Multipole density formalism
Unlike the commonly used Independent Atom Model, the Hansen-Coppens Formalism presents an aspherical approach, allowing one to model the electron distribution around a nucleus separately in different directions and therefore describe numerous chemical features of a molecule inside the unit cell of an examined crystal in detail.
The choice of the radial function used to describe this electron density is arbitrary, granted that its value at the origin is finite.
[1] Due to its simplistic approach, this method provides a straightforward model that requires no additional parameters (other than positional and Debye–Waller factors) to be refined.
[3] Although the Kappa formalism is still, strictly speaking, a spherical method, it is an important step towards understanding modern approaches as it allows one to distinguish chemically different atoms of the same element.
In the multipole model description, the charge density around a nucleus is given by the following equation: The spherical part remains almost indistinguishable from the Kappa formalism, the only difference being one parameter corresponding to the population of the inner shell.
While the singular sigma bond of the hydrogen can be described well using certain z-parallel pseudoorbitals, xy-plane oriented multipoles with a 3-fold rotational symmetry will prove more beneficial for flat aromatic structures.
[5] The primary advantage of the Hansen-Coppens formalism is its ability to free the model from spherical restraints and describe the surroundings of a nucleus far more accurately.
X-ray crystallography allows the researcher to precisely determine the position of peak electron density and to reason about the placement of nuclei based on this information.
It is possible (albeit disputable) to freely refine hydrogen atoms' positions using the Hansen-Coppens formalism, after releasing the bond lengths from any restraints derived from neutron measurements.
It may be worth approximating hydrogen atoms' anisotropic displacement parameters, e.g. using SHADE, before introducing the formalism and, possibly, discarding bond distance constraints.
Due to the complex description of the electron field provided by this aspherical model, it becomes possible to establish realistic bond paths between interacting atoms as well as to find and characterise their critical points.
Deeper insight into this data yields useful information about bond strength, type, polarity or ellipticity, and when compared with other molecules brings greater understanding about the actual electron structure of the examined compound.
In real cases, electron density flows freely through the molecule and is not bound by any restrictions resulting from the outdated Bohr atom model and found in IAM.
Therefore, through e.g. an accurate Bader analysis, net atomic charges may be estimated, which again is beneficial for deepening the understanding of systems under investigation.
Applying additional constraints resulting from local symmetry for each atom in a molecule (which decreases the number of refined multipoles)[1] or importing populational parameters from existing databases[11][12] may also be necessary to achieve a passable model.
On the other hand, the aforementioned approaches significantly reduce the amount of information required from experiments, while preserving some level of detail concerning aspherical charge distribution.
[13] Despite their similarity, individual multipoles do not correspond to atomic projections of molecular orbitals of a wavefuntion as resulting from quantum calculations.
Nevertheless, as brilliantly summarized by Stewart, "The structure of the model crystal density, as a superposition of pseudoatoms [...] does have quantitative features which are close to many results based on quantum chemical calculations".
