Calculated photoemission, inverse photoemission, and x-ray emission spectra of high-$T_c$ superconductors: ${\mathrm{Tl}}_2$${\mathrm{Ba}}_2$Ca${\mathrm{Cu}}_2$${\mathrm{O}}_8$ and ${\mathrm{Tl}}_2$${\mathrm{Ba}}_2$${\mathrm{Ca}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{10}$

Angle-integrated photoemission, inverse photoemission, and x-ray emission spectra were calculated for ${\mathrm{Tl}}_2$ ${\mathrm{Ba}}_2$Ca${\mathrm{Cu}}_2$ ${\mathrm{O}}_8$ and ${\mathrm{Tl}}_2$ ${\mathrm{Ba}}_2$ ${\mathrm{Ca}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{10}$. All spectra are based on densities of states obtained from local-density full-potential linearized augmented plane-wave (FLAPW) calculations. For the photoemission and inverse photoemission spectra, the single-scatterer final-state approximation was applied. For low photon energies mainly O $p$ states show up in the spectra, and there are appreciable differences between the two compounds. For high photon energies the spectra of both compounds are very similar and show Cu and Tl contributions. The unoccupied states probed by inverse photoemission have mainly O $p$ character with some Tl and Ba contributions. The Cu $L_{\mathrm{III}}$ x-ray emission spectra of both compounds reflect the Cu $d$ density of states (DOS) and closely resemble each other, whereas the O $K\alpha$ x-ray-emission spectra are narrower than the corresponding DOS because of differences in the core levels of inequivalent O atoms.