Scattering in the quasiperiodic and chaotic regimes

The results of a quasiclassical investigation of the scattering process for a collinear collision between a particle and two coupled anharmonic oscillators is reported. The energy transfer was found to fall at the critical energy where the oscillators made the transition from quasiperiodic to chaotic motion.     

The molecular dynamics problem in multiphoton excitation

While the molecular dynamics for weakly coupled, harmonic oscillators undergoing infinitesimal amplitude displacements are well described by normal modes, and the other extreme — strongly coupled anharmonic, large amplitude oscillating units can be treated as though they are ergodic systems, so that only state densities are needed, intermediate cases are very difficult to analyze. Here we propose that some intermediate cases, wherein we seek relative rates for reaction induced by specific frequency excitation (three-center displacement reactions or dissociation), may be approximately estimated from a normal mode analysis by calculating increment in root mean square amplitudes for a suitable internal coordinate, upon specific multiple photon absorption. A test case is presented as an illustration.     

Modified calculation scheme for evaluation and prediction of glass transition temperatures of polymers

A modified calculation scheme was developed for evaluating and predicting the glass transition temperatures of linear and cross-linked polymers. It was proposed to separate the contributions of weak (dispersion) and strong (dipole–dipole and hydrogen bonding) interactions between the same atoms and polar groups in the backbone and side chains of a polymer. The considered model is based on analyzing the system of anharmonic oscillators formed by pairs of atoms entering the intermolecular interaction. The critical temperature of destabilization of this system of oscillators on heating is the glass transition temperature T_g. As a result, without resorting to correction factors of various kinds, one can calculate the T_g of a large number of linear and cross-linked polymers of different classes and various structures with good accuracy.     

Free-energy calculations for the cubic ZrO2 crystal as an example of a system with a soft mode

We calculate the free energy for a crystalline ZrO(2) with a soft mode by the first-principles method, using the double-well energy-displacement relation. The soft-mode branch is considered as an ensemble of independent anharmonic oscillators of the parabola-plus-Gaussian or of the 2-4 polynomial forms. The anharmonic contributions are included to reproduce the cubic-to-tetragonal phase transition, however, it appears that the cubic phase does not become the most stable within the framework of the independent oscillators approach.     

Effects of anharmonicity on nonadiabatic electron transfer: a model

The effect of anharmonicity in the intramolecular modes of a model system for exothermic intramolecular nonadiabatic electron transfer is probed by examining the dependence of the transition probability on the exoergicity. The Franck-Condon factor for the Morse potential is written in terms of the Gauss hypergeometric function both for a ground initial state and for the general case, and comparisons are made between the first-order perturbation theory results for transition probability for harmonic and Morse oscillators. These results are verified with quantum dynamical simulations using wave-packet propagations on a numerical grid. The transition-probability expression incorporating a high-frequency quantum mode and low-frequency medium mode is compared for Morse and harmonic oscillators in different temperature ranges and with various coarse-graining treatments of the delta function from the Fermi golden rule expression. We find that significant deviations from the harmonic approximation are expected for even moderately anharmonic quantum modes at large values of exoergicity. The addition of a second quantum mode of opposite displacement negates the anharmonic effect at small energy change, but in the inverted regime a significantly flatter dependence on exoergicity is predicted for anharmonic modes.     

Quantum analogue of the Kolmogorov–Arnold–Moser transition in different quantum anharmonic oscillators

The results of numerical computations are presented for the Bohmian trajectories of the family of different one‐ and two‐dimensional anharmonic oscillators, which exhibit regular or chaotic motion in both classical and quantum domains, depending on the values of the parameters appearing in the respective Hamiltonians. Quantum signatures of the Kolmogorov–Arnold–Moser (KAM) transition from the regular to chaotic classical dynamics of these oscillators are studied using a quantum theory of motion (QTM) as developed by de Broglie and Bohm. A phase space distance function between two initially close Bohmian trajectories, the associated Kolmogorov–Sinai–Lyapunov (KSL) entropy, the phase space volume, the autocorrelation function, the associated power spectrum, and the nearest‐neighbor spacing distribution, clearly differentiate the quantum analogues of the corresponding regular and chaotic motions in the classical domain. These quantum anharmonic oscillators are known to be useful in several diverse branches of science. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Relation of algebraic models of coupled anharmonic oscillators

It is shown that the authors's SU(n) invariance approach to coupled anharmonic oscillators and the coupled SU(2) non-invariance approach of Levine and co-workers are complementary methods related by a transformation between the raising and lowering operators used in each.     

