Powder average with geometry-preserving treatments

For a disordered, or powder, system, the ESEEM represents the average over all possible orientations of the external magnetic field vector B₀, with respect to the molecular PAS.  The number of orientations of B₀ ^­ relative to the molecular PAS that can be evaluated is practically limited by the available computational power (time scale of the calculations) and the required accuracy.  The orientation sampling vectors can be chosen as either a set of random unit vectors, or a set of evenly-distributed points on a unit sphere, or octant, if symmetry permits.  Several methods have been used in the latter cases.  In OPTESIM, a geodesic-sphere sequence generator evenly distributes vectors on a unit sphere. The generator subdivides each of the triangular faces of an icosahedron into four smaller triangles on a unit sphere in an iterative manner, yielding geodesic-sphere sequences at different levels of angular resolution.  Users of OPTESIM also have the option to specify their own orientation vectors, as well as the weights of particular subsets of vectors, which are used in simulating orientation-selection experiments.

The overall envelope modulation when multiple nuclei are coupled to the electron is combined according to the product rule.  Specifically, for the two-pulse sequence, the modulation is as follows ,

and for the three-pulse sequence, the product is taken separately over the α and β manifolds, and the overall envelope modulation is combined, as follows,

In the case of electron spin coupling with multiple nuclei, it is computationally expedient to perform the powder, or spherical, average of individual electron–nuclear interactions prior to combination of the individual, or and , modulation terms. This method implicitly assumes that the hf PAS of all electron–nuclear interactions are coincident.  However, coincidence of all hf PAS is not generally the case, and information about the mutual geometry of the nuclei, which is contained in the order≥2 modulation product terms, is lost with this treatment.  In OPTESIM, the combination rule is specified at the level of a physical model of the geometry of electron–nuclear interactions, as defined by Euler angles that specify the mutual orientation(s) of the different hf PAS.  Therefore, the ESEEM is calculated and  combined at each powder average orientation, and subtle structural information arising from the geometry of the multiple hf PAS can be retrieved from the overall averaged spectrum.