Atomic Orbital Implementation of Extended Symmetry-Adapted Perturbation Theory (XSAPT) and Benchmark Calculations for Large Supramolecular Complexes

Substituent and Heteroatom Effects on π-π Interactions: Evidence That Parallel-Displaced π-Stacking is Not Driven by Quadrupolar Electrostatics

Stacking interactions are a recurring motif in supramolecular chemistry and biochemistry, where a persistent theme is a preference for parallel-displaced aromatic rings rather than face-to-face π-stacking. This is typically explained in terms of quadrupole-quadrupole interactions between the arene moieties but that interpretation is inconsistent with accurate calculations, which reveal that the quadrupolar picture is qualitatively wrong. At typical π-stacking distances, quadrupolar electrostatics may differ in sign from an exact calculation based on charge densities of the interacting arenes. We apply symmetry-adapted perturbation theory to dimers composed of substituted benzene and various aromatic heterocycles, which display a wide range of electrostatic interactions, and we investigate the interplay of Pauli repulsion, dispersion, and electrostatics as it pertains to parallel-displaced π-stacking. Profiles of energy components along cofacial slip-stacking coordinates support a prominent role for the "van der Waals model" (dispersion in competition with Pauli repulsion), even for polar monomers where electrostatic interactions are significant. While electrostatic interactions are necessary to explain the optimal face-to-face π-stacking distance and to account for the relative orientation of one polar arene with respect to another, we find no evidence to support continued invocation of quadrupolar electrostatics as a basis for π-stacking. Our results suggest that a driving force for offset-stacking exists even in the absence of electrostatic interactions. Consequently, tuning electrostatics via functionalization does not guarantee that slip-stacking can be avoided. This has implications for rational design of soft materials and other supramolecular architectures.

Quick-and-Easy Validation of Protein-Ligand Binding Models Using Fragment-Based Semiempirical Quantum Chemistry

Electronic structure calculations in enzymes converge very slowly with respect to the size of the model region that is described using quantum mechanics (QM), requiring hundreds of atoms to obtain converged results and exhibiting substantial sensitivity (at least in smaller models) to which amino acids are included in the QM region. As such, there is considerable interest in developing automated procedures to construct a QM model region based on well-defined criteria. However, testing such procedures is burdensome due to the cost of large-scale electronic structure calculations. Here, we show that semiempirical methods can be used as alternatives to density functional theory (DFT) to assess convergence in sequences of models generated by various automated protocols. The cost of these convergence tests is reduced even further by means of a many-body expansion. We use this approach to examine convergence (with respect to model size) of protein-ligand binding energies. Fragment-based semiempirical calculations afford well-converged interaction energies in a tiny fraction of the cost required for DFT calculations. Two-body interactions between the ligand and single-residue amino acid fragments afford a low-cost way to construct a "QM-informed" enzyme model of reduced size, furnishing an automatable active-site model-building procedure. This provides a streamlined, user-friendly approach for constructing ligand binding-site models that requires neither a priori information nor manual adjustments. Extension to model-building for thermochemical calculations should be straightforward.

Electrostatically embedded symmetry-adapted perturbation theory

Symmetry-adapted perturbation theory (SAPT) is an ab initio approach that directly computes noncovalent interaction energies in terms of electrostatics, exchange repulsion, induction/polarization, and London dispersion components. Due to its high computational scaling, routine applications of even the lowest order of SAPT are typically limited to a few hundred atoms. To address this limitation, we report here the addition of electrostatic embedding to the SAPT (EE-SAPT) and ISAPT (EE-ISAPT) methods. We illustrate the embedding scheme using water trimer as a prototype example. Then, we show that EE-SAPT/EE-ISAPT can be applied for efficiently and accurately computing noncovalent interactions in large systems, including solvated dimers and protein-ligand systems. In the latter application, particular care must be taken to properly handle the quantum mechanics/molecular mechanics boundary when it cuts covalent bonds. We investigate various schemes for handling charges near this boundary and demonstrate which are most effective in the context of charge-embedded SAPT.

QM/MM-Based Energy Decomposition Analysis Method for Large Systems

In this work, a QM/MM-based EDA method, called GKS-EDA(QM/MM), is proposed. As an extension of GKS-EDA, this scheme divides the total interaction energy into electrostatic, exchange-repulsion, polarization, and correlation/dispersion terms. GKS-EDA(QM/MM) can be applied to describe the interactions of large-scale systems combined with various QM/MM platforms. By using the examples of a hydrated hydronium ion complex in water solution, the barnase-barstar complex, and MMP-13-pyrimidinetrione in a metalloprotein, the capability of GKS-EDA(QM/MM) for various interactions in large systems is validated.

Performance of Density Functionals for Excited-State Properties of Isolated Chromophores and Exciplexes: Emission Spectra, Solvatochromic Shifts, and Charge-Transfer Character

This study assesses the performance of various meta-generalized gradient approximation (meta-GGA), global hybrid, and range-separated hybrid (RSH) density functionals in capturing the excited-state properties of organic chromophores and their excited-state complexes (exciplexes). Motivated by their uses in solar energy harvesting and photoredox CO2 reduction, we use oligo-(p-phenylenes) and their excited-state complexes with triethylamine as model systems. We focus on the fluorescence properties of these systems, specifically emission energies. We also consider solvatochromic shifts and wave function characteristics. The latter is described by using reduced quantities such as natural transition orbitals (NTOs) and exciton descriptors. The functionals are benchmarked against the experimental fluorescence spectra and the equation-of-motion coupled-cluster method with single and double excitations. Both in isolated chromophores and in exciplexes, meta-GGA functionals drastically underestimate the emission energies and exhibit significant exciton delocalization and anticorrelation between electron and hole motion. The performance of global hybrid functionals is strongly dependent on the percentage of exact exchange. Our study identifies RSH GGAs as the best-performing functionals, with ωPBE demonstrating the best agreement with experimental results. RSH meta-GGAs often overestimate emission energies in exciplexes and yield larger hole NTOs. Their performance can be improved by optimally tuning the range-separation parameter.

