Zero-dimensional descriptors

OD (zero-dimensional) descriptors such as molecular weight and atomic composition indexes. 

This is the simplest measure vdth regard to molecular size, defined as the total number of atoms in a molecule. It is a global, zero dimensional, descriptor vdth a high degeneracy. In several applications for the calculation of molecular descriptors, the atom number A refers only to nonhydrogen atoms. 

For any pD-model p 3), we can develop descriptors of variable dimensionality d. Examples of zero-dimensional descriptors are single numbers such as the radius of gyrationio (used for OD, ID, and 2D models) or the molecular volume 1 (used for 2D models). One-dimensional descriptors such as radial distribution functions or knot polynomials are used in OD and ID models, respectively. Two-dimensional descriptors include distance maps and Rama-chandran torsional-angle maps for some OD and ID models. Similarly, molecular graphs (2D descriptors) can be associated with ID models (contour lines), 2D models (molecular surfaces), or 3D models (e.g., the entire electron density function). Shape descriptors of higher dimensionality can also be constructed. 

For convenience, we shall classify the molecular models according to their topological dimensionality, p. A molecular conformation defined by the set of nuclear position vectors is a zero-dimensional (OD) model. A one-dimensional (ID) model corresponds to a molecular skeleton, defined by the set of nuclear positions and their connectivity (bond) matrix. Contour surfaces of one-particle molecular properties such as electron density or electrostatic potential are topologically two-dimensional (2D) models embedded in three dimensions. Finally, we find a true three-dimensional (3D) model whenever an entire one-electron property over all space is involved. This model can be regarded as the continuum of all 2D isoproperty surfaces. The difference among the models is summarized in Figure 1. We shall deal with pD models in this work (p = 0,1, 2, 3). Each of them requires a different type of shape descriptor. 

If the rank of M is less than M and N, then some elements of the matrix S will be zero, and the number of nonnegative elements will equal the rank of M. SVD has been used successfully to reduce the dimensionality of descriptor space of chemical libraries." Xie and co-workers report mapping chemicals with 10% of high-dimensional distances into two dimensions ranging from 30 to 100%, depending on the types of compounds and descriptors used. 

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