  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.
