Session U11: Focus Session: Few Body Problems and Long Range Interactions: Memorial for Richard Drachman

Counterintuitive long-range interaction potentials

One of the topics where Richard Drachman made strong contributions concerns the long-range interaction between a distant electron and an atom or ion, a subject that always interested me greatly.  In this presentation, I will point out one problem where  the Drachman treatment predicted an energy dependent term[1] in the long-range effective potential that differed from an analogous term that arose in other derivations including my own with Shinichi Watanabe, a puzzling discrepancy that Richard Drachman subsequently resolved by showing that such terms are nonunique.  Other examples of counterintuitive long-range interactions arise in Rydberg electron interactions with an anisotropic ionic core, where Bernard Zygelman predicted a term proportional to the scalar product of two angular momenta that arose from coupling to the open shell core and not from magnetic moment interactions.  Research carried out in my group with graduate student William Clark showed that such terms arise from an adiabatic expansion of the long-range Hamiltonian,[2] and tested experimentally by Steve Lundeen's group. A final counterintuitive type of long-range interaction effect discussed in this presentation is the surprising one that can arise in a system of three or more particles having finite range interactions, called today the Efimov effect.  In some very recent work, we see that long range adiabatic potentials computed in hyperspherical coordinates develop a modification of their long-range interaction when the scattering length or scattering volume is tuned to infinity.  The talk will show examples of this long range modification including that predicted by Vitaly Efimov for three identical bosons, which produces an infinite number of Efimovian bound or resonant states.

[1] R. J. Drachman, J. Phys. B 12, L699 (1979).

[2] W. Clark and C. H. Greene, Rev. Mod. Phys. 71, 821 (1999).   

Precision calculations of few-body atomic and molecular systems

In this talk, I will present recent advances in high-precision calculations of few-body atomic and molecular systems, including few-body structure problems, Rydberg atoms, and long-range interactions, which have been influenced by Richard Drachman.  Some of examples are QED calculations of Li⁺ to extract lithium nuclear properties, such as the nuclear charge radii and Zemach radii,  and the exotic pionic helium atom (molecule)  π^-He⁺ with the potential to extract the π^- mass at the ppb level, once experiments progress.    

Performance of Drachman's regularization in variational calculations of small atoms and molecules

Expectation values of singular operators, such as the Dirac delta function, evaluated with the wave function obtained in Rayleigh-Ritz variational calculations often exhibit poor convergence that is orders of magnitude worse than that for the energy. This is related to the fact that with such operators the expectation values sample the wave function locally rather than globally. One of the most practical approaches to deal with this issue was proposed by Richard Drachman and is based on using expectation value identities. For exact eigenfunctions, the approach gives the same expectation values as when the singular operator is used. However, in the case of approximate variational wave functions, it usually yields estimates that are much closer to the exact ones. In this talk I will review our recent studies where we investigated the performance of the regularization technique prescribed by Drachman in the case of few-electron atomic and molecular systems. These studies provide some useful insight on when the Drachman’s regularization should work very well and when it may fail.   

High precision eigenvalues for the Rydberg sates of helium up to n = 16.

In previous work, high precision eigenvalues for all states of helium up to n = 10 and L = 7 have been obtained by the use of double basis sets in Hylleraas coordinates [1].. In the present work, we show that triple basis sets using three sets of individually optimized nonlinear parameters for different distance scales yield an order of magnitude improvement in accuracy for basis sets of the same size for the Rydberg P-states states of helium.  They also allow an extension of high precision calculations for the nonelativistic energies and wave functions up to at least n = 16 with little loss of accuracy. A summary of the methods used and a comparison with asymptotic expansions [2] will be presented.

[1] G.W.F. Drake and Z.-C. Yan, Phys. Rev. A. 46, 2378 (1992).

[2] R. Drachman, Phys. Rev. A 47, 694 (1993)..

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Many-body theory and calculations of positron scattering, binding and annihilation with atoms and polyatomic molecules

Pioneering technological developments have enabled the routine trapping, accumulation and beam delivery of low-energy positrons for studies of fundamental atomic and molecular interactions, and the formation of positronium (Ps) and antihydrogen. The ability of positrons to annihilate with atomic and molecular electrons -- to release two characteristic and detectable gamma rays -- makes positrons and Ps unique probes of matter, and gives them important use across vast length scales, including as diagnositics of industrially important materials, in medical imaging (via PET, positron emission tomography), and in understanding the composition of the galaxy. 

Experiments with antimatter are costly and challenging. Their proper interpretation, and that of the positron-based material science techniques, and the development of next generation positron-based technologies (traps/beams/PET) relies on theoretical understanding of positron interactions with atoms, molecules and condensed matter. Their interactions are, however, characterised by strong correlations, e.g., polarization of the electron cloud and virtual-Ps formation, making their theoretical description a challenging many-body problem. 

I will discuss the many-body theory of positron interactions with atoms and polyatomic molecules, which recently gave the first ab initio verification of long-standing measurements of positron-molecule binding energies [1], and has since been developed to positron scattering and annihilation [2]. In doing so, I hope to convince you that positron physics remains fun [3].

[1] J. Horfierka, B. Cunningham, C. M. Rawlins, C. H. Patterson and D. G. Green, accepted to Nature (2022). 

[2] C. M. Rawlins, J. Horfierka, B. Cunningham, C. H. Patterson and D. G. Green, to be submitted (2022). 

[3] R. J. Drachman, "Why positron physics is fun", AIP Conference Proceedings 360, 369 (1996).    

Equivalent one-electron Breit-Pauli operators for relativistic corrections to two-photon decay rates in helium

Two-photon transition rates are important in determining astrophysical quantities such as population balance in planetary nebulae. We recently calculated two-photon decay rates in heliumlike ions including the finite nuclear mass effects [1]. We have now perturbatively added relativistic corrections to these results, giving the most precise and accurate calculations to date.  We first tested the relativistic corrections to dipole transition integrals for the equivalent one-electron case   We derived (and will report on) one-electron Breit-Pauli operators that are equivalent to the corresponding Dirac operators by expanding in powers of the fine-structure constant α  A continuous gauge parameter is used, and the operators are compared with the long-wavelength QED operators derived previously for few-electron atoms [2].

[1]  A. T. Bondy, D. C. Morton, and G. W. F. Drake, Phys. Rev. A 102, 052807 (2020).

[2] K. Pachucki, Phys. Rev. A 69, 052502 (2004).   

Long-range interactions of two Li atoms and a Li ion

A classic application of high-precision wave functions obtained using

variationally optimized basis sets is to calculate

coefficients appearing in the long-range potentials for atom-atom interactions.

While similar methods have been applied to the case of three ground state atoms,

applications to the case of two atoms and an ion are rare.

Here we summarize our results for two Li atoms and a Li ion, where, using           

perturbation theory and highly accurate wave functions calculated variationally in Hylleraas coordinates, long-range potentials were obtained [Phys. Rev. A 104 (2021) 022807; 101 (2020) 032702].