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

Assessing the domain-based local pair natural orbital (DLPNO) approximation for non-covalent interactions in sizable supramolecular complexes

The titular domain-based local pair natural orbital (DLPNO) approximation is the most widely used method for extending correlated wave function models to large molecular systems, yet its fidelity for intermolecular interaction energies in large supramolecular complexes has not been thoroughly vetted. Non-covalent interactions are sensitive to tails of the electron density and involve nonlocal dispersion that is discarded or approximated if the screening of pair natural orbitals (PNOs) is too aggressive. Meanwhile, the accuracy of the DLPNO approximation is known to deteriorate as molecular size increases. Here, we test the DLPNO approximation at the level of second-order Møller-Plesset perturbation theory (MP2) and coupled-cluster theory with singles, doubles, and perturbative triples [CCSD(T)] for a variety of large supramolecular complexes. DLPNO-MP2 interaction energies are within 3% of canonical values for small dimers with ≲10 heavy atoms, but for larger systems, the DLPNO approximation is often quite poor unless the results are extrapolated to the canonical limit where the threshold for discarding PNOs is taken to zero. Counterpoise correction proves to be essential in reducing errors with respect to canonical results. For a sequence of nanoscale graphene dimers up to (C96H24)2, extrapolated DLPNO-MP2 interaction energies agree with canonical values to within 1%, independent of system size, provided that the basis set does not contain diffuse functions; these cause the DLPNO approximation to behave erratically, such that results cannot be extrapolated in a meaningful way. DLPNO-CCSD(T) calculations are typically performed using looser PNO thresholds as compared to DLPNO-MP2, but this significantly impacts accuracy for large supramolecular complexes. Standard DLPNO-CCSD(T) settings afford errors of 2-6 kcal/mol for dimers involving coronene (C24H12) and circumcoronene (C54H18), even at the DLPNO-CCSD(T1) level.

How Much Electron Donation Is There In Transition Metal Complexes? A Computational Study

The "dative" covalent interactions between metals and ligands in coordination compounds, i.e., metal-to-ligand and ligand-to-metal donation, are manifestations of electron delocalization and subject to errors in approximate calculations. This work addresses the extent of dative bonding/donation in a series of closed-shell transition metal complexes. Several Kohn-Sham density functionals, representing different "rungs" of approximations, along with post-Hartree-Fock methods are assessed in comparison to CCSD(T). Two widely used nonhybrid and global hybrid density functionals (B3LYP, PBE0) tend to produce notably too strong donation. Global hybrids with elevated fractions of exact exchange (40 to 50%) and the range-separated exchange functional CAM-B3LYP tend to perform better for the description of the donation. The performance of a double-hybrid functional is found to be quite satisfactory, correcting errors seen in MP2 calculations. A fast approximate coupled-cluster model (DLPNO-CCSD) also gives a reasonable description of the donation, with a tendency to underestimate its extent.

Cluster-in-Molecule Approach with Explicitly Correlated Methods for Large Molecules

In this article, we present a series of explicitly correlated local correlation methods developed under the cluster-in-molecule (CIM) framework, including explicitly correlated second-order Møller-Plesset perturbation (MP2), coupled-cluster singles and doubles (CCSD), domain-based local pair natural orbital CCSD (DLPNO-CCSD), and DLPNO-CCSD with perturbative triples (DLPNO-CCSD(T)). In these methods, F12 correction is decomposed into contributions from each occupied local molecular orbital and then evaluated independently in a given cluster, which consists of a subset of localized orbitals. These newly developed methods allow F12 calculations of large molecules (up to 145 atoms for quasi-one-dimensional systems) on a single node. We use these methods to investigate the relative stability between extended and folded alkane C30H62, the relative stability of four secondary structures of a polyglycine Ace(Gly)10NH2, and the binding energies of two host-guest complexes. The results demonstrate that the combination of CIM with F12 methods is a promising way to investigate large molecules with small basis set errors.