Can Spin-Component Scaled MP2 Achieve kJ/mol Accuracy for Cohesive Energies of Molecular Crystals?

Attaining kJ/mol accuracy in cohesive energy for molecular crystals is a persistent challenge in computational materials science. In this study, we evaluate second-order Møller-Plesset perturbation theory (MP2) and its spin-component scaled models for calculating cohesive energies for 23 molecular crystals (X23 data set). Using periodic boundary conditions and Brillouin zone sampling, we converge results to the thermodynamic and complete basis set limits, achieving an accuracy of about 2 kJ/mol (0.5 kcal/mol), which is rarely achieved in previous MP2 calculations for molecular crystals. When compared to experimental data, our results have a mean absolute error of 12.9 kJ/mol, comparable to Density Functional Theory with the PBE functional and TS dispersion correction. By separately scaling the opposite-spin and same-spin correlation energy components, using predetermined parameters, we reduce the mean absolute error to 9.5 kJ/mol. Further fine-tuning of these scaling parameters specifically for the X23 data set brings the mean absolute error down to 7.5 kJ/mol.

Accurate three-body noncovalent interactions: the insights from energy decomposition

An impressive collection of accurate two-body interaction energies for small complexes has been assembled into benchmark databases and used to improve the performance of multiple density functional, semiempirical, and machine learning methods. Similar benchmark data on nonadditive three-body energies in molecular trimers are comparatively scarce, and the existing ones are practically limited to homotrimers. In this work, we present a benchmark dataset of 20 equilibrium noncovalent interaction energies for a small but diverse selection of 10 heteromolecular trimers. The new 3BHET dataset presents complexes that combine different interactions including π-π, anion-π, cation-π, and various motifs of hydrogen and halogen bonding in each trimer. A detailed symmetry-adapted perturbation theory (SAPT)-based energy decomposition of the two- and three-body interaction energies shows that 3BHET consists of electrostatics- and dispersion-dominated complexes. The nonadditive three-body contribution is dominated by induction, but its influence on the overall bonding type in the complex (as exemplified by its position on the ternary diagram) is quite small. We also tested the extended SAPT (XSAPT) approach which is capable of including some nonadditive interactions in clusters of any size. The resulting three-body dispersion term (obtained from the many-body dispersion formalism) is mostly in good agreement with the supermolecular CCSD(T)-MP2 values and the nonadditive induction term is similar to the three-body SAPT(DFT) data, but the overall three-body XSAPT energies are not very accurate as they are missing the first-order exchange terms.

Accurate prediction of global-density-dependent range-separation parameters based on machine learning

In this work, we develop an accurate and efficient XGBoost machine learning model for predicting the global-density-dependent range-separation parameter, ωGDD, for long-range corrected functional (LRC)-ωPBE. This ωGDDML model has been built using a wide range of systems (11 466 complexes, ten different elements, and up to 139 heavy atoms) with fingerprints for the local atomic environment and histograms of distances for the long-range atomic correlation for mapping the quantum mechanical range-separation values. The promising performance on the testing set with 7046 complexes shows a mean absolute error of 0.001 117 a0-1 and only five systems (0.07%) with an absolute error larger than 0.01 a0-1, which indicates the good transferability of our ωGDDML model. In addition, the only required input to obtain ωGDDML is the Cartesian coordinates without electronic structure calculations, thereby enabling rapid predictions. LRC-ωPBE(ωGDDML) is used to predict polarizabilities for a series of oligomers, where polarizabilities are sensitive to the asymptotic density decay and are crucial in a variety of applications, including the calculations of dispersion corrections and refractive index, and surpasses the performance of all other popular density functionals except for the non-tuned LRC-ωPBE. Finally, LRC-ωPBE (ωGDDML) combined with (extended) symmetry-adapted perturbation theory is used in calculating noncovalent interactions to further show that the traditional ab initio system-specific tuning procedure can be bypassed. The present study not only provides an accurate and efficient way to determine the range-separation parameter for LRC-ωPBE but also shows the synergistic benefits of fusing the power of physically inspired density functional LRC-ωPBE and the data-driven ωGDDML model.

Birth of the Hydrated Electron via Charge-Transfer-to-Solvent Excitation of Aqueous Iodide

A primary means to generate hydrated electrons in laboratory experiments is excitation to the charge-transfer-to-solvent (CTTS) state of a solute such as I^-(aq), but this initial step in the genesis of e^-(aq) has never been simulated directly using ab initio molecular dynamics. We report the first such simulations, combining ground- and excited-state simulations of I^-(aq) with a detailed analysis of fluctuations in the Coulomb potential experienced by the nascent solvated electron. What emerges is a two-step picture of the evolution of e^-(aq) starting from the CTTS state: I^-(aq) + hν → I^-*(aq) → I^•(aq) + e^-(aq). Notably, the equilibrated ground state of e^-(aq) evolves from I^-*(aq) without any nonadiabatic transitions, simply as a result of solvent reorganization. The methodology used here should be applicable to other photochemical electron transfer processes in solution, an important class of problems directly relevant to photocatalysis and energy transfer.

