Resolving individual atoms has always been the ultimate goal of surface microscopy. The scanning tunneling microscope images atomic-scale features on surfaces, but resolving single atoms within an adsorbed molecule remains a great challenge because the tunneling current is primarily sensitive to the local electron density of states close to the Fermi level. We demonstrate imaging of molecules with unprecedented atomic resolution by probing the short-range chemical forces with use of noncontact atomic force microscopy. The key step is functionalizing the microscope's tip apex with suitable, atomically well-defined terminations, such as CO molecules. Our experimental findings are corroborated by ab initio density functional theory calculations. Comparison with theory shows that Pauli repulsion is the source of the atomic resolution, whereas van der Waals and electrostatic forces only add a diffuse attractive background.

We show that the different bond orders of individual carbon-carbon bonds in polycyclic aromatic hydrocarbons and fullerenes can be distinguished by noncontact atomic force microscopy (AFM) with a carbon monoxide (CO)-functionalized tip. We found two different contrast mechanisms, which were corroborated by density functional theory calculations: The greater electron density in bonds of higher bond order led to a stronger Pauli repulsion, which enhanced the brightness of these bonds in high-resolution AFM images. The apparent bond length in the AFM images decreased with increasing bond order because of tilting of the CO molecule at the tip apex.

Universal fault-tolerant quantum computers will require error-free execution of long sequences of quantum gate operations, which is expected to involve millions of physical qubits. Before the full power of such machines will be available, near-term quantum devices will provide several hundred qubits and limited error correction. Still, there is a realistic prospect to run useful algorithms within the limited circuit depth of such devices. Particularly promising are optimization algorithms that follow a hybrid approach: the aim is to steer a highly entangled state on a quantum system to a target state that minimizes a cost function via variation of some gate parameters. This variational approach can be used both for classical optimization problems as well as for problems in quantum chemistry. The challenge is to converge to the target state given the limited coherence time and connectivity of the qubits. In this context, the quantum volume as a metric to compare the power of near-term quantum devices is discussed.With focus on chemistry applications, a general description of variational algorithms is provided and the mapping from fermions to qubits is explained. Coupledcluster and heuristic trial wave-functions are considered for efficiently finding molecular ground states. Furthermore, simple error-mitigation schemes are introduced that could improve the accuracy of determining ground-state energies. Advancing these techniques may lead to near-term demonstrations of useful quantum computation with systems containing several hundred qubits.PACS numbers: quantum computation, quantum chemistry, quantum algorithms

* These authors contributed equally to this workTriangulene, the smallest triplet ground state polybenzenoid (also known as Clar's hydrocarbon), has been an enigmatic molecule ever since its existence was first hypothesized isomers (2, also denoted as dihydrotriangulenes) as precursor molecules. Compound 2 was deposited on Cu(111), NaCl(100), and Xe(111) surfaces to generate triangulene (1) by means of atomic manipulation. STM/AFM is an ideal combination to study on-surface synthesis ranging from individual molecules 9, 10 to graphene nanoribbons 11,12 . The chemical structure of reactants and products can be resolved by means of AFM with functionalized tips 13 . Even molecules too elusive to be studied by other means 14,15 can be stabilized by using an ultrathin insulating film as a decoupling 2 layer. A decoupling layer also facilitates studying the frontier molecular orbitals of the free molecule by means of STM and scanning tunnelling spectroscopy (STS) 16 .Figure 2 presents STM and AFM images of four different molecular species of compound 2 adsorbed on NaCl. As expected, different isomers of dihydrotriangulene are observed. This observation can be discussed by a comparison of Fig. 2e and Fig. 2f, showing the non-equivalent isomers 2a and 2b which we found on the surface, respectively. Because the former isomer is prochiral with respect to adsorption, we also observed its surface enantiomer (see Supplementary Fig. 5). Note that 2a is about three times more abundant than the highly symmetric 2b, although two Clar sextets can be drawn for both species. This difference can be rationalized by the resonance energy of its aromatic Remarkably, the ketone 3 in Fig. 2g represents an oxidized structure of 2 that was already reported by Clar and Stewart 1. A comparison of STM and AFM data reveals that in the STM images a tiny sharp kink arises at the position of a single CH 2 group. In contrast, a ketone group leads to a fainter bulge in STM images (Figs. 2c, d) and a lower contrast of the hexagon involved. The central carbon of the three adjacent CH 2 group in 3 adopts the expected tetrahedral bond angle for sp 3 carbon leading to sharp ridges in both STM and AFM mode (Figs. 2c, g) because of strong tilting of the CO molecule at the tip apex.We dehydrogenated promising candidates (2a and 2b molecules) to obtain triangulene by means of atomic manipulation 14,15,19 . To this end, we first positioned the tip above a molecule. Then, we opened the feedback loop and retracted the tip by 0.5 to 0.7 nm to limit the tunnelling current to a few picoamperes. Finally, we increased the voltage to values ranging from 3.5 to 4.1 V for several seconds. In many cases, this procedure also resulted in a lateral displacement of the molecule. When a subsequent STM image indicated a change in the appearance of the molecule, we recorded AFM 3 images to obtain its structure. Using this procedure, we did not observe any changes in the molecular structure other than the removal of single hydrogens from a CH 2 groups throughout our experiment...

Surface energies for different GaAs surface orientations have been calculated as a function of the chemical potential. We use an energy density formalism within the first-principles pseudopotential density-functional approach. The equilibrium crystal shape has been derived from the surface energies for the ͑110͒, ͑100͒, ͑111͒, and ͑1 1 1͒ orientations. Under As-rich conditions all four considered surface orientations exist in thermodynamic equilibrium, in agreement with experimental observations. Moreover, our calculations allow us to decide on previous contradictory theoretical values for the surface energies of the ͑111͒ and ͑1 1 1͒ facets.