Nonlinear dynamical behavior of a hydrogen molecular ion and similar three-body Coulomb systems

Duan, Yuan‐Wen; Browne, C.I.; Yuan, Jian‐Min

doi:10.1103/physreva.59.238

Cited by 13 publications

(6 citation statements)

References 25 publications

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“…The screening is assumed to change abruptly whenever the electrons change places, i.e., the inner electron becomes the outer one and vice versa, which makes the classical dynamics nontrivial. Classical calculations for collinear three-particle Coulomb problems with mass ratios different from those found in two-electron atoms have been performed by Duan et al (1999).…”

Section: A Collinear Helium: the Hamiltonian And General Propertiesmentioning

confidence: 99%

The theory of two-electron atoms: between ground state and complete fragmentation

2000

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Since the first attempts to calculate the helium ground state in the early days of Bohr-Sommerfeld quantization, two-electron atoms have posed a series of unexpected challenges to theoretical physics. Despite the seemingly simple problem of three charged particles with known interactions, it took more than half a century after quantum mechanics was established to describe the spectra of two-electron atoms satisfactorily. The evolution of the understanding of correlated two-electron dynamics and its importance for doubly excited resonance states is presented here, with an emphasis on the concepts introduced. The authors begin by reviewing the historical development and summarizing the progress in measuring the spectra of two-electron atoms and in calculating them by solving the corresponding Schrö dinger equation numerically. They devote the second part of the review to approximate quantum methods, in particular adiabatic and group-theoretical approaches. These methods explain and predict the striking regularities of two-electron resonance spectra, including propensity rules for decay and dipole transitions of resonant states. This progress was made possible through the identification of approximate dynamical symmetries leading to corresponding collective quantum numbers for correlated electron-pair dynamics. The quantum numbers are very different from the independent particle classification, suitable for low-lying states in atomic systems. The third section of the review describes modern semiclassical concepts and their application to two-electron atoms. Simple interpretations of the approximate quantum numbers and propensity rules can be given in terms of a few key periodic orbits of the classical three-body problem. This includes the puzzling existence of Rydberg series for electron-pair motion. Qualitative and quantitative semiclassical estimates for doubly excited states are obtained for both regular and chaotic classical two-electron dynamics using modern semiclassical techniques. These techniques set the stage for a theoretical investigation of the regime of extreme excitation towards the three-body breakup threshold. Together with periodic orbit spectroscopy, they supply new tools for the analysis of complex experimental spectra.

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Section: A Collinear Helium: the Hamiltonian And General Propertiesmentioning

confidence: 99%

The theory of two-electron atoms: between ground state and complete fragmentation

2000

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“…In particular, a classical helium atom, a fundamental three-body system, has been extensively studied in this perspective [1,2]. Some efforts have also been made to understand classical, purely Coulombic H + 2 [3,4] and H 2 [5][6][7].…”

Section: Introductionmentioning

confidence: 99%

“…A fundamental difficulty that has to be overcome in modelling classical molecules is the stability of the ground state, having the equilibrium configuration of minimum energy. In the case of classical H + 2 , where one electron and two nuclei interact with the Coulomb force, a few stable trajectories were found [4], but no stable particle trajectories are known for H 2 or any other polyatomic molecules. A stable trajectory is defined in this context as a trajectory that would not lead to autoionization or autodissociation of the molecule when all particles, both electrons and nuclei, are allowed to move.…”

Section: Introductionmentioning

confidence: 99%

A classical model of H$_3^+$ in an intense laser field

Lötstedt¹,

Katō²,

Yamanouchi³

2013

J. Phys. B: At. Mol. Opt. Phys.

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Soft-core Coulomb potentials for the pairwise interactions among electrons and protons are introduced so that , H2, and H can take stable geometrical configurations in classical mechanics, and responses of to an ultrashort intense laser pulse are investigated by integrating the classical equations of motion. The Monte Carlo simulation shows that the single and double ionization as well as the dissociation proceed, indicating that classical mechanics can give insight into dynamics of small molecular systems interacting with an intense laser field.

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“…Pajunen 23 extends the semiclassical treatment to include third-order tunneling terms. Duan [24][25][26] has considered the classical approach to the hydrogen molecular cation, mainly from the point of view of the periodic orbits, using Gutzwiller's 27 approach. Pearson 28 treated the ground state using methods of the old quantum theory, finding that the OQT method greatest error arises from the coalescence of two classically allowed regions of motion.…”

Section: Introductionmentioning

confidence: 99%

“…A number of papers present the dimensional scaling approach, which again goes beyond a zero-order semiclassical approach. [34][35][36][37] Several authors 26,38 have looked at H + 2 without imposing the clamped-nucleus approximation; this approach must be done numerically, as analytic results are not available. A general treatise on semiclassical methods may be found in Childs' book.…”

Section: Introductionmentioning

confidence: 99%

The Old Quantum Theory for One-Electron Diatomic Ions and Molecule

Knudson

2022

Preprint

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The modern version of the Old Quantum Theory (OQT) is applied to one-electron diatomic systems in the clamped nucleus approximation. The first part reviews the theory, whose key feature is that only those trajectories whose classical action integrals are integer or half-integer multiples of Planck’s constant are allowed. The OQT is not correct, notoriously so in the case of the ground state of H+2 . Nonetheless, elements of the underlying classical structure have effects on quantum results, an understanding of which may aid in the interpretation of wave functions and energies. In particular, the classical separatracies provide an interpretation for the failure to describe the ground state of H+2 , for kinks in the electronic energy at small internuclear separation, and for quantum critical points. The OQT generates a ground-state Born-Oppenheimer potential energy minimum for the artificial neutral one-electron molecule; for such low quantum numbers, the accuracy is not high, emphasizing the importance of quantum concepts such as superposition of states in the real world. In addition, while accuracy of the classical results does increase with quantum number (Bohr Correspondence Principle), all OQT results show some disagreement with quantum values regardless of quantum number.

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Nonlinear dynamical behavior of a hydrogen molecular ion and similar three-body Coulomb systems

Cited by 13 publications

References 25 publications

The theory of two-electron atoms: between ground state and complete fragmentation

The theory of two-electron atoms: between ground state and complete fragmentation

A classical model of H$_3^+$ in an intense laser field

The Old Quantum Theory for One-Electron Diatomic Ions and Molecule

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