Morse-oscillator-like functions

An alternative approach to solving the coupled equations is to use a basis set expansion for the R coordinate as well as for the angular variables. The angular basis sets used in such calculations are generally the same as in coupled channel calculations. This approach was pioneered by Le Roy and Van Kranendonk, who used numerical basis sets for the radial (R) functions. Such basis sets are adequate for the rare gas-H2 systems, but converge very poorly for more strongly anisotropic systems. An alternative basis set, based on Morse-oscillator-like functions, has been used extensively by Tennyson and coworker... 

In our calculations, the radial parts of the wavefunction, and are represented by Morse-oscillator-like functions, with... 

The unavailability of an RKR-like inversion (hence the impossiblity of obtaining the potential energy surface, V(Q), and exact vibrational eigenfunctions directly from experimental data) makes it convenient to use products of simple harmonic or Morse-oscillator basis functions as vibrational basis states... 

However, whereas the natural collision coordinate v is a simple vibrational coordinate with a Morse oscillator-like potential energy curve throughout the course of the reaction, as can be seen by inspection of Figure 3, the hyperangular coordinate 0 is subject to a more complicated potential energy profile that changes very dramatically as a function of p (see Figures 4 and 5). 

As already stated, the Morse potential is our first example of a potential surface that describes a particular motion. The bond vibrates within the constraints imposed by this potential. One may ask, "At any given moment, what is the probability of having a particular bond length " This is similar to questions related to the probability of finding electrons at particular coordinates in space, which we will show in Chapter 14 is related to the square of the wave-function that describes the electron motion. The exact same procedure is used for bond vibrations. We square the wavefunction that describes the wave-like nature of the bond vibration. Let s explore this using the potential surface for a harmonic oscillator (such as with a normal spring), instead of an anharmonic oscillator (Morse potential). For the low energy vibrational states, the harmonic oscillator nicely mimics the anharmonic oscillator. 

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