Charge density distribution from high resolution X-ray diffraction data

Charge density distribution from high resolution X-ray diffraction data[1,2] is not only a very accurate and elaborated method of structure determination, but also provides access to the structure-reactivity relation. In standard structure determination this is limited to the comparison of geometrical parameters in a known class of compounds (bond lengths and angles, conformation, H-bonds, etc.) to indirectly deduce the reactivity or materials property from a pool of know compounds. The model is fitted to the diffraction data by optimization of nine parameters (three coordinates for the atom position and six for the anisotropic displacement parameters). The atoms are regarded to be independent (Independent Atom Model, IAM). The determined atom position and the displacement parameters can solely count for the right atom type at the correct lattice position with the right site occupation factor, but not for the interatomic region.

However, in this area the most important feature for the chemist is formed, the chemical bond. That is only described by a better adapted structural model, the multipole model (MM).[3] It assigns the gross charge density to spherical harmonics, the so-called multipoles. For example a dipole along the interatomic vector can account for the bond charge density (Fig.1). This is the reason why the residual density of the limited IAM model always shows maxima on the bonds and the MM describes the density much better and gives a flat and featureless residual map (Fig.2).

The charge density distribution modelled that way can be analysed along Bader’s quantummechanical Theory of Atoms in Molecules[4] (QTAIM) and gives beside the interatomic distance many other important descriptors of the bond. According to this theory a bond is formed when the charge density distribution between two atoms forms a saddle point, the so-called Bond Critical Point (BCP). Both atoms are connected by a path of the highest local density, proceeding from one to the other through that bond critical point. This is the bond path. A bond path in Bader’s connotation does not have to be a straight line but may be bent in any direction (bent bonds). All bond paths give the molecular graph and the charge density can be quantified along that path. This provides a more resilient model of the bonding. For instance the bond order in homoatomic bonds can be scaled to the density accumulation at the BCP. Furthermore, if the Laplacian, hence the second derivative of the density (∇2ρ(r)),is negative at the BCP the bond is covalent. If this value is positive the bond adopts high ionic contributions. Maxima in the negative Laplace function of the density stand for valence shell charge concentrations (VSCCs). Those maxima in the non-bonding regions are reminiscent to lone pair positions (Fig. 3).

Quantity and orientation of the VSCCs enable conclusions about the hybridization of the atom. The electrostatic potential of the molecule can be determined and hence the most probable region of an electrophilic or nucleophilic attack in the course of a reaction can be recognized. Beside the key-lock fitting principle the electrostatic potential is the most important recognition feature of a receptor towards a target (Fig. 4).

Hence the charge density distribution provides answers to the most basic questions in chemistry[5]: Is actually a bond formed although the atoms are close enough? To what extent the bond is covalent or ionic? Does the bond show multiple bond character independent from the distance? How has the target to be designed to be recognized and bound reversibly from the receptor? How does the packing in the solid-state affect the materials properties? What is the impact weak interactions have on macroscopic features like conductivity, colour, magnetic and photochemical behaviour?


[1] H. Ott, D. Stalke Nachr. Chem. 2008, 56, 131.

[2] D. Stalke Chem. Eur. J. (Concept) 2011, 17, 9264.

[3] P. Coppens X-ray Charge Density and Chemical Bonding, IUCr Oxford University Press, 1997.

[4] R. F. W. Bader Atoms in Molecules - A Quantum Theory, Oxford University Press, New York, 1990.

[5] U. Flierler, D. Stalke, L. J. Farrugia Chemical Information from Charge Density Studies in Modern Charge Density Analysis, eds. Carlo Gatti, Piero Macchi, Springer, Heidelberg, London, New York, 2012, 435-467.