scholarly journals The hyperfine structure of the hydrogen molecular ion

1960 ◽  
Vol 259 (1296) ◽  
pp. 100-109 ◽  
Keyword(s):  
Electron Spin ◽  
Ground State ◽  
Energy Levels ◽  
High Accuracy ◽  
Hydrogen Molecular Ion ◽  
Molecular Ion ◽  
First Order ◽  
Quantitative Results ◽  
Coupling Terms ◽  
Very High

A calculation is presented of the energy levels of the hydrogen molecular ion in the ground state for different orientations of the electron spin, taking into account the coupling of the electron spin to the nuclear spin and to the nuclear rotational angular momentum. The first-order coupling terms are evaluated to very high accuracy, and quantitative results are given for the level splitting for a number of rotational levels.

Semiclassical variational calculation of energy levels of He@C70

2006 ◽  
Vol 84 (2) ◽  
pp. 145-164
Author(s):  
G R Lee-Dadswell ◽  
C G Gray
Keyword(s):  
Variational Methods ◽  
Mechanical Energy ◽  
Energy Levels ◽  
High Accuracy ◽  
Variational Calculation ◽  
Quantum Mechanical ◽  
Simplified Models ◽  
Trial Solution ◽  
Quantum Mechanical Energy ◽  
Very High

Semiclassical variational methods are used to obtain estimates of the quantum mechanical energy levels for two simplified models of the potential seen by a helium atom trapped inside a C70 cage. We find that with the use of a simple trial solution, the calculations are simple. A more complicated trial trajectory, while improving some results of the calculation, makes the calculation prohibitively difficult. We also observe that as long as the precessional frequency of the orbits is small we can obtain very high accuracy in our results. However, the inability to accurately predict precessional frequencies results in poor prediction of energy levels when the precessional frequency is large.PACS No.: 5.45.Mt

On the vibration–rotational energy levels of the hydrogen molecular ion HD+

1985 ◽  
Vol 63 (9) ◽  
pp. 1201-1204 ◽  
Author(s):  
L. Wolniewicz ◽  
J. D. Poll
Keyword(s):  
Diatomic Molecules ◽  
Energy Levels ◽  
Rotational Energy ◽  
New Method ◽  
Dissociation Limit ◽  
Radiative Effects ◽  
Vibrational States ◽  
Hydrogen Molecular Ion ◽  
Molecular Ion ◽  
Quantum Numbers

A new method for calculating vibration–rotational energies of diatomic molecules is discussed and applied to the case of HD+. This method is designed to obtain accurate results for all vibrational states including those close to the dissociation limit. Nonadiabatic, relativistic, and radiative effects are taken into account for all the bound vibrational states with rotational quantum numbers J ≤ 5; the estimated accuracy is of the order of 0.001 cm−1.

High-accuracy calculations in the H+2 molecular ion: towards a measurement of mp/me

2007 ◽  
Vol 85 (5) ◽  
pp. 497-507 ◽  
Author(s):  
J Ph. Karr ◽  
F Bielsa ◽  
T Valenzuela ◽  
A Douillet ◽  
L Hilico ◽  
...  
Keyword(s):  
Vibrational Spectroscopy ◽  
Quantum Cascade Laser ◽  
Energy Levels ◽  
High Accuracy ◽  
Electron Mass ◽  
Mass Ratio ◽  
Molecular Ion ◽  
Quantum Cascade ◽  
Two Photon

We report on our recent advances in the calculation of the energy levels of the H+2 molecular ion, including relativistic and radiative corrections. These theoretical efforts are linked to the prospect of obtaining a new determination of the proton to electron mass ratio mp/me through precise vibrational spectroscopy of H+2. We describe the setup of our experiment, aiming at a measurement of the L = 2, υ = 0 → L = 2, υ = 1 two-photon transition at 9.166 μm using a phase-locked quantum cascade laser as excitation source.PACS Nos.: 31.15.Pf, 31.30.Jv, 32.10.Hq

Studies in jj -coupling. III. Nuclear energy levels

1952 ◽  
Vol 215 (1120) ◽  
pp. 120-132 ◽  
Keyword(s):  
Excited States ◽  
Ground State ◽  
Short Range ◽  
Nuclear Energy ◽  
Energy Levels ◽  
Wave Functions ◽  
Light Nuclei ◽  
Radial Dependence ◽  
Exchange Force ◽  
Very High

