The connected moments expansion for the zero-point energy of coupled anharmonic oscillators

The connected moment expansion (CMX) technique is used to calculate the zero-point energy of an arbitrary system of coupled anharmonic oscillators. When the anharmonic term has the form of a polynomial with respect to the normal coordinates, it is possible to calculate the zero-point energy in a completely automated way. A numerical example is presented, demonstrating the power of the new method.     

Reduced dynamics of coupled harmonic and anharmonic oscillators using higher-order perturbation theory

Several techniques to solve a hierarchical set of equations of motion for propagating a reduced density matrix coupled to a thermal bath have been developed in recent years. This is either done using the path integral technique as in the original proposal by Tanimura and Kubo [J. Phys. Soc. Jpn. 58, 101 (1998)] or by the use of stochastic fields as done by Yan et al. [Chem. Phys. Lett. 395, 216 (2004)]. Based on the latter ansatz a compact derivation of the hierarchy using a decomposition of the spectral density function is given in the present contribution. The method is applied to calculate the time evolution of the reduced density matrix describing the motion in a harmonic, an anharmonic, and two coupled oscillators where each system is coupled to a thermal bath. Calculations to several orders in the system-bath coupling with two different truncations of the hierarchy are performed. The respective density matrices are used to calculate the time evolution of various system properties and the results are compared and discussed with a special focus on the convergence with respect to the truncation scheme applied.     

The anharmonic effect study of coupled Morse oscillators for the unimolecular reaction

The importance of anharmonic effect in dissociation of molecular systems especially clusters has been noted. In this paper, we shall study the effect of coupled anharmonic oscillator of the standard bilinear form (SBF) Morse oscillator (MO) potential on unimolecular reaction. We shall use the systematic theoretical approach, YL method, proposed by Yao and Lin (YAO L, et. al. J Phys Chem A, 2007, 111(29): 6722-6729), which can evaluate anharmonic effects on the rate constants based on the transition state theory. In treating the anharmonic effect with the Morse oscillator potential on unimolecular reactions under collision-free conditions by using the RRKM (Rice-Ramsperger-Kassel-Marcus) theory, the in-verse Laplace transformation of the partition functions was used to obtain the total amount of state and density of state by using the first-order and the second-order approximations of the saddle-point method. To demonstrate the anharmonic effect of the SBF Morse model, we choose some model systems and a real reaction as examples.     

Effect of noise on the classical and quantum mechanical nonlinear response of resonantly coupled anharmonic oscillators

Multidimensional infrared spectroscopy probes coupled molecular vibrations in complex, condensed phase systems. Recent theoretical studies have focused on the analytic structure of the nonlinear response functions required to calculate experimental observables in a perturbative treatment of the radiation-matter interaction. Classical mechanical nonlinear response functions have been shown to exhibit unbounded growth for anharmonic, integrable systems, as a consequence of the nonlinearity of classical mechanics, a feature that is absent in a quantum mechanical treatment. We explore the analytic structure of the third-order vibrational response function for an exactly solvable quantum mechanical model that includes some of the important and theoretically challenging aspects of realistic models of condensed phase systems: anharmonicity, resonant coupling, fluctuations, and a well-defined classical mechanical limit.     

Vibrational-translational energy transfer from highly excited anharmonic oscillators

Model trajectory calculations reveal a simple power law behavior for anharmonic oscillator energy transfer for all bound vibrational levels.     

Application of the effective harmonic oscillator method to computing the thermal averages for a system of coupled anharmonic oscillators

By treating the Hamiltonian for coupled oscillators with polynomial anharmonicity by the Gibbs-Bogoliubov inequality, the effective harmonic oscillator (EHO) method is developed and applied to computing the thermal averages for polyatomic molecules. Practical utility is demonstrated with calculations of electron diffraction quantities, namely the distance r_a and amplitude l, and of the vibrational partition functions for CO₂, CS₂, SO₂ and H₂O from spectroscopic data on the force fields. The results are compared with those in the literature obtained by more accurate techniques. A comparison of r_a and l was also made with the results of electron diffraction measurements.     

