Wannierization using Wannier.jl or Wannier90
DFTK features an interface with the programs Wannier.jl and Wannier90, in order to compute maximally-localized Wannier functions (MLWFs) from an initial self consistent field calculation. All processes are handled by calling the routine Wannier.Model (for Wannier.jl) or run_wannier90 (for Wannier90).
This code is at an early stage and has so far not been fully tested. Bugs are likely and we welcome issues in case you find any!
This example shows how to obtain the MLWFs corresponding to the first five bands of graphene. Since the bands 2 to 11 are entangled, 15 bands are first computed to obtain 5 MLWFs by a disantanglement procedure.
using DFTK
using Plots
using Unitful
using UnitfulAtomic
d = 10u"Å"
a = 2.641u"Å" # Graphene Lattice constant
lattice = [a -a/2 0;
0 √3*a/2 0;
0 0 d]
C = ElementPsp(:C, load_psp("hgh/pbe/c-q4"))
atoms = [C, C]
positions = [[0.0, 0.0, 0.0], [1//3, 2//3, 0.0]]
model = model_DFT(lattice, atoms, positions; functionals=PBE())
basis = PlaneWaveBasis(model; Ecut=15, kgrid=[5, 5, 1])
nbandsalg = AdaptiveBands(basis.model; n_bands_converge=15)
scfres = self_consistent_field(basis; nbandsalg, tol=1e-5);n Energy log10(ΔE) log10(Δρ) Diag Δtime
--- --------------- --------- --------- ---- ------
1 -11.14943368689 -0.67 8.8 557ms
2 -11.15007308736 -3.19 -1.40 1.0 232ms
3 -11.15010737224 -4.46 -2.77 3.6 299ms
4 -11.15010940002 -5.69 -3.31 4.2 390ms
5 -11.15010943172 -7.50 -4.20 3.0 300ms
6 -11.15010943369 -8.71 -5.00 3.4 354msPlot bandstructure of the system
bands = compute_bands(scfres; kline_density=10)
plot_bandstructure(bands)
Wannierization with Wannier.jl
Now we use the Wannier.Model routine to generate a Wannier.jl model that can be used to perform the wannierization procedure. For now, this model generation produces file in the Wannier90 convention, where all files are named with the same prefix and only differ by their extensions. By default all generated input and output files are stored in the subfolder "wannierjl" under the prefix "wannier" (i.e. "wannierjl/wannier.win", "wannierjl/wannier.wout", etc.). A different file prefix can be given with the keyword argument fileprefix as shown below.
We now produce a simple Wannier model for 5 MLFWs.
For a good MLWF, we need to provide initial projections that resemble the expected shape of the Wannier functions. Here we will use:
- 3 bond-centered 2s hydrogenic orbitals for the expected σ bonds
- 2 atom-centered 2pz hydrogenic orbitals for the expected π bands
using Wannier # Needed to make Wannier.Model availableFrom chemical intuition, we know that the bonds with the lowest energy are:
- the 3 σ bonds,
- the π and π* bonds.
We provide relevant initial projections to help Wannierization converge to functions with a similar shape.
C_Z = charge_nuclear(C)
s_guess(center) = DFTK.HydrogenicWannierProjection(center, 2, 0, 0, C_Z)
pz_guess(center) = DFTK.HydrogenicWannierProjection(center, 2, 1, 0, C_Z)
