Normalization convention of WFs

There are several different conventions for Fourier transform, this page describes the normalization convention used in Wannier.jl. In short, we adopt the same convention as the Section II 3 of ^[RMP].

For a unit cell having volume $V$, the Bloch functions $| \psi \rangle$ are periodic in a super cell which is $N$ times the unit cell (thus having volume $NV$), where $N$ is determined by the discretization of the reciprocal space, i.e. the total number of kpoints.

Continuous case

If the super cell is large enough ($N \to \infty$), then the kpoint $\bm{k}$ is a continuous variable in the reciprocal cell having volume $\frac{(2\pi)^3}{V}$. The inner product between bra and ket $\langle f | g \rangle$ are integrated over the the $NV$ super cell that is $(-\infty, \infty)$ in the continuous case. If we normalize the Bloch functions over one unit cell, that is,

\[\langle \psi_{n \bm{k}} | \psi_{n \bm{k}} \rangle_V = \int_V | u_{n \bm{k}}(\bm{r}) |^2 \bm{r} = 1,\]

where $\psi_{n \bm{k}} = \exp(i \bm{k} \bm{r}) u_{n \bm{k}}(\bm{r})$, then the inner product between bra and ket is

\[\langle \psi_{n \bm{k}} | \psi_{m \bm{k^\prime}} \rangle = \frac{(2\pi)^3}{V} \delta_{n m} \delta(\bm{k} - \bm{k^\prime}).\]

This can be proven, e.g. in 1D with periodicity $\bm{R}$, by

\[\begin{aligned} & \langle \psi_{n \bm{k}} | \psi_{m \bm{k^\prime}} \rangle = \int_{-\infty}^{+\infty} \psi_{n \bm{k}}^* \psi_{m \bm{k^\prime}} d\bm{r} \\ = & \int_{-\infty}^{+\infty} \exp(i (\bm{k^\prime} - \bm{k}) \bm{r}) u_{n \bm{k}}^*(\bm{r}) u_{m \bm{k^\prime}}(\bm{r}) d\bm{r} \\ = & \sum_{l = -\infty}^{+\infty} \int_{l \bm{R}}^{(l+1) \bm{R}} d(\bm{r} + l \bm{R}) \exp(i (\bm{k^\prime} - \bm{k}) (\bm{r} + l \bm{R})) u_{n \bm{k}}^*(\bm{r} + l \bm{R}) u_{m \bm{k^\prime}}(\bm{r} + l \bm{R}) \\ = & \sum_{l = -\infty}^{+\infty} \exp(i (\bm{k^\prime} - \bm{k}) l \bm{R}) \int_{0}^{\bm{R}} d\bm{r} \exp(i (\bm{k^\prime} - \bm{k}) \bm{r}) u_{n \bm{k}}^*(\bm{r}) u_{m \bm{k^\prime}}(\bm{r}) d\bm{r} \\ = & \frac{2\pi}{\bm{R}} \operatorname{\text{Ш}}_{\frac{2\pi}{\bm{R}}}(\bm{k^\prime} - \bm{k}) \int_{0}^{\bm{R}} \psi_{n \bm{k}}^* \psi_{m \bm{k^\prime}} d\bm{r} \\ = & \frac{2\pi}{\bm{R}} \delta(\bm{k} - \bm{k^\prime}) \delta_{n m}. \end{aligned}\]

In the third line of the above equation, we replace the real space coordinate $\bm{r}$ by $\bm{r} + l \bm{R}$, such that $\bm{r} \in [0, \bm{R})$. In the fourth line, we utilize the real space periodicity of $u_{n \bm{k}}$. In the second last line, the Dirac comb $\operatorname{\text{Ш}}_{T}(t)$ having periodicity $T$ can be expanded as a Fourier series

\[\operatorname {\text{Ш}} _{T}(t) = {\frac {1}{T}} \sum _{n=-\infty }^{\infty }e^{i2\pi n{\frac {t}{T}}}.\]

In the last line, the Bloch function $| \psi \rangle$ is periodic in $\bm{k}$ thus the Dirac comb is reduced to a Dirac delta function $\delta(\bm{k} - \bm{k^\prime})$, and of course the normalization of $| \psi \rangle$ over one unit cell is used in the last line.

Thus, the Bloch functions are orthogonal w.r.t. $\bm{k}$ only in the super cell, they are not orthogonal when only integrating over one unit cell.

