Theoretical infrared line shapes of H‐bonds within the strong anharmonic coupling theory and Fermi resonances effects

In this article, we extend a previous work toward presenting a theoretical study of the effects of Fermi resonances and the fundamental anharmonic coupling parameter α between the high‐frequency mode and the H‐bond bridge. The model incorporates (i) both intrinsic anharmonicities of the fast mode (double well potential) and the H‐bond Bridge (Morse potential), (ii) strong anharmonic coupling theory, (iii) Fermi resonances by the aid of an anharmonic coupling between the fast mode and one or several harmonic bending modes, (iv) quadratic modulation of both the angular frequency and the equilibrium position of the X? …Y stretching mode on the intermonomer ? H… motions, and (v) the quantum direct (fast and bending modes) and indirect dampings (slow mode). The IR spectral density is obtained by Fourier transform of the autocorrelation function of the transition dipole moment operator of the X? H bond. The numerical calculation shows that Fermi resonances generate very complicated profiles with multisubstructure and also provide a direct evidence of Fermi resonances which were predicted to be a major feature of H‐bonds. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Phonon energy in an anharmonic quasi-one-dimensional solid

For the first time, the phonon energy per unit volume in a large anharmonic quasi-one-dimensional solid is determined by considering all polarizations of the various modes of phonon propagation and by assuming the solid as a lattice of atoms behaving as the Morse oscillators. In this context the equilibrium phonon occupation number, which is given by the Bose distribution, replaces formally the vibrational quantum number into the expression for the Morse-oscillator energy. In addition, the quasi-harmonic solid is discussed within the above framework so that the phonon energy per unit volume is calculated for a large quasi-harmonic and quasi-one-dimensional solid.     

Properties of nano-scale soliton-like excitations in two-dimensional lattice layers

Nano-scale soliton-like supersonic, intrinsic localized excitations in two-dimensional atomic anharmonic lattice layers are here considered. We study the propagation, the velocity and other soliton-like features at head-on collisions of such lattice excitations created by using suitable initial mechanical and thermal conditions. Noteworthy is that narrow, highly-energetic solitons moving along one lattice row are very robust, accompanied by weak anti-phase oscillations in the lateral direction.     

Singular densities,test particles and generalized Hamiltonian systems in general relativity

We derive the motion equations and the structure equations of neutral and charged test particles, starting from the gravitational field equations. The method consists in the application of conservation laws to singular tensor densities, which represent regions of strong matter concentration. Moreover, a Hamiltonian formulation of the particle equations is given, in the form of implicit differential equations generated by Hamiltonian Morse families.     

Existence of multiple nontrivial solutions for a Schrödinger–Poisson system

We consider a Schrödinger–Poisson system in R³${\mathbb{R}}^3$ with potential indefinite in sign and a general nonlinearity. We use the direct variational method and Morse theory to obtain the existence of multiple nontrivial solutions for this system.     

Unstable manifold,Conley index and fixed points of flows

We study dynamical and topological properties of the unstable manifold of isolated invariant compacta of flows. We show that some parts of the unstable manifold admit sections carrying a considerable amount of information. These sections enable the construction of parallelizable structures which facilitate the study of the flow. From this fact, many nice consequences are derived, specially in the case of plane continua. For instance, we give an easy method of calculation of the Conley index provided we have some knowledge of the unstable manifold and, as a consequence, a relation between the Brouwer degree and the unstable manifold is established for smooth vector fields. We study the dynamics of non-saddle sets, properties of existence or non-existence of fixed points of flows and conditions under which attractors are fixed points, Morse decompositions, preservation of topological properties by continuation and classify the bifurcations taking place at a critical point.     

Geometrical Decomposition of the Free Loop Space on a Manifold with Finitely Many Closed Geodesics

In Morse theory an isolated degenerate critical point can be resolved into a finite number of nondegenerate critical points by perturbing the totally degenerate part of the Morse function inside the domain of a generalized Morse chart. Up to homotopy we can admit pertubations within the whole characteristic manifold. Up to homotopy type a relative CW-complex is attached, which is the product of a big relative CW-complex, representing the degenerate part, and a small cell having the dimension of the Morse index.     

Systematic calculation of molecular vibrational spectra through a complete Morse expansion

We propose an accurate and efficient method to compute vibrational spectra of molecules, based on exact diagonalization of an algebraically calculated matrix based on powers of Morse coordinate. The present work focuses on the 1D potential of diatomic molecules: as typical examples, we apply this method to the standard Lennard‐Jones oscillator, and to the ab initio potential of the H₂ molecule. Global cm^?1 accuracy is exhibited through the H₂ spectrum, obtained through the diagonalization of a 30 × 30 matrix. This theory is at the root of a new method to obtain globally accurate vibrational spectral data in the context of the multi‐dimensional potential of polyatomic molecules, at an affordable computational cost. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Nontrivial solutions of Schrödinger equations with indefinite nonlinearities

We prove the existence of nontrivial solutions for the Schrödinger equation −Δu+V(x)u=aγ(x)f(u) in RN, where f is superlinear and subcritical at zero and infinity respectively, V is periodic and a(x) changes sign.