Molecular Size and Anisometry

From the purely geometrical viewpoint, the size of the distribution can be characterized by a number of molecular lengths. The simplest length is the span R. We shall define this parameter as the radius of the smallest sphere, centered at the center of mass, which completely encloses the distribution of points (where the double vertical bars indicate the norm or length of a vector)  

From the physical point of view, R is akin to an effective rotational radius, such as the hydrodynamic radius of a molecule. Despite its simplicity, R can be a useful descriptor, especially for the analysis of configurational dynamics. 

The most common descriptor for a distribution of n nuclei in space is the instantaneous radius of gyration Rq. In center-of-mass coordinates, it is defined asi°  

Equation [3] corresponds to the instantaneous radius of gyration associated with a frozen conformation. The proper radius of gyration is the configurational average of R and is a well-known measurable property for polymers. 

From structural data, Rq can be determined without knowing the nuclear connectivity. In this sense, Rq measures size, not molecular shape. However, Rq contains more information than R. The radius of gyration also provides an absolute measure of molecular compactness. Note that Rq will have small values if most nuclei are close to the center of mass. In this case, the contribution of a few far nuclei to Rq may be negligible. 

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