Harmonic oscillators numerical models

Saint-Martin H, Hernandez-Cobos J, Bernal-Uruchurtu MI, Ortega-Blake I, Berendsen HJC (2000) A mobile charge densities in harmonic oscillators (MCDHO) molecular model for numerical simulations the water-water interaction. J Chem Phys 113(24) 10899—10912... 

The chapter starts with a brief review of thermodynamic principles as they apply to the concept of the chemical equilibrium. That section is followed by a short review of the use of statistical thermodynamics for the numerical calculation of thermodynamic equilibrium constants in terms of the chemical potential (often designated as (i). Lastly, this statistical mechanical development is applied to the calculation of isotope effects on equilibrium constants, and then extended to treat kinetic isotope effects using the transition state model. These applications will concentrate on equilibrium constants in the ideal gas phase with the molecules considered in the rigid rotor, harmonic oscillator approximation. 

For a spectroscopic observation to be understood, a theoretical model must exist on which the interpretation of a spectrum is based. Ideally one would like to be able to record a spectrum and then to compare it with a spectrum computed theoretically. As is shown in the next section, the model based on the harmonic oscillator approximation was developed for interpreting IR spectra. However, in order to use this model, a complete force-constant matrix is needed, involving the calculation of numerous second derivatives of the electronic energy which is a function of nuclear coordinates. This model was used extensively by spectroscopists in interpreting vibrational spectra. However, because of the inability (lack of a viable computational method) to obtain the force constants in an accurate way, the model was not initially used to directly compute IR spectra. This situation was to change because of significant advances in computational chemistry. 

Memory effects play an important role for the description of dynamical effects in open quantum systems. As mentioned above, Meier and Tannor [32] developed a time-nonlocal scheme employing the numerical decomposition of the spectral density. The TL approach as discussed above as well as the approaches by Yan and coworkers [33-35] use similar techniques. Few systems exist for which exact solutions are available and can serve as test beds for the various theories. Among them is the damped harmonic oscillator for which a path-integral solution exists [1], In the simple model of an initially excited... 

Berendsen, J. Chem. Phys., 113, 10899-10912 (2000). A Mobile Charge Densities in Harmonic Oscillators (MCDHO) Molecular Model for Numerical Simulations The Water-Water Interaction. 

It would be of interest to apply the method of March and Murray [12] to convert C, the electron density for non-degenerate electrons, into results applicable to intermediate degeneracy governed by Fermi-Dirac statistics. Unfortunately, without switching on the model potential F(r), this is already difficult to handle by purely analytically methods, as can be seen from the case of complete degeneracy for the harmonic oscillator alone. No doubt, numerical procedures will eventually enable present results to be transformed according to the route established in [12]. 

Shapes of molecular electronic bands are studied using the methods of the statistical theory of spectra. It is demonstrated that while the Gram-Charlier and Edgeworth type expansions give a correct description of the molecular bands in the case of harmonic-oscillator-like potentials, they are inappropriate if departure from harmonicity is considerable. The cases considered include a set of analytically-solvable model potentials and the numerically exact potential of the hydrogen molecule. 

The application of this method to systems described by one-dimensional potentials is particularly simple (15, 18). Therefore, in this paper, the feasibility and the accuaracy of the approach has been illustrated by considering transitions between states described by several exactly solvable onedimensional models (20) and between X and B1 states of Henergy curves (21). It results, that with a proper choice of the functional form of the envelope, already three-moment curves give a very accurate description of the band shape. For harmonic oscillators (22), the Gram-Charlier-type expansions (23) are very accurate. They axe also rather good for the cases reasonably well approximated by harmonic-oscillator-type potentials (15,18). However, if the departure from harmonicity is considerable, these kinds of expansions are inappropriate. 

To obtain some insight into the behavior of the solutions of the Hamiltonian equation (10), we performed a numerical simulation of a model system 23 we assumed that V(s) is a symmetric double well, we coupled, v to 1000 harmonic oscillators cok with frequencies ranging from 10 to 1000 cm, and symmetrically to one oscillator Qpv. Even though the simulation is completely classical, we obtained instructive results that illustrate several of the points we have mentioned in this section. 

Note added in proof In view of the failure of the harmonic oscillator model to account for the observed rate of activation in unimolecular dissociation reactions (the dissociation lag problem) these calculations have been repeated for a Morse anharmonic oscillator with transition between nearest and next-nearest neighbor levels [S. K. Kim, /. Chem. Phys. (to be published)]. The numerical evaluation of the analytical results obtained by Kim has not yet been carried out. From the results obtained by us and our co-workers [Barley, Montroll, Rubin, and Shuler, /. Chem. Phys. in press)] on the relaxation of vibrational nonequilibrium distributions of a system of Morse anharmonic oscillators it seems clear, however, that the anharmonic oscillator model with weak interactions (i.e., adiabatic perturbation type matrix elements) does not constitute much of an improvement on the harmonic oscillator model in giving the observed rates of activation. The answer to tliis problem would seem to lie in a recalculation of the collisional matrix elements for translational-vibrational energy exchange which takes account of the strong interactions in highly energetic collisions which can lead to direct dissociation. 

A mobile charge densities in harmonic oscillators (MCDHO) molecular model for numerical simulations The water-water interaction ... 

Two forms of the cell model (CM) are then developed harmonic oscillator approximation and square-well approximation. Both forms assnme hexagonal closed packing (HCP) lattice structure for the cell geometry. The model developed by Paul and Di Benedetto [13] assumes that the chain segments interact with a cylindrical symmetric square-well potential. The FOV model discnssed in the earlier section uses a hard-sphere type repulsive potential along with a simple cubic (SC) lattice structure. The square-well cell model by Prigogine was modified by Dee and Walsh [14]. They introduced a numerical factor to decouple the potential from the choice of lattice strncture. A universal constant for several polymers was added and the modified cell model (MCM) was a three-parameter model. The Prigogine cell EOS model can be written as follows. 

As a numerical example, we investigate ET in the Marcus inverted regime to reveal the solvent effect. The potentials in the fast q) and slow (x) coordinates are modeled by two shifted harmonic oscillators, although the approaches can be straightforwardly applied to anharmonic systems. The parameters. 

Up to the present, very simple solvent models have been generally used in order to study electron transfer processes in solution. These simplified models imply a drastic reduction of the degrees of freedom of the system and in them the movement of the solvent is represented by harmonic oscillations. In the last years, statistical methods based on numerical simulation have permitted to treat explicitly many solvent molecules in a discrete representation. These statistical methods have opened very hopeful perspectives for the study of the solvent reorganization which has to be produced before electron transfer itself takes place. 

The main limitation of the TAB model is that it can only keep track of one oscillation mode, while in reality more than one mode exists. The model keeps track only of the fundamental mode, corresponding to the lowest order harmonic whose axis is aligned with the relative velocity vector between droplet and gas. This is the most important oscillation mode, but for large Weber numbers other modes are also contributing significantly to drop breakup. Despite this limitation, a rather good agreement is achieved between the numerical and experimental results for low Weber numbers. 

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