projections = [
# Note: fractional coordinates for the centers!
# 3 bond-centered 2s hydrogenic orbitals to imitate σ bonds
s_guess((positions[1] + positions[2]) / 2),
s_guess((positions[1] + positions[2] + [0, -1, 0]) / 2),
s_guess((positions[1] + positions[2] + [-1, -1, 0]) / 2),
# 2 atom-centered 2pz hydrogenic orbitals
pz_guess(positions[1]),
pz_guess(positions[2]),
]5-element Vector{DFTK.HydrogenicWannierProjection}:
DFTK.HydrogenicWannierProjection([0.16666666666666666, 0.3333333333333333, 0.0], 2, 0, 0, 6)
DFTK.HydrogenicWannierProjection([0.16666666666666666, -0.16666666666666669, 0.0], 2, 0, 0, 6)
DFTK.HydrogenicWannierProjection([-0.33333333333333337, -0.16666666666666669, 0.0], 2, 0, 0, 6)
DFTK.HydrogenicWannierProjection([0.0, 0.0, 0.0], 2, 1, 0, 6)
DFTK.HydrogenicWannierProjection([0.3333333333333333, 0.6666666666666666, 0.0], 2, 1, 0, 6)Wannierize:
wannier_model = Wannier.Model(scfres;
fileprefix="wannier/graphene",
n_bands=scfres.n_bands_converge,
n_wannier=5,
projections,
dis_froz_max=ustrip(auconvert(u"eV", scfres.εF))+1) # maximum frozen window, for example 1 eV above Fermi levellattice: Å
a1: 2.64100 0.00000 0.00000
a2: -1.32050 2.28717 0.00000
a3: 0.00000 0.00000 10.00000
atoms: fractional
C: 0.00000 0.00000 0.00000
C: 0.33333 0.66667 0.00000
n_bands: 15
n_wann : 5
kgrid : 5 5 1
n_kpts : 25
n_bvecs: 8
b-vectors:
[bx, by, bz] / Å⁻¹ weight
1 -0.47582 -0.27471 0.00000 1.10422
2 0.47582 0.27471 0.00000 1.10422
3 0.00000 -0.54943 0.00000 1.10422
4 0.00000 0.54943 0.00000 1.10422
5 0.47582 -0.27471 0.00000 1.10422
6 -0.47582 0.27471 0.00000 1.10422
7 0.00000 0.00000 -0.62832 1.26651
8 0.00000 0.00000 0.62832 1.26651Once we have the wannier_model, we can use the functions in the Wannier.jl package:
Compute MLWF:
U = disentangle(wannier_model, max_iter=200);[ Info: Initial spread
WF center [rx, ry, rz]/Šspread/Ų
1 0.00000 0.76239 -0.00000 0.62113
2 0.66025 -0.38120 -0.00000 0.62113
3 -0.66025 -0.38120 -0.00000 0.62113
4 0.00000 -0.00000 0.00000 1.19369
5 0.00000 1.52478 -0.00000 1.19369
Sum spread: Ω = ΩI + Ω̃, Ω̃ = ΩOD + ΩD
ΩI = 3.81157
Ω̃ = 0.43920
ΩOD = 0.43823
ΩD = 0.00097
Ω = 4.25077
[ Info: Initial spread (with states freezed)
WF center [rx, ry, rz]/Šspread/Ų
1 0.00000 0.76239 -0.00000 0.64218
2 0.66025 -0.38120 -0.00000 0.64218
3 -0.66025 -0.38120 -0.00000 0.64218
4 0.00000 0.00000 0.00000 1.13077
5 -0.00000 1.52478 -0.00000 1.13077
Sum spread: Ω = ΩI + Ω̃, Ω̃ = ΩOD + ΩD
ΩI = 3.46548
Ω̃ = 0.72262
ΩOD = 0.71487
ΩD = 0.00775
Ω = 4.18810
Iter Function value Gradient norm
0 4.188096e+00 1.401803e+00
* time: 0.0009210109710693359
1 3.807531e+00 8.825707e-02
* time: 0.005249977111816406
2 3.768218e+00 2.015970e-02
* time: 0.009502887725830078
3 3.765765e+00 2.446943e-03
* time: 0.013743877410888672
4 3.765725e+00 2.711466e-04
* time: 0.018002986907958984
5 3.765725e+00 3.155852e-05
* time: 0.025639057159423828
6 3.765725e+00 1.434236e-05
* time: 0.03175783157348633
7 3.765725e+00 1.359860e-06
* time: 0.03605198860168457
[ Info: Final spread
WF center [rx, ry, rz]/Šspread/Ų
1 -0.00000 0.76239 -0.00000 0.64174
2 0.66025 -0.38120 0.00000 0.64174
3 -0.66025 -0.38120 0.00000 0.64174
4 0.00000 0.00000 0.00000 0.92025
5 -0.00000 1.52478 -0.00000 0.92025
Sum spread: Ω = ΩI + Ω̃, Ω̃ = ΩOD + ΩD
ΩI = 3.02222
Ω̃ = 0.74351
ΩOD = 0.73886
ΩD = 0.00465