Smooth particle mesh Ewald-integrated stochastic Lanczos many-body dispersion algorithm

We derive and implement an alternative formulation of the Stochastic Lanczos algorithm to be employed in connection with the Many-Body Dispersion model (MBD). Indeed, this formulation, which is only possible due to the Stochastic Lanczos' reliance on matrix-vector products, introduces generalized dipoles and fields. These key quantities allow for a state-of-the-art treatment of periodic boundary conditions via the O(Nlog(N)) Smooth Particle Mesh Ewald (SPME) approach which uses efficient fast Fourier transforms. This SPME-Lanczos algorithm drastically outperforms the standard replica method which is affected by a slow and conditionally convergence rate that limits an efficient and reliable inclusion of long-range periodic boundary conditions interactions in many-body dispersion modelling. The proposed algorithm inherits the embarrassingly parallelism of the original Stochastic Lanczos scheme, thus opening up for a fully converged and efficient periodic boundary conditions treatment of MBD approaches.

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.

Repartitioned Brillouin-Wigner perturbation theory with a size-consistent second-order correlation energy

Second-order Møller-Plesset perturbation theory (MP2) often breaks down catastrophically in small-gap systems, leaving much to be desired in its performance for myriad chemical applications such as noncovalent interactions, thermochemistry, and dative bonding in transition metal complexes. This divergence problem has reignited interest in Brillouin-Wigner perturbation theory (BWPT), which is regular at all orders but lacks size consistency and extensivity, severely limiting its application to chemistry. In this work, we propose an alternative partitioning of the Hamiltonian that leads to a regular BWPT perturbation series that, through the second order, is size-extensive, size-consistent (provided its Hartree-Fock reference is also), and orbital invariant. Our second-order size-consistent Brillouin-Wigner (BW-s2) approach can describe the exact dissociation limit of H2 in a minimal basis set, regardless of the spin polarization of the reference orbitals. More broadly, we find that BW-s2 offers improvements relative to MP2 for covalent bond breaking, noncovalent interaction energies, and metal/organic reaction energies, although rivaling coupled-cluster with single and double substitutions for thermochemical properties.

A Buckycatcher in Solution-A Computational Perspective

In this work, we study the buckycatcher (C₆₀H₂₈) in solution using quantum chemical models. We investigate the conformational equilibria in several media and the effects that molecules of solvent might have in interconversion barriers between the different conformers. These are studied in a hypothetical gas phase, in the dielectric of a solvent, as well as with hybrid solvation. In the latter case, due to a disruption of π-stacking interactions, the transition states are destabilized. We also evaluate the complexation of the buckycatcher with solvent-like molecules. In most cases studied, there should be no adducts formed because the enthalpy driving force cannot overcome entropic penalties.

Toward Pair Atomic Density Fitting for Correlation Energies with Benchmark Accuracy

Pair atomic density fitting (PADF) has been identified as a promising strategy to reduce the scaling with system size of quantum chemical methods for the calculation of the correlation energy like the direct random-phase approximation (RPA) or second-order Møller-Plesset perturbation theory (MP2). PADF can however introduce large errors in correlation energies as the two-electron interaction energy is not guaranteed to be bounded from below. This issue can be partially alleviated by using very large fit sets, but this comes at the price of reduced efficiency and having to deal with near-linear dependencies in the fit set. One posibility is to use global density fitting (DF), but in this work, we introduce an alternative methodology to overcome this problem that preserves the intrinsically favorable scaling of PADF. We first regularize the Fock matrix by projecting out parts of the basis set which gives rise to orbital products that are hard to describe by PADF. After having thus obtained a reliable self-consistent field solution, we then also apply this projector to the orbital coefficient matrix to improve the precision of PADF-MP2 and PADF-RPA. We systematically assess the accuracy of this new approach in a numerical atomic orbital framework using Slater type orbitals (STO) and correlation consistent Gaussian type basis sets up to quintuple-ζ quality for systems with more than 200 atoms. For the small and medium systems in the S66 database we show the maximum deviation of PADF-MP2 and PADF-RPA relative correlation energies to DF-MP2 and DF-RPA reference results to be 0.07 and 0.14 kcal/mol, respectively. When the new projector method is used, the errors only slightly increase for large molecules and also when moderately sized fit sets are used the resulting errors are well under control. Finally, we demonstrate the computational efficiency of our algorithm by calculating the interaction energies of large, non-covalently bound complexes with more than 1000 atoms and 20000 atomic orbitals at the RPA@PBE/CC-pVTZ level of theory.

