Finite-energy $f$-sum rules for valence electrons

The number of electrons effective in optical processes up to an energy $\omega$, $n_{\mathrm{eff}}\left(\omega \right)$, may be defined in three distinct ways, a situation which had led to considerable confusion. This is a consequence of the fact that oscillator-strength sums may be constructed from the imaginary part of the dielectric function ${\epsilon }_2\left(\omega \right)$, the extinction coefficient $\kappa \left(\omega \right)$, or the energy-loss function Im[${\epsilon }^{-1\mathrm{}}\left(\omega \right)$]. Here these quantities are investigated for an electronic system embedded in a polarizable medium of dielectric constant ${\epsilon }_b$. This model closely approximates valence electrons moving in the background of polarizable ion cores in a condensed phase. The oscillator-strength sums are found to differ significantly and are not simply related at energies for which the embedded system's oscillator strength is not exhausted. In the limit in which exhaustion occurs, the sums differ only because of the shielding effects of the polarizable medium. The $f$ sum for ${\epsilon }_2\left(\omega \right)$ then yields the system's conventional oscillator strength while the $f$ sums for $\kappa \left(\omega \right)$ and Im[${\epsilon }^{-1\mathrm{}}\left(\omega \right)$] yield effective "strengths" that are reduced from the conventional value by factors of ${\epsilon }_b^{\frac{-1\mathrm{}}{2}}$ and ${\epsilon }_b^{-2\mathrm{}}$, respectively. Similar results hold for the three definitions of $n_{\mathrm{eff}}\left(\omega \right)$. The analysis of a system in terms of partial $f$ sums is shown to provide a check on the self-consistency of optical data as well as a means of determining core polarizabilities. These effects are illustrated for metallic aluminum.