Ω = 3.76572Inspect localization before and after Wannierization:
omega(wannier_model)
omega(wannier_model, U) WF center [rx, ry, rz]/Šspread/Ų
1 -0.00000 0.76239 -0.00000 0.64174
2 0.66025 -0.38120 0.00000 0.64174
3 -0.66025 -0.38120 0.00000 0.64174
4 0.00000 0.00000 0.00000 0.92025
5 -0.00000 1.52478 -0.00000 0.92025
Sum spread: Ω = ΩI + Ω̃, Ω̃ = ΩOD + ΩD
ΩI = 3.02222
Ω̃ = 0.74351
ΩOD = 0.73886
ΩD = 0.00465
Ω = 3.76572
Build a Wannier interpolation model:
kpath = irrfbz_path(model)
interp_model = Wannier.InterpModel(wannier_model; kpath=kpath)
recip_lattice: Å⁻¹
a1: 2.37909 1.37357 -0.00000
a2: 0.00000 2.74714 0.00000
a3: 0.00000 -0.00000 0.62832
kgrid : 5 5 1
n_kpts : 25
lattice: Å
a1: 2.64100 0.00000 0.00000
a2: -1.32050 2.28717 0.00000
a3: 0.00000 0.00000 10.00000
grid = 5 5 1
n_rvecs = 31
using MDRS interpolation
KPath{3} (6 points, 3 paths, 12 points in paths):
points: :M => [0.5, 0.0, 0.0]
:A => [0.0, 0.0, 0.5]
:H => [0.333333, 0.333333, 0.5]
:K => [0.333333, 0.333333, 0.0]
:Γ => [0.0, 0.0, 0.0]
:L => [0.5, 0.0, 0.5]
paths: [:Γ, :M, :K, :Γ, :A, :L, :H, :A]
[:L, :M]
[:H, :K]
basis: [1.258962, 0.726862, -0.0]
[0.0, 1.453724, 0.0]
[0.0, -0.0, 0.332492]And so on... Refer to the Wannier.jl documentation for further examples.
(Delete temporary files when done.)
rm("wannier", recursive=true)Custom initial guesses
We can also provide custom initial guesses for Wannierization, by passing a callable function in the projections array. The function receives the basis and a list of points (fractional coordinates in reciprocal space), and returns the Fourier transform of the initial guess function evaluated at each point.
For example, we could use Gaussians for the σ and pz guesses with the following code:
s_guess(center) = DFTK.GaussianWannierProjection(center)
function pz_guess(center)
# Approximate with two Gaussians offset by 0.5 Å from the center of the atom
offset = model.inv_lattice * [0, 0, austrip(0.5u"Å")]
center1 = center + offset
center2 = center - offset
# Build the custom projector
(basis, ps) -> DFTK.GaussianWannierProjection(center1)(basis, ps) - DFTK.GaussianWannierProjection(center2)(basis, ps)
end
# Feed to Wannier via the `projections` as before...pz_guess (generic function with 1 method)This example assumes that Wannier.jl version 0.3.2 is used, and may need to be updated to accommodate for changes in Wannier.jl.
Note: Some parameters supported by Wannier90 may have to be passed to Wannier.jl differently, for example the max number of iterations is passed to disentangle in Wannier.jl, but as num_iter to run_wannier90.
Wannierization with Wannier90
We can use the run_wannier90 routine to generate all required files and perform the wannierization procedure:
using wannier90_jll # Needed to make run_wannier90 available
run_wannier90(scfres;
fileprefix="wannier/graphene",
n_wannier=5,
projections,
num_print_cycles=25,
num_iter=200,
#
dis_win_max=19.0,
dis_froz_max=ustrip(auconvert(u"eV", scfres.εF))+1, # 1 eV above Fermi level
dis_num_iter=300,
dis_mix_ratio=1.0,
#
wannier_plot=true,
wannier_plot_format="cube",
wannier_plot_supercell=5,
write_xyz=true,
translate_home_cell=true,
);As can be observed standard optional arguments for the disentanglement can be passed directly to run_wannier90 as keyword arguments.
(Delete temporary files.)
rm("wannier", recursive=true)