Gram–Charlier approach for anharmonic atomic displacements in inorganic solids: A review

Abstract

All thermal vibrations of atoms are in fact anharmonic – this is the underlying reason for the manifestation of many physical properties of solids, primarily its thermal expansion. By now, over three decades have passed from the seminal works of Kuhs [Kuhs Statistical description of multimodal atomic probability densities. Acta Crystallogr A. 1983;39:148–158.Kuhs Lead Article Generalized Atomic Displacements in Crystallographic Structure Analysis. Acta Cryst. 1992;A48:80–98. Kuhs The Anharmonic Temperature Factor in Crystallographic Structure Analysis. Aust J Phys. 1988;41:369], and ‘anharmonic’ approaches are now widely used for the description of thermal displacements of the atoms. Experimental data for studies of this phenomenon can be acquired from targeted diffraction experiments, and in this mini review, we analyse the current state of this area, mostly for the inorganic structures. The modern crystallographic hardware and software permits to correct the description of anharmonicity for the objects whereof one or two decades ago this phenomenon was either neglected or beyond the power of data processing. In general, the asphericity of electron density distribution is not necessarily caused by anharmonic motions; another possible reason is the displacement of valence electrons due to the formation of chemical bonds. We analyse the hitherto reported examples of anharmonic description of thermal motions; this phenomenon is most pronounced for univalent cations (like alkalis, silver, copper and thallium) and anions (halides). We also present the new Anharmonicity program suite which permits to determine the maxima of distribution of probability density.

Keywords:

• Anharmonicity
• Single crystal diffraction
• Crystal structure
• Gram–Charlier expansion

Acknowledgements

We would like to express our thanks to Dr. Václav Petříček for fruitful discussions. We are also grateful to two anonymous reviewers for their excellent comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Subject index

Activation energy 169

Edgeworth series expansion 149, 150

Electron density asphericity 174

Gram-Charlier series expansion 151

Hirshfeld Atom Refinement (HAR) 152

Ion migration 161

Kuhs' rule 153

Multipole refinement 152

One-particle potentials 171

‘Shashlik-like’ pattern 152

Skewness vector 171, 172

Splitting of atomic positions 152

Thermal expansion 148, 149, 172

Transferable aspherical atom model (TAAM) 152

Additional information

Funding

This work was supported by the Russian Science Foundation (22-23-01133).