Analysis of hyperspherical adiabatic curves of helium: A classical dynamics study
Abstract
The hyperspherical adiabatic curves (adiabatic eigenenergies as functions of the hyperradius R) of helium for zero total angular momentum are analyzed by studying the underlying classical dynamics which in the adiabatic treatment reduces to constrained two-electron motion on a hypersphere. This dynamics supports five characteristic classical configurations which can be represented by five types of short periodic orbits: the frozen planet (FP), the inverted frozen planet (IFP), the asymmetric stretch (AS), the asynchronous (ASC), and the Langmuir periodic orbit (PO). These POs are considered as fundamental modes of the two-electron motion on a hypersphere which, after quantization, give five families of so-called adiabatic lines (adiabatic energies related to these POs as functions of R). It is found that multiplets, each of them consisting of adiabatic curves which converge to the same ionization threshold, are at large values of R delimited from the bottom and from the top by the adiabatic lines which are related to the IFP and stable AS POs and to the FP PO, respectively. At smaller values of R, where the AS PO becomes unstable, the curves move to the area between the ASC (bottom) and AS (top) lines by crossing the latter. Therefore, at different values of R the lower limiting line of the multiplet is related to the three types of PO (IFP, AS, and ASC), which are all stable in the negative-energy part of this line. As a consequence, the quantum states of helium in principle are not related individually to a single classical configuration on the hypersphere. In addition, it is demonstrated that “unstable parts” of adiabatic lines (the so-called diabatic curves) determine the positions and type of avoided and hidden crossings between hyperspherical adiabatic curves. Two clearly visible classes of avoided crossings are related to the AS and ASC POs. In addition, a number of avoided crossings of the adiabatic curves is observed at the positions where the adiabatic lines that are related to different types of PO cross mutually. Finally, a class of hidden crossings which is located near the saddle point of the potential is related to the Langmuir orbit. The large spacing between adiabatic curves at the positions of these hidden crossings is explained by high instability of the Langmuir PO compared to the AS and ASC POs.
- Publication:
-
Physical Review A
- Pub Date:
- May 2013
- DOI:
- 10.1103/PhysRevA.87.052503
- Bibcode:
- 2013PhRvA..87e2503S
- Keywords:
-
- 31.15.xg;
- 31.15.xj;
- 31.50.Gh;
- Semiclassical methods;
- Hyperspherical methods;
- Surface crossings non-adiabatic couplings