Fewer Measurements from Shadow Tomography with N-Representability Conditions.

Classical shadow tomography provides a randomized scheme for approximating the quantum state and its properties at reduced computational cost with applications in quantum computing. In this Letter we present an algorithm for realizing fewer measurements in the shadow tomography of many-body systems. Accelerated tomography of the two-body reduced density matrix (2-RDM) is achieved by combining classical shadows with necessary constraints for the 2-RDM to represent an N-body system, known as N-representability conditions. We compute the ground-state energies and 2-RDMs of hydrogen chains and the N_{2} dissociation curve. The results demonstrate a significant reduction in the number of measurements with important applications to quantum many-body simulations on near-term quantum devices.

Variable Peripheral Ligand Donation Tunes Electronic Structure and NIR II Emission in Tetrathiafulvalene Tetrathiolate Diradicaloids.

Near-infrared (NIR) lumiphores are promising candidates for numerous imaging, communication, and sensing applications, but they typically require large, conjugated scaffolds to achieve emission in this low-energy region. Due to the extended conjugation and synthetic complexity required, it is extremely difficult to tune the photophysical properties of these systems for desired applications. Here, we report facile tuning of deep NIR-emitting diradicaloid complexes through simple modification of peripheral ligands. These new lumiphores are rare examples of air-, acid-, and water-stable emissive diradicaloids. We apply a simple Hammett parameter-based strategy to tune the electron donation of the capping ligand across a series of commercially available triarylphosphines. This minor peripheral modification significantly alters the electronic structure, and consequently, the electrochemical, photophysical, and magnetic properties of the tetrathiafulvalene tetrathiolate (TTFtt)-based lumiphores. The resultant ∼100 nm absorption and emission range spans common laser lines and the desirable telecom region (ca. 1260-1550 nm). Furthermore, these lumiphores are sensitive to local dielectrics, distinguishing them as promising candidates for ratiometric imaging and/or barcoding in the deep NIR region.

Quantum simulation of conical intersections.

We explore the simulation of conical intersections (CIs) on quantum devices, setting the groundwork for potential applications in nonadiabatic quantum dynamics within molecular systems. The intersecting potential energy surfaces of H3+ are computed from a variance-based contracted quantum eigensolver. We show how the CIs can be correctly described on quantum devices using wavefunctions generated by the anti-Hermitian contracted Schrödinger equation ansatz, which is a unitary transformation of wavefunctions that preserves the topography of CIs. A hybrid quantum-classical procedure is used to locate the seam of CIs. Additionally, we discuss the quantum implementation of the adiabatic to diabatic transformation and its relation to the geometric phase effect. Results on noisy intermediate-scale quantum devices showcase the potential of quantum computers in dealing with problems in nonadiabatic chemistry.

Selective Cobalt-Mediated Formation of Hydrogen Peroxide from Water under Mild Conditions via Ligand Redox Non-Innocence.

Despite the broad importance of hydrogen peroxide (H2O2) in oxidative transformations, there are comparatively few viable routes for its production. The majority of commercial H2O2 is currently produced by the stepwise reduction of dioxygen (O2) via the anthraquinone process, but direct electrochemical formation from water (H2O) would have several advantages─namely, avoiding flammable gases or stepwise separations. However, the selective oxidation of H2O to form H2O2 over the thermodynamically favored product of O2 is a difficult synthetic challenge. Here, we present a molecular H2O oxidation system with excellent selectivity for H2O2 that functions both stoichiometrically and catalytically. We observe high efficiency for electrocatalytic H2O2 production at low overpotential with no O2 observed under any conditions. Mechanistic studies with both calculations and kinetic analyses from isolated intermediates suggest that H2O2 formation occurs in a bimolecular fashion via a dinuclear H2O2-bridged intermediate with an important role for a redox non-innocent ligand. This system showcases the ability of metal-ligand cooperativity and strategic design of the secondary coordination sphere to promote kinetically and thermodynamically challenging selectivity in oxidative catalysis.

Origin of Weak Magnetic Coupling in a Dimanganese(II) Complex Bridged by the Tetrathiafulvalene-Tetrathiolate Radical.

