Abstract:
We sketch two rigorous proofs of the stability of the hydrogen molecule in quantum mechanics. The first one is based on an extrapolation of variational estimates of the groundstate energy of a positronium molecule to arbitrary mass ratios. The second one is an extension of Heitler-London theory to nuclei of finite mass.

Abstract:
Recently Smit et al. [Nature 419, 906 (2002)] have reported that a single hydrogen molecule can form a bridge between Pt electrodes, which has a conductance close to one quantum unit, carried by a single channel. We present density functional calculations explaining these experimental findings. We show that the symmetry of the molecular orbitals selects a single conduction channel. The transmission of this channel is close to unity due to a combination of the charge transfer between hydrogen and the Pt contacts and the strong hybridization between the bonding state of the molecule and the d-band of the Pt leads.

Abstract:
The hydrogen molecule contains the basic ingredients to understand the chemical bond, i.e, a pair of electrons. We show a step to understand The Correspondence Principle for chaotic system in the Chemical World. The hydrogen molecule is studied classically as an extension of the helium atom. Several types of orbits were found for two-fixed-centers system (hydrogen molecule) starting from some known orbits for one-fixed center one (helium atom), e.g., one dimension orbits as pendulum and axial and also Bohr and Langmuir's orbits. The classical stability and the single quantization of some one-dimensional periodic orbits are shown. These orbits are used to make a global quantization of that molecular chaotic system. We discuss the importance of those periodic orbits in the comprehension of the nature of chemical bond.

Abstract:
We present a many-body calculation for the conductance of a conducting bridge of a simple hydrogen molecule between $Pt$ electrodes.The experimental results showed that the conductance $G=dI/dV$ has the maximum value near the quantum unit $G_{0}=2e^{2}/h$. The $I-V$ dependence presents peak and dip and we consider that the electron-phonon interaction is responsible for this behavior. At T=0 there is a step in this dependence for the energy of phonons $\omega_{0}$ which satisfies $eV=\omega_{0}$. We calculated the conductance at finite temperature and showed that $dG(T)/dV\propto 1/4T\cosh^{2}\frac{eV-\omega_{0}}{2T}$.

Abstract:
A recent claim that molecule H-antiH is unstable cannot be a proof as it is based on a wrong conjecture. This is illustrated with 4 examples, including observed natural hydrogen-antihydrogen oscillations never detected previously.

Abstract:
Recent years have shown steady progress in research towards molecular electronics [1,2], where molecules have been investigated as switches [3-5], diodes [6], and electronic mixers [7]. In much of the previous work a Scanning Tunnelling Microscope was employed to address an individual molecule. As this arrangement does not provide long-term stability, more recently metal-molecule-metal links have been made using break junction devices [8-10]. However, it has been difficult to establish unambiguously that a single molecule forms the contact [11]. Here, we show that a single H2 molecule can form a stable bridge between Pt electrodes. In contrast to results for other organic molecules, the bridge has a nearly perfect conductance of one quantum unit, carried by a single channel. The H2-bridge provides a simple test system and a fundamental step towards understanding transport properties of single-molecule devices.

Abstract:
In this paper we investigate some entanglement properties for the Hydrogen molecule considered as a two interacting spin 1/2 (qubit) model. The entanglement related to the $H_{2}$ molecule is evaluated both using the von Neumann entropy and the Concurrence and it is compared with the corresponding quantities for the two interacting spin system. Many aspects of these functions are examinated employing in part analytical and, essentially, numerical techniques. We have compared analogous results obtained by Huang and Kais a few years ago. In this respect, some possible controversial situations are presented and discussed.

Abstract:
a method to calculate angular distributions for electrons ejected from fixed-in-space molecular hydrogen molecules subject to ultrashort intense laser pulses is proposed, based on the ab initio solution of the time-dependent schrodinger equation. this method of solution allows for a temporal picture of interferences arising in the dissociative ionization channel (in the proton kinetic energy spectrum) due to the presence of the autoionizing double excited states in the hydrogen molecule. in particular, we show how this autoionization during the dissociative photoionization process may also induce a counterintuitive asymmetry in the angular distribution of the ionized electron with respect to nuclei inversion, in spite of dealing with an homonuclear system.

Abstract:
In the framework of nonrelativistic quantum mechanics we derive a necessary condition for four Coulomb charges $(m_{1}^+, m_{2}^-, m_{3}^+, m_{4}^-)$, where all masses are assumed finite, to form the stable system. The obtained stability condition is physical and is expressed through the required minimal ratio of Jacobi masses. In particular this provides the rigorous proof that the hydrogen-antihydrogen molecule is unstable. This is the first result of this sort for four particles.