Generalized Many-Body Dispersion Correction through Random-Phase Approximation for Chemically Accurate Density Functional Theory

We extend our recently proposed Deep Learning-aided many-body dispersion (DNN-MBD) model to quadrupole polarizability (Q) terms using a generalized Random Phase Approximation (RPA) formalism, thus enabling the inclusion of van der Waals contributions beyond dipole. The resulting DNN-MBDQ model only relies on ab initio-derived quantities as the introduced quadrupole polarizabilities are recursively retrieved from dipole ones, in turn modeled via the Tkatchenko-Scheffler method. A transferable and efficient deep-neuronal network (DNN) provides atom-in-molecule volumes, while a single range-separation parameter is used to couple the model to Density Functional Theory (DFT). Since it can be computed at a negligible cost, the DNN-MBDQ approach can be coupled with DFT functionals, such as PBE, PBE0, and B86bPBE (dispersionless). The DNN-MBQ-corrected functionals reach chemical accuracy while exhibiting lower errors compared to their dipole-only counterparts.

Molecular Understanding and Practical In Silico Catalyst Design in Computational Organocatalysis and Phase Transfer Catalysis-Challenges and Opportunities

Through the lens of organocatalysis and phase transfer catalysis, we will examine the key components to calculate or predict catalysis-performance metrics, such as turnover frequency and measurement of stereoselectivity, via computational chemistry. The state-of-the-art tools available to calculate potential energy and, consequently, free energy, together with their caveats, will be discussed via examples from the literature. Through various examples from organocatalysis and phase transfer catalysis, we will highlight the challenges related to the mechanism, transition state theory, and solvation involved in translating calculated barriers to the turnover frequency or a metric of stereoselectivity. Examples in the literature that validated their theoretical models will be showcased. Lastly, the relevance and opportunity afforded by machine learning will be discussed.

A Cost Effective Scheme for the Highly Accurate Description of Intermolecular Binding in Large Complexes

There has been a growing interest in quantitative predictions of the intermolecular binding energy of large complexes. One of the most important quantum chemical techniques capable of such predictions is the domain-based local pair natural orbital (DLPNO) scheme for the coupled cluster theory with singles, doubles, and iterative triples [CCSD(T)], whose results are extrapolated to the complete basis set (CBS) limit. Here, the DLPNO-based focal-point method is devised with the aim of obtaining CBS-extrapolated values that are very close to their canonical CCSD(T)/CBS counterparts, and thus may serve for routinely checking a performance of less expensive computational methods, for example, those based on the density-functional theory (DFT). The efficacy of this method is demonstrated for several sets of noncovalent complexes with varying amounts of the electrostatics, induction, and dispersion contributions to binding (as revealed by accurate DFT-based symmetry-adapted perturbation theory (SAPT) calculations). It is shown that when applied to dimeric models of poly(3-hydroxybutyrate) chains in its two polymorphic forms, the DLPNO-CCSD(T) and DFT-SAPT computational schemes agree to within about 2 kJ/mol of an absolute value of the interaction energy. These computational schemes thus should be useful for a reliable description of factors leading to the enthalpic stabilization of extended systems.

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.

Classifying soft self-assembled materials via unsupervised machine learning of defects

Unlike molecular crystals, soft self-assembled fibers, micelles, vesicles, etc., exhibit a certain order in the arrangement of their constitutive monomers but also high structural dynamicity and variability. Defects and disordered local domains that continuously form-and-repair in their structures impart to such materials unique adaptive and dynamical properties, which make them, e.g., capable to communicate with each other. However, objective criteria to compare such complex dynamical features and to classify soft supramolecular materials are non-trivial to attain. Here we show a data-driven workflow allowing us to achieve this goal. Building on unsupervised clustering of Smooth Overlap of Atomic Position (SOAP) data obtained from equilibrium molecular dynamics simulations, we can compare a variety of soft supramolecular assemblies via a robust SOAP metric. This provides us with a data-driven "defectometer" to classify different types of supramolecular materials based on the structural dynamics of the ordered/disordered local molecular environments that statistically emerge within them.