Overcoming Artificial Multipoles in Intramolecular Symmetry-Adapted Perturbation Theory

Intramolecular symmetry-adapted perturbation theory (ISAPT) is a method to compute and decompose the noncovalent interaction energy between two molecular fragments A and B covalently connected via a linker C. However, the existing ISAPT algorithm displays several issues for many fragmentation patterns (that is, specific assignments of atoms to the A/B/C subsystems), including an artificially repulsive electrostatic energy (even when the fragments are hydrogen-bonded) and very large and mutually cancelling induction and exchange-induction terms. We attribute those issues to the presence of artificial dipole moments at the interfragment boundary, as the atoms of A and B directly connected to C are missing electrons on one of their hybrid orbitals. Therefore, we propose several new partitioning algorithms which reassign one electron, on a singly occupied link hybrid orbital, from C to each of A/B. Once the contributions from these link orbitals are added to fragment density matrices, the computation of ISAPT electrostatic, induction, and dispersion energies proceeds exactly as normal, and the exchange energy expressions need only minor modifications. Among the link partitioning algorithms introduced, the so-called ISAPT(SIAO1) approach (in which the link orbital is obtained by a projection onto the intrinsic atomic orbitals (IAOs) of a given fragment followed by orthogonalization to this fragment's occupied space) leads to reasonable values of all ISAPT corrections for all fragmentation patterns, and exhibits a fast and systematic basis set convergence. This improvement is made possible by a significant reduction in magnitude (even though not a complete elimination) of the unphysical dipole moments at the interfragment boundaries. We demonstrate the utility of the improved ISAPT partitioning by examining intramolecular interactions in several pentanediol isomers, examples of linear and branched alkanes, and the open and closed conformations of a family of N-arylimide molecular torsion balances.

Systematic Evaluation of Counterpoise Correction in Density Functional Theory

A widespread belief persists that the Boys-Bernardi function counterpoise (CP) procedure "overcorrects" supramolecular interaction energies for the effects of basis-set superposition error. To the extent that this is true for correlated wave function methods, it is usually an artifact of low-quality basis sets. The question has not been considered systematically in the context of density functional theory, however, where basis-set convergence is generally less problematic. We present a systematic assessment of the CP procedure for a representative set of functionals and basis sets, considering both benchmark data sets of small dimers and larger supramolecular complexes. The latter include layered composite polymers with ∼150 atoms and ligand-protein models with ∼300 atoms. Provided that CP correction is used, we find that intermolecular interaction energies of nearly complete-basis quality can be obtained using only double-ζ basis sets. This is less expensive as compared to triple-ζ basis sets without CP correction. CP-corrected interaction energies are less sensitive to the presence of diffuse basis functions as compared to uncorrected energies, which is important because diffuse functions are expensive and often numerically problematic for large systems. Our results upend the conventional wisdom that CP "overcorrects" for basis-set incompleteness. In small basis sets, CP correction is mandatory in order to demonstrate that the results do not rest on error cancellation.

Physical mechanisms of intermolecular interactions from symmetry-adapted perturbation theory

Symmetry-adapted perturbation theory (SAPT) is a method for computational studies of noncovalent interactions between molecules. This method will be discussed here from the perspective of establishing the paradigm for understanding mechanisms of intermolecular interactions. SAPT interaction energies are obtained as sums of several contributions. Each contribution possesses a clear physical interpretation as it results from some specific physical process. It also exhibits a specific dependence on the intermolecular separation R. The four major contributions are the electrostatic, induction, dispersion, and exchange energies, each due to a different mechanism, valid at any R. In addition, at large R, SAPT interaction energies are seamlessly connected with the corresponding terms in the asymptotic multipole expansion of interaction energy in inverse powers of R. Since such expansion explicitly depends on monomers' multipole moments and polarizabilities, this connection provides additional insights by rigorously relating interaction energies to monomers' properties.

Coupled Cluster Benchmarking of Large Noncovalent Complexes in L7 and S12L as Well as the C60 Dimer, DNA-Ellipticine, and HIV-Indinavir

In this work, we report the benchmark binding energies of the seven complexes within the L7 data set, six host-guest complexes from the S12L data set, a C₆₀ dimer, the DNA-ellipticine intercalation complex, and the largest system of the study, the HIV-indinavir system, which contained 343 atoms or 139 heavy atoms. The high-quality values reported were obtained via a focal point method that relies on the canonical form of second-order Møller-Plesset theory and the domain-based local pair natural orbital scheme for the coupled cluster with single double and perturbative triple excitations [DLPNO-CCSD(T)] extrapolated to the complete basis set (CBS) limit. The results in this work not only corroborate but also improve upon some previous benchmark values for large noncovalent complexes albeit at a relatively steep cost. Although local CCSD(T) and the largely successful fixed-node diffusion Monte Carlo (FN-DMC) have been shown to generally agree for small- to medium-size systems, a discrepancy in their reported binding energy values arises for large complexes, where the magnitude of the disagreement is a definite cause for concern. For example, the largest deviation in the L7 data set was 2.8 kcal/mol (∼10%) on the low end in C3GC. Such a deviation only grows worse in the S12L set, which showed a difference of up to 10.4 kcal/mol (∼25%) by a conservative estimation in buckycatcher-C₆₀. The DNA-ellipticine complex also generated a disagreement of 4.4 kcal/mol (∼10%) between both state-of-the-art methods. The disagreement between local CCSD(T) and FN-DMC in large noncovalent complexes shows that it is urgently needed to have the canonical CCSD(T), the Monte Carlo CCSD(T), or the full configuration interaction quantum Monte Carlo approaches available to large systems on the hundred-atom scale to solve this dilemma. In addition, the performances of cheaper popular computational methods were assessed for the studied complexes with respect to DLPNO-CCSD(T)/CBS. r²SCAN-3c, B97M-V, and PBE0+D4 work well in large noncovalent complexes in this work, and GFN2-xTB performs well in π-π stacking complexes. B97M-V is the most reliable computationally efficient approach to predicting noncovalent interactions for large complexes, being the only one to have binding errors within the so-called 1 kcal/mol "chemical accuracy". The benchmark interaction energies of these host-guest complexes, molecular materials, and biological systems with electronic and medicinal implications provide crucial reference data for the improvement of current and future lower-cost methods.

