At its core, this is a story about electrons. Electrons drive the interactions of matter at the nanoscale, so an understanding of electron behavior offers significant insight into the behavior of nanoscale materials.

Atomic force microscopy (AFM) has demonstrated great success as a tool for probing matter at the nanoscale, and recent reports suggest that it may even be capable of mapping electron clouds on atomic surfaces. The most recent of these claims came in 2004, when Hembacher <italic>et al</italic>. [<italic>Science</italic> <bold>305</bold>] observed subatomic features while imaging a graphite surface with a tungsten tip using higher-harmonics frequency modulation AFM (FM-AFM). The authors' interpretation of these features as the footprint of the electron density at the tungsten tip's apex atom has been met with much skepticism. But despite the potential significance of the results, a detailed theoretical study has not been performed. 

In this work, a computational method based in density functional theory (DFT) is developed in order to simulate the imaging process and draw fundamental conclusions regarding the feasibility of subatomic imaging with higher harmonics FM-AFM. The application of this method to the tungsten/graphite system reveals that the bonding lobes of increased charge density are in fact present at the tungsten tip's apex atom and that the corresponding higher harmonics images can exhibit subatomic features similar to those observed experimentally. 

We further show that the filtering process used to experimentally measure the harmonics does not introduce imaging artifacts but that harmonics averaging is not an appropriate method for enhancing contrast. We then suggest an alternate approach: the individual mapping of the first two harmonics, which are expected to dominate the contrast under the experimental conditions studied.

Finally, we demonstrate the important role played by the surface atom used to probe the AFM tip. We find that a small, non-reactive atom is necessary for resolving subatomic features. Most importantly, we show that the observed features are <italic>not a direct reflection</italic> of the electron density at the AFM tip's front atom. Instead, they represent a measure of the bonding stiffness between the tip's front atom and the atoms in the layer above.