Discretized case

For finite sampling of reciprocal space, e.g. a 1D cell of periodicity $\bm{R}$, sampled by $N$ kpoints, then $\bm{k} = \frac{2\pi}{\bm{R}}\frac{n}{N}, n = 0, 1, ..., N-1$. The inner product between bra and ket is

\[\begin{aligned} & \langle \psi_{n \bm{k}} | \psi_{m \bm{k^\prime}} \rangle = \int_{0}^{N \bm{R}} \psi_{n \bm{k}}^* \psi_{m \bm{k^\prime}} d\bm{r} \\ = & \sum_{l = 0}^{N-1} \exp(i (\bm{k^\prime} - \bm{k}) l \bm{R}) \int_{0}^{\bm{R}} \psi_{n \bm{k}}^* \psi_{m \bm{k^\prime}} d\bm{r} \\ = & N \delta_{\bm{k},\bm{k^\prime}} \delta_{n m}. \end{aligned}\]

The second last line uses the fact that the discrete Fourier transform of Dirac delta function is 1,

\[{\mathcal {F}}(\delta_{n})_{k} = \sum_{n=0}^{N-1} \delta_{n} \cdot e^{-{\frac{i2\pi }{N}} kn} = 1, k = 0, 1, ..., N-1.\]

so that the delta function can be represented by

\[\delta_{n} = \frac{1}{N} \sum_{k=0}^{N-1} e^{{\frac{i2\pi }{N}} kn}\]

Real space representation

In the discrete case, the WFs are the discrete Fourier transform of the Bloch function,

\[| \bm{R} n \rangle = \frac{1}{N} \sum_{\bm{k}} e^{-i \bm{k} \bm{R}} | \psi_{n \bm{k}} \rangle.\]

Note the factor $\frac{1}{N}$ is multiplied in the forward transform, while the inverse transform is

\[| \psi_{n \bm{k}} \rangle = \sum_{\bm{k}} e^{i \bm{k} \bm{R}} | \bm{R} n \rangle.\]

Then the orthonormalization relation of WFs are (note the inner product is integrated over the super cell $NV$)

\[\begin{aligned} & \langle \bm{R} n | \bm{R^\prime} m \rangle \\ = & \frac{1}{N^2} \sum_{\bm{k},\bm{k^\prime}} e^{i \bm{k} \bm{R} - i \bm{k^\prime} \bm{R^\prime}} \langle \psi_{n \bm{k}} | \psi_{m \bm{k^\prime}} \rangle \\ = & \frac{1}{N^2} \sum_{\bm{k},\bm{k^\prime}} e^{i \bm{k} \bm{R} - i \bm{k^\prime} \bm{R^\prime}} N \delta_{\bm{k},\bm{k^\prime}} \delta_{n m} \\ = & \frac{1}{N} \delta_{n m} \sum_{\bm{k}} e^{i \bm{k} (\bm{R} - \bm{R^\prime})} \\ = & \frac{1}{N} \delta_{n m} N \delta_{\bm{R}, \bm{R^\prime}} = \delta_{n m} \delta_{\bm{R}, \bm{R^\prime}}. \end{aligned}\]

Implementation details

• The UNK files (generated by QE pw2wannier90.x) are not normalized to 1 in the unit cell (instead to $N = N_x \times N_y \times N_z$, where $N_x, N_y, N_z$ are the number of points of the real space grid), so we need to multiply a factor of $\frac{1}{\sqrt{N}}$. This is done in the read_realspace_wf function.

  Warning

  This means for other DFT codes to interface with Wannier.jl, the output UNKs need to be normalized to $N$ in the unit cell, otherwise the real space computations, e.g., center(rgrid::RGrid, W::AbstractArray), are wrong!

• Then calculating the WF center in real space can be easily achieved in center(rgrid::RGrid, W::AbstractArray).

  Note

  This is not as accurate as computing WF center in reciprocal space, unless you have a dense real space sampling (usually determined by the energy cutoff in plane-wave code), and generate the real space WF in the $NV$-sized super cell (the WF lives in the super cell!).

• However, Wannier90 does not consider this factor when writing xsf or cube files, so we remove the factor in write_realspace_wf function, such that

  • the output file is consistent with Wannier90 output.
  • another advantage is that WF is normalized in one unit cell, so the numbers in the xsf are not too small.
  Note

  Somehow the Wannier90 output xsf or cube is on a slightly shifted grid, in write_realspace_wf we just output the WF on the unshifted grid so that the origin of the WF 3D mesh is the same as the lattice origin (or its periodic replicas translated by lattice vectors).