Morse

P. M. Morse, K. U. Ingard, Theoretical Acoustics, (Me Graw-Hill Book Comp., 1968). 

Morse, PM, Feshbach, H 1953 Methods of Theoretical Physics, (Me. Grow- HillBook Comp., New York). 

Morse [119] introduced a potential energy model for tire vibrations of bound molecules... 

Morse P M 1929 Diatomic molecules according to the wave mechanics II. Vibrational levels Phys. Rev. 34 57... 

The above expressions are empirical approaches, with m and D. as parameters, for including an anliamionic correction in the RRKM rate constant. The utility of these equations is that they provide an analytic fomi for the anliamionic correction. Clearly, other analytic fomis are possible and may be more appropriate. For example, classical sums of states for Fl-C-C, F1-C=C, and F1-C=C hydrocarbon fragments with Morse stretching and bend-stretch coupling anhamionicity [M ] are fit accurately by the exponential... 

Heather R and Metiu H 1985 Some remarks concerning the propagation of a Gaussian wave packet trapped in a Morse potential Chem. Phys. Lett. 118 558-63... 

Morse M D 1986 Clusters of transition-metal atoms Chem. Rev. 86 1049... 

Behm J M and Morse M D 1994 Spectroscopy of]et-cooled AIMn and trends in the electronic structure of the 3d transition metal aluminides J. Chem. Rhys. 101 6500... 

Arrington C A, Morse M D and Doverstal M 1995 Spectroscopy of mixed early-late transition metal diatomics ScNi, YPd, and ZrCo J. Chem. Rhys. 102 1895... 

Morse T 1979 Handbook of Organic Additives for Use in Ceramic Body Formuiation (Butte, MA Montana Energy and MHD Research and Deveiopment institute)... 

During initialization and final analysis of the QCT calculations, the numerical values of the Morse potential paiameters that we have used aie given as... 

P, M, Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1953, pp. 52-54, 1763,... 

The center of the wavepacket thus evolves along the trajectory defined by classical mechanics. This is in fact a general result for wavepackets in a hannonic potential, and follows from the Ehrenfest theorem [147] [see Eqs. (154,155) in Appendix C]. The equations of motion are straightforward to integrate, with the exception of the width matrix, Eq. (44). This equation is numerically unstable, and has been found to cause problems in practical applications using Morse potentials [148]. As a result, Heller inboduced the P-Z method as an alternative propagation method [24]. In this, the matrix A, is rewritten as a product of matrices... 

Next, we replace the stiff spring potential a(r — 1) /2 by the Morse potential... 

To calculate the bonded interaction of two atoms, a Morse function is often used. It has the form described by Eq, (19). 

Figure 7-9. Variation of the potential energy of the bonded interaction of two atoms with the distance between them. The solid line comes close to the experimental situation by using a Morse function the broken line represents the approximation by a harmonic potential.

For each pair of interacting atoms (/r is their reduced mass), three parameters are needed D, (depth of the potential energy minimum, k (force constant of the par-tictilar bond), and l(, (reference bond length). The Morse ftinction will correctly allow the bond to dissociate, but has the disadvantage that it is computationally very expensive. Moreover, force fields arc normally not parameterized to handle bond dissociation. To circumvent these disadvantages, the Morse function is replaced by a simple harmonic potential, which describes bond stretching by Hooke s law (Eq. (20)). 

Compared with the Morse potential, Hooke s law performs reasonably well in the equilibrium area near If, where the shape of the Morse function is more or less quadratic (see Figure 7-9 in the minimum-energy region). To improve the performance of the harmonic potential for non-equilibrium bond lengths also, higher-order terms can be added to the potential according to Eq. (21). 

Additionally to and a third adjustable parameter a was introduced. For a-values between 14 and 15, a form very similar to the Lennard-Jones [12-6] potential can be obtained. The Buckingham type of potential has the disadvantage that it becomes attractive for very short interatomic distances. A Morse potential may also be used to model van der Waals interactions in a PEF, assuming that an adapted parameter set is available. 

D MoRSE desaiptor, radial distribution function (RDF code), WHIM descriptors, GETAWAY descriptors,... 

D Molecule Representation of Structures Based on Electron Diffraction Code (3D MoRSE Code)... 

The 3D MoRSE code is closely related to the molecular transform. The molecular transform is a generalized scattering function. It can be used to predict the intensity of the scattered radiation i for a known molecular structure in X-ray and electron diffraction experiments. The general molecular transform is given by Eq. (22), where i(s) is the intensity of the scattered radiation caused by a collection of N atoms located at points r. ... 

Steinhauer and Gasteiger [30] developed a new 3D descriptor based on the idea of radial distribution functions (RDFs), which is well known in physics and physico-chemistry in general and in X-ray diffraction in particular [31], The radial distribution function code (RDF code) is closely related to the 3D-MoRSE code. The RDF code is calculated by Eq. (25), where/is a scaling factor, N is the number of atoms in the molecule, p/ and pj are properties of the atoms i and/ B is a smoothing parameter, and Tij is the distance between the atoms i and j g(r) is usually calculated at a number of discrete points within defined intervals [32, 33]. 

A Morse fun Cl ion best approxini ales a boiul polcn iial. One of ihc obvious clifl erciices between a Morse and harinoiiic poleiUial is lhal only (he Morse potential can describe a dissociating bond. 

The Morse function rises more steeply ihan ihe harmonic potential at short bonding distances. This difference can be important especially during molecular dynamics simulations, where thermal energy takes a molecule away from a potential minimum. ... 

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