Editing Diatomic molecule (section)

== Energy levels ==
The [[molecular term symbol]] is a shorthand expression of the angular momenta that characterize the electronic quantum states of a diatomic molecule, which are also [[eigenstate]]s of the electronic molecular [[Hamiltonian (quantum mechanics)|Hamiltonian]]. It is also convenient, and common, to represent a diatomic molecule as two-point masses connected by a massless spring. The energies involved in the various motions of the molecule can then be broken down into three categories: the translational, rotational, and vibrational energies. The theoretical study of the rotational energy levels of the diatomic molecules can be described using the below description of the rotational energy levels. While the study of vibrational energy level of the diatomic molecules can be described using the harmonic oscillator approximation or using the quantum vibrational interaction potentials.<ref name="Swati">{{Cite journal|doi = 10.1016/j.ctta.2022.100073|title = Temperature guided behavioral transitions in confined helium: Gas-wall interaction effects on dynamics and transport in the cryogenic limit |year = 2022|last =  Mishra|first = Swati|journal = Chemical Thermodynamics and Thermal Analysis |volume = 7|issue = August |pages = 100073|doi-access = free}}</ref><ref name="Al-Raeei">{{Cite journal|doi = 10.1088/1361-648X/ac6a9b|title = Morse potential specific bond volume: a simple formula with applications to dimers and soft–hard slab slider |year = 2022|last = Al-Raeei|first = Marwan |journal = Journal of Physics: Condensed Matter |volume = 34|issue = 28|pages = 284001|pmid = 35544352 |doi-access = free|bibcode = 2022JPCM...34B4001A }}</ref> These potentials give more accurate energy levels because they take multiple vibrational effects into account.

Concerning history, the first treatment of diatomic molecules with quantum mechanics was made by [[Lucy Mensing]] in 1926.<ref>{{cite journal |last=Mensing |first=Lucy |title=Die Rotations-Schwingungsbanden nach der Quantenmechanik |journal=Zeitschrift für Physik |volume=36 |issue=11 |date=1926-11-01 |issn=0044-3328 |pages=814–823 |doi=10.1007/BF01400216 |bibcode=1926ZPhy...36..814M |s2cid=123240532 |language=German }}</ref>

=== Translational energies ===
The translational energy of the molecule is given by the [[kinetic energy]] expression:
<math display="block">E_\text{trans} = \frac{1}{2}mv^2</math>
where <math>m</math> is the mass of the molecule and <math>v</math> is its velocity.

=== Rotational energies ===
Classically, the kinetic energy of rotation is
<math display="block">E_\text{rot} = \frac{L^2}{2 I} </math>
where
* <math>L \,</math> is the [[angular momentum]]
* <math>I \,</math> is the [[moment of inertia]] of the molecule

For microscopic, atomic-level systems like a molecule, angular momentum can only have specific discrete values given by
<math display="block">L^2 = \ell(\ell+1) \hbar^2 </math>
where <math>\ell</math> is a non-negative integer and <math>\hbar</math> is the [[reduced Planck constant]].

Also, for a diatomic molecule the moment of inertia is
<math display="block">I = \mu r_0^2 </math>
where
* <math>\mu \,</math> is the [[reduced mass]] of the molecule and
* <math>r_0 \,</math> is the average distance between the centers of the two atoms in the molecule.

So, substituting the angular momentum and moment of inertia into {{math|''E''<sub>rot</sub>}}, the rotational energy levels of a diatomic molecule are given by:
<math style="" display="block">E_\text{rot} = \frac{\ell (\ell + 1) \hbar^2}{2 \mu r_0^2}, \quad \ell = 0, 1, 2, \dots</math>

=== Vibrational energies ===
Another type of motion of a diatomic molecule is for each atom to oscillate—or [[vibration|vibrate]]—along the line connecting the two atoms. The vibrational energy is approximately that of a [[quantum harmonic oscillator]]:
<math display="block">E_\text{vib} = \left(n + \tfrac{1}{2} \right)\hbar \omega, \quad n = 0, 1, 2, \dots,</math>
where
* <math>n</math> is an integer
* <math>\hbar</math> is the [[reduced Planck constant]] and
* <math>\omega</math> is the [[angular frequency]] of the vibration.

=== Comparison between rotational and vibrational energy spacings ===
The spacing, and the energy of a typical spectroscopic transition, between vibrational energy levels is about 100 times greater than that of a typical transition between [[rotational energy]] levels.