Distribution of Electron Charge Centres: A Picture of Bonding Based on   Geometric Phases
  In the past two decades, geometric phases have provided a powerful new way of looking at quantum mechanical systems, manifesting themselves in subtle but observable ways. Here, we use them to define a versatile function ("distribution of electron charge centres" or DECC) which can be easily evaluated and interpreted, providing information about electronic structure in real space. Its utility is illustrated by application to a large variety of insulators, metals and molecules, treated here within the framework of density functional theory. The DECC is shown to provide a precise and compact description of bonding. Unshared-electron (ionic) and shared-electron (covalent, metallic) bonds are shown to present clearly distinct signatures: the former are uni-centred while the latter are $n$-centred (n > 1). Moreover, the charge contained in the DECC peaks gives either the ionic charge or the number of shared electrons, which is an even integer for covalent bonds. One obtains revealing insight into the microscopic chemical origins of macroscopic phenomena such as ferroelectricity in PbTi0$_3$ and the anomalous mechanical behaviour of bulk Al relative to that of Cu. 
Joydeep Bhattacharjee, Shobhana Narasimhan and Umesh V Waghmare