How to Catch the Ball: Fullerene Binding to the Corannulene Pincer

The corannulene pincer (also known in the literature as the buckycatcher) is a fascinating system that may encapsulate, among other molecules, the C60 and C70 fullerenes. These complexes are held together by strong π-stacking interactions. Although these are quantum mechanical effects, their description by quantum chemical methods has proved very hard. We used three semi-empirical methods, PM6-D3H4X, PM6-D3H+ and GFN2-xTB, to model the interactions. Binding to fullerenes was extended to all open conformations of the buckycatcher, and with the proper choice of solvation model and partition functions, we obtained Gibbs free energies of binding that deviated by 1.0-1.5 kcal/mol from the experimental data. Adding three-body dispersion to PM6-D3H+ led to even better agreement. These results agree better with the experimental data than calculations using higher-level methods at a significantly lower fraction of the computational cost. Furthermore, the formation of adducts with C60 was studied using dynamical simulations, which helped to build a more complete picture of the behavior of the corannulene pincer with fullerenes. We also investigated the use of exchange-binding models to recover more information on this system in solution. Though the final Gibbs free energies in solution were worsened, gas-phase enthalpies and entropies better mirrored the experimental data.

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.

Role of van der Waals forces in the metal-insulator transition of transition metal oxides

Transition metal oxides (TMOs) exhibit great potential in technological applications due to their ability to undergo a rapid metal-insulator transition (MIT). However, the phase stability of TMOs, which models the on/off voltages of electronic devices, remains controversial due to the incomplete knowledge of the determinants of its stability. Herein, we study the effect of van der Waals (vdW) interactions on the phase stability of TMOs by employing the pairwise and screened vdW methods. Our calculations manifest that the vdW interactions are crucial to the TMOs' phase stability and tend to stabilize the insulating phase. Furthermore, the long-range electrodynamic screening interactions correct the TMOs' phase stability by revising the vdW term.

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.

Addressing the System-Size Dependence of the Local Approximation Error in Coupled-Cluster Calculations

Over the last two decades, the local approximation has been successfully used to extend the range of applicability of the "gold standard" singles and doubles coupled-cluster method with perturbative triples CCSD(T) to systems with hundreds of atoms. The local approximation error grows in absolute value with the increasing system size, i.e., by increasing the number of electron pairs in the system. In this study, we demonstrate that the recently introduced two-point extrapolation scheme for approaching the complete pair natural orbital (PNOs) space limit in domain-based pair natural orbital CCSD(T) calculations drastically reduces the dependence of the error on the system size, thus opening up unprecedented opportunities for the calculation of benchmark quality relative energies for large systems.

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.

Characterization of cisplatin/membrane interactions by QM/MM energy decomposition analysis

We extend for the first time a quantum mechanical energy decomposition analysis scheme based on deformation electron densities to a hybrid electrostatic embedding quantum mechanics/molecular mechanics framework. The implemented approach is applied to characterize the interactions between cisplatin and a dioleyl-phosphatidylcholine membrane, which play a key role in the permeation mechanism of the drug inside the cells. The interaction energy decomposition into electrostatic, induction, dispersion and Pauli repulsion contributions is performed for ensembles of geometries to account for conformational sampling. It is evidenced that the electrostatic and repulsive components are predominant in both polar and non-polar regions of the bilayer.

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.

