F = −kx

k is a proportionality constant called the force constant and x is the displacement from the equilibrium position. This relationship is called Hooke's law. For the spring example, k will be large for a stiff spring and smaller for springs that are weaker. Similarly, if you stretch a spring twice as far, it "springs back" with twice the force. Of course this law is valid for limited values of x. Try stretching a spring too far and you'll find that the restoring force is no longer directly proportional to displacement!

The potential energy, V, for a one-dimensional system is equal to the negative of the force integrated over x:

V(x) = -òFdx = k∫ xdx = ½ kx² + constant

The constant of integration depends on the physical system being modeled.
For the ground state of a diatomic molecule, as modeled below, we can set
it to zero.

V(x) = ½ kx²

where k is the force constant as above and the constant of integration
is zero. We can make this expression more useful by changing x to R-R_{e},
where R is the internuclear distance (the distance between atoms) and R_{e}
is the equilibrium internuclear distance (the bond length):

V(R) = ½ k(R − R_{e})²

The following figure shows the ground-state potential energy curve (called
a potential well) for the H_{2} molecule using the harmonic oscillator
model. R_{e} for H_{2} is 0.7412 Å. There is one
obvious deficiency in the model, it does not show the energy at which the
two atoms dissociate, which occurs at 4.748 eV for the H_{2} molecule
(1 eV = 8065.48 cm^{-1}). At some internuclear distance the atoms
are far enough apart so that they do not "feel" each other. That is, they
are isolated and the bond is broken. A more realistic model of the potential
well of a diatomic molecule is the Morse potential,
which does model the dissociation energy.

The solid blue horizontal lines show the energy levels that are calculated using the harmonic oscillator model:

E_{v} = (v + ½) ν_{e}

where v is the vibrational quantum number (v = 0,1,2,...). The v = 0 level is the vibrational ground state and is the lowest horizontal line in the plot.

ν_{e }is called the vibrational constant:

ν_{e} = ½
πc
Ö(^{k}/_{m})

where µ is the reduced mass (m_{1}m_{2}/m_{1}+m_{2}).
The simple harmonic oscillator provides a good fit to energies for the
lowest energy levels, but fails at higher energies.

The dotted red lines shows the energy levels calculated from:

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

where v and ν_{e} are the same as
above and x_{e} and y_{e} are the first and second anharmonicity
constants respectively. These correction terms provide much better match
of the calculated energies to the energies that are observed experimentally.

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