#
Ground State of a Diatomic Molecule

##
Introduction

Two atoms that can form a bond will do so to create a diatomic molecule
when they approach each other closely. We describe the energy minimum with
a potential energy curve, called a potential well.

##
Morse Potential

The potential energy, V(R), of a diatomic molecule can be described by
the Morse potential:
V(R) = D_{e} (1 − 2^{-b(R-Re)})

where D_{e} is the well depth, R is internuclear distance, R_{e}
is the equlibrium internuclear distance (bond length), and

β = pn_{e}Ö(2m/D_{e})

ν_{e} is the vibrational constant
and µ is the reduced mass.

The following figure shows the ground state potential well of the H_{2}
molecule. The curve is calculated from the Morse potential and the energy
levels are calculated using the harmonic oscillator model with the first
anharmonic correction.

D_{o} is the dissociation energy, which is slightly different
from the well depth, D_{e}. The vibrational energy levels in this
plot are calculated from:

Ev = (v + ½) ν_{e}
- (v + ½)² ν_{e}x_{e}
+ (v + ½)³ ν_{e}y_{e}
+ higher terms

where v is the vibrational quantum number, v = 0,1,2,..., and x_{e}
and y_{e} are the first and second anharmonicity constants, respectively.
The v = 0 level is the vibrational ground state.

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