Structure Euler angle computation

Technical advances in computing and use of quaternions for the Euler angle variables enabled less constrained approaches and allowed vacancies and impurities to be studied in their effect on structures. Distances between molecules were varied, giving good agreement with experiment [150],... 

As one moves up in dimensions in this generalization, common mathematical laws gradually get lost. Quarternions, for example, do not obey the commutative law (q qb qvq, while octonions (8-vectors) in addition do not obey the associative law q q qc a( b c)). Quarternions are encountered for example in relativistic (4-component) quantum mechanics, and they also form a more natural basis for parameterizing the rotation of a three-dimensional structure, rather than the traditional three Euler angles. The latter involve trigonometric functions that are both computationally expensive to evaluate and display singularities. Furthermore, the quartemion formulation treats all the coordinate axes as equivalent, while the Euler parameterization makes the z-axis a special direction. 

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