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# Oscillator strength

An atom or a molecule can absorb light and undergo a transition from one quantum state to another. The oscillator strength is a dimensionless quantity to express the strength of the transition. The oscillator strength f12 of a transition from a lower state $|1 m_1\rangle$ to an upper state $|2 m_2\rangle$ may be defined by

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

where me is the mass of an electron and $\hbar$ is the reduced Planck constant. The quantum states $|n m_n\rangle, n=$ 1,2, are assumed to have several degenerate sub-states, which are labeled by mn. "Degenerate" means that that they all have the same energy En. The operator Rx is the sum of the x-coordinates ri,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 $|1 m_1\rangle$.

## 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:

 ∑ fij = N. j