Reducing the Runtime of Fault-Tolerant Quantum Simulations in Chemistry through Symmetry-Compressed Double Factorization.

Quantum phase estimation based on qubitization is the state-of-the-art fault-tolerant quantum algorithm for computing ground-state energies in chemical applications. In this context, the 1-norm of the Hamiltonian plays a fundamental role in determining the total number of required iterations and also the overall computational cost. In this work, we introduce the symmetry-compressed double factorization (SCDF) approach, which combines a CDF of the Hamiltonian with the symmetry shift technique, significantly reducing the 1-norm value. The effectiveness of this approach is demonstrated numerically by considering various benchmark systems, including the FeMoco molecule, cytochrome P450, and hydrogen chains of different sizes. To compare the efficiency of SCDF to other methods in absolute terms, we estimate Toffoli gate requirements, which dominate the execution time on fault-tolerant quantum computers. For the systems considered here, SCDF leads to a sizable reduction of the Toffoli gate count in comparison to other variants of DF or even tensor hypercontraction, which is usually regarded as the most efficient approach for qubitization.

Estimation of Electrostatic Interaction Energies on a Trapped-Ion Quantum Computer.

We present the first hardware implementation of electrostatic interaction energies by using a trapped-ion quantum computer. As test system for our computation, we focus on the reduction of NO to N2O catalyzed by a nitric oxide reductase (NOR). The quantum computer is used to generate an approximate ground state within the NOR active space. To efficiently measure the necessary one-particle density matrices, we incorporate fermionic basis rotations into the quantum circuit without extending the circuit length, laying the groundwork for further efficient measurement routines using factorizations. Measurements in the computational basis are then used as inputs for computing the electrostatic interaction energies on a classical computer. Our experimental results strongly agree with classical noise-less simulations of the same circuits, finding electrostatic interaction energies within chemical accuracy despite hardware noise. This work shows that algorithms tailored to specific observables of interest, such as interaction energies, may require significantly fewer quantum resources than individual ground state energies would require in the straightforward supermolecular approach.

Accurate non-covalent interaction energies on noisy intermediate-scale quantum computers via second-order symmetry-adapted perturbation theory.

The calculation of non-covalent interaction energies on noisy intermediate-scale quantum (NISQ) computers appears to be challenging with straightforward application of existing quantum algorithms. For example, the use of the standard supermolecular method with the variational quantum eigensolver (VQE) would require extremely precise resolution of the total energies of the fragments to provide for accurate subtraction to the interaction energy. Here we present a symmetry-adapted perturbation theory (SAPT) method that may provide interaction energies with high quantum resource efficiency. Of particular note, we present a quantum extended random-phase approximation (ERPA) treatment of the SAPT second-order induction and dispersion terms, including exchange counterparts. Together with previous work on first-order terms (Chem. Sci., 2022, 13, 3094), this provides a recipe for complete SAPT(VQE) interaction energies up to second order, which is a well established truncation. The SAPT interaction energy terms are computed as first-level observables with no subtraction of monomer energies invoked, and the only quantum observations needed are the VQE one- and two-particle density matrices. We find empirically that SAPT(VQE) can provide accurate interaction energies even with coarsely optimized, low circuit depth wavefunctions from a quantum computer, simulated through ideal statevectors. The errors of the total interaction energy are orders of magnitude lower than the corresponding VQE total energy errors of the monomer wavefunctions. In addition, we present heme-nitrosyl model complexes as a system class for near term quantum computing simulations. They are strongly correlated, biologically relevant and difficult to simulate with classical quantum chemical methods. This is illustrated with density functional theory (DFT) as the predicted interaction energies exhibit a strong sensitivity with respect to the choice of functional. Thus, this work paves the way to obtain accurate interaction energies on a NISQ-era quantum computer with few quantum resources. It is the first step in alleviating one of the major challenges in quantum chemistry, where in-depth knowledge of both the method and system is required a priori to reliably generate accurate interaction energies.

Towards the simulation of large scale protein-ligand interactions on NISQ-era quantum computers.

