Variational Principles and Their Applications

Authors

  • A. K. Bhatia Heliophysics Division, NASA/Goddard Space Flight Center, Greenbelt, MD 20701, USA

DOI:

https://doi.org/10.26713/jamcnp.v12i1.3018

Keywords:

Variational principles, Variational calculations, Ritz variational method, Rydberg states, Photoionization, Electron-hydrogen scattering, Positron-hydrogen scattering

Abstract

There are many variational calculations of energies of various systems which have applications in Rydberg states and polarizabilities of these systems. There are variational calculations of scattering functions which have applications in calculations of excitation, photoabsorption, and radiative attachment cross sections. A few of these applications are mentioned.

Downloads

Download data is not yet available.

References

J. F. Babb, Effective oscillator strengths and transition energies for the hydrogen molecular ion, Molecular Physics 81(1), 17 – 29 (1994), DOI: 10.1080/00268979400100021.

A. K. Bhatia (unpublished).

A. K. Bhatia and A. Temkin, Reviews of Modern Physics 36, 1050 (1964), DOI: 10.1103/RevModPhys.36.1050.

A. K. Bhatia and J. Sucher, New approach to hyperfine structure: Application to the Li ground state, Journal of Physics B: Atomic and Molecular Physics 13, L409 (1980), DOI: 10.1088/0022-3700/13/13/002.

A. K. Bhatia and R. J. Drachman, Comments on Atomic and Molecular Physics 21, 281 (1989).

A. K. Bhatia and R. J. Drachman, Energy levels of triply ionized carbon (C IV): Polarization method, Physical Review A 60, 2848 (1999), DOI: 10.1103/PhysRevA.60.2848.

A. K. Bhatia and R. J. Drachman, New calculation of the properties of the positronium ion, Physical Review A 28, 2523 (1983), DOI: 10.1103/PhysRevA.28.2523.

A. K. Bhatia and R. J. Drachman, Polarizability of the ground state of the hydrogen molecular ion, Physical Review A 59, 205 (1999), DOI: 10.1103/PhysRevA.59.205.

A. K. Bhatia and R. J. Drachman, Properties of two-electron systems in an electric field, Canadian Journal of Physics 75, 11 – 18 (1997), DOI: 10.1139/p96-132.

A. K. Bhatia, Applications of the hybrid theory to the scattering of electrons from He+ and Li2+ and resonances in these systems, Physical Review A 77, 052707 (2008), DOI: 10.1103/PhysRevA.77.052707.

A. K. Bhatia, Excitation of the 2P state of atomic hydrogen by electron impact, Atoms 10(4), 113 (2022), DOI: 10.3390/atoms10040113.

A. K. Bhatia, Excitation of the 2S state of atomic hydrogen by electron impact, Atoms 6(1), 7 (2018), DOI: 10.3390/atoms6010007.

A. K. Bhatia, Hybrid theory of P-wave electron-Li2+ elastic scattering and photoabsorption in two-electron systems, Physical Review A 87, 042705 (2013), DOI: 10.1103/PhysRevA.87.042705.

A. K. Bhatia, Hybrid theory of electron-hydrogen elastic scattering, Physical Review A 75, 032713 (2007), DOI: 10.1103/PhysRevA.75.032713.

A. K. Bhatia, Photoejection from various systems and radiative-rate coefficients, Atoms 10, 9 (2022), DOI: 10.3390/atoms10010005.

A. K. Bhatia, Positron impact excitation of the 2S state of atomic hydrogen, Atoms 7(3), 69 (2019), DOI: 10.3390/atoms7030069.

A. K. Bhatia, Positron impact excitation of the nS, nP, and nD states of atomic hydrogen, Modern Concepts in Material Science 3(1), 3 pages (2020), DOI: 10.33552/MCMS.2020.03.000553.

A. K. Bhatia, Properties of the ground state of the hydrogen molecular ion, Physical Review A 58, 2787 (1998), DOI: 10.1103/PhysRevA.58.2787.

D. M. Bishop and B. Lam, An analysis of the interaction between a distant point charge and H+2, Molecular Physics 65(3), 679 – 688 (1980), DOI: 10.1080/00268978800101331.

D. M. Bishop and L. M. Chung, Accurate expectation values for H+ 2 (and its isotopes) and H2, Molecular Physics 36(2), 501 – 507 (1978), DOI: 10.1080/00268977800101721

K. Bockaten, Mean values of powers of the radius for hydrogenic electron orbits, Physical Review A 9, 1087 (1974), DOI: 10.1103/PhysRevA.9.1087.

R. J. Drachman and A. K. Bhatia, Rydberg levels of lithium, Physical Review A 51, 2926 (1995), DOI: 10.1103/PhysRevA.51.2926.

R. J. Drachman and J. Sucher, Annihilation in positron—atom collisions: A new approach, Physical Review A 20, 442 (1979), DOI: 10.1103/PhysRevA.20.442.

R. J. Drachman, High Rydberg states of two-electron atoms in perturbation theory, in: LongRange Casimir Forces: Theory and Recent Experiments on Atomic Systems (Finite Systems and Multiparticle Dynamics), F. S. Levin and D. A. Micha (editors), Springer-Verlag, New York Inc., p. 219 (2013), DOI: 10.1007/978-1-4899-1228-2_4.

R. J. Drachman, Rydberg states of helium: An optical-potential analysis, Physical Review A 26, 1228 (1982), DOI: 10.1103/PhysRevA.26.1228.

