# Attractive interatomic force as a tunnelling phenomenon

## Abstract

Based on time-dependent perturbation theory, the author established a fundamental equality between Bardeen's tunnelling matrix element and Heisenberg's resonance energy. Applying this equality to the hydrogen molecular ion, he derived a simple analytic expression for the potential curves in the attractive-force regime, which is found to be accurate to better than 1*10-4 au throughout the entire regime. By extending it to the many-body case, he presents a unified view of scanning tunnelling microscopy (STM) and atomic force microscopy (AFM). The fundamental equality then has a measurable consequence: for metals, the observed attractive atomic force F and the observed tunnelling conductance G should conform to the general equation F=-f kappa epsilon (GRK)12/, where kappa is the inverse decay length of the surface wavefunction near the Fermi level, epsilon is the width of the conduction band of the metal, RK is von Klitzing's constant and f is a dimensionless factor of the order of unity, which depends on tip geometry. The equation is found to be in quantitative agreement with recent results of combined experiments of AFM and STM. Conceptually, it means that the imaging process in STM is a sequence of bond forming and bond rupturing. From the computational point of view. the equality between Bardeen's tunnelling matrix element and Heisenberg's resonance energy may open a new first-principles method for calculating potential curves of molecules and the exchange coupling responsible for magnetism.