The Energetic Origins of Pi-Pi Contacts in Proteins

Accurate potential energy models of proteins must describe the many different types of noncovalent interactions that contribute to a protein's stability and structure. Pi-pi contacts are ubiquitous structural motifs in all proteins, occurring between aromatic and nonaromatic residues and play a nontrivial role in protein folding and in the formation of biomolecular condensates. Guided by a geometric criterion for isolating pi-pi contacts from classical molecular dynamics simulations of proteins, we use quantum mechanical energy decomposition analysis to determine the molecular interactions that stabilize different pi-pi contact motifs. We find that neutral pi-pi interactions in proteins are dominated by Pauli repulsion and London dispersion rather than repulsive quadrupole electrostatics, which is central to the textbook Hunter-Sanders model. This results in a notable lack of variability in the interaction profiles of neutral pi-pi contacts even with extreme changes in the dielectric medium, explaining the prevalence of pi-stacked arrangements in and between proteins. We also find interactions involving pi-containing anions and cations to be extremely malleable, interacting like neutral pi-pi contacts in polar media and like typical ion-pi interactions in nonpolar environments. Like-charged pairs such as arginine-arginine contacts are particularly sensitive to the polarity of their immediate surroundings and exhibit canonical pi-pi stacking behavior only if the interaction is mediated by environmental effects, such as aqueous solvation.

Accurate Non-Covalent Interaction Energies on Noisy Intermediate-Scale Quantum Computers via Second-Order Symmetry-Adapted Perturbation Theory

The computation 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, use of the standard supermolecular method...

Templating Bicarbonate in the Second Coordination Sphere Enhances Electrochemical CO2 Reduction Catalyzed by Iron Porphyrins

Bicarbonate-based electrolytes are ubiquitous in aqueous electrochemical CO₂ reduction, particularly in heterogenous catalysis, where they demonstrate improved catalytic performance relative to other buffers. In contrast, the presence of bicarbonate in organic electrolytes and its roles in homogeneous electrocatalysis remain underexplored. Here, we investigate the influence of bicarbonate on iron porphyrin-catalyzed electrochemical CO₂ reduction. We show that bicarbonate is a viable proton donor in organic electrolyte (pK_a = 20.8 in dimethyl sulfoxide) and that urea pendants in the second coordination sphere can be used to template bicarbonate in the vicinity of a molecular iron porphyrin catalyst. The templated binding of bicarbonate increases its acidity, resulting in a 1500-fold enhancement in catalytic rates relative to unmodified parent iron porphyrin. This work emphasizes the importance of bicarbonate speciation in wet organic electrolytes and establishes second-sphere bicarbonate templating as a design strategy to harness this adventitious acid and enhance CO₂ reduction catalysis.

Revisiting the Bonding Model for Gold(I) Species: The Importance of Pauli Repulsion Revealed in a Gold(I)-Cyclobutadiene Complex

Understanding the bonding of gold(I) species has been central to the development of gold(I) catalysis. Herein, we present the synthesis and characterization of the first gold(I)-cyclobutadiene complex, accompanied with bonding analysis by state-of-the-art energy decomposition analysis methods. Analysis of possible coordination modes for the new species not only confirms established characteristics of gold(I) bonding, but also suggests that Pauli repulsion is a key yet hitherto overlooked element. Additionally, we obtain a new perspective on gold(I)-bonding by comparison of the gold(I)-cyclobutadiene to congeners stabilized by p-, d-, and f-block metals. Consequently, we refine the gold(I) bonding model, with a delicate interplay of Pauli repulsion and charge transfer as the key driving force for various coordination motifs. Pauli repulsion is similarly determined as a significant interaction in AuI -alkyne species, corroborating this revised understanding of AuI bonding.

