# Oscillator strength

In spectroscopy, oscillator strength is a dimensionless quantity that expresses the probability of absorption or emission of electromagnetic radiation in transitions between energy levels of an atom or molecule.[1][2][3]

## Theory

An atom or a molecule can absorb light and undergo a transition from one quantum state to another.

The oscillator strength $f_{12}$ of a transition from a lower state $|1\rangle$ to an upper state $|2\rangle$ may be defined by

$f_{12} = \frac{2 }{3}\frac{m_e}{\hbar^2}(E_2 - E_1) \sum_{\alpha=x,y,z} | \langle 1 m_1 | R_\alpha | 2 m_2 \rangle |^2,$

where $m_e$ is the mass of an electron and $\hbar$ is the reduced Planck constant. The quantum states $|n\rangle, n=$ 1,2, are assumed to have several degenerate sub-states, which are labeled by $m_n$. "Degenerate" means that they all have the same energy $E_n$. The operator $R_x$ is the sum of the x-coordinates $r_{i,x}$ of all $N$ electrons in the system, etc.:

$R_\alpha = \sum_{i=1}^N r_{i,\alpha}.$

The oscillator strength is the same for each sub-state $|n m_n\rangle$.

## Thomas–Reiche–Kuhn sum rule

The sum of the oscillator strength from one sub-state $|i m_i\rangle$ to all other states $|j m_j\rangle$ is equal to the number of electrons $N$:

$\sum_j f_{ij} = N.$[4]