Abstract
The energy of any bound state of the hydrogen molecule ion has an expansion in inverse powers of the internuclear distance R of the form Rayleigh-Schrödinger perturbation theory (RSPT) gives the coefficients but is otherwise unable to treat the exponentially small series, which in part are characteristic of the double-well aspect of . (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 ∼-(N+1)!/ for large N. The asymptotics are governed via a dispersion relation by the imaginary series, which itself is given by the square of the series times a ‘‘normalization integral.’’ That the expansion itself contains imaginary terms might seem inconsistent with the reality of the 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 (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 ) alternating-sign contribution to , 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