Deciphering Distinct Overpotential-Dependent Pathways for Electrochemical CO2 Reduction Catalyzed by an Iron-Terpyridine Complex

[Fe(tpyPY2Me)]²⁺ ([Fe]²⁺) is a homogeneous electrocatalyst for converting CO₂ into CO featuring low overpotentials of <100 mV, near-unity selectivity, and high activity with turnover frequencies faster than 100 000 s^-1. To identify the origins of its exceptional performance and inform future catalyst design, we report a combined computational and experimental study that establishes two distinct mechanistic pathways for electrochemical CO₂ reduction catalyzed by [Fe]²⁺ as a function of applied overpotential. Electrochemical data shows the formation of two catalytic regimes at low (η_TOF/2 of 160 mV) and high (η_TOF/2 of 590 mV) overpotential plateaus. We propose that at low overpotentials [Fe]²⁺ undergoes a two-electron reduction, two-proton-transfer mechanism (electrochemical-electrochemical-chemical-chemical, EECC), where turnover occurs through the dicationic iron complex, [Fe]²⁺. Computational analysis supports the importance of the singlet ground-state electronic structure for CO₂ binding and that the rate-limiting step is the second protonation in this low-overpotential regime. When more negative potentials are applied, an additional electron-transfer event occurs through either a stepwise or proton-coupled electron-transfer (PCET) pathway, enabling catalytic turnover from the monocationic iron complex ([Fe]⁺) via an electrochemical-chemical-electrochemical-chemical (ECEC) mechanism. Comparison of experimental kinetic data obtained from variable controlled potential electrolysis (CPE) experiments with direct product detection with calculated rates obtained from the energetic span model supports the PCET pathway as the most likely mechanism. Moreover, we build upon this mechanistic understanding to propose the design of an improved ligand framework that is predicted to stabilize the key transition states identified in our study and explore their electronic structures using an energy decomposition analysis. Taken together, this work highlights the value of synergistic computational/experimental approaches to decipher mechanisms of new electrocatalysts and direct the rational design of improved platforms.

Regularized Second-Order Møller-Plesset Theory: A More Accurate Alternative to Conventional MP2 for Noncovalent Interactions and Transition Metal Thermochemistry for the Same Computational Cost

Second-order Møller-Plesset theory (MP2) notoriously breaks down for π-driven dispersion interactions and dative bonds in transition metal complexes. Herein, we investigate three physically justified forms of single-parameter, energy-gap dependent regularization which can yield high and transferable accuracy for a variety of noncovalent interactions (including S22, S66, and L7 test sets) and (mostly closed shell) transition metal thermochemistry. Regularization serves to damp overestimated pairwise additive contributions, renormalizing first-order amplitudes such that the effects of higher-order correlations are incorporated. The optimal parameter values for the noncovalent and transition metal sets are 1.1, 0.7, and 0.4 for κ, σ, and σ² regularizers, respectively. However, such regularization slightly degrades the accuracy of conventional MP2 for some small-molecule test sets, most of which have relatively large average frontier energy gaps. Our results suggest that appropriately regularized MP2 models may improve double hybrid density functionals, at no additional cost over conventional MP2.

Exploring the Limits of Second- and Third-Order Møller-Plesset Perturbation Theories for Noncovalent Interactions: Revisiting MP2.5 and Assessing the Importance of Regularization and Reference Orbitals

This work systematically assesses the influence of reference orbitals, regularization, and scaling on the performance of second- and third-order Møller-Plesset perturbation theory wave function methods for noncovalent interactions (NCIs). Testing on 19 data sets (A24, DS14, HB15, HSG, S22, X40, HW30, NC15, S66, AlkBind12, CO2Nitrogen16, HB49, Ionic43, TA13, XB18, Bauza30, CT20, XB51, and Orel26rad) covers a wide range of different NCIs including hydrogen bonding, dispersion, and halogen bonding. Inclusion of potential energy surfaces from different hydrogen bonds and dispersion-bound complexes gauges accuracy for nonequilibrium geometries. Fifteen methods are tested. In notation where nonstandard choices of orbitals are denoted as methods:orbitals, these are MP2, κ-MP2, SCS-MP2, OOMP2, κ-OOMP2, MP3, MP2.5, MP3:OOMP2, MP2.5:OOMP2, MP3:κ-OOMP2, MP2.5:κ-OOMP2, κ-MP3:κ-OOMP2, κ-MP2.5:κ-OOMP2, MP3:ωB97X-V, and MP2.5:ωB97X-V. Furthermore, we compare these methods to the ωB97M-V and B3LYP-D3 density functionals, as well as CCSD. We find that the κ-regularization (κ = 1.45 au was used throughout) improves the energetics in almost all data sets for both MP2 (in 17 out of 19 data sets) and OOMP2 (16 out of 19). The improvement is significant (e.g., the root-mean-square deviation (RMSD) for the S66 data set is 0.29 kcal/mol for κ-OOMP2 versus 0.67 kcal/mol for MP2) and for interactions between stable closed-shell molecules, not strongly dependent on the reference orbitals. Scaled MP3 (with a factor of 0.5) using κ-OOMP2 reference orbitals (MP2.5:κ-OOMP2) provides significantly more accurate results for NCIs across all data sets with noniterative O(N⁶) scaling (S66 data set RMSD: 0.10 kcal/mol). Across the entire data set of 356 points, the improvement over standard MP2.5 is approximately a factor of 2: RMSD for MP3:κ-OOMP2 is 0.25 vs 0.50 kcal/mol for MP2.5. The use of high-quality density functional reference orbitals (ωB97X-V) also significantly improves the results of MP2.5 for NCI over a Hartree-Fock orbital reference. All our assessments and conclusions are based on the use of the medium-sized aug-cc-pVTZ basis to yield results that are directly compared against complete basis set limit reference values.