Comprehensive Basis-Set Testing of Extended Symmetry-Adapted Perturbation Theory and Assessment of Mixed-Basis Combinations to Reduce Cost

Hybrid or "extended" symmetry-adapted perturbation theory (XSAPT) replaces traditional SAPT's treatment of dispersion with better performing alternatives while at the same time extending two-body (dimer) SAPT to a many-body treatment of polarization using a self-consistent charge embedding procedure. The present work presents a systematic study of how XSAPT interaction energies and energy components converge with respect to the choice of Gaussian basis set. Errors can be reduced in a systematic way using correlation-consistent basis sets, with aug-cc-pVTZ results converged within <0.1 kcal/mol. Similar (if slightly less systematic) behavior is obtained using Karlsruhe basis sets at much lower cost, and we introduce new versions with limited augmentation that are even more efficient. Pople-style basis sets, which are more efficient still, often afford good results if a large number of polarization functions are included. The dispersion models used in XSAPT afford much faster basis-set convergence as compared to the perturbative description of dispersion in conventional SAPT, meaning that "compromise" basis sets (such as jun-cc-pVDZ) are no longer required and benchmark-quality results can be obtained using triple-ζ basis sets. The use of diffuse functions proves to be essential, especially for the description of hydrogen bonds. The "δ(Hartree-Fock)" correction for high-order induction can be performed in double-ζ basis sets without significant loss of accuracy, leading to a mixed-basis approach that offers 4× speedup over the existing (cubic scaling) XSAPT approach.

Interaction Energy Analysis of Monovalent Inorganic Anions in Bulk Water Versus Air/Water Interface

Soft anions exhibit surface activity at the air/water interface that can be probed using surface-sensitive vibrational spectroscopy, but the structural implications of this surface activity remain a matter of debate. Here, we examine the nature of anion-water interactions at the air/water interface using a combination of molecular dynamics simulations and quantum-mechanical energy decomposition analysis based on symmetry-adapted perturbation theory. Results are presented for a set of monovalent anions, including Cl-, Br-, I-, CN-, OCN-, SCN-, NO2-, NO3-, and ClOn- (n=1,2,3,4), several of which are archetypal examples of surface-active species. In all cases, we find that average anion-water interaction energies are systematically larger in bulk water although the difference (with respect to the same quantity computed in the interfacial environment) is well within the magnitude of the instantaneous fluctuations. Specifically for the surface-active species Br-(aq), I-(aq), ClO4-(aq), and SCN-(aq), and also for ClO-(aq), the charge-transfer (CT) energy is found to be larger at the interface than it is in bulk water, by an amount that is greater than the standard deviation of the fluctuations. The Cl-(aq) ion has a slightly larger CT energy at the interface, but NO3-(aq) does not; these two species are borderline cases where consensus is lacking regarding their surface activity. However, CT stabilization amounts to <20% of the total induction energy for each of the ions considered here, and CT-free polarization energies are systematically larger in bulk water in all cases. As such, the role of these effects in the surface activity of soft anions remains unclear. This analysis complements our recent work suggesting that the short-range solvation structure around these ions is scarcely different at the air/water interface from what it is in bulk water. Together, these observations suggest that changes in first-shell hydration structure around soft anions cannot explain observed surface activities.

Predicting and Understanding Non-Covalent Interactions Using Novel Forms of Symmetry-Adapted Perturbation Theory

