ABSTRACT
As noted in the Introduction, there are two conceptually different approaches
to define an atom within a molecule, leading to the “Hilbert space” analy-
sis and the analysis in the three-dimensional (3D) physical space. Classical
quantum chemistry was based on the LCAO concept which later developed
into the use of (larger and larger) atom-centered basis sets as the standard
tools of calculations. Therefore-started with the simple Hu¨ckel theory-the
methods of Hilbert-space analysis were first developed historically. Hilbert
space analysis also has the advantage that it usually does not require calcu-
lation of any new quantities, only some combination of those that are calcu-
lated anyway when computing the wave function. Accordingly, Hilbert space
analysis got the main attention in the present book, too. However, as noted on
page 14, one may expect adequate results only if some “well-balanced” basis
sets are used. Applying methods of Hilbert space analysis, one may meet ir-
recoverable difficulties if the basis set contains diffuse functions lacking any
true atomic character. For anions, in particular, the use of diffuse functions
is mandatory, and their treatment by using Hilbert-space based indices is vir-
tually impossible. (The same holds to a large extent also to intermolecular
interactions, for which obtaining any quantitative results requires the use of
diffuse functions.) From a conceptual point of view, it also represents a prob-
lem that the results of Hilbert space analysis are lacking any basis set limits.
Last but not least, the methods of Hilbert space analysis are not applicable to
the calculations using plane wave basis, which recently became widespread.∗
