A formula for the contribution of a resonance to the canonical partition function

Abstract

An explicit expression for the Laplace transform of a Lorentzian function is applied to evaluate the contribution to the canonical partition function of ideal-gas molecules stemming from resonances. This formula can be used instead of numerical integration. A discussion of potential errors in replacing this contribution by the conventional formula for the Boltzmann weight is given. When generalised to the complex plane, the known expression is shown to be discontinuous and a new formulation is proposed that removes the discontinuity. As a result, an explicit expression is obtained for the Fourier transform of a Lorentzian function that is truncated at a lower energy bound.

GRAPHICAL ABSTRACT

Keywords:

• Partition function
• resonance
• Laplace Transform: Lorentzian

Acknowledgments

The author thanks Dr. Killian Deur for carefully reading the manuscript and for his comments. He dedicates this work to Prof. Nimrod Moiseyev on the occasion of his 70th birthday.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

The author gratefully acknowledges the support received from the Lady Davis Fellowship Trust of the Hebrew University of Jerusalem for a leave at the Israel Institute of Technology, during which this work was carried out in the context of investigations about the quantum dynamics of the diffusion of adsorbates.