Although sometimes derided as "weak" interactions, non-covalent forces play a critical role in ligand binding and crystal packing and in determining the conformational landscape of flexible molecules. Symmetry-adapted perturbation theory (SAPT) provides a framework for accurate ab initio calculation of intermolecular interactions and furnishes a natural decomposition of the interaction energy into physically meaningful components: semiclassical electrostatics (rigorously obtained from monomer charge densities), Pauli or steric repulsion, induction (including both polarization and charge transfer), and dispersion. This decomposition helps to foster deeper understanding of non-covalent interactions and can be used to construct transferable, physics-based force fields. Separability of the SAPT interaction energy also provides the flexibility to construct composite methods, a feature that we exploit to improve the description of dispersion interactions. These are challenging to describe accurately because they arise from nonlocal electron correlation effects that appear for the first time at second order in perturbation theory but are not quantitatively described at that level.As with all quantum-chemical methods, a major limitation of SAPT is nonlinear scaling of the computational cost with respect to system size. This cost can be significantly mitigated using "SAPT0(KS)", which incorporates monomer electron correlation by means of Kohn-Sham (KS) molecular orbitals from density functional theory (DFT), as well as by an "extended" theory called XSAPT, developed by the authors. XSAPT generalizes traditional dimer SAPT to many-body systems, so that a ligand-protein interaction (for example) can be separated into contributions from individual amino acids, reducing the cost of the calculation below that of even supramolecular DFT while retaining the accuracy of high-level ab initio quantum chemistry.This Account provides an overview of the SAPT0(KS) approach and the XSAPT family of methods. Several low-cost variants are described that provide accuracy approaching that of the best ab initio benchmarks yet are affordable enough to tackle ligand-protein binding and sizable host-guest complexes. These variants include SAPT+aiD, which uses ab initio atom-atom dispersion potentials ("+aiD") in place of second-order SAPT dispersion, and also SAPT+MBD, which incorporates many-body dispersion (MBD) effects that are important in the description of nanoscale materials. Applications to drug binding highlight the size-extensive nature of dispersion, which is not a weak interaction in large systems. Other applications highlight how a physics-based analysis can sometimes upend conventional wisdom regarding intermolecular forces. In particular, careful reconsideration of π-π interactions makes clear that the quadrupolar electrostatics (or "Hunter-Sanders") model of π-π stacking should be replaced by a "van der Waals model" in which conformational preferences arise from a competition between dispersion and Pauli repulsion. Our analysis also suggests that molecular shape, rather than aromaticity per se, is the key factor driving strong stacking interactions. Looking forward, we anticipate that XSAPT-based methods can play a role in screening of drug candidates and in materials design.

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.

Neat, Simple, and Wrong: Debunking Electrostatic Fallacies Regarding Noncovalent Interactions

Multipole moments such as charge, dipole, and quadrupole are often invoked to rationalize intermolecular phenomena, but a low-order multipole expansion is rarely a valid description of electrostatics at the length scales that characterize nonbonded interactions. This is illustrated by examining several common misunderstandings rooted in erroneous electrostatic arguments. First, the notion that steric repulsion originates in Coulomb interactions is easily disproved by dissecting the interaction potential for Ar₂. Second, the Hunter-Sanders model of π-π interactions, which is based on quadrupolar electrostatics, is shown to have no basis in accurate calculations. Third, curved "buckybowls" exhibit unusually large dipole moments, but these are ancillary to the forces that control their intermolecular interactions, as illustrated by two examples involving corannulene. Finally, the assumption that interactions between water and small anions are dictated by the dipole moment of H₂O is shown to be false in the case of binary halide-water complexes. These examples present a compelling case that electrostatic explanations based on low-order multipole moments are very often counterfactual for nonbonded interactions at close range and should not be taken seriously in the absence of additional justification.

Coupled cluster benchmarks of large noncovalent complexes: The L7 dataset as well as DNA-ellipticine and buckycatcher-fullerene

In this work, benchmark binding energies for dispersion-bound complexes in the L7 dataset, the DNA-ellipticine intercalation complex, and the buckycatcher-C₆₀ complex with 120 heavy atoms using a focal-point method based on the canonical form of second-order Møller-Plesset theory (MP2) and the domain based local pair natural orbital scheme for the coupled cluster with single, double, and perturbative triple excitations [CCSD(T)] extrapolated to the complete basis set (CBS) limit are reported. This work allows for increased confidence given the agreement with respect to values recently obtained using the local natural orbital CCSD(T) for L7 and the canonical CCSD(T)/CBS result for the coronene dimer (C2C2PD). Therefore, these results can be considered pushing the CCSD(T)/CBS binding benchmark to the hundred-atom scale. The disagreements between the two state-of-the-art methods, CCSD(T) and fixed-node diffusion Monte Carlo, are substantial with at least 2.0 (∼10%), 1.9 (∼5%), and 10.3 kcal/mol (∼25%) differences for C2C2PD in L7, DNA-ellipticine, and buckycatcher-C₆₀, respectively. Such sizable discrepancy above "chemical accuracy" for large noncovalent complexes indicates how challenging it is to obtain benchmark binding interactions for systems beyond small molecules, although the three up-to-date density functionals, PBE0+D4, ωB97M-V, and B97M-V, agree better with CCSD(T) for these large systems. In addition to reporting these values, different basis sets and various CBS extrapolation parameters for Hartree-Fock and MP2 correlation energies were tested for the first time in large noncovalent complexes with the goal of providing some indications toward optimal cost effective routes to approach the CBS limit without substantial loss in quality.

Simplified tuning of long-range corrected density functionals for use in symmetry-adapted perturbation theory

Long considered a failure, second-order symmetry-adapted perturbation theory (SAPT) based on Kohn-Sham orbitals, or SAPT0(KS), can be resurrected for semiquantitative purposes using long-range corrected density functionals whose asymptotic behavior is adjusted separately for each monomer. As in other contexts, correct asymptotic behavior can be enforced via "optimal tuning" based on the ionization energy theorem of density functional theory, but the tuning procedure is tedious, expensive for large systems, and comes with a troubling dependence on system size. Here, we show that essentially identical results are obtained using a fast, convenient, and automated tuning procedure based on the size of the exchange hole. In conjunction with "extended" (X)SAPT methods that improve the description of dispersion, this procedure achieves benchmark-quality interaction energies, along with the usual SAPT energy decomposition, without the hassle of system-specific tuning.

