Basis set convergence of the coupled-cluster correction, δMP2CCSD(T): Best practices for benchmarking non-covalent interactions and the attendant revision of the S22, NBC10, HBC6, and HSG databases

Optimization of damping function parameters for -D3 and -D4 dispersion models for Hartree-Fock based symmetry-adapted perturbation theory

Symmetry-adapted perturbation theory (SAPT) directly computes intermolecular interaction energy in terms of electrostatics, exchange-repulsion, induction/polarization, and London dispersion components. In SAPT based on Hartree-Fock ("SAPT0") or based on density functional theory, the most time-consuming step is the computation of the dispersion terms. Previous work has explored the replacement of these expensive dispersion terms with simple damped asymptotic models. We recently examined [Schriber et al. J. Chem. Phys. 154, 234107 (2021)] the accuracy of SAPT0 when replacing its dispersion term with Grimme's popular -D3 correction, reducing the computational cost scaling from O(N5) to O(N3). That work optimized damping function parameters for SAPT0-D3/jun-cc-pVDZ using estimates of the coupled-cluster complete basis set limit [CCSD(T)/CBS] on a 8299 dimer dataset. Here, we explore the accuracy of SAPT0-D3 with additional basis sets, along with an analogous model using -D4. Damping parameters are rather insensitive to basis sets, and the resulting SAPT0-D models are more accurate on average for total interaction energies than SAPT0. Our results are surprising in several respects: (1) improvement of -D4 over -D3 is negligible for these systems, even charged systems where -D4 should, in principle, be more accurate; (2) addition of Axilrod-Teller-Muto terms for three-body dispersion does not improve error statistics for this test set; and (3) SAPT0-D is even more accurate on average for total interaction energies than the much more computationally costly density functional theory based SAPT [SAPT(DFT)] in an aug-cc-pVDZ basis. However, SAPT0 and SAPT0-D3/D4 interaction energies benefit from significant error cancellation between exchange and dispersion terms.

Kohn-Sham fragment energy decomposition analysis

We introduce the concept of Kohn-Sham fragment localized molecular orbitals (KS-FLMOs), which are Kohn-Sham molecular orbitals (MOs) localized in specific fragments constituting a generic molecular system. In detail, we minimize the local electronic energies of various fragments, while maximizing the repulsion between them, resulting in the effective localization of the MOs. We use the developed KS-FLMOs to propose a novel energy decomposition analysis, which we name Kohn-Sham fragment energy decomposition analysis, which allows for rationalizing the main non-covalent interactions occurring in interacting systems both in vacuo and in solution, providing physical insights into non-covalent interactions. The method is validated against state-of-the-art energy decomposition analysis techniques and with high-level calculations.

Accurate and efficient open-source implementation of domain-based local pair natural orbital (DLPNO) coupled-cluster theory using a t1-transformed Hamiltonian

We present an efficient, open-source formulation for coupled-cluster theory through perturbative triples with domain-based local pair natural orbitals [DLPNO-CCSD(T)]. Similar to the implementation of the DLPNO-CCSD(T) method found in the ORCA package, the most expensive integral generation and contraction steps associated with the CCSD(T) method are linear-scaling. In this work, we show that the t1-transformed Hamiltonian allows for a less complex algorithm when evaluating the local CCSD(T) energy without compromising efficiency or accuracy. Our algorithm yields sub-kJ mol-1 deviations for relative energies when compared with canonical CCSD(T), with typical errors being on the order of 0.1 kcal mol-1, using our TightPNO parameters. We extensively tested and optimized our algorithm and parameters for non-covalent interactions, which have been the most difficult interaction to model for orbital (PNO)-based methods historically. To highlight the capabilities of our code, we tested it on large water clusters, as well as insulin (787 atoms).

MP2-Based Composite Extrapolation Schemes Can Predict Core-Ionization Energies for First-Row Elements with Coupled-Cluster Level Accuracy

X-ray photoelectron spectroscopy (XPS) measures core-electron binding energies (CEBEs) to reveal element-specific insights into the chemical environment and bonding. Accurate theoretical CEBE prediction aids XPS interpretation but requires proper modeling of orbital relaxation and electron correlation upon core-ionization. This work systematically investigates basis set selection for extrapolation to the complete basis set limit of CEBEs from ΔMP2 and ΔCC energies across 94 K-edges in diverse organic molecules. We demonstrate that an alternative composite scheme using ΔMP2 in a large basis corrected by ΔCC-ΔMP2 difference in a small basis can quantitatively recover optimally extrapolated ΔCC CEBEs within 0.02 eV. Unlike ΔCC, MP2 calculations do not suffer from convergence issues and are computationally cheaper, and thus, the composite ΔMP2/ΔCC scheme balances accuracy and cost, overcoming limitations of solely using either method. We conclude by providing a comprehensive analysis of the choice of small and large basis sets for the composite schemes and provide practical recommendations for highly accurate (within 0.10-0.15 eV MAE) ab initio prediction of XPS data.

