Excited $Σ$ states of the hydrogen-antihydrogen molecule

TL;DR

This work addresses excited Sigma-state spectroscopy of the hydrogen-antihydrogen system within the Born-Oppenheimer framework, using explicitly correlated Kołos-Wolniewicz basis functions and exploiting Q symmetry to separate the leptonic spectrum into Q-even and Q-odd sectors. The authors compute a large set of leptonic Born-Oppenheimer curves V_m(R) by solving a generalized eigenproblem for non-orthogonal bases, employing a dual-base optimization to capture both molecular and discretized positronium characters, and extrapolate below the critical distance Rc ≈ 0.744 a0. They find a dense cluster of rovibrational states near the ground-state dissociation threshold and demonstrate numerous avoided crossings with discretized positronium channels, implying that excited leptonic states can resonantly couple to ground-state HbarH scattering. A cross-check with a full four-body calculation corroborates the BO results within a few parts in 10^3 and confirms the relevance of including excited leptonic channels in scattering descriptions of H–Hbar collisions.

Abstract

Adopting explicitly correlated Kolos-Wolniewicz-type basis functions, the Born-Oppenheimer potential curves of a number of excited $Σ$ states of the hydrogen-antihydrogen system ($\bar{\rm H}$) were calculated for both, even and odd, Q symmetries, including also free positronium states. It is demonstrated that the excited leptonic states support ro-vibrational states with energies close to the ground-state dissociation threshold. As a consequence, the excited leptonic states need to be considered in theoretical treatments of ground-state H-$\bar{\mathrm{H}}$ collisions.

Figures (11)

• Figure 1: Structure of the Hamiltonian matrix for a dual-base basis set, i. e. a basis set comprising of two basis sets $\{\phi^{(0)}\}$ and $\{\phi^{(1)}\}$ with the different sets of exponential parameters $(y_0, x_0, u_0, w_0)$ and $(y_1, x_1, u_1, w_1)$, respectively. This leads to three types of matrix elements: $\langle\phi_i^{(0)}|\hat{H}|\phi_j^{(0)}\rangle = \langle\phi_j^{(0)}|\hat{H}|\phi_i^{(0)}\rangle$ (red), $\langle\phi_i^{(1)}|\hat{H}|\phi_j^{(1)}\rangle = \langle\phi_j^{(1)}|\hat{H}|\phi_i^{(1)}\rangle$ (blue), and $\langle\phi_i^{(0)}|\hat{H}|\phi_j^{(1)}\rangle \neq \langle\phi_i^{(1)}|\hat{H}|\phi_j^{(0)}\rangle$ (purple).
• Figure 2: Lowest-lying Born-Oppenheimer potential curves of $\mathrm{H}\bar{\mathrm{H}}$ (in Hartree) as a function of the inter-hadronic separation $R$ (in Bohr) for the $\Sigma$ states that possess Q-even (left, red) or Q-odd (right, blue) symmetry. The inset shows a zoom that more clearly reveals the asymptotic behavior of the low-lying Q-odd states. (The data points are connected by lines for guiding the eye. The basis-sets described in Sec. \ref{['sec:optimization']} with $\Omega=10$ were adopted.)
• Figure 3: As Fig. \ref{['fig:bo_overview']}, but showing the leptonic energies $E_{\rm lep}$ instead of the potential curves $V$. Also shown is the ground state obtained by Strasburger anti:stra02 (left, black dashed) and two diabatized states, sketched manually for guiding the eye (right, black solid).
• Figure 4: Convergence study for the 10 lowest-lying Q-even $\Sigma$ states and $R=0.8\,a_0$. The basis-set is sytematically improved by increasing $\Omega$, see Eq. \ref{['eq:omega']}. Shown is the relative error with respect to the results obtained with $\Omega=10$.
• Figure 5: As Fig. \ref{['fig:conv08']}, but for $R=5.0\,a_0$.