The wave functions for the configurations j n , with j = 3/2, 5/2 and 7/2, obtained in part II of this series according to the particular scheme of part I, are here used to obtain the central force energy matrices for nuclear configurations. The ordering of energy levels is studied as a function of the nature of the force with especial reference to its range, and the results are presented in the form of energy-level diagrams. The ordering of levels predicted to hold in parts I and II when forces are of short range is shown to occur with practical ranges; but for light nuclei in particular it is demonstrated that levels of very high angular momentum may sometimes occur near to the ground state. The results are found to be largely independent of the precise nature of the forces provided they are of reasonably short range and are attractive in the 1 S state of two like particles; for definiteness, however, the diagrams have been drawn for a Rosenfeld symmetrical exchange force of Gaussian radial dependence. The effect of using other types of central interaction is discussed qualitatively. The lowest excited states of even-even nuclei are seen to have a simple interpretation in jj -coupling.

wave functions

1965 ◽  
Vol 61 (1) ◽  
pp. 207-209 ◽  
Author(s):  
A. R. Holt
Keyword(s):  
Wave Function ◽  
Spherical Harmonics ◽  
Energy Levels ◽  
Wave Functions ◽  
Hydrogen Molecular Ion ◽  
Molecular Ion ◽  
Polar Representation ◽  
Image Position

Cohen and Coulson (4) and Cohen (5) have obtained wave-functions and energy levels for the hydrogen molecular ion in a spherical polar representation. They expanded the wave-function in terms of spherical harmonics, and their expansion may be written aswhere S = 2k for even states, and S = 2k + 1 for odd states.

scholarly journals Cryptosystem based on a key function of a real variable

2021 ◽  
Vol 2 (2) ◽  
pp. 2576-2591
Author(s):  
Viktor Avramenko ◽  
Volodymyr Demianenko
Keyword(s):  
Cauchy Problem ◽  
Real Variable ◽  
High Accuracy ◽  
Integral Function ◽  
Inverse Transformation ◽  
First Order ◽  
The Cauchy Problem ◽  
Text Information ◽  
Symmetric Key ◽  
Very High

Using the function of a real variable in cryptosystems as a key allows you to increase its cryptographic strength, because it is more difficult to pick up such key. Therefore, the development of such systems is relevant. A cryptosystem with a symmetric key is offered. This key is some function of a real variable that satisfies some restrictions. It can be either continuous or discrete. The transmitting and receiving parties select the key-function, the first transmitted character or the first transmitted value for the analog message, the function area of the key function, and the step of changing the function argument. A Disproportion over first-order derivative is used to encrypt an analog message. The Cauchy problem is solving for decrypting this message. Discrete messages are encrypted using the first-order disproportionality integral function. Decryption is performed by the inverse transformation of the formula for integral disproportion. Algorithms for encrypting and decrypting messages are presented. The ability to encrypt and decrypt text information, 2D graphic images, as well as analog messages are shown. The examples show the complexity to pick up the key function and the cryptographic strength of the proposed cryptosystem. A cryptosystem, in which the function of a real variable is used as a key and as well as disproportion functions are used, is suitable for encryption of both discrete and continuous messages. To “crack” such a system, it is required to pick up the form of the key function and to find the values of its parameters with very high accuracy. That is, the system has high cryptographic strength.

Polarizability of the ground state of the hydrogen molecular ion

1999 ◽  
Vol 59 (1) ◽  
pp. 205-208 ◽  
Author(s):  
A. K. Bhatia ◽  
Richard J. Drachman
Keyword(s):  
Ground State ◽  
Hydrogen Molecular Ion ◽  
Molecular Ion

The Dipole Polarizability of the Hydrogen Molecular Ion: A Variational Two-Center Calculation

1974 ◽  
Vol 52 (20) ◽  
pp. 3463-3467 ◽  
Author(s):  
R. P. McEachran ◽  
Sharon Smith ◽  
M. Cohen
Keyword(s):  
Electric Dipole ◽  
Hydrogen Molecule ◽  
Moment Function ◽  
Polarizability Tensor ◽  
Dipole Polarizability ◽  
Hydrogen Molecular Ion ◽  
Molecular Ion ◽  
First Order ◽  
Wide Range ◽  
Good Agreement

The electric dipole polarizability tensor of the hydrogen molecule-ion has been calculated variationally over a wide range of internuclear separations, using the accurate two center James (ref. 3) orbital as the unperturbed wavefunction. Our most accurate values, obtained with first order functions containing between three and five variational parameters, are in good agreement with some recent moment function values calculated by Bates (ref. 7).

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