Adiabatic and post-adiabatic hyperspherical treatment of the huge ungerade proton-hydrogen scattering length

TL;DR

This work analyzes the ungerade electronic state of the H$_2^+$ molecular ion near the dissociation threshold using a hyperspherical adiabatic framework. By diagonalizing the adiabatic Hamiltonian at fixed $R$ and employing a streamlined R-matrix method, the authors quantify adiabatic potentials, bound-state energies, and the $s$-wave scattering length, finding a near-threshold bound state with $E_0\approx -1.56625\times10^{-5}$ a.u. and a large scattering length $a\approx 765.5\,a_0$. The post-adiabatic theory of Klar and Fano is tested and shown to capture the main nonadiabatic corrections via effective, energy-dependent potentials, validating the adiabatic hyperspherical description as an efficient and accurate single-channel-like representation for near-threshold physics. These results underscore the robustness of the hyperspherical adiabatic approach for few-body near-threshold systems and provide a precise benchmark for nonadiabatic corrections in proton-hydrogen scattering.

Abstract

While the hydrogen molecular ion is the simplest molecule in nature and very well studied in all of its properties, it remains an interesting system to use for explorations of fundamental questions. One such question treated in this study relates to finding an optimal adiabatic representation of the physics, i.e. the best adiabatic description that minimizes the role of nonadiabatic effects. As a test case explored here in detail, we consider the ungerade symmetry of H$_2^+$, which is known to have a huge scattering length of order 750 bohr radii, and an incredibly weakly bound excited state. We show that a hyperspherical adiabatic description does an excellent job of capturing the main physics. Our calculation yields a competitive scattering length and shows that nonadiabatic corrections are small and can even be adequately captured using the postadiabatic theory of Klar and Fano.

Figures (5)

• Figure 1: The lowest six potential curves for L=0 in the ungerade symmetry as a function of hyperradius $R$. The zero of the energy scale is set to the energy of the hydrogen atom $E_h = -0.4997278397$ in atomic units. Notice that two curves will approach the $n=2$ threshold of hydrogen and three the $n=3$ threshold. The inset shows the lowest potential curve in the ungerade symmetry. The shallow well is responsible for the extremely weakly bound state.
• Figure 2: The figures show the lowest potential curve $U_1(R)$ plotted against $1/n$ at two different R values. $n$ is the number of basis functions included in each dimension; the plots contain the same number of basis functions in each dimension. Figures (a) and (b) are for $U_1(R)$ at $R=12$ a.u. and $R=30$ a.u. respectively.
• Figure 3: This shows a log plot of $Res(R)^{(n)}$. $Res(R)^{(n)}=U^{eff}_1(R,0)^{(n)}-U^{eff}_1(R,0)^{(8)}$, which clearly shows the convergence of $U^{eff}_1(R)^{(n)}$ as n increases.
• Figure 4: The figure shows the variation of the bound state energy of the first excited state with the inverse box size of the R-matrix $1/R_o$. The bound state energy starts converging as the box size $R_o$ increases.
• Figure 5: The variation of the scattering length is shown as a function of the inverse of the box radius, $1/R_0$, for the H$^+$ + H(1s) collision with 1, 2 or 3 channels. It can be seen that the scattering length starts converging as the box size $R_0$ is increased.