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.

References

[1]

J. Pipek, P. G. Mezey A fast intrinsic localization procedure applicable for ab initio and semiempirical linear combination of atomic orbital wave functions, J. Chem. Phys., (1989)

[2]

E. Ö. Jónsson, S. Lethola, M. Puska, H. Jónsson Theory and Application of Generalized Pipek-Mezey Wannier Functions, J. Chem. Theory Comput., (2017)

[3]

S. Lethola, H. Jónsson Pipek-Mezey Orbital Localization Using Various Partial Charge Estimates J. Chem. Theory Comput., (2014)