.024468183517456055 25 1.144139e+00 2.574190e-02 * time: 0.025199174880981445 26 1.144084e+00 1.957682e-02 * time: 0.026006221771240234 27 1.144044e+00 1.027799e-02 * time: 0.026789188385009766 28 1.144040e+00 4.777109e-03 * time: 0.02761220932006836 29 1.144038e+00 2.389800e-03 * time: 0.028562068939208984 30 1.144037e+00 1.594167e-03 * time: 0.02924323081970215 31 1.144037e+00 1.154946e-03 * time: 0.03010106086730957 32 1.144037e+00 9.267397e-04 * time: 0.030907154083251953 33 1.144037e+00 5.423600e-04 * time: 0.03166913986206055 34 1.144037e+00 4.033488e-04 * time: 0.032476186752319336 35 1.144037e+00 3.783241e-04 * time: 0.033306121826171875 36 1.144037e+00 2.676693e-04 * time: 0.03416800498962402 37 1.144037e+00 1.821987e-04 * time: 0.03498411178588867 38 1.144037e+00 1.222264e-04 * time: 0.035822153091430664 39 1.144037e+00 7.712673e-05 * time: 0.036646127700805664 40 1.144037e+00 5.254643e-05 * time: 0.037484169006347656 41 1.144037e+00 2.344311e-05 * time: 0.038346052169799805 42 1.144037e+00 2.212903e-05 * time: 0.03913617134094238 43 1.144037e+00 1.542135e-05 * time: 0.03997921943664551 44 1.144037e+00 9.354198e-06 * time: 0.04082822799682617 45 1.144037e+00 6.800788e-06 * time: 0.041671037673950195 46 1.144037e+00 4.920860e-06 * time: 0.04252910614013672 47 1.144037e+00 3.296588e-06 * time: 0.04340004920959473 48 1.144037e+00 1.754460e-06 * time: 0.04426121711730957 49 1.144037e+00 1.175943e-06 * time: 0.04513120651245117 50 1.144037e+00 8.524324e-07 * time: 0.04598808288574219 51 1.144037e+00 5.671743e-07 * time: 0.04689311981201172 52 1.144037e+00 3.244964e-07 * time: 0.047750234603881836 53 1.144037e+00 1.671850e-07 * time: 0.048693180084228516 54 1.144037e+00 9.836016e-08 * time: 0.04957318305969238 55 1.144037e+00 1.091886e-07 * time: 0.05047917366027832
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))
9.290622208760838e-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)
1.8411086231969123e-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))
0.00022342725151536436