We explore the use of symmetry-adapted perturbation theory (SAPT) as a simple and efficient means to compute interaction energies between large molecular systems with a hybrid method combining NISQ-era quantum and classical computers. From the one- and two-particle reduced density matrices of the monomer wavefunctions obtained by the variational quantum eigensolver (VQE), we compute SAPT contributions to the interaction energy [SAPT(VQE)]. At first order, this energy yields the electrostatic and exchange contributions for non-covalently bound systems. We empirically find from ideal statevector simulations that the SAPT(VQE) interaction energy components display orders of magnitude lower absolute errors than the corresponding VQE total energies. Therefore, even with coarsely optimized low-depth VQE wavefunctions, we still obtain sub kcal mol-1 accuracy in the SAPT interaction energies. In SAPT(VQE), the quantum requirements, such as qubit count and circuit depth, are lowered by performing computations on the separate molecular systems. Furthermore, active spaces allow for large systems containing thousands of orbitals to be reduced to a small enough orbital set to perform the quantum portions of the computations. We benchmark SAPT(VQE) (with the VQE component simulated by ideal statevector simulators) against a handful of small multi-reference dimer systems and the iron center containing human cancer-relevant protein lysine-specific demethylase 5 (KDM5A).

Probing Molecular Excited States by Atomic Force Microscopy.

By employing single charge injections with an atomic force microscope, we investigated redox reactions of a molecule on a multilayer insulating film. First, we charged the molecule positively by attaching a single hole. Then we neutralized it by attaching an electron and observed three channels for the neutralization. We rationalize that the three channels correspond to transitions to the neutral ground state, to the lowest energy triplet excited states and to the lowest energy singlet excited states. By single-electron tunneling spectroscopy we measured the energy differences between the transitions obtaining triplet and singlet excited state energies. The experimental values are compared with density functional theory calculations of the excited state energies. Our results show that molecules in excited states can be prepared and that energies of optical gaps can be quantified by controlled single-charge injections. Our work demonstrates the access to, and provides insight into, ubiquitous electron-attachment processes related to excited-state transitions important in electron transfer and molecular optoelectronics phenomena on surfaces.

Sub-cycle atomic-scale forces coherently control a single-molecule switch.

Scanning probe techniques can leverage atomically precise forces to sculpt matter at surfaces, atom by atom. These forces have been applied quasi-statically to create surface structures1-7 and influence chemical processes8,9, but exploiting local dynamics10-14 to realize coherent control on the atomic scale remains an intriguing prospect. Chemical reactions15-17, conformational changes18,19 and desorption20 have been followed on ultrafast timescales, but directly exerting femtosecond forces on individual atoms to selectively induce molecular motion has yet to be realized. Here we show that the near field of a terahertz wave confined to an atomically sharp tip provides femtosecond atomic-scale forces that selectively induce coherent hindered rotation in the molecular frame of a bistable magnesium phthalocyanine molecule. Combining lightwave-driven scanning tunnelling microscopy21-24 with ultrafast action spectroscopy10,13, we find that the induced rotation modulates the probability of the molecule switching between its two stable adsorption geometries by up to 39 per cent. Mapping the response of the molecule in space and time confirms that the force acts on the atomic scale and within less than an optical cycle (that is, faster than an oscillation period of the carrier wave of light). We anticipate that our strategy might ultimately enable the coherent manipulation of individual atoms within single molecules or solids so that chemical reactions and ultrafast phase transitions can be manipulated on their intrinsic spatio-temporal scales.

Charge-Induced Structural Changes in a Single Molecule Investigated by Atomic Force Microscopy.

Intramolecular structural relaxations occurring upon electron transfer are crucial in determining the rate of redox reactions. Here, we demonstrate that subangstrom structural changes occurring upon single-electron charging can be quantified by means of atomically resolved atomic force microscopy (AFM) for the case of single copper(II)phthalocyanine (CuPc) molecules deposited on an ultrathin NaCl film. Imaging the molecule in distinct charge states (neutral and anionic) reveals characteristic differences in the AFM contrast. In comparison to density functional theory simulations these changes in contrast can be directly related to relaxations of the molecule's geometric structure upon charging. The dominant contribution arises from a nonhomogeneous vertical relaxation of the molecule, caused by a change in the electrostatic interaction with the surface.