Natural Charge-Transfer Analysis: Eliminating Spurious Charge-Transfer States in Time-Dependent Density Functional Theory via Diabatization, with Application to Projection-Based Embedding

For many types of vertical excitation energies, linear-response time-dependent density functional theory (LR-TDDFT) offers a useful degree of accuracy combined with unrivaled computational efficiency, although charge-transfer excitation energies are often systematically and dramatically underestimated, especially for large systems and those that contain explicit solvent. As a result, low-energy electronic spectra of solution-phase chromophores often contain tens to hundreds of spurious charge-transfer states, making LR-TDDFT needlessly expensive in bulk solution. Intensity borrowing by these spurious states can affect intensities of the valence excitations, altering electronic bandshapes. At higher excitation energies, it is difficult to distinguish spurious charge-transfer states from genuine charge-transfer-to-solvent (CTTS) excitations. In this work, we introduce an automated diabatization that enables fast and effective screening of the CTTS acceptor space in bulk solution. Our procedure introduces "natural charge-transfer orbitals" that provide a means to isolate orbitals that are most likely to participate in a CTTS excitation. Projection of these orbitals onto solvent-centered virtual orbitals provides a criterion for defining the most important solvent molecules in a given excitation and be used as an automated subspace selection algorithm for projection-based embedding of a high-level description of the CTTS state in a lower-level description of its environment. We apply this method to an ab initio molecular dynamics trajectory of I^-(aq) and report the lowest-energy CTTS band in the absorption spectrum. Our results are in excellent agreement with the experiment, and only one-third of the water molecules in the I^-(H₂O)₉₆ simulation cell need to be described with LR-TDDFT to obtain excitation energies that are converged to <0.1 eV. The tools introduced herein will improve the accuracy, efficiency, and usability of LR-TDDFT in solution-phase environments.

Electrostatics, Charge Transfer, and the Nature of the Halide-Water Hydrogen Bond

Binary halide-water complexes X^-(H₂O) are examined by means of symmetry-adapted perturbation theory, using charge-constrained promolecular reference densities to extract a meaningful charge-transfer component from the induction energy. As is known, the X^-(H₂O) potential energy surface (for X = F, Cl, Br, or I) is characterized by symmetric left and right hydrogen bonds separated by a C_2v-symmetric saddle point, with a tunneling barrier height that is <2 kcal/mol except in the case of F^-(H₂O). Our analysis demonstrates that the charge-transfer energy is correspondingly small (<2 kcal/mol except for X = F), considerably smaller than the electrostatic interaction energy. Nevertheless, charge transfer plays a crucial role determining the conformational preferences of X^-(H₂O) and provides a driving force for the formation of quasi-linear X··· H-O hydrogen bonds. Charge-transfer energies correlate well with measured O-H vibrational redshifts for the halide-water complexes and also for OH^-(H₂O) and NO₂^-(H₂O), providing some indication of a general mechanism.

Many-Body Dispersion

A broad range of approaches to many-body dispersion are discussed, including empirical approaches with multiple fitted parameters, augmented density functional-based approaches, symmetry adapted perturbation theory, and a supermolecule approach based on coupled cluster theory. Differing definitions of "body" are considered, specifically atom-based vs molecule-based approaches.

Reinterpreting π-stacking

The nature of π-π interactions has long been debated. The term "π-stacking" is considered by some to be a misnomer, in part because overlapping π-electron densities are thought to incur steric repulsion, and the physical origins of the widely-encountered "slip-stacked" motif have variously been attributed to either sterics or electrostatics, in competition with dispersion. Here, we use quantum-mechanical energy decomposition analysis to investigate π-π interactions in supramolecular complexes of polycyclic aromatic hydrocarbons, ranging in size up to realistic models of graphene, and for comparison we perform the same analysis on stacked complexes of polycyclic saturated hydrocarbons, which are cyclohexane-based analogues of graphane. Our results help to explain the short-range structure of liquid hydrocarbons that is inferred from neutron scattering, trends in melting-point data, the interlayer separation of graphene sheets, and finally band gaps and observation of molecular plasmons in graphene nanoribbons. Analysis of intermolecular forces demonstrates that aromatic π-π interactions constitute a unique and fundamentally quantum-mechanical form of non-bonded interaction. Not only do stacked π-π architectures enhance dispersion, but quadrupolar electrostatic interactions that may be repulsive at long range are rendered attractive at the intermolecular distances that characterize π-stacking, as a result of charge penetration effects. The planar geometries of aromatic sp² carbon networks lead to attractive interactions that are "served up on a molecular pizza peel", and adoption of slip-stacked geometries minimizes steric (rather than electrostatic) repulsion. The slip-stacked motif therefore emerges not as a defect induced by electrostatic repulsion but rather as a natural outcome of a conformational landscape that is dominated by van der Waals interactions (dispersion plus Pauli repulsion), and is therefore fundamentally quantum-mechanical in its origins. This reinterpretation of the forces responsible for π-stacking has important implications for the manner in which non-bonded interactions are modeled using classical force fields, and for rationalizing the prevalence of the slip-stacked π-π motif in protein crystal structures.

Electrostatics does not dictate the slip-stacked arrangement of aromatic π-π interactions

Benzene dimer has long been an archetype for π-stacking. According to the Hunter–Sanders model, quadrupolar electrostatics favors an edge-to-face CH⋯π geometry but competes with London dispersion that favors cofacial π-stacking, with a compromise “slip-stacked” structure emerging as the minimum-energy geometry. This model is based on classical electrostatics, however, and neglects charge penetration. A fully quantum-mechanical analysis, presented here, demonstrates that electrostatics actually exerts very little influence on the conformational landscape of (C₆H₆)₂. Electrostatics also cannot explain the slip-stacked arrangement of C₆H₆⋯C₆F₆, where the sign of the quadrupolar interaction is reversed. Instead, the slip-stacked geometry emerges in both systems due to competition between dispersion and Pauli repulsion, with electrostatics as an ambivalent spectator. This revised interpretation helps to rationalize the persistence of offset π-stacking in larger polycyclic aromatic hydrocarbons and across the highly varied electrostatic environments that characterize π–π interactions in proteins.

