Dipole transitions in the bound rotational-vibrational spectrum of the heteronuclear molecular ion HD<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mrow /><mml:mo>+</mml:mo></mml:msup></mml:math>

Pilón, Horacio Olivares; Baye, Daniel Jean

doi:10.1103/physreva.88.032502

Cited by 18 publications

(13 citation statements)

References 24 publications

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“…An important advantage of the LMM is that the approximate wave functions are both precise and easy to use, specially for matrix elements calculated with the associated Gauss quadrature. These wave functions have been used to calculate mean values [8,187,186], transition probabilities [205,199,200] and polarizabilities [188]. Let me again take the example of the polarizabilities.…”

Section: Hydrogen Molecular Ionmentioning

confidence: 99%

The Lagrange-mesh method

2015

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The Lagrange-mesh method is an approximate variational method taking the form of equations on a grid thanks to the use of a Gauss-quadrature approximation. The variational basis related to this Gauss quadrature is composed of Lagrange functions which are infinitely differentiable functions vanishing at all mesh points but one. This method is quite simple to use and, more importantly, can be very accurate with small number of mesh points for a number of problems. The accuracy may however be destroyed by singularities of the potential term. This difficulty can often be overcome by a regularization of the Lagrange functions which does not affect the simplicity and accuracy of the method.The principles of the Lagrange-mesh method are described, as well as various generalizations of the Lagrange functions and their regularization. The main existing meshes are reviewed and extensive formulas are provided which make the numerical calculations simple. They are in general based on classical orthogonal polynomials. The extensions to non-classical orthogonal polynomials and periodic functions are also presented.Applications start with the calculations of energies, wave functions and some observables for bound states in simple solvable models which can rather easily be used as exercises by the reader. The Dirac equation is also considered. Various problems in the continuum can also simply and accurately be solved with the Lagrange-mesh technique including multichannel scattering or scattering by non-local potentials. The method can be applied to three-body systems in appropriate systems of coordinates. Simple atomic, molecular and nuclear systems are taken as examples. The applications to the timedependent Schrödinger equation, to the Gross-Pitaevskii equation and to Hartree-Fock calculations are also discussed as well as translations and rotations on a Lagrange mesh.

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Section: Hydrogen Molecular Ionmentioning

confidence: 99%

The Lagrange-mesh method

2015

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“…Indeed, rovibrational states supported by the ground 1sσ electronic curve have long lifetimes of the order of days [26,27] in H + 2 and tens of milliseconds [28,29] in HD + . This is a significant advantage with respect to atomic hydrogen, where only the 1S-2S transition has a small natural width and can be measured with the highest accuracy [30], and a second measurement on a much wider transition such as 2S-8S(D) or 2S-12D [31,32], involving an intricate analysis of systematic effects, is required for a joint determination of R ∞ and r p .…”

Section: And Hdmentioning

confidence: 99%

Hydrogen molecular ions for improved determination of fundamental constants

et al. 2016

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The possible use of high-resolution rovibrational spectroscopy of the hydrogen molecular ions H + 2 and HD+ for an independent determination of several fundamental constants is analyzed. While these molecules had been proposed for metrology of nuclear-to-electron mass ratios, we show that they are also sensitive to the radii of the proton and deuteron and to the Rydberg constant at the level of the current discrepancies colloquially known as the proton size puzzle. The required level of accuracy, in the 10 −12 range, can be reached both by experiments, using Doppler-free two-photon spectroscopy schemes, and by theoretical predictions. It is shown how the measurement of several well-chosen rovibrational transitions may shed new light on the proton-radius puzzle, provide an alternative accurate determination of the Rydberg constant, and yield new values of the proton-toelectron and deuteron-to-proton mass ratios with one order of magnitude higher precision. From Bohr's model of the atom to the advent of quantum electrodynamics (QED), precision spectroscopy of atomic hydrogen has played a key role in our understanding of matter and its interaction with light. Since the first measurement of the Lamb shift in 1947 [1, 2], the predictions of QED have been verified with an increasing level of accuracy which, together with stringent tests in other areas of physics, led to assume the validity of this theory and use it to extract the values of fundamental physical constants from experimental data [3]. Specifically, available data on the hydrogen (H) and deuterium (D) atoms are used to extract the Rydberg constant R ∞ and the charge radii of the proton (r p ) and deuteron (r d ). Data from electron-proton and electron-deuteron scattering experiments also contribute in this determination.Recently, the undisputed status of these results has been challenged by the measurement of the Lamb shift in muonic hydrogen [4,5]. The very precise value of r p deduced from this experiment is in strong disagreement with previous determinations. The discrepancy with the CODATA adjustment [6] amounts to 5.6σ, or to 4.5σ if only the H and D data are taken into account [3]. Similar discrepancies on the deuteron radius can be inferred through the very precise determination of r 2 d − r 2 p from the 1S-2S H/D isotopic shift [7]. Although many efforts have been undertaken in the last few years, no convincing solution of the "proton size puzzle" has been found so far (see [8] for a review). One of the possible explanations is that the error bars, both of hydrogen spectroscopy and scattering experiments [9] were underestimated. New scattering experiments are in preparation or underway, including electron-proton [10], electron-deuteron [11] and muon-proton scattering [12]. In atomic hydrogen, the 1S-3S(D) [13,14], 2S-2P [15], and 2S-4P [16] transitions are under study in order to cross-check and improve previous results. An independent determination of R ∞ , which is strongly correlated to r p and r d , is another way to shed new light on this prob...

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“…For instance, 12 s.d. have been obtained in the rotational-vibrational energies of the ground state for the three-body system H + 2 [6] (see also [10]) and some of their isotopomers: D + 2 [7], HD + [8] and T + 2 [9] in full dynamics (beyond Born-Oppenheimer approximation). In this approach, in the center-of-mass coordinate system, the wave function is presented as a sum of products of the Wigner matrix elements with coefficient functions depending only on the relative http://dx.doi.org/10.1016/j.physleta.2014.12.029 0375-9601/© 2015 Elsevier B.V. All rights reserved.…”

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confidence: 99%

Nuclear critical charge for two-electron ion in Lagrange mesh method

Pilón

Turbiner

2015

Physics Letters A

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The Schroedinger equation for two electrons in the field of a charged fixed center $Z$ is solved with the Lagrange mesh method for charges close to the critical charge $Z_{cr}$. We confirm the value of the nuclear critical charge $Z_{cr}$ recently calculated in Estienne et al. {\em Phys. Rev. Lett. \bf 112}, 173001 (2014) to 11 decimal digits using an inhomogeneous (non-uniform) three-dimensional lattice of size $70 \times 70 \times 20$. We show that the ground state energy for H$^-$ is accurate to 14 decimals on the lattice $50 \times 50 \times 40$ in comparison with the highly accurate result by Nakashima-Nakatsuji, {\it J. Chem. Phys. \bf 127}, 224104 (2007).Comment: 6 pages, no figure

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Dipole transitions in the bound rotational-vibrational spectrum of the heteronuclear molecular ion HD+

Cited by 18 publications

References 24 publications

The Lagrange-mesh method

The Lagrange-mesh method

Hydrogen molecular ions for improved determination of fundamental constants

Nuclear critical charge for two-electron ion in Lagrange mesh method

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