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.
Of the three methods applied in adf to compute charges (Mulliken, Hirshfeld, Voronoi) we recommend the Hirshfeld analysis [125, 126] and the analysis based on Voronoi deformation density (VDD) charges [109, 127], see below. The fragments to which the Hirshfeld charges apply are enumerated in the early geometry part of the output file, where for each fragment the numbers of the atoms are given that belong to the fragment. The sum of the Hirshfeld charges may not add up to the analytical net total charge of the molecule. Any deviation from this is caused by numerical integration precision (small effect) and the neglect of long-distance terms that adf uses to speed up the integral evaluations. This approximation does not affect very much the energy and molecular orbital properties, but it does show up in the sum-of-charges somewhat more. It does not indicate an error (unless the deviation is really large, say in the order of 1‰ of the total number of electrons).
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 VDD method is based on the deformation density and a rigorous partitioning of space into non-overlapping atomic areas, the so-called Voronoi cells [109, 127, 128]. The Voronoi cell of an atom A is the region in space closer to nucleus A than to any other nucleus (cf. Wigner-Seitz cells in crystals). The VDD charge of an atom A monitors the flow of charge into, or out of the atomic Voronoi cell as a result of 'turning on' the chemical interactions between the atoms. The VDD method summarizes the three-dimensional deformation density on a per-atom basis. It is conceptually simple and affords a transparent interpretation based on the plausible notion of charge redistribution due to chemical bonding, i.e. the gain or loss of charge in well-defined geometrical compartments of space. For the use of VDD in analyzes involving molecular fragments, see Ref. [129].
In the same fashion as for the Hirshfeld analysis, a summation over all atoms is given which should yield zero (for a neutral molecule). The deviation from zero is caused by numerical integration and by neglect-of-long-distance-terms; the same remarks apply as for the Hirshfeld analysis above.
The 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).
