Geometrical complexity of conformations of ring polymers under topological constraints

One measure of geometrical complexity of a spatial curve is the number of crossings in a planar projection of the curve. For $N$-noded ring polymers with a fixed knot type, we evaluate numerically the average of the crossing number over some directions. We find that the average crossing number under the topological constraint are less than that of no topological constraint for large $N$. The decrease of the geometrical complexity is significant when the thickness of polymers is small. The simulation with or without a topological constraint also shows that the average crossing number and the average size of ring polymers are independent measures of conformational complexity.Comment: 8 pages, 4figure

Calculation of the hydrogen molecule by means of a parametrization obtained from the hydrogen ion-molecule

The electronic energy function of the hydrogen molecule is calculated on the basis of the two simplest and most typical wave functions (MO-LCAO and VB) by introducing a reduction factor for the interelectronic repulsion and other functional parameters modifying the expectation values of both kinetic and potential energies. Such factors are associated to the CI contribution and some of them are calculated from the hydrogen ion-molecule. The comparison with the exact results shows a satisfactory agreement in binding energy, bond length and energy minimum values for the ground state. Some possible extensions of the method are discussed.L'énergie électronique de la molécule d'hydrogène est calculée à partir des deux fonctions d'onde les plus simples et les plus typiques (MO-LCAO et VB), en introduisant un facteur de réduction pour la répulsion interélectronique et d'autres paramètres (fonction de la distance internucléaire) modifiant les valeurs moyennes de l'énergie cinétique et de l'énergie potentielle. De tels facteurs sont associés à l'interaction de configuration et certains d'entre eux sont calculés à partir de l'ion moléculaire hydrogène. La comparaison avec les résultats exacts montre un accord satisfaisant concernant le minimum d'énergie, la longueur de liaison et l'énergie de dissociation pour l'état fondamental. Les auteurs discutent quelques extensions possibles de la méthode