Interactions between large molecules pose a puzzle for reference quantum mechanical methods

Quantum-mechanical methods are used for understanding molecular interactions throughout the natural sciences. Quantum diffusion Monte Carlo (DMC) and coupled cluster with single, double, and perturbative triple excitations [CCSD(T)] are state-of-the-art trusted wavefunction methods that have been shown to yield accurate interaction energies for small organic molecules. These methods provide valuable reference information for widely-used semi-empirical and machine learning potentials, especially where experimental information is scarce. However, agreement for systems beyond small molecules is a crucial remaining milestone for cementing the benchmark accuracy of these methods. We show that CCSD(T) and DMC interaction energies are not consistent for a set of polarizable supramolecules. Whilst there is agreement for some of the complexes, in a few key systems disagreements of up to 8 kcal mol^-1 remain. These findings thus indicate that more caution is required when aiming at reproducible non-covalent interactions between extended molecules.

Intermolecular Charge-Transfer-Induced Strong Optical Emission from Herringbone H-Aggregates

Luminescence in molecular aggregates can be quenched either by intermolecular charge transfer or by forming a dipole-forbidden lower Frenkel exciton in H-aggregate. Taking intermolecular charge transfer and excitonic coupling into a three-state model through localized diabatization, we demonstrate that the low-lying intermolecular charge-transfer state could couple with the upper bright Frenkel exciton to form dipole-allowed S₁ that lies below the dark state, which accounts for the recent experimentally discovered strong luminescence in organic light-emitting transistors (OLETs) system with DPA and dNaAnt herringbone aggregates. The condition of forming such bright state is that the electron and hole transfer integrals, t_e and t_h, are of the same sign, and should be notably larger than the excitonic coupling (J), that is , t_e × t_h > 2J². This theoretical finding not only rationalizes recent experiments but unravels an exciting scenario where strong luminescence and high charge mobilities become compatible, which is a preferable condition for both OLETs and electrically pumped lasing.

Accuracy of DLPNO-CCSD(T): Effect of Basis Set and System Size

The DLPNO-CCSD(T) method is designed to study large molecular systems at significantly reduced cost relative to its canonical counterpart. However, the error in this approach is also size-extensive and relies on cancellation of errors for the calculation of relative energies. This work provides a direct comparison of canonical CCSD(T) and TightPNO DLPNO-CCSD(T) calculations of reaction energies and barriers of a broad range of chemical reactions. The dataset includes acidities, anion binding affinities, enolization, Diels-Alder, nucleophilic substitution, and atom transfer reactions and complements existing theoretical datasets in terms of system size as well as new reaction types (e.g., anion binding affinities and chlorine atom transfer reactions). The performance of DLPNO-CCSD(T) was further examined with respect to systematic variation of basis set and system size and amounts of nonbonded interaction present in the system. The errors in the DLPNO-CCSD(T) were found to be relatively insensitive to the choice of basis set for small systems but increase monotonically with system size. Additionally, calculations of barriers appear to be more challenging than reaction energies with errors exceeding 5 kJ mol^-1 for many Diels-Alder reactions. Further tests on three realistic organic reactions reveal the impact of the DLPNO approximation in calculating absolute and relative barriers that are important for predictions such as stereoselectivity.

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.

Molecular Design Strategy for Simultaneously Strong Luminescence and High Mobility: Multichannel CH-π Interaction

It is a big challenge to achieve high-performance organic semiconductor materials integrating both high luminescence efficiency and carrier mobility, because they are commonly regarded as a pair of contradiction. Here, combining a tight-binding model and density functional theory/time-dependent density functional theory, we propose a theoretical protocol to characterize the luminescence efficiency via an excitonic effective mass and charge transport ability via charge effective mass at the same level. Applying this protocol to a series of organic semiconductor materials, we find that the multichannel CH-π interaction can induce a heavy excitonic effective mass and light charge effective mass, which effectively balance the light-emitting efficiency and carrier mobility. Thus, a practical molecular design strategy is figured out to exploit novel organic semiconductor materials with strong luminescence and fast carrier transport simultaneously.

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.

