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Theory and implementation

The SD can be seen as an generalization of the pair distribution function. Indeed, pair distribution functions are defined as orientionally averaged distribution functions. are in the sense that, Altough these correlation functions reflects many key features of the short-range order in molecular systems, it should be realized that an average spatial assembly of non-spherical particles can not be uniquely characterized from these one-dimensionals functions. So, structural models postulated for the molecular ordering in nonsimple systems based only on one-dimensional PDF will always be somewhat ambiguous. The goal of SD analysis is to provide greater clarity in the structual analysis of molecular systems by utilizing distribution function which span both the radial and angular coordinates of the separation vector. This can provide useful information about the average local structure in a complex system.

nMOLDYN allows one to compute the SD in spherical coordinates on a set of concentrics shells surrounding the centers of mass of selected triplets of atoms using the formula:

| (4.194)

where and are respectively the number of triplets and groups, , and are the spherical coordinates at which the SD is evaluated, , and are respectively the number of discrete r, and values and is the number of group of atoms of type g whose centers of mass is found to be in the volume element defined by , and in the spherical coordinates basis centered on the center of mass of triplet t.

So technically, nMOLDYN proceeds more or less on the following way:

- defines the center of mass for each triplet of atoms,
- defines the center of mass for each group of atoms,
- constructs an oriented orthonormal basis centered on each , this basis is defined from the three vectors , , ,
  - where and are respectively the normalized vectors in (a1,a2) and (a1,a3) directions where (a1,a2,a3) are the three atoms of the triplet t,
  - is defined as the clockwise normal vector orthogonal to that belongs to the plane defined by a1, a2 and a3 atoms,
- expresses the cartesian coordinates of each in each ,
- transforms these coordinates in spherical coordinates,
- discretizes the spherical coordinates in , and ,
- does .

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