Large-Scale Chirality Measures and General Symmetry Deficiency Measures for Functional Group Polyhedra of Proteins

Functional Group Polyhedra provide a simplified representation of the most essential spatial features of macromolecules, especially, of globular proteins. Since the functional group polyhedron model focuses on large scale features, the chirality and other symmetry deficiency measures of these molecules, when adapted to these polyhedra, should also be based on the characterization of large scale shape features. Two new approaches for the evaluation of such symmetry deficiency and chirality measures are presented.     

Logical Models of Molecular Shapes and Their Families

A new discrete mathematical model of molecular shape is proposed, making use of the partition property of a representation of molecular shape. According to its geometrical and topological structure, a molecular surface can be partitioned into unbounded two-dimensional subsets (domains) and some common subsets of closures of two or more domains. The sets of these domains as a base of a finite topology, containing the Boolean n-cube as a lower Boolean sub-lattice of this topology, defines the domain of the proposed logical model. A logical function can be obtained that reflects the properties of the topological domains as well as the interrelations on the set of domains. Based on classical or quantum-chemical representations of molecular shape, these models allow one the implementation of methods of logical diagnostics in chemistry, and the definition of a metric on the set of molecular shape equivalence classes. The families of molecular shapes can be considered as sets of logical models. The proposed model is unified in the sense that the structures of differentiable and non-differentiable surfaces can be represented in the same mathematical framework. These logical models will also work for interpenetrations of the above types of surfaces.     

Symmetry and periodicity of potential surfaces: a test for multicenter interactions

Symmetry and periodicity of potential energy surfaces of chemical reactions and conformational changes are determined by the symmetry properties of the nuclear frameworks of all possible nuclear configurations of the given overall stoichiometry. For example, a mirror plane of a nuclear configuration implies a mirror plane of the potential surface (or that of the potential energy hypersurface in higher dimensions), and a local rotational symmetry of substituents implies a translational symmetry, that is, periodicity of the potential surface, if the latter is defined in terms of the usual bond length/bond angle internal coordinates. Such symmetry relations on potential surfaces are rather trivial consequences of molecular symmetry properties; however, when taken collectively for entire domains of nuclear configurations, they lead to nontrivial conclusions. Whereas symmetry properties and energy contents of individual conformations can be studied locally within limited domains of the potential surface, a global analysis of the potential surface may reveal significantly more. In this note, some consequences of the above approach are explored, and a simple test is proposed for the detection and evaluation of the importance of multicenter interactions in conformers related to one another by bond rotations.Dedicated to Professor J. Koutecký on the occasion of his 65th birthday     

Stability and properties of polyhelicenes and annelated fused-ring carbon helices: models toward helical graphites

The geometrical structures and properties of conjugated polyhelicenes and annelated fused-ring carbon helices with analogous frameworks were theoretically studied at the HF/6-31G and B3LYP/6-31G levels. These studies focused on the stability of the fused-ring structures with special emphasis on the helical geometrical arrangements. To elucidate bonding patterns, the orbitals, electron density contours, and the electrostatic potential of these helical compounds were analyzed. The structure of fused polynaphthalenes arranged in a helical spiral can be regarded as part of a locally helical graphite lattice that is expected to give rise to special electronic properties along the helically layered conjugated single sheet that can be regarded as a single extended pi system but also involving local layer-to-layer pi-pi interactions that are typical in ordinary graphite. This dual feature might lead to novel materials.     

On the relative importance of core and valence shell representations in the calculation of conformational energies using small Gaussian basis sets

Several “core-deficient” small Gaussian basis sets were constructed and analyzed in terms of the balance requirements of functions that contribute predominantly to the core. Variations in the conformational energy barriers and geometrical parameters for ammonia and ethane, calculated with these basis sets, were analyzed with a gradient technique. A scheme for the reduction of the size of molecular basis sets is proposed.