Asymptotic evaluation of the Keesom integral

Semiclassical WUG corrections to the interaction energy of two coupled dipoles at low temperatures

“…(23) for qða; bÞ, leads to the result (25). This is tantamount to ignoring the proportionality factor originating from integration over the momenta, so as to match the definition of K(a) that was given in our previous articles [5,6]. The following result is then obtained…”

“…[5,6]. The diagonal elements of the Wigner-Uhlenbeck-Gropper (WUG) density matrix in the coordinate representation q have the form [9] q q / exp À …”

“…The Keesom integral over angular variables may be defined through the asymptotic series in inverse powers of a [5,6]:…”

“…Since Keesom formula (1) looses its validity at low temperatures and for high dipole strengths (large values of the parameter a) owing to the more pronounced orientational effect, Battezzati and Magnasco [5] derived a two-term asymptotic representation in inverse powers of a % large allowing to overcome this difficulty, obtaining for the Keesom integral the result:…”

“…This may be of interest in the physics and chemistry of cold [7] and ultracold [8] molecules. In this Letter, the appropriate mapping from spherical to planar coordinates introduced by us previously [5,6] is used to simplify the calculations by transforming all integrals into simple exponentials (or Gaussians). This coordinate system for the angle variables is embedded into a larger system by relaxing the constraints over the bead length.…”