Shape Distributions for Gaussian Molecules: Circular and Linear Chains in Two Dimensions

The shape distribution functions P(S_1,S₂) for the principal components of the gyration tensors for Gaussian molecules in 2-dimensional space have been numerically evaluated for both circular and linear chains. It is found that the most probable configurations are quite compact in comparison with the mean squared dimensions. Fluctuations away from the most probable configurations are achieved primarily by extension in one direction only, rather than by a uniform inflation. The theory for disks is given, and these distribution functions are also evaluated. Although disks are improbable when compared with all ellipsoids, their distributions may be formulated on appropriate subspaces. They complete the description of two-dimensional configurations.