How to interpret

Before the more chemical aspects of interpretation are treated we feel (watching the newer literature with ELF applications) that a section of the physical interpretation may be useful. While in section Quantum Mechanics the quantum mechanical definition(s) of ELF was given, we shall be concerned now with some common pitfalls and some remarks on what ELF is not.

The relevant kernel of ELF is the ratio χ(r) = D(r) ⁄ Dh(r) and χ(r) = tP(r) ⁄ tP,h(r), respectively. All the other mathematical constructions serves just for the purpose to confine the values within [0, 1] and to give it a suitable sharpness in the η-region of interest for most applications. It is important to note, that these mathematical transformations do not alter the topology of the kernel χ(r), i.e. the critcal points of χ(r)-1 and &eta(r) occur at the same positions in space and have identical rank and signature (r, s).

In the seminal paper of Becke and Edgecombe [BECKE1990] it was argued that "electron localization information" (in the authors' sense) is already contained in D(r), however in an inverse relationship: "...The smaller the probability of finding a second like-spin electron near the reference point, the more highly localized is the reference electron." As it was felt that a direct relationship between "electron localizability" [BECKE1990] (perhaps the better term) and the values of the newly defined "Electron Localization Function" is more desirable, D(r) appears in a reciprocal manner in the final formula. The last but not the least step in the definition was introduced as the "calibration" of D(r) "with respect to the uniform electron gas reference" Dh(r). It is important to recognize, that this last step was the decisive one, as only χ(r) but not D(r) shows all the exciting structuring in direct space that makes ELF such a valuable tool.
The interpretation of &eta(r) distributions for chemical systems in terms of the local effect of the Pauli principle is not as straightforward as it seems from the literature. Surely, D(r) and tP(r) are guided by the Pauli principle, but, as has been mentioned before, they do not reveal the same spatial structuring as the ELF kernel χ(r) [KOHOUT1997]. As ELF shows only the ratio D(r) ⁄ Dh(r) and tP(r) ⁄ tP,h(r), respectively, unless proved otherwise (which is not the case until now), it cannot be expected to directly give information on the local effect of the Pauli principle.

Concerning the interpretation of the absolute values of ELF in the original paper [BECKE1990] the following hint was given: "... the upper limit ELF = 1 corresponding to perfect localization and the value ELF=½ corresponding to electron-gas-like pair probability". Two remarks can be made:
1.) Concerning the upper limit it may be justified that the authors interpret this value "corresponding to perfect localization" as they have given before their definition of "localization". However, in order to avoid confusion, e.g. with the much older physical concept of localized and itinerant electrons used in modern solid state electronic structure theory, the analysis of ELF should usually be performed in terms "high/low values of η" instead of "high/low electron localization". A simple example may serve as a demonstration: a single-determinant (Hartree-Fock or Kohn-Sham) solution for a H2 molecule gives one doubly occupied &sigma bonding orbital. The &eta(r) distribution is equal to 1.0 everywhere in space. Thus, in the sense of the above definition, each electron is "perfectly localized" at every point in space. Concluding this remark there certainly is a relation (to be found in the future) between the physical concept of localized and itinerant electrons and ELF but it seems to be a subtle one and the terms should not be intermixed.
2.) For the value η = ½ the authors give the absolutly correct interpretation. Any further interpretation of the &eta = ½ and lower values has to be justified strictly. Clearly, points or regions with &eta(r) = 0.5 for a chemical system do not imply "perfectly delocalized electrons".

A common pitfall is to associate ELF with the electron density ρ infering that η is small (large), where ρ is small (large). But, for example, the spherical ELF separatrix between the 1st and the 2nd shell for a heavy atom has an η value much below 0.5 although ρ in these regions is extremely high (much higher than in the valence regions, where η values close to 1 may appear).

ELF is a tool to describe chemical bonding including the Lewis picture and going beyond it (see, e.g. [NOURY2002]). It can in principle be derived from experiment, but is usually obtained from quantum mechanical calculation (even if it is sometimes claimed to be "determined" from experimental electron density, see section How to calculate).

The correlation between ELF and chemical bonding is (yet) a topological and not an energetical one. So ELF can be said to represent the organization of chemical bonding in direct space. Although it has been termed "electron localization function" its relation with the physical concept of localized and itinerant (delocalized) electrons (orbital picture) seems to be more subtle (see above). The absolute value of η at critical points does not (yet) play a general role. Instead, the topology is analyzed (see section Topological Analysis). Chemical information can be obtained from ELF attractors taking the other topological elements into account as well. A suitable way to characterize the η(r) representation of chemical bonding for a compound is the construction of a "bifurcation diagram" or "basin interconnection diagram". The attractors can be attributed to
I) bonds,
II) lone pairs,
III) atomic shells and
IV) other elements of chemical bonding.
The attribution is done in an empirical way, as there is no direct proof that relates ELF with these conceptual aspects of chemistry.

Changes of ELF topology on varying a control parameter (e.g. the structure) can be chemically significant or not. This is not very well understood and has to be thoroughly investigated in each case where it occurs.

Polycentric bonding has been related to the synapticity of valence basins (see section Topological Analysis). While this seems to be working well for molecules it seems to be more subtle for solids. Additionally, its relation with orbital-derived pictures of polycentric bonding is not known. There are indications that the basin fluctuations (see section Quantum Mechanics) are useful quantities to discuss polycentric bonding.

The physical meaning of an ELF basin is unknown, as there is no quantum mechanical motivation yet for the definition of a surface of zero flux in the gradient vectors of ELF. The integrated electron density in an ELF basin (electronic basin population) does not correlate in an obvious manner with the energetical aspects of the bonding. However, the electronic basin population characterizes the spatial organization of the bonding in terms of ELF and the electron density. It may not be expected to resemble the bonding analysis in Hilbert space ("population analysis"). An investigation about the relative basin population with respect to a suitable standard bond [CHESNUT2001A] revealed a correlation with the chemical bond order for a selected set of test examples.

last update: 25.07.2002