# 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**) ⁄ D_{h}(**r**) and
χ(**r**) = t_{P}(**r**) ⁄ t_{P,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 **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"
**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" D_{h}(**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
t_{P}(**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**)
**r**) ⁄ D_{h}(**r**) and t_{P}(**r**) ⁄ t_{P,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

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 H_{2} 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 1^{st}
and the 2^{nd} 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. *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

last update: 25.07.2002