Basic Ideas of Hilbert Space Analysis
István Mayer · 2016
As noted in the introduction, one of the main avenues of analyzing the results of quantum chemical calculations makes use of the fact that most of the prac- tical calculations apply atom-centered basis functions. This permits to define the atom within a molecule as a nucleus and the set of the basis orbitals cen- tered on that atom.∗ This approach can be traced back to the early days of quantum chemistry, when the interaction between the individual atoms has been described by using the atomic orbitals of the free atoms to build up an approximate wave function. Hence comes the term “linear combination of atomic orbitals” (LCAO), which is also often used simply to denote the situ- ations in which finite basis sets are used. A significant part of our “chemical thinking” is based on this approach-it is sufficient to recall here the dis- tinction between sp, sp2, and sp3 carbon atoms; this characterization of the hybridization state of a minimal atomic basis tells us much about the atoms in question. Almost all of our qualitative considerations of the behavior (i.e., electronic state) of an atom in a molecule is based on the concept of a minimal atomic basis.