Accessing a Charged Intermediate State Involved in the Excitation of Single Molecules.

Intermediate states are elusive to many experimental techniques due to their short lifetimes. Here, by performing single-electron alternate charging scanning tunneling microscopy of molecules on insulators, we accessed a charged intermediate state involved in the rapid toggling of individual metal phthalocyanines deposited on NaCl films. By stabilizing the transient species, we reveal how electron injection into the lowest unoccupied molecular orbital leads to a pronounced change in the adsorption geometry, characterized by a different azimuthal orientation. This observation allows clarifying the nature of the toggling process, unveiling the role of transient ionic states involved into fundamental processes occurring at interfaces.

Molecular structure elucidation with charge-state control.

The charge state of a molecule governs its physicochemical properties, such as conformation, reactivity, and aromaticity, with implications for on-surface synthesis, catalysis, photoconversion, and applications in molecular electronics. On insulating, multilayer sodium chloride (NaCl) films, we controlled the charge state of organic molecules and resolved their structures in neutral, cationic, anionic, and dianionic states by atomic force microscopy, obtaining atomic resolution and bond-order discrimination using carbon monoxide (CO)-functionalized tips. We detected changes in conformation, adsorption geometry, and bond-order relations for azobenzene, tetracyanoquinodimethane, and pentacene in multiple charge states. Moreover, for porphine, we investigate the charge state-dependent change of aromaticity and conjugation pathway in the macrocycle. This work opens the way to studying chemical-structural changes of individual molecules for a wide range of charge states.

Controlled Fragmentation of Single Molecules with Atomic Force Microscopy by Employing Doubly Charged States.

By atom manipulation we performed on-surface chemical reactions of a single molecule on a multilayer insulating film using noncontact atomic force microscopy. The single-electron sensitivity of atomic force microscopy allows us to follow the addition of single electrons to the molecule and the investigation of the reaction products. By performing a novel strategy based on long-lived doubly charged states a single molecule is fragmented. The fragmentation can be reverted by again changing the charge state of the system, characterizing a reversible reaction. The experimental results in addition to density-functional theory provide insight into the charge states of the different products and reaction pathways. Similar molecular systems could be used as charge-transfer units and to induce reversible chemical reactions.

Studying an antiaromatic polycyclic hydrocarbon adsorbed on different surfaces.

Antiaromatic and open-shell molecules are attractive because of their distinct electronic and magnetic behaviour. However, their increased reactivity creates a challenge for probing their properties. Here, we describe the on-surface and in-solution generation and characterisation of a highly reactive antiaromatic molecule: indeno[1,2-b]fluorene (IF). In solution, we generated IF by KI-induced dehalogenation of a dibromo-substituted precursor molecule and found that IF survives for minutes at ambient conditions. Using atom manipulation at low temperatures we generated IF on Cu(111) and on bilayer NaCl. On these surfaces, we characterised IF by bond-order analysis using non-contact atomic force microscopy with CO-functionalised tips and by orbital imaging using scanning tunnelling microscopy. We found that the closed-shell configuration and antiaromatic character predicted for gas-phase IF are preserved on the NaCl film. On Cu(111), we observed significant bond-order reorganisation within the s-indacene moiety because of chemisorption, highlighting the importance of molecule surface interactions on the π-electron distribution.

Atomic Force Microscopy for Molecular Structure Elucidation.