Magnetic exchange coupling (J) between different spin centers plays a crucial role in molecule-based magnetic materials. Direct exchange coupling between an organic radical and a metal is frequently stronger than superexchange through diamagnetic ligands, and the strategy of using organic radicals to engender desirable magnetic properties has been an area of active investigation. Despite significant advances and exciting bulk properties, the magnitude of J for radical linkers bridging paramagnetic centers is still difficult to rationally predict. It is thus important to elucidate the features of organic radicals that govern this parameter. Here, we measure J for the tetrathiafulvalene-tetrathiolate radical (TTFtt3-•) in a dinuclear Mn(II) complex. Magnetometry studies show that the antiferromagnetic coupling in this complex is much weaker than that in related Mn(II)-radical compounds, in contrast to what might be expected for the S-based chelating donor atoms of TTFtt. Experimental and computational analyses suggest that this small J coupling may be attributed to poor overlap between Mn- and TTFtt-based magnetic orbitals coupled with insignificant spin density on the coordinating S-atoms. These factors override any expected increase in J from the comparatively strong S-donors. This work elucidates the magnetic coupling properties of the TTFtt3-• radical for the first time and also demonstrates how multiple competing factors must be considered in rationally designing organic radical ligands for molecular-based magnetic compounds.

Qubit Condensation for Assessing Efficacy of Molecular Simulation on Quantum Computers.

Quantum computers may demonstrate significant advantages over classical devices, as they are able to exploit a purely quantum-mechanical phenomenon known as entanglement in which a single quantum state simultaneously populates two-or-more classical configurations. However, due to environmental noise and device errors, elaborate quantum entanglement can be difficult to prepare on modern quantum computers. In this paper, we introduce a metric based on the condensation of qubits to assess the ability of a quantum device to simulate many-electron systems. Qubit condensation occurs when the qubits on a quantum computer condense into a single, highly correlated particle-hole state. While conventional metrics like gate errors and quantum volume do not directly target entanglement, the qubit-condensation metric measures the quantum computer's ability to generate an entangled state that achieves nonclassical long-range order across the device. To demonstrate, we prepare qubit condensations on various quantum devices and probe the degree to which qubit condensation is realized via postmeasurement analysis. We show that the predicted ranking of the quantum devices is consistent with the errors obtained from molecular simulations of H2 using a contracted quantum eigensolver.

Quantum Many-Body Theory from a Solution of the N-Representability Problem.

Here we present a many-body theory based on a solution of the N-representability problem in which the ground-state two-particle reduced density matrix (2-RDM) is determined directly without the many-particle wave function. We derive an equation that re-expresses physical constraints on higher-order RDMs to generate direct constraints on the 2-RDM, which are required for its derivation from an N-particle density matrix, known as N-representability conditions. The approach produces a complete hierarchy of 2-RDM constraints that do not depend explicitly upon the higher RDMs or the wave function. By using the two-particle part of a unitary decomposition of higher-order constraint matrices, we can solve the energy minimization by semidefinite programming in a form where the low-rank structure of these matrices can be potentially exploited. We illustrate by computing the ground-state electronic energy and properties of the H_{8} ring.

Universal Generalization of Density Functional Theory for Static Correlation.

A major challenge for density functional theory (DFT) is its failure to treat static correlation, yielding errors in predicted charges, band gaps, van der Waals forces, and reaction barriers. Here we combine one- and two-electron reduced density matrix (1- and 2-RDM) theories with DFT to obtain a universal O(N^{3}) generalization of DFT for static correlation. Using the lowest unitary invariant of the cumulant 2-RDM, we generate a 1-RDM functional theory that corrects the convexity of any DFT functional to capture static correlation in its fractional orbital occupations. Importantly, the unitary invariant yields a predictive theory by revealing the dependence of the correction's strength upon the trace of the two-electron repulsion matrix. We apply the theory to the barrier to rotation in ethylene, the relative energies of the benzynes, as well as an 11-molecule, dissociation benchmark. By inheriting the computational efficiency of DFT without sacrificing the treatment of static correlation, the theory opens new possibilities for the prediction and interpretation of significant quantum molecular effects and phenomena.

Reducing the Quantum Many-Electron Problem to Two Electrons with Machine Learning.

