Long-range interactions of hydrogen atoms in excited states. III. nS−1S interactions for n≥3

The long-range interaction of excited neutral atoms has a number of interesting and surprising properties such as the prevalence of long-range oscillatory tails and the emergence of numerically large van der Waals ${C}_{6}$ coefficients. Furthermore, the energetically quasidegenerate $nP$ states require special attention and lead to mathematical subtleties. Here we analyze the interaction of excited hydrogen atoms in $nS$ states ($3\ensuremath{\le}n\ensuremath{\le}12$) with ground-state hydrogen atoms and find that the ${C}_{6}$ coefficients roughly grow with the fourth power of the principal quantum number and can reach values in excess of $240\phantom{\rule{0.16em}{0ex}}000$ (in atomic units) for states with $n=12$. The nonretarded van der Waals result is relevant to the distance range $R\ensuremath{\ll}{a}_{0}/\ensuremath{\alpha}$, where ${a}_{0}$ is the Bohr radius and $\ensuremath{\alpha}$ is the fine-structure constant. The Casimir-Polder range encompasses the interatomic distance range ${a}_{0}/\ensuremath{\alpha}\ensuremath{\ll}R\ensuremath{\ll}\ensuremath{\hbar}c/\mathcal{L}$, where $\mathcal{L}$ is the Lamb shift energy. In this range, the contribution of quasidegenerate excited $nP$ states remains nonretarded and competes with the $1/{R}^{2}$ and $1/{R}^{4}$ tails of the pole terms, which are generated by lower-lying $mP$ states with $2\ensuremath{\le}m\ensuremath{\le}n\ensuremath{-}1$, due to virtual resonant emission. The dominant pole terms are also analyzed in the Lamb shift range $R\ensuremath{\gg}\ensuremath{\hbar}c/\mathcal{L}$. The familiar $1/{R}^{7}$ asymptotics from the usual Casimir-Polder theory is found to be completely irrelevant for the analysis of excited-state interactions. The calculations are carried out to high precision using computer algebra in order to handle a large number of terms in intermediate steps of the calculation for highly excited states.