Vibrational spacing, Morse-like

The bend overtone progression (0, V2, 0) exhibits surprisingly regular Morse-like vibrational spacings [5, 12, 19], described by... 

This equation assumes harmonic behavior where the vibrational levels are equally spaced (separated by an energy equivalent to np). However, bonds do not behave exactly like springs (for example, the restoring force is weaker for a bond than is predicted for a spring at greatly stretched bond lengths). The actual potential energy curve (called the Morse potential) for a bond is represented by the blue curve in Fig. 14.58 and is shown in detail in Fig. 14.59. 

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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