An outstanding challenge in chemical computation is the many-electron problem where computational methodologies scale prohibitively with system size. The energy of any molecule can be expressed as a weighted sum of the energies of two-electron wave functions that are computable from only a two-electron calculation. Despite the physical elegance of this extended "aufbau" principle, the determination of the distribution of weights─geminal occupations─for general molecular systems has remained elusive. Here we introduce a new paradigm for electronic structure where approximate geminal-occupation distributions are "learned" via a convolutional neural network. We show that the neural network learns the N-representability conditions, constraints on the distribution for it to represent an N-electron system. By training on hydrocarbon isomers with only 2-7 carbon atoms, we are able to predict the energies for isomers of octane as well as hydrocarbons with 8-15 carbons. The present work demonstrates that machine learning can be used to reduce the many-electron problem to an effective two-electron problem, opening new opportunities for accurately predicting electronic structure.

Intrinsic glassy-metallic transport in an amorphous coordination polymer.

Conducting organic materials, such as doped organic polymers1, molecular conductors2,3 and emerging coordination polymers4, underpin technologies ranging from displays to flexible electronics5. Realizing high electrical conductivity in traditionally insulating organic materials necessitates tuning their electronic structure through chemical doping6. Furthermore, even organic materials that are intrinsically conductive, such as single-component molecular conductors7,8, require crystallinity for metallic behaviour. However, conducting polymers are often amorphous to aid durability and processability9. Using molecular design to produce high conductivity in undoped amorphous materials would enable tunable and robust conductivity in many applications10, but there are no intrinsically conducting organic materials that maintain high conductivity when disordered. Here we report an amorphous coordination polymer, Ni tetrathiafulvalene tetrathiolate, which displays markedly high electronic conductivity (up to 1,200 S cm-1) and intrinsic glassy-metallic behaviour. Theory shows that these properties are enabled by molecular overlap that is robust to structural perturbations. This unusual set of features results in high conductivity that is stable to humid air for weeks, pH 0-14 and temperatures up to 140 °C. These findings demonstrate that molecular design can enable metallic conductivity even in heavily disordered materials, raising fundamental questions about how metallic transport can exist without periodic structure and indicating exciting new applications for these materials.

Comparison of Density-Matrix Corrections to Density Functional Theory.

Density functional theory (DFT), one of the most widely utilized methods available to computational chemistry, fails to describe systems with statically correlated electrons. To address this shortcoming, in previous work, we transformed DFT into a one-electron reduced density matrix theory (1-RDMFT) via the inclusion of a quadratic one-electron reduced density matrix (1-RDM) correction. Here, we combine our 1-RDMFT approach with different DFT functionals as well as Hartree-Fock to elucidate the method's dependence on the underlying functional selection. Furthermore, we generalize the information density matrix functional theory (iDMFT), recently developed as a correction to the Hartree-Fock method, by incorporating density functionals in place of the Hartree-Fock functional. We relate iDMFT mathematically to our approach and benchmark the two with a common set of functionals and systems.

Accelerated Convergence of Contracted Quantum Eigensolvers through a Quasi-Second-Order, Locally Parameterized Optimization.

A contracted quantum eigensolver (CQE) finds a solution to the many-electron Schrödinger equation by solving its integration (or contraction) to the two-electron space─a contracted Schrödinger equation (CSE)─on a quantum computer. When applied to the anti-Hermitian part of the CSE (ACSE), the CQE iterations optimize the wave function, with respect to a general product ansatz of two-body exponential unitary transformations that can exactly solve the Schrödinger equation. In this work, we accelerate the convergence of the CQE and its wave function ansatz via tools from classical optimization theory. By treating the CQE algorithm as an optimization in a local parameter space, we can apply quasi-second-order optimization techniques, such as quasi-Newton approaches or nonlinear conjugate gradient approaches. Practically, these algorithms result in superlinear convergence of the wave function to a solution of the ACSE. Convergence acceleration is important because it can both minimize the accumulation of noise on near-term intermediate-scale quantum (NISQ) computers and achieve highly accurate solutions on future fault-tolerant quantum devices. We demonstrate the algorithm, as well as some heuristic implementations relevant for cost-reduction considerations, comparisons with other common methods such as variational quantum eigensolvers, and a Fermionic-encoding-free form of the CQE.