According to the Hunter–Sanders model, geometries in π–π systems arise from competition between quadrupolar electrostatics (favoring an edge-to-face geometry) and London dispersion (favoring stacking), but this model misrepresents the molecular physics.

SAPT codes for calculations of intermolecular interaction energies

Symmetry-adapted perturbation theory (SAPT) is a method for calculations of intermolecular (noncovalent) interaction energies. The set of SAPT codes that is described here, the current version named SAPT2020, includes virtually all variants of SAPT developed so far, among them two-body SAPT based on perturbative, coupled cluster, and density functional theory descriptions of monomers, three-body SAPT, and two-body SAPT for some classes of open-shell monomers. The properties of systems governed by noncovalent interactions can be predicted only if potential energy surfaces (force fields) are available. SAPT is the preferred approach for generating such surfaces since it is seamlessly connected to the asymptotic expansion of interaction energy. SAPT2020 includes codes for automatic development of such surfaces, enabling generation of complete dimer surfaces with a rigid monomer approximation for dimers containing about one hundred atoms. These codes can also be used to obtain surfaces including internal degrees of freedom of monomers.

Fantasy versus reality in fragment-based quantum chemistry

Since the introduction of the fragment molecular orbital method 20 years ago, fragment-based approaches have occupied a small but growing niche in quantum chemistry. These methods decompose a large molecular system into subsystems small enough to be amenable to electronic structure calculations, following which the subsystem information is reassembled in order to approximate an otherwise intractable supersystem calculation. Fragmentation sidesteps the steep rise (with respect to system size) in the cost of ab initio calculations, replacing it with a distributed cost across numerous computer processors. Such methods are attractive, in part, because they are easily parallelizable and therefore readily amenable to exascale computing. As such, there has been hope that distributed computing might offer the proverbial "free lunch" in quantum chemistry, with the entrée being high-level calculations on very large systems. While fragment-based quantum chemistry can count many success stories, there also exists a seedy underbelly of rarely acknowledged problems. As these methods begin to mature, it is time to have a serious conversation about what they can and cannot be expected to accomplish in the near future. Both successes and challenges are highlighted in this Perspective.

HFLD: A Nonempirical London Dispersion-Corrected Hartree-Fock Method for the Quantification and Analysis of Noncovalent Interaction Energies of Large Molecular Systems †

A nonempirical quantum mechanical method for the efficient and accurate quantification and analysis of intermolecular interactions is presented and tested on existing benchmark sets. The leading idea here is to focus on the intermolecular part of the correlation energy that contains the all-important London dispersion (LD) interaction. To keep the cost of the method low, essentially at the level of a Hartree-Fock (HF) calculation, the intramolecular part of the correlation energy is neglected. We also neglect the nondispersive parts of the intermolecular correlation energy. This scheme that we denote as Hartree-Fock plus London dispersion (HFLD) can be readily realized on the basis of the recently reported multilevel implementation of the domain-based local pair natural orbital coupled-cluster (DLPNO-CC) theory in conjunction with the well-established local energy decomposition (LED) analysis. The accuracy and efficiency of the HFLD method are evaluated on rare gas dimers, on the S66 and L7 benchmark sets of noncovalent interactions, and on an additional set (LP14) consisting of bulky Lewis pairs held together by intermolecular interactions of various strengths, with interaction energies ranging from -8 to -107 kcal/mol. It is first shown that the LD energy calculated with this approach is essentially identical to that obtained from the full DLPNO-CCSD(T)/LED calculation, with a mean absolute error of 0.2 kcal/mol on the S66 benchmark set. Moreover, in terms of the overall interaction energies, the HFLD method shows an efficiency that is comparable to that of the HF method, while retaining an accuracy between that of the DLPNO-CCSD and DLPNO-CCSD(T) schemes. Since the underlying DLPNO-CCSD method is linear scaling with respect to the system size, the HFLD approach also does not lead to new bottlenecks for large systems. As an illustrative example of its efficiency, the HFLD scheme was applied to the interaction between the substrate and the residues in the active site of the cyclohexanone monooxygenase enzyme. The excellent cost/performance ratio indicates that the HFLD method opens new avenues for the accurate calculation and analysis of noncovalent interaction energies in large molecular systems.

Symmetry-Adapted Perturbation Theory Decomposition of the Reaction Force: Insights into Substituent Effects Involved in Hemiacetal Formation Mechanisms

The decomposition of the reaction force based on symmetry-adapted perturbation theory (SAPT) has been proposed. This approach was used to investigate the substituent effects along the reaction coordinate pathway for the hemiacetal formation mechanism between methanol and substituted aldehydes of the form CX₃CHO (X = H, F, Cl, and Br), providing a quantitative evaluation of the reaction-driving and reaction-retarding force components. Our results highlight the importance of more favorable electrostatic and induction effects in the reactions involving halogenated aldehydes that leads to lower activation energy barriers. These substituent effects are further elucidated by applying the functional-group partition of symmetry-adapted perturbation theory (F-SAPT). The results show that the reaction is largely driven by favorable direct noncovalent interactions between the CX₃ group on the aldehyde and the OH group on methanol.

