Energy spectrum of the long-range Lennard-Jones potential

TL;DR

This paper derives an analytical formula for the discrete energy spectra of long-range inverse power-law potentials, aiding understanding in molecular physics, polymers, and quantum interactions of bosons.

Contribution

It introduces a compact analytical expression for the energy spectra of a class of long-range potentials, derived via a functional matching method.

Findings

Derived a compact analytical formula for energy spectra

Applicable to polarized molecules, polymers, and bosonic quantum models

Provides insights into highly-excited bound-state resonances

Abstract

The discrete energy spectra of composite inverse power-law binding potentials of the form $V(r;\alpha,\beta,n)=-\alpha/r^2+\beta/r^n$ with $n>2$ are studied {\it analytically}. In particular, using a functional matching procedure for the eigenfunctions of the radial Schr\"odinger equation, we derive a remarkably compact analytical formula for the discrete spectra of binding energies $\{E(\alpha,\beta,n;k)\}^{k=\infty}_{k=1}$ which characterize the highly-excited bound-state resonances of these long-range binding potentials. Our results are of practical importance for the physics of polarized molecules, the physics of composite polymers, and also for physical models describing the quantum interactions of bosonic particles.