Software for the frontiers of quantum chemistry: An overview of developments in the Q-Chem 5 package

This article summarizes technical advances contained in the fifth major release of the Q-Chem quantum chemistry program package, covering developments since 2015. A comprehensive library of exchange-correlation functionals, along with a suite of correlated many-body methods, continues to be a hallmark of the Q-Chem software. The many-body methods include novel variants of both coupled-cluster and configuration-interaction approaches along with methods based on the algebraic diagrammatic construction and variational reduced density-matrix methods. Methods highlighted in Q-Chem 5 include a suite of tools for modeling core-level spectroscopy, methods for describing metastable resonances, methods for computing vibronic spectra, the nuclear-electronic orbital method, and several different energy decomposition analysis techniques. High-performance capabilities including multithreaded parallelism and support for calculations on graphics processing units are described. Q-Chem boasts a community of well over 100 active academic developers, and the continuing evolution of the software is supported by an "open teamware" model and an increasingly modular design.

Revealing the nature of electron correlation in transition metal complexes with symmetry breaking and chemical intuition

In this work, we provide a nuanced view of electron correlation in the context of transition metal complexes, reconciling computational characterization via spin and spatial symmetry breaking in single-reference methods with qualitative concepts from ligand-field and molecular orbital theories. These insights provide the tools to reliably diagnose the multi-reference character, and our analysis reveals that while strong (i.e., static) correlation can be found in linear molecules (e.g., diatomics) and weakly bound and antiferromagnetically coupled (monometal-noninnocent ligand or multi-metal) complexes, it is rarely found in the ground-states of mono-transition-metal complexes. This leads to a picture of static correlation that is no more complex for transition metals than it is, e.g., for organic biradicaloids. In contrast, the ability of organometallic species to form more complex interactions, involving both ligand-to-metal σ-donation and metal-to-ligand π-backdonation, places a larger burden on a theory's treatment of dynamic correlation. We hypothesize that chemical bonds in which inter-electron pair correlation is non-negligible cannot be adequately described by theories using MP2 correlation energies and indeed find large errors vs experiment for carbonyl-dissociation energies from double-hybrid density functionals. A theory's description of dynamic correlation (and to a less important extent, delocalization error), which affects relative spin-state energetics and thus spin symmetry breaking, is found to govern the efficacy of its use to diagnose static correlation.

Controlled Single-Electron Transfer via Metal-Ligand Cooperativity Drives Divergent Nickel-Electrocatalyzed Radical Pathways

Electrocatalysis enables the construction of C-C bonds under mild conditions via controlled formation of carbon-centered radicals. For sequences initiated by alkyl halide reduction, coordinatively unsaturated Ni complexes commonly serve as single-electron transfer agents, giving rise to the foundational question of whether outer- or inner-sphere electron transfer oxidative addition prevails in redox mediation. Indeed, rational design of electrochemical processes requires the discrimination of these two electron transfer pathways, as they can have outsized effects on the rate of substrate bond activation and thus impact radical generation rates and downstream product selectivities. We present results from combined synthetic, electroanalytical, and computational studies that examine the mechanistic differences of single electron transfer to alkyl halides imparted by Ni metal-ligand cooperativity. Electrogenerated reduced Ni species, stabilized by delocalized spin density onto a redox-active tpyPY2Me polypyridyl ligand, activates alkyl iodides via outer-sphere electron transfer, allowing for the selective activation of alkyl iodide substrates over halogen atom donors and the controlled generation and sequestration of electrogenerated radicals. In contrast, the Ni complex possessing a redox-innocent pentapyridine congener activates the substrates in an inner-sphere fashion owning to a purely metal-localized spin, thereby activating both substrates and halogen atom donors in an indiscriminate fashion, generating a high concentration of radicals and leading to unproductive dimerization. Our data establish that controlled electron transfer via Ni-ligand cooperativity can be used to limit undesired radical recombination products and promote selective radical processes in electrochemical environments, providing a generalizable framework for designing redox mediators with distinct rate and potential requirements.

