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 1.597445e+02 1.635877e+02 * time: 0.0006971359252929688 1 1.540207e+02 1.529545e+02 * time: 0.002201080322265625 2 1.168600e+02 1.706955e+02 * time: 0.003958940505981445 3 4.719435e+01 1.074110e+02 * time: 0.005813121795654297 4 1.990498e+01 6.233995e+01 * time: 0.0075130462646484375 5 1.446872e+01 5.470987e+01 * time: 0.008764028549194336 6 1.028559e+01 3.463054e+01 * time: 0.009814977645874023 7 9.239520e+00 1.018030e+01 * time: 0.010827064514160156 8 6.585533e+00 1.100819e+01 * time: 0.011834144592285156 9 3.948380e+00 1.142955e+01 * time: 0.012946128845214844 10 2.122278e+00 2.391355e+00 * time: 0.01401209831237793 11 1.416845e+00 1.553289e+00 * time: 0.015066146850585938 12 1.369622e+00 1.573589e+00 * time: 0.01585698127746582 13 1.250320e+00 7.940685e-01 * time: 0.01666712760925293 14 1.184392e+00 8.198353e-01 * time: 0.017460107803344727 15 1.158106e+00 4.092584e-01 * time: 0.018252134323120117 16 1.148319e+00 1.811436e-01 * time: 0.019036054611206055 17 1.145951e+00 1.127054e-01 * time: 0.019865989685058594 18 1.145023e+00 7.041936e-02 * time: 0.02069711685180664 19 1.144439e+00 4.518698e-02 * time: 0.021502017974853516 20 1.144231e+00 6.553943e-02 * time: 0.022083044052124023 21 1.144195e+00 6.369244e-02 * time: 0.022686004638671875 22 1.144130e+00 3.549207e-02 * time: 0.02433609962463379 23 1.144091e+00 1.907598e-02 * time: 0.02518606185913086 24 1.144058e+00 1.124082e-02 * time: 0.0260009765625 25 1.144040e+00 7.059753e-03 * time: 0.026803970336914062 26 1.144040e+00 4.578568e-03 * time: 0.027383089065551758 27 1.144038e+00 4.604402e-03 * time: 0.028222084045410156 28 1.144037e+00 2.416074e-03 * time: 0.029050111770629883 29 1.144037e+00 1.055905e-03 * time: 0.02986311912536621 30 1.144037e+00 6.336158e-04 * time: 0.03067493438720703 31 1.144037e+00 7.362609e-04 * time: 0.03147411346435547 32 1.144037e+00 2.860822e-04 * time: 0.03205394744873047 33 1.144037e+00 1.964097e-04 * time: 0.03290200233459473 34 1.144037e+00 1.304958e-04 * time: 0.03381705284118652 35 1.144037e+00 1.078917e-04 * time: 0.0346379280090332 36 1.144037e+00 4.127986e-05 * time: 0.03550004959106445 37 1.144037e+00 6.999419e-05 * time: 0.036366939544677734 38 1.144037e+00 3.934010e-05 * time: 0.03724193572998047 39 1.144037e+00 2.635852e-05 * time: 0.03808093070983887 40 1.144037e+00 1.961053e-05 * time: 0.03891801834106445 41 1.144037e+00 1.166020e-05 * time: 0.03971505165100098 42 1.144037e+00 7.174118e-06 * time: 0.04052400588989258 43 1.144037e+00 6.365032e-06 * time: 0.0413510799407959 44 1.144037e+00 4.842403e-06 * time: 0.04215693473815918 45 1.144037e+00 3.860060e-06 * time: 0.04297900199890137 46 1.144037e+00 3.154550e-06 * time: 0.044181108474731445 47 1.144037e+00 2.319926e-06 * time: 0.0450289249420166 48 1.144037e+00 1.045393e-06 * time: 0.04588508605957031 49 1.144037e+00 6.560508e-07 * time: 0.04670095443725586 50 1.144037e+00 2.975209e-07 * time: 0.04751706123352051
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))
8.338746851893258e-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)
3.326664185386434e-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))
0.00022348749912162638