SPAHM(a,b): Encoding the Density Information from Guess Hamiltonian in Quantum Machine Learning Representations.

Recently, we introduced a class of molecular representations for kernel-based regression methods─the spectrum of approximated Hamiltonian matrices (SPAHM)─that takes advantage of lightweight one-electron Hamiltonians traditionally used as a self-consistent field initial guess. The original SPAHM variant is built from occupied-orbital energies (i.e., eigenvalues) and naturally contains all of the information about nuclear charges, atomic positions, and symmetry requirements. Its advantages were demonstrated on data sets featuring a wide variation of charge and spin, for which traditional structure-based representations commonly fail. SPAHM(a,b), as introduced here, expand the eigenvalue SPAHM into local and transferable representations. They rely upon one-electron density matrices to build fingerprints from atomic and bond density overlap contributions inspired from preceding state-of-the-art representations. The performance and efficiency of SPAHM(a,b) is assessed on the predictions for data sets of prototypical organic molecules (QM7) of different charges and azoheteroarene dyes in an excited state. Overall, both SPAHM(a) and SPAHM(b) outperform state-of-the-art representations on difficult prediction tasks such as the atomic properties of charged open-shell species and of π-conjugated systems.

OSCAR: an extensive repository of chemically and functionally diverse organocatalysts.

The automated construction of datasets has become increasingly relevant in computational chemistry. While transition-metal catalysis has greatly benefitted from bottom-up or top-down strategies for the curation of organometallic complexes libraries, the field of organocatalysis is mostly dominated by case-by-case studies, with a lack of transferable data-driven tools that facilitate both the exploration of a wider range of catalyst space and the optimization of reaction properties. For these reasons, we introduce OSCAR, a repository of 4000 experimentally derived organocatalysts along with their corresponding building blocks and combinatorially enriched structures. We outline the fragment-based approach used for database generation and showcase the chemical diversity, in terms of functions and molecular properties, covered in OSCAR. The structures and corresponding stereoelectronic properties are publicly available (https://archive.materialscloud.org/record/2022.106) and constitute the starting point to build generative and predictive models for organocatalyst performance.

SPAHM: the spectrum of approximated Hamiltonian matrices representations.

Physics-inspired molecular representations are the cornerstone of similarity-based learning applied to solve chemical problems. Despite their conceptual and mathematical diversity, this class of descriptors shares a common underlying philosophy: they all rely on the molecular information that determines the form of the electronic Schrödinger equation. Existing representations take the most varied forms, from non-linear functions of atom types and positions to atom densities and potential, up to complex quantum chemical objects directly injected into the ML architecture. In this work, we present the spectrum of approximated Hamiltonian matrices (SPAHM) as an alternative pathway to construct quantum machine learning representations through leveraging the foundation of the electronic Schrödinger equation itself: the electronic Hamiltonian. As the Hamiltonian encodes all quantum chemical information at once, SPAHM representations not only distinguish different molecules and conformations, but also different spin, charge, and electronic states. As a proof of concept, we focus here on efficient SPAHM representations built from the eigenvalues of a hierarchy of well-established and readily-evaluated "guess" Hamiltonians. These SPAHM representations are particularly compact and efficient for kernel evaluation and their complexity is independent of the number of different atom types in the database.

Local Kernel Regression and Neural Network Approaches to the Conformational Landscapes of Oligopeptides.

The application of machine learning to theoretical chemistry has made it possible to combine the accuracy of quantum chemical energetics with the thorough sampling of finite-temperature fluctuations. To reach this goal, a diverse set of methods has been proposed, ranging from simple linear models to kernel regression and highly nonlinear neural networks. Here we apply two widely different approaches to the same, challenging problem: the sampling of the conformational landscape of polypeptides at finite temperature. We develop a local kernel regression (LKR) coupled with a supervised sparsity method and compare it with a more established approach based on Behler-Parrinello type neural networks. In the context of the LKR, we discuss how the supervised selection of the reference pool of environments is crucial to achieve accurate potential energy surfaces at a competitive computational cost and leverage the locality of the model to infer which chemical environments are poorly described by the DFTB baseline. We then discuss the relative merits of the two frameworks and perform Hamiltonian-reservoir replica-exchange Monte Carlo sampling and metadynamics simulations, respectively, to demonstrate that both frameworks can achieve converged and transferable sampling of the conformational landscape of complex and flexible biomolecules with comparable accuracy and computational cost.