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.     

A holomorphic representation approach to the regularization of model field theories in coupled cluster form

Summary We explain in detail the so-called Bargmann or holomorphic representation, and apply it to the general class of single-mode bosonic field theories. Since these model field theories have no attribute of separability and are, in some sense, maximally nonlocal, they are an especially severe test of the capability of coupled cluster methods to parametrize them satisfactorily. They include the cases of anharmonic oscillators of order 2K (K=2, 3,...), for which ordinary perturbation theory is known to diverge, and we therefore make a special study of such systems. We demonstrate for the first time for any quantum-mechanical problem with infinite Hilbert space that both the normal and extended coupled cluster methods (NCCM and ECCM) have phase spaces which rigorously exist. We analyze completely the asymptotic properties of the complete sets of the NCCM and ECCM amplitudes, either of which fully characterizes the system. It is thereby shown how the holomorphic representation can be used to regularize completely all otherwise formally divergent series that appear. We demonstrate in detail how the entire NCCM and ECCM programmes can be carried through for these systems, including the diagonalization of the classically mapped Hamilitonians in the respective classical NCCM and ECCM phase spaces.     

Benchmark calculations for dissipative dynamics of a system coupled to an anharmonic bath with the multiconfiguration time-dependent Hartree method

In this paper, we present benchmark results for dissipative dynamics of a harmonic oscillator coupled to an anharmonic bath of Morse oscillators. The microscopic Hamiltonian has been chosen so that the anharmonicity can be adjusted as a free parameter, and its effect can be isolated. This leads to a temperature dependent spectral density of the bath, which is studied for ohmic and lorentzian cases. Also, we compare numerically exact multiconfiguration time-dependent Hartree results with approximate solutions using continuous configuration time-dependent self-consistent field and local coherent state approximation.     

A vibrational CASSCF study of stretch-bend interactions and their influence on infrared intensities in the water molecule

Summary An extension of the multiconfigurational SCF approach for the resolution of the vibrational problem is presented; it follows the philosophy of the CASSCF method developed in Quantum Chemistry. The new method allows a more complete treatment of anharmonic mode couplings, converges much faster and gives a clearer physical insight of vibrational interactions. This is exemplified by the calculation of infrared transition moments in the H₂O and D₂O isotopomers of the water molecule. It is shown how this property varies with the quality of the wave function when vibrational resonances occur. A detailed analysis by means of this new VCASSCF method demonstrates the crucial importance of excited bending oscillators in the intensity of some pure stretching transitions.Boursier F.R.I.A.     

Revealing defects hampering the formation of epoxy networks with extremely high thermal properties: Theory and experiments

We present a combined experimental and theoretical investigation of thermal properties of cycloaliphatic epoxy networks. The networks are prepared from 3,4-epoxycyclohexylmethyl-3,4-epoxycyclohexanecarboxylate ERL-4221 as a monomer and 4-methylhexahydrophthalic anhydride as a curing agent and their glass transition temperature T_g is evaluated by dynamic mechanical and thermal mechanical analyses as well as by differential scanning calorimetry. It is found that the cured epoxy networks have high T_g values reaching 233–238 °C. The method of anharmonic oscillators is first proposed to simulate the effect of network structure on the thermal properties. It suggests that further increase of T_g values is not attained because of the formation of intramolecular cyclic structures. Studies of model reaction by mass-spectrometry confirm the formation of such structures at curing.     

Classical and quantum mechanical infrared echoes from resonantly coupled molecular vibrations

The nonlinear response function associated with the infrared vibrational echo is calculated for a quantum mechanical model of resonantly coupled, anharmonic oscillators at zero temperature. The classical mechanical response function is determined from the quantum response function by setting variant Planck's over 2pi-->0, permitting the comparison of the effects of resonant vibrational coupling among an arbitrary number of anharmonic oscillators on quantum and classical vibrational echoes. The quantum response function displays a time dependence that reflects both anharmonicity and resonant coupling, while the classical response function depends on anharmonicity only through a time-independent amplitude, and shows a time dependence controlled only by the resonant coupling. In addition, the classical response function grows without bound in time, a phenomenon associated with the nonlinearity of classical mechanics, and absent in quantum mechanics. This unbounded growth was previously identified in the response function for a system without resonant vibrational energy transfer, and is observed to persist in the presence of resonant coupling among vibrations. Quantitative agreement between classical and quantum response functions is limited to a time scale of duration inversely proportional to the anharmonicity.     

