# 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

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 \(n\)

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

Formulated in this way the atomic charge matrix is defined as

where the number of electrons localized on atom \(a\) follows

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.