Morse case

In the Morse Case (7), one of the claims had been worded so that it claimed the effect—the principle of nature by which the inventor s apparatus operated—rather than the process or machinery necessary to produce the desired effect. The court ruling in this instance can be said to be no more than a ruling that the abstract principle discovered is not of itself the patentable part of invention. It is the concrete embodiment or the manner of application, either as a process or in a machine— the how to use to the benefit of mankind—which is patentable. 

A Mathematica calculation of Franck-Condon factors that determine electronic transition intensities of I2 is presented in Chapter III, and program statements for this are illustrated for I2 in Fig. III-6. In this fignre, note the dramatic differences between the intensity patterns predicted for the harmonic oscillator and Morse cases and compare these patterns with those seen in your absorption spectra. If yon have access to this software, yon might examine the changes in the harmonic-oscillator and Morse-oscillator wavefnnctions for different v, v" choices. A calcnlation of the relative emission intensities from the v = 25, 40, or 43 level conld also be done for comparison with emission spectra obtained with a mercury lamp or with a krypton- or argon-ion laser, hi contrast to the smooth variation in the intensity factors seen in the absorption spectra, wide variations are observed in relative emission to v" odd and even valnes, and this can be contrasted with the calcnlated intensities. Note that, if accnrate relative comparisons are to be made with experimental intensities, the theoretical intensity factor from the Mathematica program for each transition of wavennmber valne v shonld be mnltiphed by v for absorption and for emission. ... 

Consider Fig. 5e, which is a uniform asymptotic action distribution for the coupled Morse case. If one extracts the > 103 component of this distribution... 

Bond stretching is most often described by a harmonic oscillator equation. It is sometimes described by a Morse potential. In rare cases, bond stretching will be described by a Leonard-Jones or quartic potential. Cubic equations have been used for describing bond stretching, but suffer from becoming completely repulsive once the bond has been stretched past a certain point. 

Acute-Duration Exposure. There are numerous case reports of human fatalities (Adelson and Sunshine 1966 Allyn 1931 Breysse 1961 Campanya et al. 1989 Deng and Chang 1987 Freireich 1946 Hagley and South Morse et al. 1981 Osbem and Crapo 1981 Parra et al. 1991) or survivors who developed immediate as well as delayed neurological effects (Deng and Chang 1987 Kilbum 1993,... 

There is thus an apparent continuity between the kinetics of an electron transfer leading to a stable product and a dissociative electron transfer. The reason for this continuity is the use of a Morse curve to model the stretching of a bond in a stable product in the first case and the use of a Morse curve also to model a weak charge-dipole interaction in the second case. We will come back later to the distinction between stepwise and concerted mechanisms in the framework of this continuity of kinetic behavior. 

Case reports are available regarding lethal effects of acute exposure to arsine (Pinto et al. 1950 Morse and Setterlind 1950 Hesdorffer et al. 1986). However, no definitive quantitative exposure data accompany these reports. Signs and symptoms varied depending on the exposure situation but usually included abdominal and muscle pain, nausea and diarrhea, hematuria, and oliguria. Delayed lethality, common in arsine poisoning, varied considerably. 

The multiplicity of excitations possible are shown more clearly in Figure 9.16, in which the Morse curves have been omitted for clarity. Initially, the electron resides in a (quantized) vibrational energy level on the ground-state Morse curve. This is the case for electrons on the far left of Figure 9.16, where the initial vibrational level is v" = 0. When the electron is photo-excited, it is excited vertically (because of the Franck-Condon principle) and enters one of the vibrational levels in the first excited state. The only vibrational level it cannot enter is the one with the same vibrational quantum number, so the electron cannot photo-excite from v" = 0 to v = 0, but must go to v = 1 or, if the energy of the photon is sufficient, to v = 1, v = 2, or an even higher vibrational state. 

In the stepwise case, the intermediate ion radical cleaves in a second step. Adaptation of the Morse curve model to the dynamics of ion radical cleavages, viewed as intramolecular dissociative electron transfers. Besides the prediction of the cleavage rate constants, this adaptation opens the possibility of predicting the rate constants for the reverse reaction (i.e., the reaction of radicals with nucleophiles). The latter is the key step of SrnI chemistry, in which electrons (e.g., electrons from an electrode) may be used as catalysts of a chemical reaction. A final section of the chapter deals... 

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)... 

This is a Dunham-like expansion but done around the anharmonic solution. It converges very quickly to the exact solution if the potential is not too different from that of a Morse oscillator (Figure 2.3). This will not, however, be the case for the highest-lying vibrational states just below the dissociation threshold. The inverse power dependence of the potential suggests that fractional powers of n must be included (LeRoy and Bernstein, 1970). 

This is identical to Eq. (6.45) with A = D and X - 1 = N/2. One also notes that the spectrum of the Poschl-Teller potential in one dimension is identical to that of the Morse potential in one dimension. These two potentials are therefore called isospectral. This identity arises from the fact that, as mentioned in Chapter 3, the two algebras 0(2) and U(l) are isomorphic. The situation is different in three dimensions, where this is no longer the case. 

For a later purpose (Chapter 7), we shall explain the perfectness of the Morse function given by the moment map of a torus action on a general symplectic manifold. However, when the fixed points of a torus action are all isolated, such as the case of the... 

Notice that our argument also gives the proof of the perfectness of the Morse function in the case of a noncompact symplectic manifold if the appropriate conditions on / are satisfied. For example, the condition that / ((—oo, c]) is compact for all c G IR is sufficient, and this is the case for as will be shown later. 

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