Atom charges, fragment charges, and bond orders
Mulliken populations can be summarized to yield atomic charges. Alternative methods exist to deduce atom charges from the self-consistent results of a molecular calculation. Three of those alternatives are provided by adf: Hirshfeld analysis, Voronoi analysis, and multipole derived charges.
The Hirshfeld analysis produces a charge value per fragment, computed as the integral of the SCF charge density over space, in each point weighted by the relative fraction of the (initial) density of that fragment in the total initial (sum-of-fragments) density:
Qfrag(i) = ∫ ρSCF ρinitial frag(i)/(∑j ρinitial frag(j)) (5.1.1)
The Voronoi charge analysis consists of assigning the charge density in a point in space to the nearest atom. The Voronoi cell of an atom is the region in space closer to that atom than to any other.
This partitioning of space, using mid-way separation planes, is inappropriate to produce useful absolute numbers when neighboring atoms have very different sizes, for instance, Hydrogen and a heavy metal. However, changes in the density analyzed in this way do give a reasonable general insight in the effect of bonding on the location of charge densities, in particular because the Voronoi data per atom are split up in contributions within the atomic sphere and the rest of its Voronoi cell.
Hirshfeld and Voronoi charge analyzes are printed at the end of the SCF (of the last geometry, in case of an Optimization).
The Hirshfeld analysis in adf produces charges per fragment, so that atomic charges are obtained only if single-atom fragments are used. This limitation does not apply to Voronoi charges (data per atom). Mulliken charges are given both per atom and per fragment.
In the printout of charges per fragment (as for the Hirshfeld analysis), you have to be aware of the ordering of fragments. A complete list of fragments is printed in the early GEOMETRY section of standard output, where you also find which atom(s) correspond(s) to which fragment. Note that even when you use single-atom fragments only, the order of fragments is usually quite different from the order of atoms in your input file. Typically (but not necessarily exactly in each case), when you use single-atom fragments: consider the first non-dummy atom in your ATOMS block. This defines the first atom type. Then browse the ATOMS list until you find an atom of a different type. This defines the second atom type, and so on. The single-atom fragment list will often be such that you first get all atoms of the first atom type, then all atoms of the second type, and so on. Check the printed list-of-fragments always, to avoid mistakes in assigning Hirshfeld charges to atoms (fragments).
The multipole derived charges (MDC) analysis [170] uses the atomic multipoles (obtained from the fitted density) up to some level X, and reconstructs these multipoles exactly (up to level X) by distributing charges over all atoms. This is achieved by using Lagrange multipliers and a weight function to keep the multipoles local. Since the atomic multipoles are reconstructed up to level X, the molecular multipoles are represented also up to level X. The recommended level is to reconstruct up to quadrupole: MDC-q charges. The SCF should have converged for a meaningful MDC analysis.
Mayer bond orders are printed at the end of the SCF, see the key EXTENDEDPOPAN.
