Dipole sum rules of a hydrogen atom in Debye-Huckel plasma

In this work, we carry out a study of the dipole oscillator strength sum rule, S_k, and the logarithmic, L_k, version, for a hydrogen atom embedded in a Debye-Huckel plasma. We solve the time-independent Schroedinger equation by means of a finite-differences approach to obtain the discrete and continuum spectrum of the atom embedded in the plasma, as a function of the screening parameter, λ_D. We find excellent agreement for the S_k and L_k sum rules for λ_D → ∞, i.e., no plasma, in comparison with exact theoretical results. For plasma interaction, we obtain the conditions for the delocalization of the hydrogen atom energy levels finding its effects on the S_k sum rules. We find that for k ≤ 0, S_k increases as the screening constant decreases until λ ∼ 0.84 a.u., where the 1s ground state becomes delocalized due to the plasma weakening. For the static dipole polarizability, S_-2, we find excellent agreement between our results for the discrete and continuum spectrum and available data in the literature for selected values of screening parameters. In the logarithmic counterpart, L_k, the sum rules decrease for lower values of the screening length for k ≤ 0. We observe a decrease of the mean excitation energy, I₀ = exp(L₀/S₀), as the screening length decreases finding a sudden change for λ_D ∼ 2 a.u. Furthermore, we obtain closure relations for the S_k sum rule with k = -1, 1, 2, 3 as a function of the Debye-Huckel screening and find excellent agreement with the sum over excited states for k = -1, 0, 1, and 2 and find that S₃ sum rule diverges for a hydrogen atom immersed in a Debye-Huckel plasma. We conclude that the fulfillment of the closure relations requires a good description of the excitation states. (authors)