Using scanning probe microscopy techniques, at low temperatures and in ultrahigh vacuum, individual molecules adsorbed on surfaces can be probed with ultrahigh resolution to determine their structure and details of their conformation, configuration, charge states, aromaticity, and the contributions of resonance structures. Functionalizing the tip of an atomic force microscope with a CO molecule enabled atomic-resolution imaging of single molecules, and measurement of their adsorption geometry and bond-order relations. In addition, by using scanning tunneling microscopy and Kelvin probe force microscopy, the density of the molecular frontier orbitals and the electric charge distribution within molecules can be mapped. Combining these techniques yields a high-resolution tool for the identification and characterization of individual molecules. The single-molecule sensitivity and the possibility of atom manipulation to induce chemical reactions with the tip of the microscope open up unique applications in chemistry, and differentiate scanning probe microscopy from conventional methods for molecular structure elucidation. Besides being an aid for challenging cases in natural product identification, atomic force microscopy has been shown to be a powerful tool for the investigation of on-surface reactions and the characterization of radicals and molecular mixtures. Herein we review the progress that high-resolution scanning probe microscopy with functionalized tips has made for molecular structure identification and characterization, and discuss the challenges it will face in the years to come.

Synthesis and characterization of triangulene.

Triangulene, the smallest triplet-ground-state polybenzenoid (also known as Clar's hydrocarbon), has been an enigmatic molecule ever since its existence was first hypothesized. Despite containing an even number of carbons (22, in six fused benzene rings), it is not possible to draw Kekulé-style resonant structures for the whole molecule: any attempt results in two unpaired valence electrons. Synthesis and characterization of unsubstituted triangulene has not been achieved because of its extreme reactivity, although the addition of substituents has allowed the stabilization and synthesis of the triangulene core and verification of the triplet ground state via electron paramagnetic resonance measurements. Here we show the on-surface generation of unsubstituted triangulene that consists of six fused benzene rings. The tip of a combined scanning tunnelling and atomic force microscope (STM/AFM) was used to dehydrogenate precursor molecules. STM measurements in combination with density functional theory (DFT) calculations confirmed that triangulene keeps its free-molecule properties on the surface, whereas AFM measurements resolved its planar, threefold symmetric molecular structure. The unique topology of such non-Kekulé hydrocarbons results in open-shell π-conjugated graphene fragments that give rise to high-spin ground states, potentially useful in organic spintronic devices. Our generation method renders manifold experiments possible to investigate triangulene and related open-shell fragments at the single-molecule level.

Identical Binding Energies and Work Functions for Distinct Adsorption Structures: Olympicenes on the Cu(111) Surface.

Reliability is one of the major concerns and challenges in designing organic/inorganic interfaces for (opto)electronic applications. Even small structural differences for molecules on substrates can result in a significant variation in the interface functionality, due to the strong correlation between geometry, stability, and electronic structure. Here, we employed state-of-the-art first-principles calculations with van der Waals interactions, in combination with atomic force microscopy experiments, to explore the interaction mechanism for three structurally related olympicene molecules adsorbed on the Cu(111) surface. The substitution of a single atom in the olympicene molecule switches the nature of adsorption from predominantly physisorptive character [olympicene on Cu(111)], to an intermediate state [olympicene-derived ketone on Cu(111)], then to chemisorptive character [olympicene radical on Cu(111)]. Despite the remarkable difference in adsorption structures (by up to 0.9 Å in adsorption height) and different nature of bonding, the olympicene, its ketone, and its radical derivatives have essentially identical binding energies and work functions upon interaction with the metal substrate. Our findings suggest that the stability and work functions of molecular adsorbates could be rendered insensitive to their adsorption structures, which could be a useful property for (opto)electronic applications.

Reversible Bergman cyclization by atomic manipulation.

The Bergman cyclization is one of the most fascinating rearrangements in chemistry, with important implications in organic synthesis and pharmacology. Here we demonstrate a reversible Bergman cyclization for the first time. We induced the on-surface transformation of an individual aromatic diradical into a highly strained ten-membered diyne using atomic manipulation and verified the products by non-contact atomic force microscopy with atomic resolution. The diyne and diradical were stabilized by using an ultrathin NaCl film as the substrate, and the diyne could be transformed back into the diradical. Importantly, the diradical and the diyne exhibit different reactivity, electronic, magnetic and optical properties associated with the changes in the bond topology, and spin multiplicity. With this reversible, triggered Bergman cyclization we demonstrated switching on demand between the two reactive intermediates by means of selective C-C bond formation or cleavage, which opens up the field of radical chemistry for on-surface reactions by atomic manipulation.