Impact of quantum-chemical metrics on the machine learning prediction of electron density.

Machine learning (ML) algorithms have undergone an explosive development impacting every aspect of computational chemistry. To obtain reliable predictions, one needs to maintain a proper balance between the black-box nature of ML frameworks and the physics of the target properties. One of the most appealing quantum-chemical properties for regression models is the electron density, and some of us recently proposed a transferable and scalable model based on the decomposition of the density onto an atom-centered basis set. The decomposition, as well as the training of the model, is at its core a minimization of some loss function, which can be arbitrarily chosen and may lead to results of different quality. Well-studied in the context of density fitting (DF), the impact of the metric on the performance of ML models has not been analyzed yet. In this work, we compare predictions obtained using the overlap and the Coulomb-repulsion metrics for both decomposition and training. As expected, the Coulomb metric used as both the DF and ML loss functions leads to the best results for the electrostatic potential and dipole moments. The origin of this difference lies in the fact that the model is not constrained to predict densities that integrate to the exact number of electrons N. Since an a posteriori correction for the number of electrons decreases the errors, we proposed a modification of the model, where N is included directly into the kernel function, which allowed lowering of the errors on the test and out-of-sample sets.

Learning the Exciton Properties of Azo-dyes.

The ab initio determination of electronic excited state (ES) properties is the cornerstone of theoretical photochemistry. Yet, traditional ES methods become impractical when applied to fairly large molecules, or when used on thousands of systems. Machine learning (ML) techniques have demonstrated their accuracy at retrieving ES properties of large molecular databases at a reduced computational cost. For these applications, nonlinear algorithms tend to be specialized in targeting individual properties. Learning fundamental quantum objects potentially represents a more efficient, yet complex, alternative as a variety of molecular properties could be extracted through postprocessing. Herein, we report a general framework able to learn three fundamental objects: the hole and particle densities, as well as the transition density. We demonstrate the advantages of targeting those outputs and apply our predictions to obtain properties, including the state character and the exciton topological descriptors, for the two bands (nπ* and ππ*) of 3427 azoheteroarene photoswitches.

Learning on-top: Regressing the on-top pair density for real-space visualization of electron correlation.

The on-top pair density [Πr] is a local quantum-chemical property that reflects the probability of two electrons of any spin to occupy the same position in space. Being the simplest quantity related to the two-particle density matrix, the on-top pair density is a powerful indicator of electron correlation effects, and as such, it has been extensively used to combine density functional theory and multireference wavefunction theory. The widespread application of Π(r) is currently hindered by the need for post-Hartree-Fock or multireference computations for its accurate evaluation. In this work, we propose the construction of a machine learning model capable of predicting the complete active space self-consistent field (CASSCF)-quality on-top pair density of a molecule only from its structure and composition. Our model, trained on the GDB11-AD-3165 database, is able to predict with minimal error the on-top pair density of organic molecules, bypassing completely the need for ab initio computations. The accuracy of the regression is demonstrated using the on-top ratio as a visual metric of electron correlation effects and bond-breaking in real-space. In addition, we report the construction of a specialized basis set, built to fit the on-top pair density in a single atom-centered expansion. This basis, cornerstone of the regression, could be potentially used also in the same spirit of the resolution-of-the-identity approximation for the electron density.

Learning (from) the Electron Density: Transferability, Conformational and Chemical Diversity.

Machine-learning in quantum chemistry is currently booming, with reported applications spanning all molecular properties from simple atomization energies to complex mathematical objects such as the many-body wavefunction. Due to its central role in density functional theory, the electron density is a particularly compelling target for non-linear regression. Nevertheless, the scalability and the transferability of the existing machine-learning models of ρ(r) are limited by its complex rotational symmetries. Recently, in collaboration with Ceriotti and coworkers, we combined an efficient electron density decomposition scheme with a local regression framework based on symmetry-adapted Gaussian process regression able to accurately describe the covariance of the electron density spherical tensor components. The learning exercise is performed on local environments, allowing high transferability and linear-scaling of the prediction with respect to the number of atoms. Here, we review the main characteristics of the model and show its predictive power in a series of applications. The scalability and transferability of the trained model are demonstrated through the prediction of the electron density of Ubiquitin.

Machine learning models of the energy curvature vs particle number for optimal tuning of long-range corrected functionals.

