1/R expansion for H2+: Calculation of exponentially small terms and asymptotics

Jiří Cížek, Robert J. Damburg, Sandro Graffi, Vincenzo Grecchi, Evans M. Harrell, II, Johathan G. Harris, Sachiko Nakai, Josef Paldus, Rafail Kh. Propin, and Harris J. Silverstone
Phys. Rev. A 33, 12 – Published 1 January 1986
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Abstract

The energy of any bound state of the hydrogen molecule ion H2+ has an expansion in inverse powers of the internuclear distance R of the form Rayleigh-Schrödinger perturbation theory (RSPT) gives the coefficients E(N) but is otherwise unable to treat the exponentially small series, which in part are characteristic of the double-well aspect of H2+. (Here n denotes the hydrogenic principal quantum number.) We develop a quasisemiclassical method for solving the Schrödinger equation that gives all the exponentially small subseries.

The RSPT series diverges: for the ground state E(N)∼-(N+1)!/e2 for large N. The E(N) asymptotics are governed via a dispersion relation by the imaginary e2R/n series, which itself is given by the square of the eR/n series times a ‘‘normalization integral.’’ That the expansion itself contains imaginary terms might seem inconsistent with the reality of the H2+ eigenvalues. In fact, the RSPT series is Borel summable for R complex. The Borel sum has a cut on the real R axis, and its limit from above or below the positive R axis is complex. The imaginary e2R/n (and higher) series consist of just the counterterms to cancel the imaginary part of the Borel sum.

Extensive numerical examples are given. Of interest is a weak (down by a factor N6) alternating-sign contribution to E(N), which is uncovered both theoretically and numerically. Also of interest is the identification of the Borel sum of the RSPT series with nonphysical boundary conditions. This too is illustrated both theoretically and numerically.

  • Received 9 May 1985

DOI:https://doi.org/10.1103/PhysRevA.33.12

©1986 American Physical Society

Authors & Affiliations

Jiří Cížek

  • Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Robert J. Damburg

  • Institute of Physics, Latvian Academy of Sciences, Riga, Salaspils, Union of Soviet Socialist Republics

Sandro Graffi

  • Dipartimento di Matematica, Università di Bologna, 40127 Bologna, Italy

Vincenzo Grecchi

  • Dipartimento di Matematica, Università di Modena, 41100 Modena, Italy

Evans M. Harrell, II

  • Department of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332

Johathan G. Harris and Sachiko Nakai

  • Department of Chemistry, The Johns Hopkins University, Baltimore, Maryland 21218

Josef Paldus

  • Department of Applied Mathematics, University of Waterloo, Waterloo Ontario, Canada N2L 3G1

Rafail Kh. Propin

  • Institute of Physics, Latvian Academy of Sciences, Riga, Salaspils, Union of Soviet Socialist Republics

Harris J. Silverstone

  • Department of Chemistry, The Johns Hopkins University, Baltimore, Maryland 21218

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Vol. 33, Iss. 1 — January 1986

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