From Intermolecular Interaction Energies and Observable Shifts to Component Contributions and Back Again: A Tale of Variational Energy Decomposition Analysis

Quantum chemistry in the form of density functional theory (DFT) calculations is a powerful numerical experiment for predicting intermolecular interaction energies. However, no chemical insight is gained in this way beyond predictions of observables. Energy decomposition analysis (EDA) can quantitatively bridge this gap by providing values for the chemical drivers of the interactions, such as permanent electrostatics, Pauli repulsion, dispersion, and charge transfer. These energetic contributions are identified by performing DFT calculations with constraints that disable components of the interaction. This review describes the second-generation version of the absolutely localized molecular orbital EDA (ALMO-EDA-II). The effects of different physical contributions on changes in observables such as structure and vibrational frequencies upon complex formation are characterized via the adiabatic EDA. Example applications include red- versus blue-shifting hydrogen bonds; the bonding and frequency shifts of CO, N₂, and BF bound to a [Ru(II)(NH₃)₅]^{2 +} moiety; and the nature of the strongly bound complexes between pyridine and the benzene and naphthalene radical cations. Additionally, the use of ALMO-EDA-II to benchmark and guide the development of advanced force fields for molecular simulation is illustrated with the recent, very promising, MB-UCB potential.

Mechanistic Insights into Co and Fe Quaterpyridine-Based CO2 Reduction Catalysts: Metal-Ligand Orbital Interaction as the Key Driving Force for Distinct Pathways

Both [Co^II(qpy)(H₂O)₂]²⁺ and [Fe^II(qpy)(H₂O)₂]²⁺ (with qpy = 2,2':6',2″:6'',2‴-quaterpyridine) are efficient homogeneous electrocatalysts and photoelectrocatalysts for the reduction of CO₂ to CO. The Co catalyst is more efficient in the electrochemical reduction, while the Fe catalyst is an excellent photoelectrocatalyst ( ACS Catal. 2018, 8, 3411-3417). This work uses density functional theory to shed light on the contrasting catalytic pathways. While both catalysts experience primarily ligand-based reductions, the second reduction in the Co catalyst is delocalized onto the metal via a metal-ligand bonding interaction, causing a spin transition and a distorted ligand framework. This orbital interaction explains the experimentally observed mild reduction potential and slow kinetics of the second reduction. The decreased hardness and doubly occupied d_z²-orbital facilitate a σ-bond with the CO₂-π* in an η¹-κC binding mode. CO₂ binding is only possible after two reductions resulting in an EEC mechanism (E = electron transfer, C = chemical reaction), and the second protonation is rate-limiting. In contrast, the Fe catalyst maintains a Lewis acidic metal center throughout the reduction process because the metal orbitals do not strongly mix with the qpy-π* orbitals. This allows binding of the activated CO₂ in an η²-binding mode. This interaction stabilizes the activated CO₂ via a π-type interaction of a Fe-t_2g orbital and the CO₂-π* and a dative bond of the oxygen lone pair. This facilitates CO₂ binding to a singly reduced catalyst resulting in an ECE mechanism. The barrier for CO₂ addition and the second protonation are higher than those for the Co catalyst and rate-limiting.