The average energy curvature as a function of the particle number is a molecule-specific quantity, which measures the deviation of a given functional from the exact conditions of density functional theory. Related to the lack of derivative discontinuity in approximate exchange-correlation potentials, the information about the curvature has been successfully used to restore the physical meaning of Kohn-Sham orbital eigenvalues and to develop non-empirical tuning and correction schemes for density functional approximations. In this work, we propose the construction of a machine-learning framework targeting the average energy curvature between the neutral and the radical cation state of thousands of small organic molecules (QM7 database). The applicability of the model is demonstrated in the context of system-specific gamma-tuning of the LC-ωPBE functional and validated against the molecular first ionization potentials at equation-of-motion coupled-cluster references. In addition, we propose a local version of the non-linear regression model and demonstrate its transferability and predictive power by determining the optimal range-separation parameter for two large molecules relevant to the field of hole-transporting materials. Finally, we explore the underlying structure of the QM7 database with the t-SNE dimensionality-reduction algorithm and identify structural and compositional patterns that promote the deviation from the piecewise linearity condition.

Balancing Density Functional Theory Interaction Energies in Charged Dimers Precursors to Organic Semiconductors.

The pursuit of an increasingly accurate description of intermolecular interactions within the framework of Kohn-Sham density functional theory (KS-DFT) has motivated the construction of numerous benchmark databases over the past two decades. By far, the largest efforts have been spent on closed-shell, neutral dimers for which today the interaction energies and geometries can be accurately reproduced by various combinations of dispersion-corrected density functional approximations (DFAs). In sharp contrast, charged, open-shell dimers remain a challenge as illustrated by the analysis of the OREL26rad benchmark set, composed of π-dimer radical cations. Aside from the methodological aspect, achieving a proper description of radical cationic complexes is appealing due to their role as models for charge carriers in organic semiconductors. In the interest of providing an assessment of more realistic dimer systems, we construct a data set of large radical cationic dimers (CryOrel9) and jointly train the 19 parameters of a dispersion corrected, range-separated hybrid density functional (ωB97X-dDsC). The main objective of ωB97X-dDsC is to provide the maximum balance between the treatment of long-range London dispersion and reduction of the delocalization error, which are essential conditions to obtain accurate energy profiles and binding energies of charged, open-shell dimers. The performance of ωB97X-dDsC, its parent ωB97X functional series, and a selection of wave function-based methods is reported for the CryOrel9 data set. The robustness of the reoptimized variant (ωB97X-dDsC) is also tested on other GMTKN30 data sets.

Hamiltonian-Reservoir Replica Exchange and Machine Learning Potentials for Computational Organic Chemistry.

This work combines a machine learning potential energy function with a modular enhanced sampling scheme to obtain statistically converged thermodynamical properties of flexible medium-size organic molecules at high ab initio level. We offer a modular environment in the python package MORESIM that allows custom design of replica exchange simulations with any level of theory including ML-based potentials. Our specific combination of Hamiltonian and reservoir replica exchange is shown to be a powerful technique to accelerate enhanced sampling simulations and explore free energy landscapes with a quantum chemical accuracy unattainable otherwise (e.g., DLPNO-CCSD(T)/CBS quality). This engine is used to demonstrate the relevance of accessing the ab initio free energy landscapes of molecules whose stability is determined by a subtle interplay between variations in the underlying potential energy and conformational entropy (i.e., a bridged asymmetrically polarized dithiacyclophane and a widely used organocatalyst) both in the gas phase and in solution (implicit solvent).

Data-powered augmented volcano plots for homogeneous catalysis.

Given the computational resources available today, data-driven approaches can propel the next leap forward in catalyst design. Using a data-driven inspired workflow consisting of data generation, statistical analysis, and dimensionality reduction algorithms we explore trends surrounding the thermodynamics of a model hydroformylation reaction catalyzed by group 9 metals bearing phosphine ligands. Specifically, we introduce "augmented volcano plots" as a means to easily visualize the similarity of each catalyst's complete catalytic cycle energy profile to that of a hypothetical ideal reference profile without relying upon linear scaling relationships. In addition to quickly identifying catalysts that most closely match the ideal thermodynamic catalytic cycle energy profile, these maps also enable a more refined comparison of closely lying species in standard volcano plots. For the reaction studied here, they inherently uncover the presence of multiple sets of scaling relationships differentiated by metal type, where iridium catalysts follow distinct relationships from cobalt/rhodium catalysts and have profiles that more closely match the ideal thermodynamic profile. Reconstituted molecular volcano plots confirm the findings of the augmented volcanoes by showing that hydroformylation thermodynamics are governed by two distinct volcano shapes, one for iridium catalysts and a second for cobalt/rhodium species.