Generation and Aerobic Oxidative Catalysis of a Cu(II) Superoxo Complex Supported by a Redox-Active Ligand.

Cu systems feature prominently in aerobic oxidative catalysis in both biology and synthetic chemistry. Metal ligand cooperativity is a common theme in both areas as exemplified by galactose oxidase and by aminoxyl radicals in alcohol oxidations. This has motivated investigations into the aerobic chemistry of Cu and specifically the isolation and study of Cu-superoxo species that are invoked as key catalytic intermediates. While several examples of complexes that model biologically relevant Cu(II) superoxo intermediates have been reported, they are not typically competent aerobic catalysts. Here, we report a new Cu complex of the redox-active ligand tBu,TolDHP (2,5-bis((2-t-butylhydrazono)(p-tolyl)methyl)-pyrrole) that activates O2 to generate a catalytically active Cu(II)-superoxo complex via ligand-based electron transfer. Characterization using ultraviolet (UV)-visible spectroscopy, Raman isotope labeling studies, and Cu extended X-ray absorption fine structure (EXAFS) analysis confirms the assignment of an end-on κ1 superoxo complex. This Cu-O2 complex engages in a range of aerobic catalytic oxidations with substrates including alcohols and aldehydes. These results demonstrate that bioinspired Cu systems can not only model important bioinorganic intermediates but can also mediate and provide mechanistic insight into aerobic oxidative transformations.

Bright, Modular, and Switchable Near-Infrared II Emission from Compact Tetrathiafulvalene-Based Diradicaloid Complexes.

Near-infrared (NIR)-emitting molecules are promising candidates for biological sensing and imaging applications; however, many NIR dyes are large conjugated systems which frequently have issues with stability, solubility, and tunability. Here, we report a novel class of compact and tunable fluorescent diradicaloid complexes which are air-, water-, light-, and temperature-stable. These properties arise from a compressed π manifold which promotes an intense ligand-centered π-π transition in the NIR II (1000-1700 nm) region and which subsequently emits at ∼1200 nm. This emission is among the brightest known for monomolecular lumiphores with deep NIR II (>1100 nm) emission, nearly an order of magnitude brighter than the commercially available NIR II dye IR 26. Furthermore, this fluorescence is electrochemically sensitive, with efficient switching upon addition of redox agents. The brightness, stability, and modularity of this system distinguish it as a promising candidate for the development of new technologies built around NIR emission.

Interplay of Electronic and Geometric Structure Tunes Organic Biradical Character in Bimetallic Tetrathiafulvalene Tetrathiolate Complexes.

The synthesis and design of organic biradicals with tunable singlet-triplet gaps have become the subject of significant research interest, owing to their possible photochemical applications and use in the development of molecular switches and conductors. Recently, tetrathiafulvalene tetrathiolate (TTFtt) has been demonstrated to exhibit such organic biradical character in doubly ionized bimetallic complexes. In this article we use high-level ab initio calculations to interrogate the electronic structure of a series of TTFtt-bridged metal complexes, resolving the factors governing their biradical character and singlet-triplet gaps. We show that the degree of biradical character correlates with a readily measured experimental predictor, the central TTFtt C-C bond length, and that it may be described by a one-parameter model, providing valuable insight for the future rational design of TTFtt based biradical compounds and materials.

Elucidating the molecular orbital dependence of the total electronic energy in multireference problems.

The accurate resolution of the chemical properties of strongly correlated systems, such as biradicals, requires the use of electronic structure theories that account for both multi-reference and dynamic correlation effects. A variety of methods exist that aim to resolve the dynamic correlation in multi-reference problems, commonly relying on an exponentially scaling complete-active-space self-consistent-field (CASSCF) calculation to generate reference molecular orbitals (MOs). However, while CASSCF orbitals provide the optimal solution for a selected set of correlated (active) orbitals, their suitability in the quest for the resolution of the total correlation energy has not been thoroughly investigated. Recent research has shown the ability of Kohn-Shan density functional theory to provide improved orbitals for coupled cluster (CC) and Møller-Plesset perturbation theory (MP) calculations. Here, we extend the search for optimal and more cost effective MOs to post-configuration-interaction [post-(CI)] methods, surveying the ability of the MOs obtained with various density functional theory (DFT) functionals, as well as Hartree-Fock and CC and MP calculations to accurately capture the total electronic correlation energy. Applying the anti-Hermitian contracted Schrödinger equation to the dissociation of N2, the calculation of biradical singlet-triplet gaps, and the transition states of bicylobutane isomerization, we demonstrate that DFT provides a cost-effective alternative to CASSCF in providing reference orbitals for post-CI dynamic correlation calculations.

