Subsections

• The attractor of the $ELF$ gradient field
• The hierarchy of the basins.
• Localization domain analysis.
• The synaptic order.
• Superbasins (cwm)

Technique of analysis.

The attractor of the $ELF$ gradient field

$ELF$ is totally symmetric therefore the attractors of its gradient field are single points except for atoms and linear molecules. In an atom the K-shell attractor is located at the nucleus, the spherical symmetry prescribes that the attractors assigned to the other shells are degenerated spheres. These attractor are non-hyperbolic since two of their characteristic exponents are zero. The structural instability of the dynamical system arises from the fact that any non spherical weak perturbation breaks the spherical symmetry. In fact, for an atom the convenient graphical representation of the$ELF$ function is the curve $\eta(r)$.

In a linear molecule, the attractors which are not located on the $C(\phi)$axis are degenerated on circles centered on this axis and lying in planes perpendicular to the axis direction. The dynamical system of the electron localization function of a linear molecule is structurally unstable if it possesses such circular attractors. The convenient graphical representation is a map in any plane containing the internuclear axis.

There is two types of attractors: the core attractors and the valence attractors. The core basins correspond to the core shells of atoms with $Z>2$, they are concentrically organized around each nucleus with $Z>2$. The other basins are valence basins.

The hierarchy of the basins.

Any subset of the molecular space bounded by an external closed isosurface $\eta(\mathbf r) = f$ is a domain. A $f$-localization domain is such a subset with the restriction that each point satisfies $\eta(\mathbfr) > f$. A localization domain surrounds at least one attractor, in this case it is called irreducible. If it contains more than one attractor it is reducible. An irreducible domain is a subset of a basin whereas a reducible one is the union of subsets of different basins. Except for atoms and linear molecules, the irreducible domains are always filled volumes whereas the reducible ones can be either filled volumes or hollowed volumes.
 

 Localization domain analysis.

The hierarchy of the electron localization basins is a powerful tool of analysis of the bonding in molecules and solids within the ``elfological'' framework. It is a generalization of the molecular isodensity contour analysis originally proposed by Mezey. In this approach the basins are ordered with respect to the $ELF$ values at the critical points which determine the reduction of the reducible localization domains.

Upon the increase of the value of $\eta(\mathbf r)$ defining the bounding isosurface, a reducible domain split into several domains each containing less attractors than the parent domain. The reduction of localization occurs for at turning point which are critical points of index 1 located on the separatrix of two basins involved in the parent domain. Ordering these turning points (localization nodes) by increasing $\eta(\mathbf r)$ enables to build tree-diagrams reflecting the hierarchy of the basins. Three types of domains can be distinguished according to the nature of the attractors within them. A core domain contains the core attractor(s) of a given atoms, a valence domain only valence attractors and a composite domain both valence and core ones. For any system there exist low values of $\eta(\mathbf r) = f$ defining a unique composite parent domain. The successive reductions of localization will split this parent domain. Every child which is a composite domain corresponds to one or more chemical species. A chemical unit is the union of the basins of the last appearing composite domain of a branch provided it is a filled volume.

                                                       0.05                                                                         0.07

                                                     0.25                                                                               0.8

                            Reduction of localization of the CO₂ HF complex.

        
 

the first reduction yields two composite domains corresponding to the interacting moieties, therefore such a complex cannot be considered as being chemically bonded. In the same way in an ionic pair the first reduction yields composite domains corresponding to the cation and to the anion. In a molecule the initial parent domain first splits into core domains and a single valence domain which contains all the valence attractors. The shape of this latter domain is that of a hollowed volume with as many holes as atomic cores in the molecule. Each hole contain a core domain. The first reductions correspond to the separation of the core domains from the valence domains which yields two filled volumes (the core domains) encompassed in a hollowed volume (the valence domain). In this case there is a unique chemical object.

                                   0.07                                                0.20                                           0.55

                                             0.66                                                 0.70                                         0.85
 

The synaptic order.

The synaptic order characterizes the valence basins which belong to a given chemical unit. The synaptic order of a valence basin is the number of core basins which have a common boundary (separatrix) with it. If there is a proton within this valence basin it counts one for the synaptic order. According to their synaptic order the valence basins can be asynaptic (synaptic order zero), monosynaptic, disynaptic or polysynaptic. Asynaptic basins correspond to unusual chemical situations such as F centre in surface chemistry, monosynaptic basins are the signature of lone pairs, disynaptic basins of two centre bonds and polysynaptic basin of multicenter bonds. This description is complementary to the valence view point, instead of counting the neighbors from a given atomic centre, one considers the number of centres which are linked to a given valence basin. For example in the ethanol molecule  (above) there are thirteen basins: three core basins one C(O) and two C(C), two disynaptic basins V(C, C), one V(C,O),  five protonated disynaptic basins V(C,H) , one V(O,H) and two monosynaptic ones V(O).
 
 
 

                                             There are seven valence basins in the CH₃F molecules: three protonated
                                    disynaptic V(C,H), one disynaptic V(C,F) and three monosynaptic V(F).

                                                    In diborane there are two protonated trisynaptic basins
                                        (in navy blue and yellow) which correspond to Pitzer's
                                        protonated double bond picture.

                                                        The agostic hydrogen is also represented by a protonated
                                            trisynaptic basin (in navy blue).

                                        Trisynaptic basins are responsible for anomalous coordination.
                                         Here two pentacoordinated carbon in the Al₂H₄(CH₃)₂ molecule.
Superbasins (cwm)
It often happens that irreducible domains corresponding to basins of the same type, for example V(A) monosynaptic basins, are not well separated at the end of a branch. In this case the value of $ELF$ at the turning point is very close to the values at the attractors and it is more chemically meaningful to consider the union of the corresponding basins rather than the individual basins themselves all the more so that the number of such basins might be dependent of the quality of the wave function.

2002-04-01