Quantum Chemistry Meets Machine Learning.

In this account, we demonstrate how statistical learning approaches can be leveraged across a range of different quantum chemical areas to transform the scaling, nature, and complexity of the problems that we are tackling. Selected examples illustrate the power brought by kernel-based approaches in the large-scale screening of homogeneous catalysis, the prediction of fundamental quantum chemical properties and the free-energy landscapes of flexible organic molecules. While certainly non-exhaustive, these examples provide an intriguing glimpse into our own research efforts.

Tuning the structure, reactivity and magnetic communication of nitride-bridged uranium complexes with the ancillary ligands.

Molecular uranium nitride complexes were prepared to relate their small molecule reactivity to the nature of the U[double bond, length as m-dash]N[double bond, length as m-dash]U bonding imposed by the supporting ligand. The U4+-U4+ nitride complexes, [NBu4][{(( t BuO)3SiO)3U}2(µ-N)], [NBu4]-1, and [NBu4][((Me3Si)2N)3U}2(µ-N)], 2, were synthesised by reacting NBu4N3 with the U3+ complexes, [U(OSi(O t Bu)3)2(µ-OSi(O t Bu)3)]2 and [U(N(SiMe3)2)3], respectively. Oxidation of 2 with AgBPh4 gave the U4+-U5+ analogue, [((Me3Si)2N)3U}2(µ-N)], 4. The previously reported methylene-bridged U4+-U4+ nitride [Na(dme)3][((Me3Si)2)2U(µ-N)(µ-κ2-C,N-CH2SiMe2NSiMe3)U(N(SiMe3)2)2] (dme = 1,2-dimethoxyethane), [Na(dme)3]-3, provided a versatile precursor for the synthesis of the mixed-ligand U4+-U4+ nitride complex, [Na(dme)3][((Me3Si)2N)3U(µ-N)U(N(SiMe3)2)(OSi(O t Bu)3)], 5. The reactivity of the 1-5 complexes was assessed with CO2, CO, and H2. Complex [NBu4]-1 displays similar reactivity to the previously reported heterobimetallic complex, [Cs{(( t BuO)3SiO)3U}2(µ-N)], [Cs]-1, whereas the amide complexes 2 and 4 are unreactive with these substrates. The mixed-ligand complexes 3 and 5 react with CO and CO2 but not H2. The nitride complexes [NBu4]-1, 2, 4, and 5 along with their small molecule activation products were structurally characterized. Magnetic data measured for the all-siloxide complexes [NBu4]-1 and [Cs]-1 show uncoupled uranium centers, while strong antiferromagnetic coupling was found in complexes containing amide ligands, namely 2 and 5 (with maxima in the χ versus T plot of 90 K and 55 K). Computational analysis indicates that the U(µ-N) bond order decreases with the introduction of oxygen-based ligands effectively increasing the nucleophilicity of the bridging nitride.

Synthesis of aminyl biradicals by base-induced Csp3-Csp3 coupling of cationic azo dyes.

The synthesis of the industrially important polymer parylene is achieved by polymerization of p-quinodimethane (p-QDM). The polymerization is thought to proceed via a biradical p-QDM dimer, but isolation or characterization of such a biradical has remained elusive. Here, we describe the synthesis of an aza-analogue of this p-QDM dimer. The biradical is formed by base-induced dimerization of an azoimidazolium dye. Due to the presence of sterically shielded aminyl radicals instead of terminal H2C groups, the stability of this dimer is sufficient for analyses by ESR spectroscopy and X-ray crystallography. A similar Csp3-Csp3 coupling was observed for an azotriazolium dye, suggesting that base-induced C-C coupling reactions can be realized for different types of azo dyes.

Open-Shell Nonbenzenoid Nanographenes Containing Two Pairs of Pentagonal and Heptagonal Rings.

