We analyse the influence of the attractive tail of the p–H two-body potential on the energy progression of the near-threshold rotationless bound states of the H molecular ion. For the 1sσ g configuration we study different definitions of the reference potential describing the potential tail. In each case, the scattering length consistent with the energies of the highly excited vibrational states, as calculated by Hilico et al. [Eur. Phys. J. D 12, 449 (2000)], is found to lie in the range between −30.5 and −31 au. In the 2pσ u configuration, for which there is an extremely weakly bound v = 1 vibrational state [Carbonell et al., Europhys. Lett. 64, 316 (2003)], we obtain a scattering length near 762.8 au. In contrast to previously conveyed impressions, the weakly bound v = 1 state can be satisfactorily described as the excited state in a distant shallow potential well of the p–H two-body potential, provided that a sufficient number of terms is included in the representation of the large-distance behaviour of this potential.
Ab initio calculations of the energy level structure of H2+ that include relativistic and radiative corrections to nonrelativistic energies and the diagonal part of the hyperfine interaction have predicted the existence of four bound rovibrational levels [(v = 0, N = 0 − 2) and (v = 1, N = 0)] of the first electronically excited (A+ Σu+2) state of H2+, the (v = 1, N = 0) level having a calculated binding energy of only Eb = 1.082 219 8(4)·10−9 Eh and leading to an extremely large scattering length of 750(5) a0 for the H+ + H collision [J. Carbonell et al., J. Phys. B: At., Mol. Opt. Phys. 37, 2997 (2004)]. We present an investigation of the nonadiabatic coupling between the first two electronic states (X+ Σg+2 and A+ Σu+2) of H2+ induced by the Fermi-contact term of the hyperfine-coupling Hamiltonian. This interaction term, which mixes states of total spin quantum number G = 1/2, is rigorously implemented in a close-coupling approach to solve the spin-rovibronic Schrödinger equation. We show that it mixes states of gerade and ungerade electronic symmetry, that it shifts the positions of all weakly bound rovibrational states of H2+, and that it affects both the positions and widths of its shape resonances. The calculations demonstrate that the G = 1/2 hyperfine component of the A+ (v = 1, N = 0) state does not exist and that, for G = 1/2, the s-wave scattering lengths of the H+ + H(1s) collision are −578(6) a0 and −43(4) a0 for the F = 0 and F = 1 hyperfine components of the H(1s) atom, respectively. The binding energy of the G = 3/2 hyperfine component of the A+ (v = 1, N = 0) state is not significantly affected by the hyperfine interaction and the corresponding scattering length for the H+ + H(1s, F = 1) collision is 757(7) a0.
The interactions between atoms and molecules with each other and with surfaces are typically deep potential wells with attractive tails behaving asymptotically as an inverse power of the distance. In such potential wells, bound state energies
En are determined by a quantization function
F(E) according to ${n}_{\mathrm{th}}\ensuremath{-}n=F({E}_{n})$, and
F(E) is dominantly determined by the singular potential tail for near-threshold states. In this paper we formulate a general theory for the contribution
Ftail(E) of the singular potential tail to the quantization function. The general expression for
Ftail(E) contains two terms: a difference of action integrals and a difference of outer reflection phases, taken at threshold and at energy
E. Close to threshold,
E=0, strongly energy dependent and nonanalytic contributions of both terms cancel, so
Ftail(E) acquires a universal form determined by a threshold length and an effective length which is related to a subthreshold effective range. For a homogeneous potential tail proportional to $\ensuremath{-}1∕{r}^{\ensuremath{\alpha}}$, one universal expression for
Ftail(E) caters for all potential strengths. We give an explicit analytical expression for the important case $\ensuremath{\alpha}=6$ and present applications involving the derivation of atom-atom scattering lengths from the binding energies of high-lying bound states of the associated diatomic molecule. We also demonstrate how the dissociation energy of a diatomic molecule can be determined from spectroscopic energies of high-lying states, and we make a quantitative comparison with the performance of the LeRoy-Bernstein formula, which fails near threshold, because the strongly energy dependent and nonanalytic contribution from the action integrals is not, as it should be, compensated by terms coming from the corresponding energy dependence of the outer reflection phase.
We formulate the quantization rule ${n}_{\mathrm{th}}\ensuremath{-}n=F({E}_{n})$ for bound-state energies
En in a deep potential well with a long-range attractive tail for an extended range of near-threshold energies. This makes it possible to accurately read off the scattering length
a from the energy of a high-lying (not necessarily the highest-lying) bound state via $a=\overline{a}+b∕\mathrm{tan}[\ensuremath{\pi}F({E}_{n})]$. For a potential tail proportional to $\ensuremath{-}1∕{r}^{\ensuremath{\alpha}}$, $\ensuremath{\alpha}>3$, the constants
a and
b are known and an explicit and universally valid expression for the function
F(E) is given analytically.
By making use of the quantization rule of Raab and Friedrich [Phys. Rev. A 78, 022707 (2008)], we derive simple and accurate formulae for the number of rotational states supported by a weakly bound vibrational level of a diatomic molecule and the rotational constants of any such levels up to the threshold, and provide a criterion for determining whether a given weakly bound vibrational level is rotationless. The results depend solely on the long-range part of the molecular potential and are applicable to halo molecules.
For potential wells with long-range attractive tails proportional to $\ensuremath{-}1/{r}^{3}$, as occur in the resonant dipole-dipole interaction in homonuclear alkali-metal dimers, we present a highly accurate analytical expression for the tail contribution to the quantization function
F(E). This quantization function determines the near-threshold bound-state energies via the quantization rule ${n}_{\mathrm{th}}\ensuremath{-}n=F({E}_{n})$. The performance of the quantization function derived in this paper is demonstrated by applying it to a model Lennard-Jones potential and to vibrational bound-state spectra of sodium dimers (Na
2). These results are compared with those obtained via the semiclassical LeRoy-Bernstein formula which neglects quantum effects that are important in the near-threshold regime.
We discuss quantum enhancement of the quasiclassical vibrational predissociation (VP) rate of an atom (A)-diatom (BC) van der Waals (vdW) complex $A\ensuremath{-}\mathrm{BC}$ from a state lying close to the dissociation threshold. The enhancement is due to the accumulation of a noticeable fraction of the state population near the outer turning point of the $A\ensuremath{-}\mathrm{BC}$ bond. For potentials behaving as power-laws at large separations, the enhancement manifests itself in the variation of the energy dependence of the VP frequency factor from the classical frequency to its quantum counterpart. We show that it is related to the complex scattering length associated with the vibrational relaxation of BC in collisions with A. We also discuss the corrected quasiclassical quantization condition for energy levels of a diatom lying very close to the dissociation threshold. Our results generalize those of J. Trost, C. Eltshka, and H. Friedrich [J. Phys. B 31, 361 (1998)] and C. Boisseau et al. [Eur. Phys. J. D 12, 199 (2000)] for the calculation of complex energy levels of an anharmonic oscillator that dissociates from highly excited states under the action of a weak high-frequency perturbation.