Beginnings of exciton condensation in coronene analog of graphene double layer.

Exciton condensation, a Bose-Einstein condensation of excitons into a single quantum state, has recently been achieved in low-dimensional materials including twin layers of graphene and van der Waals heterostructures. Here, we computationally examine the beginnings of exciton condensation in a double layer composed of coronene, a seven-benzene-ring patch of graphene. As a function of interlayer separation, we compute the exciton population in a single coherent quantum state, showing that the population peaks around 1.8 at distances near 2 Å. Visualization reveals interlayer excitons at the separation distance of the condensate. We determine the exciton population as a function of the twist angle between two coronene layers to reveal the magic angles at which the condensation peaks. As with previous recent calculations showing some exciton condensation in hexacene double layers and benzene stacks, the present two-electron reduced-density-matrix calculations with coronene provide computational evidence for the ability to realize exciton condensation in molecular-scale analogs of extended systems such as the graphene double layer.

Density Functional Theory Transformed into a One-Electron Reduced-Density-Matrix Functional Theory for the Capture of Static Correlation.

Density Functional Theory (DFT), the most widely adopted method in modern computational chemistry, fails to describe accurately the electronic structure of strongly correlated systems. Here we show that DFT can be formally and practically transformed into a one-electron reduced-density-matrix (1-RDM) functional theory, which can address the limitations of DFT while retaining favorable computational scaling compared to wave function based approaches. In addition to relaxing the idempotency restriction on the 1-RDM in the kinetic energy term, we add a quadratic 1-RDM-based term to DFT's density-based exchange-correlation functional. Our approach, which we implement by quadratic semidefinite programming at DFT's computational scaling of O(r3), yields substantial improvements over traditional DFT in the description of static correlation in chemical structures and processes such as singlet biradicals and bond dissociations.

Exciton Condensation in Molecular-Scale van der Waals Stacks.

Recent experiments have realized the Bose-Einstein condensation of excitons, known as exciton condensation, in extended systems such as bilayer graphene and van der Waals heterostructures. Here we computationally demonstrate the beginnings of exciton condensation in multilayer, molecular-scale van der Waals stacks composed of benzene subunits. The populations of excitons, which are computed from the largest eigenvalue of the particle-hole reduced density matrix (RDM) through advanced variational RDM calculations, are shown to increase with the length of the stack. The large eigenvalue indicates a nonclassical long-range ordering of the excitons that can support the frictionless flow of energy. Moreover, we use chemical substitutions and geometric modifications to tune the extent of the condensation. Results suggest exciton condensation in a potentially large family of molecular systems with applications to energy-efficient transport.

Correlation-driven phenomena in periodic molecular systems from variational two-electron reduced density matrix theory.

Correlation-driven phenomena in molecular periodic systems are challenging to predict computationally not only because such systems are periodically infinite but also because they are typically strongly correlated. Here, we generalize the variational two-electron reduced density matrix (2-RDM) theory to compute the energies and properties of strongly correlated periodic systems. The 2-RDM of the unit cell is directly computed subject to necessary N-representability conditions such that the unit-cell 2-RDM represents at least one N-electron density matrix. Two canonical but non-trivial systems, periodic metallic hydrogen chains and periodic acenes, are treated to demonstrate the methodology. We show that while single-reference correlation theories do not capture the strong (static) correlation effects in either of these molecular systems, the periodic variational 2-RDM theory predicts the Mott metal-to-insulator transition in the hydrogen chains and the length-dependent polyradical formation in acenes. For both hydrogen chains and acenes, the periodic calculations are compared with previous non-periodic calculations with the results showing a significant change in energies and increase in the electron correlation from the periodic boundary conditions. The 2-RDM theory, which allows for much larger active spaces than are traditionally possible, is applicable to studying correlation-driven phenomena in general periodic molecular solids and materials.