Variational Formulation of the Generalized Many-Body Expansion with Self-Consistent Charge Embedding: Simple and Correct Analytic Energy Gradient for Fragment-Based ab Initio Molecular Dynamics

The many-body expansion (MBE) and its extension to overlapping fragments, the generalized (G)MBE, constitute the theoretical basis for most fragment-based approaches for large-scale quantum chemistry. We reformulate the GMBE for use with embedding charges determined self-consistently from the fragment wave functions, in a manner that preserves the variational nature of the underlying self-consistent field method. As a result, the analytic gradient retains the simple "sum of fragment gradients" form that is often assumed in practice, sometimes incorrectly. This obviates (without approximation) the need to solve coupled-perturbed equations, and we demonstrate stable, fragment-based ab initio molecular dynamics simulations using this technique. Energy conservation fails when charge-response contributions to the Fock matrix are neglected, even while geometry optimizations and vibrational frequency calculations may yet be accurate. Stable simulations can be recovered by means of straightforward modifications introduced here, providing a general paradigm for fragment-based ab initio molecular dynamics.

Accurate and Efficient ab Initio Calculations for Supramolecular Complexes: Symmetry-Adapted Perturbation Theory with Many-Body Dispersion

Symmetry-adapted perturbation theory (SAPT) provides a chemically meaningful energy decomposition scheme for nonbonded interactions that is useful for interpretive purposes. Although formally a dimer theory, we have previously introduced an "extended" version (XSAPT) that incorporates many-body polarization via self-consistent charge embedding. Here, we extend the XSAPT methodology to include nonadditive dispersion, using a modified form of the many-body dispersion (MBD) method of Tkatchenko and co-workers. Dispersion interactions beyond the pairwise atom-atom approximation improve total interaction energies even in small systems, and for large π-stacked complexes these corrections can amount to several kilocalories per mole. The XSAPT+MBD method introduced here achieves errors of ≲1 kcal/mol (as compared to high-level ab initio benchmarks) for the L7 data set of large dispersion-bound complexes and ≲4 kcal/mol (as compared to experiment) for the S30L data set of host-guest complexes. This is superior to the best contemporary density functional methods for noncovalent interactions, at comparable or lower cost. XSAPT+MBD represents a promising method for application to supramolecular assemblies, including protein-ligand binding.

Spin splittings from first-order symmetry-adapted perturbation theory without single-exchange approximation

The recently proposed spin-flip symmetry-adapted perturbation theory (SF-SAPT) first-order exchange energy [Patkowski et al., J. Chem. Phys. 148, 164110 (2018)] enables the standard open-shell SAPT approach to treat arbitrary spin states of the weakly interacting complex. Here, we further extend first-order SF-SAPT beyond the single-exchange approximation to a complete treatment of the exchanges of electrons between monomers. This new form of the exchange correction replaces the single-exchange approximation with a more moderate single-spin-flip approximation. The newly developed expressions are applied to a number of small test systems to elucidate the quality of both approximations. They are also applied to the singlet-triplet splittings in pancake bonded dimers. The accuracy of the single-exchange approximation deteriorates at short intermolecular separations, especially for systems with few electrons and for the high-spin state of the complex. In contrast, the single-spin-flip approximation is exact for interactions involving a doublet molecule and remains highly accurate for any number of unpaired electrons. Because the single-exchange approximation affects the high-spin and low-spin states of pancake bonded complexes evenly, the resulting splitting values are of similar accuracy to those produced by the formally more accurate single-spin-flip approximation.

Second-Order Dispersion Energy Based on Multireference Description of Monomers

We propose a method for calculating a second-order dispersion energy for weakly interacting multireference systems in arbitrary electronic states. It is based on response properties obtained from extended random phase approximation equations. The introduced formalism is general and requires only one- and two-particle reduced density matrices of monomers. We combine the new method with either generalized valence bond perfect pairing (GVB) or complete active space (CAS) self-consistent field description of the interacting systems. In addition to a general scheme, three approximations, leading to significant reduction of the computational cost, are developed by exploiting Dyall partitioning of the monomer Hamiltonians. For model multireference systems (H₂···H₂ and Be···Be) the method is accurate, unlike its single-reference-based counterpart. Neither GVB nor CAS description of single-reference monomers improves the dispersion energy with respect to the Hartree-Fock-based results.

Understanding non-covalent interactions in larger molecular complexes from first principles

Non-covalent interactions pervade all matter and play a fundamental role in layered materials, biological systems, and large molecular complexes. Despite this, our accumulated understanding of non-covalent interactions to date has been mainly developed in the tens-of-atoms molecular regime. This falls considerably short of the scales at which we would like to understand energy trends, structural properties, and temperature dependencies in materials where non-covalent interactions have an appreciable role. However, as more reference information is obtained beyond moderately sized molecular systems, our understanding is improving and we stand to gain pertinent insights by tackling more complex systems, such as supramolecular complexes, molecular crystals, and other soft materials. In addition, accurate reference information is needed to provide the drive for extending the predictive power of more efficient workhorse methods, such as density functional approximations that also approximate van der Waals dispersion interactions. In this perspective, we discuss the first-principles approaches that have been used to obtain reference interaction energies for beyond modestly sized molecular complexes. The methods include quantum Monte Carlo, symmetry-adapted perturbation theory, non-canonical coupled cluster theory, and approaches based on the random-phase approximation. By considering the approximations that underpin each method, the most accurate theoretical references for supramolecular complexes and molecular crystals to date are ascertained. With these, we also assess a handful of widely used exchange-correlation functionals in density functional theory. The discussion culminates in a framework for putting into perspective the accuracy of high-level wavefunction-based methods and identifying future challenges.