Three-dimensional moments of molecular property fields

Descriptors that capture certain three-dimensional molecular features and do not require molecular superposition or alignment for the assignment of molecular similarity have recently been proposed and investigated. Among these, moments of molecular features have been utilized in the QSAR of several molecular series. The present work examines certain formal aspects of the use of moment expansions for which the zero-order moment of a property field is nonvanishing. The first-order term of such moment expansion is then dependent upon the origin of expansion, and it is pointed out that expansion about the molecular centroid is descriptive of first-order differences about the property-field mean. For a hydrophobic property field, this first-order term is just the Eisenberg hydrophobic moment. Second-order moments about the centroid can be written as components of the WHIM covariant matrix. Moment expansions are also performed about the property-field center in analogy with expansions about the center of mass. For such expansion, a set of descriptors consisting of moments of the molecular density and of a molecular hydrophobic property field as well as of related quantities are used in a QSAR of the binding of 74 polyhalogenated aromatic molecules to the Ah cystolic receptor. This QSAR has been named CoMMA2 to distinguish it from CoMMA, for which the zero-order moment of the expansion vanishes.