Graphical Representation

After the two calculation steps described in section How to calculate one has obtained a file, which contains the necessary information for the graphical representation: positions and associated η-values. The easiest type of data for the graphical system to handle are those with a structured, regular grid, i.e. the neighbouring points of a reference point are already implictly defined. On the other side of the difficulty scale are unstructured, irregular grids, where just the positions and values are given in arbitrary order and the positions are unrelated to each other. For such a type of mesh the graphics system (or a suitable preprocessing program) has to perform a triangulation of the data.

The Electron Localization Function η(r) for a chemical systemis a scalar field in 3D space. So, in order to show all information in one diagram one would need a 4D-graphic device suited for the human observer. Therefore, different lower dimensional graphical representations have been developed serving special purposes. The most frequently used are the following:

1.) 2D-slices of η(r), where the η-values are coded in a special colourmap similar to the one in a map (this kind of colourmap for ELF has actually become some kind of a standard), or alternatively in form of isolines. They have the advantage that within the plane all the information is given, however nothing in the perpendicular direction, e.g. a point that looks like an ELF attractor in a 2D representation may equally well be a (3, -1) saddle point in three dimensions. In order to clarify the situation another, perpendicular slice through the interesting point is often given in the same diagram.


Fig. 1a: structure of the P4S3 molecule (C3v symmetry) given in a ball-and-stick representation
green spheres: P
red spheres: S


Fig. 1b: ELF diagram for P4S3 coded in standard colormap; only the vertical slice contains a mirror plane; note, that the diagram seems to indicate that the S atoms have thre e ELF attractors for in the lone pair region, which is not true: careful inspection of the topology reveals that the maximum within the vertical plane is in fact a (3, -1) saddle point connecting the two (monosynaptic) attractors above the S atoms' plane.

2.) A 3D-representation of η(r), where only one value of η is shown. These ELF isosurfaces with η= f exactly represent f-localization domains. The colouring of the localization domains can be used to refer to some other property. However, it is clear that focusing on just one single η-value usually hides a lot of details, which may be important or at least of interest, too. So, more than one isosurface diagram may be necessary in order to characterize a special situation.

Fig. 2: ELF isosurface with ηiso = 0.819; the atom cores are already encapsulated by a separate isosurface; the disynaptic attractors between P and S V2(P,S) are contained within irreducible localization domains, while those between two P atoms V2(P,P) have already merged with the monosynaptic V1(P) into a large reducible localization domain.

3.) A combined representation of the electron density and ELF, where an electron density isosurface is used to define a complex surface, on which η(r) is projected. The fact, that there exist very different values of ELF for one and the same electron density value in a molecule clearly demonstrates that there is no correpondence between these two quantities. Also, in the early times of ELF 2D slices with a black background were shown, on which randomly located η(r)-coloured points were projected in such a way that the local point density was proportional to the electron density (log ρ(r)).

Fig. 3: An all-electron density isosurface (ρiso = 0.04 electrons/bohr3) on which η(r) was projected.

4.) As the ELF basins represent important quantities for the discussion of chemical bonding, graphical representations for them are also in use.


Fig. 4:
Visualization of two disynaptic valence basins:

All these representations may also occur intermixed within one single diagram.

last update: 26.06.2002