Inter-Atomic Vibration, Interaction, and Bonding Localization

2 INTER-ATOMIC VIBRATION, INTERACTION, AND BONDING LOCALIZATION 

A direct application of the Schodinger eqrration consists in determining the harmonic oscillator spectrum, specific for the biatomic molecular vibrations (with the typical case of the molecule). For this case, the quantum 

However, although we will not directly solve this equation, we will determine the solutions by the test functions method. Thus, based on the stationary wave functions properties, to be continuous, derivable and tend to zero when the variable tends to infinite, one will try the wave function with the right form, which corresponds to the first vibration mode  

The quality of the absolute energetic minimum resides in the fact that the wave function associated with Eq. (2.2) has no nodes, it is not annu-lated anywhere in the space, thus corresponding to the fundamental energy of the quantum harmonic oscillator. 

Further considering the wave function on the first excited state 

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Inter-Atomic Vibration, Interaction, and Bonding Localization. 97... 

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