We study entropies caused by the unstable part of partially hyperbolic systems. We define unstable metric entropy and unstable topological entropy, and establish a variational principle for partially hyperbolic diffeomorphsims, which states that the unstable topological entropy is the supremum of the unstable metric entropy taken over all invariant measures. The unstable metric entropy for an invariant measure is defined as a conditional entropy along unstable manifolds, and it turns out to be the same as that given by Ledrappier-Young, though we do not use increasing partitions. The unstable topological entropy is defined equivalently via separated sets, spanning sets and open covers along a piece of unstable leaf, and it coincides with the unstable volume growth along unstable foliation. We also obtain some properties for the unstable metric entropy such as affineness, upper semi-continuity and a version of Shannon-McMillan-Breiman theorem.

Charge‐shift bonds (CSBs) constitute a new class of bonds different than covalent/polar‐covalent and ionic bonds. Bonding in CSBs does not arise from either the covalent or the ionic structures of the bond, but rather from the resonance interaction between the structures. This Essay describes the reasons why the CSB family was overlooked by valence‐bond pioneers and then demonstrates that the unique status of CSBs is not theory‐dependent. Thus, valence bond (VB), molecular orbital (MO), and energy decomposition analysis (EDA), as well as a variety of electron density theories all show the distinction of CSBs vis‐à‐vis covalent and ionic bonds. Furthermore, the covalent–ionic resonance energy can be quantified from experiment, and hence has the same essential status as resonance energies of organic molecules, e.g., benzene. The Essay ends by arguing that CSBs are a distinct family of bonding, with a potential to bring about a Renaissance in the mental map of the chemical bond, and to contribute to productive chemical diversity.

Does, or doesn't C2 break the glass ceiling of triple bonding? This work provides an overview on the bonding conundrum in C2 and on the recent discussions regarding our proposal that it possesses a quadruple bond. As such, we focus herein on the main point of contention, the 4th bond of C2, and discuss the main views. We present new data and an overview of the nature of the 4th bond--its proposed antiferromagnetically coupled nature, its strength, and a derivation of its bond energy from experimentally based thermochemical data. We address the bond-order conundrum of C2 arising from generalized VB (GVB) calculations by comparing it to HC≡CH, and showing that the two molecules behave very similarly, and C2 is in no way an exception. We analyse the root cause of the deviation of C2 from the Badger Rule, and demonstrate that the reason for the smaller force constant (FC) of C2 relative to HC≡CH has nothing to do with the bond energies, or with the number of bonds in the two molecules. The FC is determined primarily by the bond length, which is set by the balance between the bond length preferences of the σ- versus π-bonds in the two molecules. This interplay in the case of C2 clearly shows the fingerprints of the 4th bond. Our discussion resolves the points of contention and shows that the arguments used to dismiss the quadruple bond nature of C2 are not well founded.

Ab initio valence bond calculations are performed for the allyl cation, radical, and anion with 6-31G* basis set. Delocalized and hypothetically localized structures of these systems are thoroughly optimized and analyzed. The delocalization energies, defined as the energy difference between the delocalized structure and its hypothetically localized one, for the three allyl systems are −55.7, −28.4, and −52.3 kcal/mol, respectively. Our results clarify the recent debate on whether the allyl anion has little or comparable resonance stabilization with the allyl cation. The methylene rotation barriers of the allyl cation, radical, and anion are successfully explained in terms of resonance, hyperconjugation, and rehybridization. For the allyl radical, its resonance energy is only about half those of the allyl cation and anion; thus, it has the lowest rotation barrier. The twisted allyl cation, in which the rotating methylene group is perpendicular to the C−C−C plane, has the highest hyperconjugation energy (−6.8 kcal/mol), while, in its twisted form, the allyl anion has a negligible hyperconjugation effect. As the allyl anion assumes its twisted form, the carbon atom in the rotating methylene experiences a remarkable rehybridization from sp2 mode in the planar form to sp3 mode. This process decreases the total energy of the twisted allyl anion as much as 14.3 kcal/mol and eventually makes its rotation barrier smaller.