Generalized Morse potentials

The generalized Morse potentials (37) in which the molecules dissociate to their lowest neutral asymptotes, are shown in Figures 2 and 3 and illustrate the significantly larger Dq of CaF (an ionic effect) and the significantly smaller Tq which is a consequence of the much smaller S, separation in Ca relative to Zn. This small separation also permits the state in CaF to be bound relative to the ground state atoms. ... 

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... 

A third functional form, which has an exponential dependence and the correct general shape, is the Morse potential, eq. (2.5). It does not have the dependence at long range, but as mentioned above, in reality there are also R, /f etc. terms. The D and a parameters of a Morse function describing vdw will of course be much smaller than for str, and Ro will be longer. 

It is interesting to note that the simple Morse potential model, when employed with appropriate values for the parameters a and D (a = 2.3 x 1010 m 1, D = 5.6 x 10 19 J as derived from spectroscopic and thermochemical data), gives fb = 6.4 nN and eb = 20%, which are quite comparable to the results obtained with the more sophisticated theoretical techniques [89]. The best experimental data determined on highly oriented UHMWPE fibers give values which are significantly lower than the theoretical estimates (fb 2 nN, b = 4%), the differences are generally explained by the presence of faults in the bulk sample [72, 90] or by the phonon concept of thermomechanical strength [15]. 

The expressions of the Sections 1.5 and 1.6 are general and apply to any solution of the Schrodinger equation. In the special case of a Morse potential, the radial integrals in Eq. (1.34) can be evaluated, with some approximations, in closed form. The approximation consists in replacing the lower limit of integration by -oo. This approximation is similar to that used in Section 1.3 when obtaining the wave functions. Thus... 

The typical behavior of M0 v is shown in Figure 1.6. One should note that, for the Morse potential, and in lowest approximation, the radial wave functions and thus v are independent of /. This is no longer the case for more general potentials and for the exact solution of the Morse problem. 

The double degeneracy of the 0(2) case corresponds to the fact that the algebraic method describes in this case two Morse potentials related to each other by a reflection around x = 0. This is a peculiar feature of one-dimensional problems, and it does not appear in the general case of three dimensions. If one uses the 0(2) basis for calculations, this peculiarity can be simply dealt with by considering only the positive branch of M. 

A general potential V(r) corresponds to a generic algebraic Hamiltonian (2.29). In the most general case the solution cannot be obtained in explicit form but requires the diagonalization of a matrix. The matrix is (N + 1) dimensional. An alternative approach, useful in the case in which the potential does not deviate too much from a case with dynamical symmetry, is to expand it in terms of the limiting potential. For the Morse potential, this implies an expansion of the type (1.7)... 

Matsumoto, A. (1988), Generalized Matrix Elements in Discrete and Continuum States for the Morse Potential, J. Phys. B 21, 2863. 

The Lennard-Jones potential continues to be used in many force fields, particularly those targeted for use in large systems, e.g., biomolecular force fields. In more general force fields targeted at molecules of small to medium size, slightly more complicated functional forms, arguably having more physical justification, tend to be used (computational times for small molecules are so short dial the efficiency of the Lennard-Jones potential is of little consequence). Such forms include the Morse potential [Eq. (2.5)] and the Hill potential... 

The potential energy curves for the states of most diatomics are generally known with high accuracy from the analysis of spectroscopic or scattering data. Many analytical functions have been proposed which reproduce the main features of attractive and repulsive potentials and we have already mentioned the Morse potentials of equations (32) and (33) as typical of these. Comprehensive reviews of other functions have been given by Varshni 121) and Goodisman 122). 

The commonest form of general potential function is one in which the Morse potential of BC, corresponding to V with A infinitely separated from BC, is converted smoothly along the reaction path to the Morse potential for AB [318, 305, 319, 298]. The inclusion of adjustable parameters allows systematic changes to be made in the position of the potential barrier (or col) on the surface, the curvature of the path passing from reactants with minimum energy, and the steepness of the potential along this reaction path. 

Jordan ( ) predicted five electronic levels (X E, A a, B e, D n, B e) based on the reported value (2) of 29560 cm" for the C n level (this level is designated A II by Herzberg (2)). Several qualitative spectroscopic investigations of PH(g) have been reported and are in general accord with the predictions of Jordan (1 ). Ishaq and Pearse (3 ) reported the values of the rotational constant B and the fundamental vibrational frequency The value of is calculated from the Morse potential function. The... 

Abstract A generalization of the Landau-Teller model for vibrational relaxation of diatoms in collisions with atoms at very low energies is presented. The extrapolation of the semiclassical Landau-Teller approach to the zero-energy Bethe-Wigner limit is based on the quasiclassical Landau method for calculation of transition probabilities, and the recovery of the Landau exponent from the classical collision time. The quantum suppression-enhancement probabilities are calculated for a general potential well, which supports several bound states, and for a Morse potential with arbitrary number of states. The model is applied to interpretation of quantum scattering calculations for the vibrational relaxation of H2 in collisions with He. 

Morse Clusters. - As another generalization one may consider clusters consisting of atoms that are interacting via Morse potentials. Assuming that we have only one type of atoms, this gives... 

Thus the transfer probability depends on (i) the way in which the population is split and recombined and (ii) the difference of the dynamical phases. In the context of complete transfer, the process has been named generalized or multiphoton 71-pulse and has been tested numerically for a five-photon resonance in a Morse potential [60]. This formula (264) also displays generalized Rabi oscillations. 

In 1963 negative-ion Morse parameters for the ground-state anions of Br2 and I2 were obtained by estimating D, re, and v from the VEa measured from charge transfer spectra and properties of the excited states of the neutral. Multiple excited states of I2(—) were characterized by D. R. Herschbach in 1966. He presented general forms for ionic Morse potential energy curves (HIMPEC). Nine total groups... 

All the curves discussed in this chapter have been assigned to the general Herschbach ionic Morse potential curves classification. While many of these curves are speculative, they reflect the data. As stated by Robert Sanderson Mulliken in offering curves for I2 in 1971, While the curves shown cannot possibly be quantitatively correct, they should be useful as forming a sort of zeroth approximation to the true curves. [128]. 

---