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Some notes on the interpolation

In wannier90 v.2.1, a new flag use_ws_distance has been introduced (and it is set to .true. by default since version v3.0). Setting it to .false. reproduces the "standard" behavior of wannier90 in v.2.0.1 and earlier, while setting it to .true. changes the interpolation method as described below. In general, this allows a smoother interpolation, helps reducing (a bit) the number of \(k-\)points required for interpolation, and reproduces the band structure of large supercells sampled at \(\Gamma\) only (setting it to .false. produces instead flat bands, which might instead be the intended behaviour for small molecules carefully placed at the centre of the cell).

The core idea rests on the fact that the Wannier functions \(w_{n\bm{\mathrm{R}}}(\bm{\mathrm{r}})\) that we build from \(N\times M\times L\) \(k-\)points are actually periodic over a supercell of size \(N\times M\times L\), but when you use them to interpolate you want them to be zero outside this supercell. In 1D it is pretty obvious want we mean here, but in 3D what you really want that they are zero outside the Wigner--Seitz cell of the \(N\times M\times L\) superlattice.

The best way to impose this condition is to check that every real-space distance that enters in the \(R\to k\) Fourier transform is the shortest possible among all the \(N\times M\times L-\)periodic equivalent copies.

If the distances were between unit cells, this would be trivial, but the distances are between Wannier functions which are not centred on \(\bm{\mathrm{R}}=0\). Hence, when you want to consider the matrix element of a generic operator \(\bm{\mathrm{O}}\) (i.e., the Hamiltonian) \(\langle w_{i\bm{\mathrm{0}}}(\bm{\mathrm{r}})|\bm{\mathrm{O}}|w_{j\bm{\mathrm{R}}}(\bm{\mathrm{r}})\rangle\) you must take in account that the centre \(\bm{\mathrm{\tau}}_i\) of \(w_{i\bm{\mathrm{0}}}(\bm{\mathrm{r}})\) may be very far away from \(\bm{\mathrm{0}}\) and the centre \(\bm{\mathrm{\tau}}_j\) of \(w_{j\bm{\mathrm{R}}}(\bm{\mathrm{r}})\) may be very far away from \(\bm{\mathrm{R}}\).

There are many way to find the shortest possible distance between \(w_{i\bm{\mathrm{0}}}(\bm{\mathrm{r}})\) and \(w_{j\bm{\mathrm{R}}}(\bm{\mathrm{r}}-\bm{\mathrm{R}})\), the one used here is to consider the distance \(\bm{\mathrm{d}}_{ij\bm{\mathrm{R}}} = \bm{\mathrm{\tau}}_i - (\bm{\mathrm{\tau}}_j+\bm{\mathrm{R}})\) and all its superlattice periodic equivalents \(\bm{\mathrm{d}}_{ij\bm{\mathrm{R}}}+ \bm{\mathrm{\tilde R}}_{nml}\), with \(\bm{\mathrm{\tilde R}}_{nml} = (Nn\bm{\mathrm{a}}_1 + Mm\bm{\mathrm{a}}_2 + Ll\bm{\mathrm{a}}_3)\) and \(n,l,m = {-L,-L+1,...0,...,L-1,L}\), with \(L\) controlled by the parameter ws_search_size.

Then,

  1. if \(\bm{\mathrm{d}}_{ij\bm{\mathrm{R}}}+ \bm{\mathrm{\tilde R}}_{nml}\) is inside the \(N\times M \times L\) super-WS cell, then it is the shortest, take it and quit

  2. if it is outside the WS, then it is not the shortest, throw it away

  3. if it is on the border/corner of the WS then it is the shortest, but there are other choices of \((n,m,l)\) which are equivalent, find all of them

In all distance comparisons, a small but finite tolerance is considered, which can be controlled with the parameter ws_distance_tol.

Because of how the Fourier transform is defined in the wannier90 code (not the only possible choice) it is only \(\bm{\mathrm{R}}+\bm{\mathrm{\tilde R}}_{nml}\) that enters the exponential, but you still have to consider the distance among the actual centres of the Wannier functions. Using the centres of the unit-cell to which the Wannier functions belong is not enough (but is easier, and saves you one index).

Point 3 is not stricly necessary, but using it helps enforcing the symmetry of the system in the resulting band structure. You will get some small but evident symmetry breaking in the band plots if you just pick one of the equivalent \(\bm{\mathrm{\tilde R}}\) vectors.

Note that in some cases, all this procedure does absolutely nothing, for instance if all the Wannier function centres are very close to 0 (e.g., a molecule carefully placed in the periodic cell).

In some other cases, the effect may exist but be imperceptible. E.g., if you use a very fine grid of \(k-\)points, even if you don't centre each functions perfectly, the periodic copies will still be so far away that the change in centre applied with \(\tt use\_ws\_distance\) does not matter.

When instead you use few \(k-\)points, activating the \(\tt use\_ws\_distance\) may help a lot in avoiding spurious oscillations of the band structure even when the Wannier functions are well converged.