# Pipek-Mezey Wannier Functions¶

## Introduction¶

Pipek-Mezey [1] Wannier functions (PMWF) is an alternative to the maximally localized (Foster-Boys) Wannier functions (MLWF). PMWFs are higly localized orbitals with chemical intuition where a distinction is maintained between $$\sigma$$ and $$\pi$$ type orbitals. The PMWFs are as localized as the MLWFs as measured by spread function, whereas the MLWFs frequently mix chemically distinct orbitals [2].

## Theoretical Background¶

In PMWFs the objective function which is maximized is

$\mathcal{P}(\mathbf{W}) = \sum^{N_\mathrm{occ}}_n \sum_{a}^{N_a} \mid Q^a_{nn}(\mathbf{W}) \mid^p$

where the quantity $$Q^a_{nn}$$ is the atomic partial charge matrix of atom $$a$$. $$\mathbf{W}$$ is a unitary matrix which connects the canonical orbitals $$R$$ to the localized orbitals $$n$$

$\psi_n(\mathbf{r}) = \sum_R W_{Rn}\phi_R(\mathbf{r})$

The atomic partial charge is defined by partitioning the total electron density, in real-space, with suitable atomic centered weight functions

$n_a(\mathbf{r}) = w_a(\mathbf{r})n(\mathbf{r})$

Formulated in this way the atomic charge matrix is defined as

$Q^a_{mn} = \int \psi^*_m(\mathbf{r})w_a(\mathbf{r})\psi_n(\mathbf r)d^3r$

where the number of electrons localized on atom $$a$$ follows

$\sum_n^{N_\mathrm{occ}}Q^a_{nn}=n_a$

A choice of Wigner-Seitz or Hirshfeld weight functions is provided, but the orbital localization is insensitive to the choice of weight function [3].

### Localization¶

The PMWFs is applicable to LCAO, PW and FD mode, and to both open and periodic boundary conditions. For periodic simulations a uniform Monkhorst-Pack grid must be used.