Statistical approaches for the treatment of anharmonic motion in crystals. I. A comparison of the most frequently used formalisms of anharmonic thermal vibrations

Several statistical approaches to anharmonic motion are known, but there exists no comparison between different formalisms. Therefore the most common formalisms (potential, Edgeworth and Gram-Charlier expansions and α formalism) are compared with each other in part I of this series of papers. Mathematical shortcomings of the potential and Edgeworth expansions may lead to doubtful interpretations of the corresponding probability densities and the potentials. The a formalism works inefficiently in conventional crystallographic refinement programs; it tends to converge badly and its computation is very time consuming. The best results of all are obtained for the Gram-Charlier expansion, which can be easily expanded to tensors up to sixth rank.