The temperature dependence of the d-electron dipole strengths of cobalt(II) tetrahalides

The dipole strengths of the ⁴A₂ → ⁴T₁(F), ⁴T₁(P), d-electron transitions of the cobalt(II) tetrahalides are found to decrease with an increase in temperature, in agreement with a dynamic ligand-polarisation model for the absorption mechanism. The reduction in dipole strength is accounted for by a decrease in the coulombic coupling between the quadrupole moment of the d-electron transition of the metal ion and the transient electric dipole moment induced in the ligands, due to the anharmonic increase in the mean bond length and the progressively larger mean-square amplitudes of the bending modes of the complex ion in its ground electronic state as the temperature is raised.     

Anharmonic effect on unimolecular reactions with application to the photodissociation of ethylene

The importance of anharmonic effect on dissociation of molecular systems, especially clusters, has been noted. In this paper, we shall present a theoretical approach that can carry out the first principle calculations of anharmonic canonical and microcanonical rate constants of unimolecular reactions within the framework of transition state theory. In the canonical case, it is essential to calculate the partition function of anharmonic oscillators; for convenience, the Morse oscillator potential will be used for demonstration in this paper. In the microcanical case, which involves the calculation of the total number of states for the activated complex and the density of states for the reactant, we make use of the fact that both the total number of states and the density of states can be expressed in the inverse Laplace transformation of the partition functions and that the inverse Laplace transformation can in turn be carried out by using the saddle-point method. We shall also show that using the theoretical approach presented in this paper the total number of states and density of states can be determined from thermodynamic properties and the difference between the method used in this paper and the thermodynamic model used by Krems and Nordholm will be given. To demonstrate the application of our theoretical approach, we chose the photodissociation of ethylene at 157 and 193 nm as an example.     

The identification of normal mode behavior in the resonance picture of local mode dynamics: normal mode splitting,stability of periodic orbits and energy transfer rates for coupled morse oscillators

The resonance model of coupled anharmonic local mode oscillators is extended to include second-order variations of the Fourier amplitude. Using this extension, all of the usual normal mode behavior including normal mode splittings, stability of periodic motions, and the energy transfer rate for degenerate stretching modes is recovered. A method is developed to predict the identity and energy of fundamental periodic orbits which become unstable. This method is more accurate and more general than previous analytic methods. These results may be of use in fitting or interpreting vibrational spectra and in studies of intramolecular vibrational energy flow.     

Effects of permanent dipole moments in transient four-wave mixing experiments

A two-pulse degenerate four-wave mixing experiment is analyzed in the case where the medium under investigation can be modeled by two-level systems having unequal permanent dipole moments. By modeling the light pulses by double exponentials [exp(-Gamma/t/)], we give an analytical expression of the third-order nonlinear polarization of the medium. We apply this result to simulate the measured signal in such experiment. We show that in the case of a two-photon transition, a signal can be detected if the pump pulse interacts with the medium before the probe pulse contrary to what is observed for excitations in the resonance region. An attempt to explain this behavior is made and the detected signal is analyzed in terms of pure coherent processes. This effect appears as a signature of the presence of permanent dipole moments. To test this property on a more realistic system, we then have considered a one-dimensional frequency-selected infrared degenerate four-wave mixing experiment on a molecular anharmonic vibrational mode modeled by a Morse potential and coupled to a dissipative bath of harmonic oscillators. We show that the two-photon transitions allowed by the presence of permanent dipole moments enable to analyze the multilevel system dynamics as if they were the one of a two-level system. Our results can also be extended to the case of inhomogeneous broadening and are of interest to study the infrared photon-echo response of anharmonic vibrational modes.     

A novel combined electrochemical oscillator from cathode and anode in IO3−/Fe(CN)64− system

Experimental studies have been performed for the simultaneous potential or current oscillations occurring both on cathode and anode in the IO₃⁻/Fe(CN)₆⁴⁻ system. The combined oscillator exhibits very different behavior from that of the individual oscillators on each electrode owing to the different waveforms and incommensurate oscillating frequencies of the individual oscillators. The potential oscillation of the combined oscillator is essentially a sum of those of the individual oscillators for the mixed system, whereas the sum coming from the single systems is slightly different, mainly due to the difference in ionic strength between the single system and the mixed system. The separate current oscillators also seem to interfere with each other, probably through the restricted distribution of the applied voltage in both electrodes and the ohmic drop in addition to the effect of ionic strength.