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Title: Detecting level crossings without solving the Hamiltonian. II. Applications to atoms and molecules

Abstract

A number of interesting phenomena occur at points where the energy levels of an atom or a molecule (anti) cross as a function of some parameter such as an external field. In a previous paper [M. Bhattacharya and C. Raman, Phys. Rev. Lett. 97, 140405 (2006)] we have outlined powerful mathematical techniques useful in identifying the parameter values at which such (avoided) crossings occur. In the accompanying article [M. Bhattacharya and C. Raman, Phys. Rev A 75, 033405 (2007)] we have developed the mathematical basis of these algebraic techniques in some detail. In this article we apply these level-crossing methods to the spectra of atoms and molecules in a magnetic field. In the case of atoms the final result is the derivation of a class of invariants of the Breit-Rabi Hamiltonian of magnetic resonance. These invariants completely describe the parametric symmetries of the Hamiltonian. In the case of molecules we present an indicator which can tell when the Born-Oppenheimer approximation breaks down without using any information about the molecular potentials other than the fact that they are real. We frame our discussion in the context of Feshbach resonances in the atom-pair {sup 23}Na-{sup 85}Rb which are of current interest.

Authors:
;  [1]
  1. School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332 (United States)
Publication Date:
OSTI Identifier:
20982354
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. A; Journal Volume: 75; Journal Issue: 3; Other Information: DOI: 10.1103/PhysRevA.75.033406; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
74 ATOMIC AND MOLECULAR PHYSICS; ATOMS; BORN-OPPENHEIMER APPROXIMATION; ENERGY LEVELS; HAMILTONIANS; MAGNETIC FIELDS; MAGNETIC RESONANCE; MATHEMATICAL SOLUTIONS; MOLECULES; POTENTIAL ENERGY; POTENTIALS; RUBIDIUM 85; SODIUM 23; SPECTRA; SYMMETRY

Citation Formats

Bhattacharya, M., and Raman, C. Detecting level crossings without solving the Hamiltonian. II. Applications to atoms and molecules. United States: N. p., 2007. Web. doi:10.1103/PHYSREVA.75.033406.
Bhattacharya, M., & Raman, C. Detecting level crossings without solving the Hamiltonian. II. Applications to atoms and molecules. United States. doi:10.1103/PHYSREVA.75.033406.
Bhattacharya, M., and Raman, C. Thu . "Detecting level crossings without solving the Hamiltonian. II. Applications to atoms and molecules". United States. doi:10.1103/PHYSREVA.75.033406.
@article{osti_20982354,
title = {Detecting level crossings without solving the Hamiltonian. II. Applications to atoms and molecules},
author = {Bhattacharya, M. and Raman, C.},
abstractNote = {A number of interesting phenomena occur at points where the energy levels of an atom or a molecule (anti) cross as a function of some parameter such as an external field. In a previous paper [M. Bhattacharya and C. Raman, Phys. Rev. Lett. 97, 140405 (2006)] we have outlined powerful mathematical techniques useful in identifying the parameter values at which such (avoided) crossings occur. In the accompanying article [M. Bhattacharya and C. Raman, Phys. Rev A 75, 033405 (2007)] we have developed the mathematical basis of these algebraic techniques in some detail. In this article we apply these level-crossing methods to the spectra of atoms and molecules in a magnetic field. In the case of atoms the final result is the derivation of a class of invariants of the Breit-Rabi Hamiltonian of magnetic resonance. These invariants completely describe the parametric symmetries of the Hamiltonian. In the case of molecules we present an indicator which can tell when the Born-Oppenheimer approximation breaks down without using any information about the molecular potentials other than the fact that they are real. We frame our discussion in the context of Feshbach resonances in the atom-pair {sup 23}Na-{sup 85}Rb which are of current interest.},
doi = {10.1103/PHYSREVA.75.033406},
journal = {Physical Review. A},
number = 3,
volume = 75,
place = {United States},
year = {Thu Mar 15 00:00:00 EDT 2007},
month = {Thu Mar 15 00:00:00 EDT 2007}
}
  • When the parameters of a physical system are varied, the eigenvalues of observables can undergo crossings and avoided crossings among themselves. It is relevant to be aware of such points since important physical processes often occur there. In a recent paper [M. Bhattacharya and C. Raman, Phys. Rev. Lett. 97, 140405 (2006)] we introduced a powerful algebraic solution to the problem of finding (avoided) crossings in atomic and molecular spectra. This was done via a mapping to the problem of locating the roots of a polynomial in the parameters of interest. In this article we describe our method in detail.more » Given a physical system that can be represented by a matrix, we show how to find a bound on the number of (avoided) crossings in its spectrum, the scaling of this bound with the size of the Hilbert space and the parametric dependencies of the Hamiltonian, the interval in which the (avoided) crossings all lie in parameter space, the number of crossings at any given parameter value, and the minimum separation between the (avoided) crossings. We also show how the crossings can reveal the symmetries of the physical system, how (avoided) crossings can always be found without solving for the eigenvalues, how they may sometimes be found even in case the Hamiltonian is not fully known, and how crossings may be visualized in a more direct way than displayed by the spectrum. In the accompanying paper [M. Bhattacharya and C. Raman, Phys. Rev. A 75, 033406 (2007)] we detail the application of these techniques to atoms and molecules.« less
  • A local Schroedinger equation (LSE) method is proposed for solving the Schroedinger equation (SE) of general atoms and molecules without doing analytic integrations over the complement functions of the free ICI (iterative-complement-interaction) wave functions. Since the free ICI wave function is potentially exact, we can assume a flatness of its local energy. The variational principle is not applicable because the analytic integrations over the free ICI complement functions are very difficult for general atoms and molecules. The LSE method is applied to several 2 to 5 electron atoms and molecules, giving an accuracy of 10{sup -5} Hartree in total energy.more » The potential energy curves of H{sub 2} and LiH molecules are calculated precisely with the free ICI LSE method. The results show the high potentiality of the free ICI LSE method for developing accurate predictive quantum chemistry with the solutions of the SE.« less
  • The author points out that experiments to measure parity nonconservation in metastable atoms, such as hydrogen in the 2S state, are not, in general, made more sensitive by operating near level crossings in a magnetic field. New possibilities related to this observation are mentioned.
  • In general, extended Runge–Kutta–Nyström (ERKN) methods are more effective than traditional Runge–Kutta–Nyström (RKN) methods in dealing with oscillatory Hamiltonian systems. However, the theoretical analysis for ERKN methods, such as the order conditions, the symplectic conditions and the symmetric conditions, becomes much more complicated than that for RKN methods. Therefore, it is a bottleneck to construct high-order ERKN methods efficiently. In this paper, we first establish the ERKN group Ω for ERKN methods and the RKN group G for RKN methods, respectively. We then rigorously show that ERKN methods are a natural extension of RKN methods, that is, there exists anmore » epimorphism η of the ERKN group Ω onto the RKN group G. This epimorphism gives a global insight into the structure of the ERKN group by the analysis of its kernel and the corresponding RKN group G. Meanwhile, we establish a particular mapping φ of G into Ω so that each image element is an ideal representative element of the congruence class in Ω. Furthermore, an elementary theoretical analysis shows that this map φ can preserve many structure-preserving properties, such as the order, the symmetry and the symplecticity. From the epimorphism η together with its section φ, we may gain knowledge about the structure of the ERKN group Ω via the RKN group G. In light of the theoretical analysis of this paper, we obtain high-order structure-preserving ERKN methods in an effective way for solving oscillatory Hamiltonian systems. Numerical experiments are carried out and the results are very promising, which strongly support our theoretical analysis presented in this paper.« less