Variational calculations of the Wigner distribution function for selected anharmonic oscillators

The quantum (? =1) and semiclassical (? → 0) Wigner distribution functions were calculated for some anharmonic oscillators using a direct phase-space variational method.     

Quasi-bound states of coupled Morse oscillators

Quasi-bound (resonance) states are present in the continuous spectrum of the Hamiltonian of two coupled Morse oscillators. Two different methods for approximating these as localized states are compared. The algebraic approach is shown to be in very good accord with the other method which is formulated in coordinate space and hence is differential in character. For these highly excited states an intermultiplet mixing term is included in the algebraic Hamiltonian.     

Complex behavior in coupled bromate oscillators

In this study, coupled bromate-oscillators constructed by adding 1,4-cyclohexanedione (1,4-CHD) to the ferroin-catalyzed Belousov-Zhabotinsky reaction are investigated in a batch reactor under anaerobic conditions. Various complex behaviors such as sequential oscillations and bursting phenomena are observed. At low concentrations of ferroin or malonic acid (MA), the development of sequential oscillations is found to depend on the ratio of [1,4-CHD]/[ferroin] and [1,4-CHD]/[MA] rather than their absolute concentrations. As the concentration of MA or ferroin was increased gradually, however, the minimum 1,4-CHD concentration required to induce complex oscillations reaches a plateau. Perturbations by light illustrate that the first oscillatory window is governed by the ferroin-MA-BZ mechanism, whereas the 1,4-CHD-bromate oscillator plays a prominent role during the non-oscillatory evolution and the second oscillatory window. Our conclusion is further supported by numerical simulations in which sequential oscillations observed in experiments are qualitatively reproduced by a modified FKN mechanism.     

The adiabatic approximation for coupled oscillators

The adiabatic approximation is applied to determine the quantum states of coupled oscillators described by a generalized Hénon-Heiles hamiltonian. Comparison with exact quantum and other results show that numerically calculated adiabatic energy levels are accurate even for excited states.     

Experimental and theoretical studies of coupled chemical oscillators

Chemical oscillators may be coupled together in a variety of ways. Two of the most important forms of coupling are physical (via transport) and chemical (via common species). Such coupling can result in new phenomena. Here we focus on rhythmogenesis, the onset of oscillations when two steady state systems are coupled, and oscillator death, the cessation of oscillations when two oscillatory systems are coupled. We also discuss briefly a biological example, the crustacean stomatogastric ganglion, and the important role of delay, which may be brought on by coupling, in chemical oscillation.     

Analytical results for the classical and semiclassical coordinate probability distribution function of anharmonic oscillators

The classical and Wigner—Kirkwood semiclassical distribution functions for single and double-well anharmonic oscillators with ? = p?²/2m + k₂x?² + k_2nx?²ⁿ are given in closed form. Applications in the theory of electron diffraction are discussed briefly.     

New unitary transformation for the three coupled oscillators

A new unitary operator U, which can transform the Fock space of a three-dimensional isotropic harmonic oscillator into the space in which the Hamiltion of three coupled oscillators is diagonized, is found. The coordinate representation of U is presented and is used to directly derive the wave function of the energy eigenstate of the coupled oscillators.     

Internal stochastic resonance in two coupled chemical oscillators

The dynamics of two coupled chemical oscillators was investigated numerically when the first subsystem was subjected to external parametric noise. The signal-to-noise ratio (SNR) of the response of each subsystem to external noise shows internal stochastic resonance (SR). In addition, the SNR also shows resonance behavior with the variation of coupling strength.     

Symmetry broken self-consistent field approximation for coupled oscillators

A self-consistent field approximation allowing for a change of coordinates is applied to a model of two coupled oscillators considered by Davis and Heller [J. Chem. Phys. 75 , 246 (1981)]. The model accounts well for the unsymmetrical nonstationary states involved in the quantum dynamical tunneling phenomenon.