Computational screen-out strategy for electrically pumped organic laser materials

Electrically pumped organic lasing is one of the most challenging issues in organic optoelectronics. We present a systematic theoretical investigation to screen out electrical pumping lasing molecules over a wide range of organic materials. With the electronic structure information obtained from time-dependent density functional theory, we calculate multiple photophysical parameters of a set of optical pumping organic laser molecules in our self-developed molecular material property prediction package (MOMAP) to judge whether the electrically pumped lasing conditions can be satisfied, namely, to avoid reabsorption from excitons and/or polarons, and the accumulation of triplet excitons. In addition, a large oscillator strength of S₁ and weak intermolecular π–π interaction are preferred. With these criteria, we are able to conclude that BP3T, BSBCz, and CzPVSBF compounds are promising candidates for electrically pumped lasing, and the proposed computational strategy could serve as a general protocol for molecular design of organic lasing materials.

Though the goal of current organic solid-state laser research remains the realization of electrically pumped lasing, identifying organic semiconductors with ideal properties remains a challenge. Here, the authors report a computational strategy for screening electrical pumping lasing molecules.

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.

Toward a less costly but accurate calculation of the CCSD(T)/CBS noncovalent interaction energy

The popular method of calculating the noncovalent interaction energies at the coupled-cluster single-, double-, and perturbative triple-excitations [CCSD(T)] theory level in the complete basis set (CBS) limit was to add a CCSD(T) correction term to the CBS second-order Møller-Plesset perturbation theory (MP2). The CCSD(T) correction term is the difference between the CCSD(T) and MP2 interaction energies evaluated in a medium basis set. However, the CCSD(T) calculations with the medium basis sets are still very expensive for systems with more than 30 atoms. Comparatively, the domain-based local pair natural orbital coupled-cluster method [DLPNO-CCSD(T)] can be applied to large systems with over 1,000 atoms. Considering both the computational accuracy and efficiency, in this work, we propose a new scheme to calculate the CCSD(T)/CBS interaction energies. In this scheme, the MP2/CBS term keeps intact and the CCSD(T) correction term is replaced by a DLPNO-CCSD(T) correction term which is the difference between the DLPNO-CCSD(T) and DLPNO-MP2 interaction energies evaluated in a medium basis set. The interaction energies of the noncovalent systems in the S22, HSG, HBC6, NBC10, and S66 databases were recalculated employing this new scheme. The consistent and tight settings of the truncation parameters for DLPNO-CCSD(T) and DLPNO-MP2 in this noncanonical CCSD(T)/CBS calculations lead to the maximum absolute deviation and root-mean-square deviation from the canonical CCSD(T)/CBS interaction energies of less than or equal to 0.28 kcal/mol and 0.09 kcal/mol, respectively. The high accuracy and low cost of this new computational scheme make it an excellent candidate for the study of large noncovalent systems.

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.

Theory and practice of modeling van der Waals interactions in electronic-structure calculations

The accurate description of long-range electron correlation, most prominently including van der Waals (vdW) dispersion interactions, represents a particularly challenging task in the modeling of molecules and materials. vdW forces arise from the interaction of quantum-mechanical fluctuations in the electronic charge density. Within (semi-)local density functional approximations or Hartree-Fock theory such interactions are neglected altogether. Non-covalent vdW interactions, however, are ubiquitous in nature and play a key role for the understanding and accurate description of the stability, dynamics, structure, and response properties in a plethora of systems. During the last decade, many promising methods have been developed for modeling vdW interactions in electronic-structure calculations. These methods include vdW-inclusive Density Functional Theory and correlated post-Hartree-Fock approaches. Here, we focus on the methods within the framework of Density Functional Theory, including non-local van der Waals density functionals, interatomic dispersion models within many-body and pairwise formulation, and random phase approximation-based approaches. This review aims to guide the reader through the theoretical foundations of these methods in a tutorial-style manner and, in particular, highlight practical aspects such as the applicability and the advantages and shortcomings of current vdW-inclusive approaches. In addition, we give an overview of complementary experimental approaches, and discuss tools for the qualitative understanding of non-covalent interactions as well as energy decomposition techniques. Besides representing a reference for the current state-of-the-art, this work is thus also designed as a concise and detailed introduction to vdW-inclusive electronic structure calculations for a general and broad audience.