The Electric Field of CO Tips and Its Relevance for Atomic Force Microscopy.

Metal tips decorated with CO molecules have paved the way for an impressively high resolution in atomic force microscopy (AFM). Although Pauli repulsion and the associated CO tilting play a dominant role at short distances, experiments on polar and metallic systems show that electrostatic interactions are necessary to understand the complex contrast observed and its distance evolution. Attempts to describe those interactions in terms of a single electrostatic dipole replacing the tip have led to contradictory statements about its nature and strength. Here, we solve this puzzle with a comprehensive experimental and theoretical characterization of the AFM contrast on Cl vacancies. Our model, based on density functional theory (DFT) calculations, reproduces the complex evolution of the contrast between both the Na cation and Cl anion sites, and the positively charged vacancy as a function of tip height, and highlights the key contribution of electrostatic interactions for tip-sample distances larger than 500 pm. For smaller separations, Pauli repulsion and the associated CO tilting start to dominate the contrast. The electrostatic field of the CO-metal tip can be represented by the superposition of the fields from the metal tip and the CO molecule. The long-range behavior is defined by the metal tip that contributes the field of a dipole with its positive pole at the apex. At short-range, the CO exhibits an opposite field that prevails. The interplay of these fields, with opposite sign and rather different spatial extension, is crucial to describe the contrast evolution as a function of the tip height.

On-surface generation and imaging of arynes by atomic force microscopy.

Reactive intermediates are involved in many chemical transformations. However, their characterization is a great challenge because of their short lifetimes and high reactivities. Arynes, formally derived from arenes by the removal of two hydrogen atoms from adjacent carbon atoms, are prominent reactive intermediates that have been hypothesized for more than a century. Their rich chemistry enables a widespread use in synthetic chemistry, as they are advantageous building blocks for the construction of polycyclic compounds that contain aromatic rings. Here, we demonstrate the generation and characterization of individual polycyclic aryne molecules on an ultrathin insulating film by means of low-temperature scanning tunnelling microscopy and atomic force microscopy. Bond-order analysis suggests that a cumulene resonance structure is the dominant one, and the aryne reactivity is preserved at cryogenic temperatures. Our results provide important insights into the chemistry of these elusive intermediates and their potential application in the field of on-surface synthesis.

Image distortions of a partially fluorinated hydrocarbon molecule in atomic force microscopy with carbon monoxide terminated tips.

The underlying mechanisms of image distortions in atomic force microscopy (AFM) with CO-terminated tips are identified and studied in detail. AFM measurements of a partially fluorinated hydrocarbon molecule recorded with a CO-terminated tip are compared with state-of-the-art ab initio calculations. The hydrogenated and fluorinated carbon rings in the molecule appear different in size, which primarily originates from the different extents of their π-electrons. Further, tilting of the CO at the tip, induced by van der Waals forces, enlarges the apparent size of parts of the molecule by up to 50%. Moreover, the CO tilting in response to local Pauli repulsion causes a significant sharpening of the molecule bonds in AFM imaging.

Adsorption geometry determination of single molecules by atomic force microscopy.

We measured the adsorption geometry of single molecules with intramolecular resolution using noncontact atomic force microscopy with functionalized tips. The lateral adsorption position was determined with atomic resolution, adsorption height differences with a precision of 3 pm, and tilts of the molecular plane within 0.2°. The method was applied to five π-conjugated molecules, including three molecules from the olympicene family, adsorbed on Cu(111). For the olympicenes, we found that the substitution of a single atom leads to strong variations of the adsorption height, as predicted by state-of-the-art density-functional theory, including van der Waals interactions with collective substrate response effects.

Bond-order discrimination by atomic force microscopy.

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.