The hydrogen molecules H 2 and ( ) H 2 2 are analyzed with electronic correlations taken into account between the s 1 electrons in an exact manner. The optimal single-particle Slater orbitals are evaluated in the correlated state of H 2 by combining their variational determination with the diagonalization of the full Hamiltonian in the second-quantization language. All electron-ion coupling constants are determined explicitly and their relative importance is discussed. Sizable zero-point motion amplitude and the corresponding energy are then evaluated by taking into account the anharmonic contributions up to the ninth order in the relative displacement of the ions from their static equilibrium value. The applicability of the model to solid molecular hydrogen is briefly analyzed by calculating intermolecular microscopic parameters for the × H 2 2 rectangular configuration, as well its ground state energy.Keywords: hydrogen molecules, electron-proton coupling for hydrogen molecule, electronic correlations for hydrogen molecule, intermolecular hopping and interaction parameters MotivationThe few-site models of correlated fermions play an important role in singling out, in an exact manner, the role of various local intra-and inter-site interactions against hopping (i.e., containing both covalent and the ionic factors) and thus, in establishing the optimal correlated state of fermions [1-8] on a local (nanoscopic) scale. The model has also been used to obtain a realistic analytic estimate of the hydrogen-molecule energies of the ground and the excited states in the correlated state [9]. For this purpose, we have developed the so-called EDABI method, which combines Exact Diagonalization in the Fock space with a concomitant Ab Initio determination of the single-particle basis in the Hilbert space. So far, the method has been implemented by taking only s 1 Slater orbitals, one per site [10]. The method contains no parameters; the only approximation made is taking a truncated single-particle basis (i.e., one Slater orbital per site) when constructing the field operator, that in turn is used to derive the starting Hamiltonian in the second-quantization representation. This Hamiltonian represents an extended Hubbard Hamiltonian, with all two-site interactions taken into account and the solution comprises not only the exact eigenvalues of the few-site Hamiltonian, but also at the same time an evaluation of the adjustable single-particle wave functions in the correlated state. Also, the calculated thermodynamic properties rigorously exemplify [12,11] the low-and highenergy scales, corresponding to spin and local charge fluctuations, respectively. The former represents the precursory magnetic-ordering effect whereas the latter represents local effects accompanying the Mott-Hubbard transition. In general, our approach follows the tradition of accounting for interelectronic correlations via the second-quantization procedure, with the adjustment of single-particle wave functions, contained in microscopic parameters of the startin...

The boson-fermion model, describing a mixture of fermions and bosons is analyzed on a small, two-site lattice. The model includes all the on-site Coulomb-type interactions between bosons, between fermions and between fermions and bosons. Additionally, the Hamiltonian includes a term that describes a conversion of a pair of opposite-spin fermions into a boson and vice versa. We show how this conversion depends on the model parameters.