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Theory and implementation

The PDF is an example of a pair correlation function, which describes how, on average, the atoms in a system are radially packed around each other. This proves to be a particularly effective way of describing the average structure of disordered molecular systems such as liquids. Also in systems like liquids, where there is continual movement of the atoms and a single snapshot of the system shows only the instantaneous disorder, it is extremely useful to be able to deal with the average structure.

The PDF is useful in other ways. For example, it is something that can be deduced experimentally from x-ray or neutron diffraction studies, thus providing a direct comparison between experiment and simulation. It can also be used in conjunction with the interatomic pair potential function to calculate the internal energy of the system, usually quite accurately.

Mathematically, the PDF can be computed using the following formula:

| (4.186)

where is the number of selected species, and are respectively the numbers of atoms of species I and J, and respectively the weights for species I and J (see Section 4.2.1 for more details) and is the partial PDF for and species that can be defined as:

| (4.187)

where is the density of atom of specie J and is the mean number of atoms of specie J in a shell of width dr at distance r of the atom of specie I.

From the computation of PDF, two related quantities are computed in nMOLDYN, the Radial-Distribution Function (RDF) defined as:

| (4.188)

and the Total-Correlation Function (TCF) defined as:

| (4.189)

where is the average atomic density defined as:

| (4.190)

where N is the total number of atoms of the system and the volume of the simulation box.

In nMOLDYN, the PDF, the RDF and the TCF are further splitted into an intra-and inter-molecular parts which added together give respectively the total PDF, RDF and TCF.

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