A New Euler's Formula for DNA Polyhedra

DNA polyhedra are cage-like architectures based on interlocked and interlinked DNA strands. We propose a formula which unites the basic features of these entangled structures. It is based on the transformation of the DNA polyhedral links into Seifert surfaces, which removes all knots. The numbers of components , of crossings , and of Seifert circles are related by a simple and elegant formula: . This formula connects the topological aspects of the DNA cage to the Euler characteristic of the underlying polyhedron. It implies that Seifert circles can be used as effective topological indices to describe polyhedral links. Our study demonstrates that, the new Euler's formula provides a theoretical framework for the stereo-chemistry of DNA polyhedra, which can characterize enzymatic transformations of DNA and be used to characterize and design novel cages with higher genus

The permutational symmetry of the icosahedral orbital quintuplet and its implication for vibronic interactions

The vibronic coupling between the fivefold degenerate orbital representation of the icosahedral point group and vibrational modes which are also fivefold degenerate is characterised by the existence of two equisymmetric but independent channels. It is shown that this coupling multiplicity of the quintuplet Jahn-Teller Hamiltonian can be resolved using the symmetric group of permutations of six elements, S6, of which the group of rotations of an icosahedron is a subgroup. After the application of this symmetry in atomic physics by Judd and Lo (Phys. Rev. Lett., 82 (1999) 3224), this report forms the first molecular application of this extension of the icosahedral symmetry group

Investigations of the Boron Buckyball B₈₀: Bonding Analysis and Chemical Reactivity

The boron fullerene B80 is a spherical network of 80 boron atoms, which has a shape similar to the celebrated C60. The 80 Bs span two orbits: while the first contains 60 atoms localised on the vertices of a truncated icosahedron like C60, the second includes 20 extra B atoms capping the hexagons of the frame. Quantum chemical calculations showed that B80 is unusually stable and has interesting physical and chemical properties. Its geometry is slightly distorted from I h to T h symmetry. However, the boron buckyball is only observed in silico, so far the synthesis of this molecule is only a remote possibility. Using DFT at the B3LYP/SVP level, we have analyzed the chemical bonding in B80, the possibility of methyne substitution and the stability of endohedral boron buckyball complexes. A symmetry analysis revealed a perfect match between the occupied molecular orbitals in B80 and C60. The cap atoms transfer their electrons to the truncated icosahedral frame, and they contribute essentially to the formation of σ bonds. The frontier MOs have π character and are localised on the B60 truncated icosahedral frame. The boron cap atoms can be replaced by other chemical groups, such as methyne (CH), which are also able to introduce three electrons in the cage. Symmetrical substitutions of the boron cap atoms by methyne groups in T and T h symmetries revealed two stable endo methyne boron buckyballs, endo- B80−x(CH)x , with x = 4, 8. The stability of these compounds seems to be due to the formation of six boron 4-centre bonding motifs in between the substituted hexagons. These localized bonding motifs are at the basis of the observed symmetry lowering, via a pseudo-Jahn-Teller effect. The methyne hydrogen atoms in the two endohedral fullerenes can be replaced by other atoms, which can lead to cubane or tetrahedral endohedral boron fullerenes. Theoretical study on encapsulated small bases molecules, tetrahedral and cubane like clusters of Group V atoms, showed that the boron buckyball is a hard acid and prefers hard bases like NH3 or N2H4, to form stables off-centred complexes with B80. Tetrahedral and cubane like clusters of this family are usually metastable in the encapsulated state, due to steric strain. The most favorable clusters are mixed tetrahedral and cubane clusters formed by nitrogen and phosphorus atoms such as P2N2@B80, P3N@B80 and P4N4@B80. The boron cap atoms act as electrophilic centres, which react with nucleophilic sites rich in electrons