# Rotational Transitions

## Rotational Transitions

### 1 Order of Magnitude Energies

Molecules can rotate, and have rotational energies that depend on their angular momentum. If ${\displaystyle J}$ is the quantum angular momentum number,

${\displaystyle E_{rot}\sim {\frac {\hbar ^{2}J(J+1)}{2I}}\,\!}$
${\displaystyle I\sim ML^{2}\,\!}$

For small ${\displaystyle J}$,

${\displaystyle E_{rot}\sim {\frac {m}{M}}E_{elec}\,\!}$

So the electronic, vibrational, and rotational energy states have contributions that scale with the electron-to-nucleus mass ratio:

${\displaystyle E_{elec}:E_{vib}:E_{rot}\sim 1:\left({\frac {m}{M}}\right)^{1/2}:\left({\frac {m}{M}}\right)\,\!}$

Rotational energies can be described using the angular momentum number ${\displaystyle J}$:

${\displaystyle E_{rot}={\frac {\hbar ^{2}}{2I}}J(J+1)\,\!}$

where ${\displaystyle J=0,\ 1,\ 2,\ ...}$, ${\displaystyle I}$ is the moment of inertia, and for diatomic molecules, we can use the reduced mass ${\displaystyle \mu }$:

${\displaystyle I=\mu L^{2}\,\!}$
${\displaystyle \mu ={\frac {m_{1}m_{2}}{m_{1}+m_{2}}}\,\!}$