Consistent inclusion of continuum solvation in energy decomposition analysis: theory and application to molecular CO2 reduction catalysts

To facilitate computational investigation of intermolecular interactions in the solution phase, we report the development of ALMO-EDA(solv), a scheme that allows the application of continuum solvent models within the framework of energy decomposition analysis (EDA) based on absolutely localized molecular orbitals (ALMOs). In this scheme, all the quantum mechanical states involved in the variational EDA procedure are computed with the presence of solvent environment so that solvation effects are incorporated in the evaluation of all its energy components. After validation on several model complexes, we employ ALMO-EDA(solv) to investigate substituent effects on two classes of complexes that are related to molecular CO₂ reduction catalysis. For [FeTPP(CO₂-κC)]^2- (TPP = tetraphenylporphyrin), we reveal that two ortho substituents which yield most favorable CO₂ binding, -N(CH₃)₃ ⁺ (TMA) and -OH, stabilize the complex via through-structure and through-space mechanisms, respectively. The coulombic interaction between the positively charged TMA group and activated CO₂ is found to be largely attenuated by the polar solvent. Furthermore, we also provide computational support for the design strategy of utilizing bulky, flexible ligands to stabilize activated CO₂ via long-range Coulomb interactions, which creates biomimetic solvent-inaccessible "pockets" in that electrostatics is unscreened. For the reactant and product complexes associated with the electron transfer from the p-terphenyl radical anion to CO₂, we demonstrate that the double terminal substitution of p-terphenyl by electron-withdrawing groups considerably strengthens the binding in the product state while moderately weakens that in the reactant state, which are both dominated by the substituent tuning of the electrostatics component. These applications illustrate that this new extension of ALMO-EDA provides a valuable means to unravel the nature of intermolecular interactions and quantify their impacts on chemical reactivity in solution.

Probing radical-molecule interactions with a second generation energy decomposition analysis of DFT calculations using absolutely localized molecular orbitals

Intermolecular interactions between radicals and closed-shell molecules are ubiquitous in chemical processes, ranging from the benchtop to the atmosphere and extraterrestrial space. While energy decomposition analysis (EDA) schemes for closed-shell molecules can be generalized for studying radical-molecule interactions, they face challenges arising from the unique characteristics of the electronic structure of open-shell species. In this work, we introduce additional steps that are necessary for the proper treatment of radical-molecule interactions to our previously developed unrestricted Absolutely Localized Molecular Orbital (uALMO)-EDA based on density functional theory calculations. A "polarize-then-depolarize" (PtD) scheme is used to remove arbitrariness in the definition of the frozen wavefunction, rendering the ALMO-EDA results independent of the orientation of the unpaired electron obtained from isolated fragment calculations. The contribution of radical rehybridization to polarization energies is evaluated. It is also valuable to monitor the wavefunction stability of each intermediate state, as well as their associated spin density profiles, to ensure the EDA results correspond to a desired electronic state. These radical extensions are incorporated into the "vertical" and "adiabatic" variants of uALMO-EDA for studies of energy changes and property shifts upon complexation. The EDA is validated on two model complexes, H₂O˙F and FH˙OH. It is then applied to several chemically interesting radical-molecule complexes, including the sandwiched and T-shaped benzene dimer radical cation, complexes of pyridine with benzene and naphthalene radical cations, binary and ternary complexes of the hydroxyl radical with water (˙OH(H₂O) and ˙OH(H₂O)₂), and the pre-reactive complexes and transition states in the ˙OH + HCHO and ˙OH + CH₃CHO reactions. These examples suggest that this second generation uALMO-EDA is a useful tool for furthering one's understanding of both energetic and property changes associated with radical-molecule interactions.

Two New Methods To Generate Internal Coordinates for Molecular Wave Packet Dynamics in Reduced Dimensions

The curse of dimensionality still remains as the central challenge of molecular quantum dynamical calculations. Either compromises on the accuracy of the potential landscape have to be made or methods must be used that reduce the dimensionality of the configuration space of molecular systems to a low dimensional one. For dynamic approaches such as grid-based wave packet dynamics that are confined to a small number of degrees of freedom this dimensionality reduction can become a major part of the overall problem. A common strategy to reduce the configuration space is by selection of a set of internal coordinates using chemical intuition. We devised two methods that increase the degree of automation of the dimensionality reduction as well as replace chemical intuition by more quantifiable criteria. Both methods reduce the dimensionality linearly and use the intrinsic reaction coordinate as guidance. The first one solely relies on the intrinsic reaction coordinate (IRC), whereas the second one uses semiclassical trajectories to identify the important degrees of freedom.