Nonbenzenoid carbocyclic rings are postulated to serve as important structural elements toward tuning the chemical and electronic properties of extended polycyclic aromatic hydrocarbons (PAHs, or namely nanographenes), necessitating a rational and atomically precise synthetic approach toward their fabrication. Here, using a combined bottom-up in-solution and on-surface synthetic approach, we report the synthesis of nonbenzenoid open-shell nanographenes containing two pairs of embedded pentagonal and heptagonal rings. Extensive characterization of the resultant nanographene in solution shows a low optical gap, and an open-shell singlet ground state with a low singlet-triplet gap. Employing ultra-high-resolution scanning tunneling microscopy and spectroscopy, we conduct atomic-scale structural and electronic studies on a cyclopenta-fused derivative on a Au(111) surface. The resultant five to seven rings embedded nanographene displays an extremely narrow energy gap of 0.27 eV and exhibits a pronounced open-shell biradical character close to 1 (y0 = 0.92). Our experimental results are supported by mean-field and multiconfigurational quantum chemical calculations. Access to large nanographenes with a combination of nonbenzenoid topologies and open-shell character should have wide implications in harnessing new functionalities toward the realization of future organic electronic and spintronic devices.

Highly Substituted Δ3 -1,2,3-Triazolines: Solid-State Emitters with Electrofluorochromic Behavior.

Highly substituted Δ3 -1,2,3-triazolines can be prepared by reaction of triarylvinyl Grignard reagents with functionalized organic azides. The heterocycles are fluorescent in the solid state, and-depending on the substituents-they can display aggregation-induced emission. Upon oxidation, the triazolines form stable radical cations with altered photophysical properties. Therefore, they represent rare examples of solid-state emitters with intrinsic electrofluorochromic behavior.

Transferable Machine-Learning Model of the Electron Density.

The electronic charge density plays a central role in determining the behavior of matter at the atomic scale, but its computational evaluation requires demanding electronic-structure calculations. We introduce an atom-centered, symmetry-adapted framework to machine-learn the valence charge density based on a small number of reference calculations. The model is highly transferable, meaning it can be trained on electronic-structure data of small molecules and used to predict the charge density of larger compounds with low, linear-scaling cost. Applications are shown for various hydrocarbon molecules of increasing complexity and flexibility, and demonstrate the accuracy of the model when predicting the density on octane and octatetraene after training exclusively on butane and butadiene. This transferable, data-driven model can be used to interpret experiments, accelerate electronic structure calculations, and compute electrostatic interactions in molecules and condensed-phase systems.

Electron density learning of non-covalent systems.

Chemists continuously harvest the power of non-covalent interactions to control phenomena in both the micro- and macroscopic worlds. From the quantum chemical perspective, the strategies essentially rely upon an in-depth understanding of the physical origin of these interactions, the quantification of their magnitude and their visualization in real-space. The total electron density ρ( r ) represents the simplest yet most comprehensive piece of information available for fully characterizing bonding patterns and non-covalent interactions. The charge density of a molecule can be computed by solving the Schrödinger equation, but this approach becomes rapidly demanding if the electron density has to be evaluated for thousands of different molecules or very large chemical systems, such as peptides and proteins. Here we present a transferable and scalable machine-learning model capable of predicting the total electron density directly from the atomic coordinates. The regression model is used to access qualitative and quantitative insights beyond the underlying ρ( r ) in a diverse ensemble of sidechain-sidechain dimers extracted from the BioFragment database (BFDb). The transferability of the model to more complex chemical systems is demonstrated by predicting and analyzing the electron density of a collection of 8 polypeptides.

The role of bridging ligands in dinitrogen reduction and functionalization by uranium multimetallic complexes.

Cooperativity between metal centres is identified as a crucial step in dinitrogen reduction both for the industrial Haber-Bosch process and for the natural fixation of nitrogen by nitrogenase enzymes, but the mechanism of N2 reduction remains poorly understood. This is in large part because multimetallic complexes that reduce and functionalize dinitrogen in the absence of strong alkali reducing agents are crucial to establish a structure-activity relationship, but remain extremely rare. Recently, we reported a multimetallic nitride-bridged diuranium(III) complex capable of reducing and functionalizing dinitrogen. Here we show that an analogous complex assembled with an oxo instead of a nitride linker also effects the four-electron reduction of dinitrogen, but the reactivity of the resulting oxo-(N2) complex differs significantly from that of the nitride-(N2). Computational studies show a different bonding scheme for the dinitrogen where the bridging nitride does participate in the binding and consequent activation of N2, while the oxide does not.