G. W. F. Drake, High-precision calculations for the Rydberg states of helium, in: Long-Range Casimir Forces: Theory and Recent Experiments on Atomic Systems, F. S. Levin, D. A. Micha and P. W. Milonni, Plenum Press, New York, p. 107, DOI: 10.1007/978-1-4899-1228-2_3.

G. W. F. Drake, Variational eigenvalues for the Rydberg states of helium: Comparison with experiment and with asymptotic expansions, Physical Review Letters 65, 2769 (1990), DOI: 10.1103/PhysRevLett.65.2769.

Z. W. Fu, E. A. Hessels and S. R. Lundeen, Determination of the hyperfine structure of H2+ (ν = 0, R = 1) by microwave spectroscopy of high-L n = 27 Rydberg states of H2, Physical Review A 46, R5313 (1992), DOI: 10.1103/PhysRevA.46.R5313.

E. A. Hassels, F. J. Deck, P. W. Arcuni and S. R. Lundeen, Precision spectroscopy of high-L, n = 10 Rydberg helium: An improved test of relativistic, radiative, and retardation effects, Physical Review Letters 65, 2765 (1990); 66, 2544 (1991) [Erratum], DOI: 10.1103/PhysRevLett.65.2765.

E. A. Hylleraas, Über den Grundzustand des Heliumatoms, Zeitschrift für Physik 48, 469 – 494 (1928), DOI: 10.1007/BF01340013.

P. L. Jacobson, D. S. Fisher, C. W. Fehrenbach, W. G. Sturus and S. R. Lundeen, Determination of the dipole polarizabilities of H+ 2 (0,0) and D+ 2 (0,0) by microwave spectroscopy of high-L Rydberg states of H2 and D2, Physical Review A 56, R4361(R) (1997), DOI: 10.1103/PhysRevA.56.R4361.

P. Kusch and H. Taub, On the gJ values of the alkali atoms, Physical Review Journals Archive 75, 1477 (1949), DOI: 10.1103/PhysRev.75.1477,

S. Larsson, Calculations on the 2S ground state of the lithium atom using wave functions of Hylleraas type, Physical Review Journals Archive 169, 49 (1968), DOI: 10.1103/PhysRev.169.49.

P. M. Morse and W. P. Allis, The effect of exchange on the scattering of slow electrons from atoms, Physical Review Journals Archive 44, 269 (1933), DOI: 10.1103/PhysRev.44.269.

S. N. Nahar, Photoionization, recombination, and radiative transitions of atoms and ions, in: New Quests in Steller Astrophysics II. The Ultraviolet Properties of Evolved Stellar Populations, M. C. Dagostino (editor), Springer Science + Business Media, LLC, New York, p. 245 (2009), DOI: 10.1007/978-0-387-87621-4_34.

T. Ohmura and H. Ohmura, Electron-hydrogen scattering at low energies, Physical Review Journals Archive 118, 154 (1960), DOI: 10.1103/PhysRev.118.154.

C. L. Pekeris, Ground state of two-electron atoms, Physical Review Journals Archive 112, 1649 (1958), DOI: 10.1103/PhysRev.112.1649.

N. E. Rothery, C. H. Storry and E. A. Hessel, Precision radio-frequency measurements of the high-L Rydberg states of lithium, Physical Review A 51, 2919 (1995), DOI: 10.1103/PhysRevA.51.2919.

J. A. R. Samson, Z. X. He, I. Yin and G. N. Haddad, Precision measurements of the absolute photoionization cross sections of He, Journal of Physics B: Atomic, Molecular and Optical Physics 27, 887 (1994), DOI: 10.1088/0953-4075/27/5/008,

J. Shertzer and A. Temkin, Direct calculation of the scattering amplitude without partialwave analysis. III. Inclusion of correlation effects, Physical Review A 74, 052701 (2006), DOI: 10.1103/PhysRevA.74.052701.

J. Shertzer and C. H. Greene, Nonadiabatic dipole polarizabilities of H+ 2 and D+ 2 ground states, Physical Review A 58, 1082 (1998), DOI: 10.1103/PhysRevA.58.1082.

W. G. Strurus, E. A. Hessels, P. W. Arcuni and S. R. Lundeen, Laser spectroscopy of (ν = 0, R = 1)10F and (ν = 0, R = 1)10G states of H2: A test of the polarization model, Physical Review A 38, 135 (1988), DOI: 10.1103/PhysRevA.38.135.

J. Sucher and R. J. Drachman, Global operators for δ functions: Extension to scattering states and the inclusion of spin, Physical Review A 20, 424 (1979), DOI: 10.1103/PhysRevA.20.424.

A. Temkin and J. C. Lamkin, Application of the method of polarized orbitals to the scattering of electrons from hydrogen, Physical Review Journals Archive 121, 788 (1961), DOI: 10.1103/PhysRev.121.788.

A. Temkin, A note on the scattering of electrons from atomic hydrogen, Physical Review Journals Archive 116, 358 (1959), DOI: 10.1103/PhysRev.116.358.

J. B. West and G. V. Mars, The absolute photoionization cross sections of helium, neon, argon and krypton in the extreme vacuum ultraviolet region of the spectrum, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 349, 397 – 421 (1976), DOI: 10.1098/rspa.1976.0081.

R. Wildt, Electron affinity in astrophysics, Astrophysical Journal 89, 295 – 301 (1939), DOI: 10.1086/144048.

Downloads

Published

2025-10-26
CITATION

How to Cite

Bhatia, A. K. (2025). Variational Principles and Their Applications. Journal of Atomic, Molecular, Condensed Matter and Nano Physics, 12(1), 9–26. https://doi.org/10.26713/jamcnp.v12i1.3018

Issue

Section

Research Article