Benchmarking Quantum Mechanical Levels of Theory for Valence Parametrization in Force Fields

A wide range of density functional methods and basis sets are available to derive the electronic structure and properties of molecules. Quantum mechanical calculations are too computationally intensive for routine simulation of molecules in the condensed phase, prompting the development of computationally efficient force fields based on quantum mechanical data. Parametrizing general force fields, which cover a vast chemical space, necessitates the generation of sizable quantum mechanical data sets with optimized geometries and torsion scans. To achieve this efficiently, choosing a quantum mechanical method that balances computational cost and accuracy is crucial. In this study, we seek to assess the accuracy of quantum mechanical theory for specific properties such as conformer energies and torsion energetics. To comprehensively evaluate various methods, we focus on a representative set of 59 diverse small molecules, comparing approximately 25 combinations of functional and basis sets against the reference level coupled cluster calculations at the complete basis set limit.

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.

Beryllium as a Base: Complexes of Be(CO)3 with HX (X=F, Cl, Br, CN, NC, CCH, OH)

Beryllium chemistry is typically governed by its electron deficient character, but in some compounds it can act as a base. In order to understand better the unusual basicity of Be, we have systematically explored the complexes of one such compound, Be(CO)3, towards several hydrogen bond donors HX (X=F, Cl, Br, CN, NC, CCH, OH). For all complexes we find three different minima, two hydrogen bonded minima (to the Be or O atoms), and one weak beryllium bonded minimum. Further characterization of the interactions using a topological analysis of the electron density and Symmetry Adapted Perturbation Theory (SAPT) provide insight into the nature of these interactions. Overall these results highlight the capability of certain beryllium compounds to act as either a weak Lewis acid or, unconventionally, a Lewis base whose basicity towards hydrogen bonding is comparable to that of π systems.

Development of a multi-step screening procedure for redox active molecules in organic radical polymer anodes and as redox flow anolytes

Benzo[d]-X-zolyl-pyridinyl (XO, S, NH) radicals represent a promising class of redox-active molecules for organic batteries. We present a multistep screening procedure to identify the most promising radical candidates. Experimental investigations and highly correlated wave function-based calculations are performed to determine benchmark redox potentials. Based on these, the accuracies of different methods (semi-empirical, density functional theory, wave function-based), solvent models, dispersion corrections, and basis sets are evaluated. The developed screening procedure consists of three steps: First, a conformer search is performed with CREST. The molecules are selected based on the redox potentials calculated using GFN2-xTB. Second, HOMO energies calculated with reparametrized B3LYP-D3(BJ) and the def2-SVP basis set are used as selection criteria. The final molecules are selected based on the redox potentials calculated from Gibbs energies using BP86-D3(BJ)/def2-TZVP. With this multistep screening approach, promising molecules can be suggested for synthesis, and structure-property relationships can be derived.

The p-block challenge: assessing quantum chemistry methods for inorganic heterocycle dimerizations

The elements of the p-block of the periodic table are of high interest in various chemical and technical applications like frustrated Lewis-pairs (FLP) or opto-electronics. However, high-quality benchmark data to assess approximate density functional theory (DFT) for their theoretical description are sparse. In this work, we present a benchmark set of 604 dimerization energies of 302 "inorganic benzenes" composed of all non-carbon p-block elements of main groups III to VI up to polonium. This so-called IHD302 test set comprises two classes of structures formed by covalent bonding and by weaker donor-acceptor (WDA) interactions, respectively. Generating reliable reference data with ab initio methods is challenging due to large electron correlation contributions, core-valence correlation effects, and especially the slow basis set convergence. To compute reference values for these dimerization reactions, after thorough testing, we applied a computational protocol using state-of-the-art explicitly correlated local coupled cluster theory termed PNO-LCCSD(T)-F12/cc-VTZ-PP-F12(corr.). It includes a basis set correction at the PNO-LMP2-F12/aug-cc-pwCVTZ level. Based on these reference data, we assess 26 DFT methods in combination with three different dispersion corrections and the def2-QZVPP basis set, five composite DFT approaches, and five semi-empirical quantum mechanical methods. For the covalent dimerizations, the r2SCAN-D4 meta-GGA, the r2SCAN0-D4 and ωB97M-V hybrids, and the revDSD-PBEP86-D4 double-hybrid functional are found to be the best-performing methods among the evaluated functionals of the respective class. However, since def2 basis sets for the 4th period are not associated to relativistic pseudo-potentials, we obtained significant errors in the covalent dimerization energies (up to 6 kcal mol-1) for molecules containing p-block elements of the 4th period. Significant improvements were achieved for systems containing 4th row elements by using ECP10MDF pseudopotentials along with re-contracted aug-cc-pVQZ-PP-KS basis sets introduced in this work with the contraction coefficients taken from atomic DFT (PBE0) calculations. Overall, the IHD302 set represents a challenge to contemporary quantum chemical methods. This is due to a large number of spatially close p-element bonds which are underrepresented in other benchmark sets, and the partial covalent bonding character for the WDA interactions. The IHD302 set may be helpful to develop more robust and transferable approximate quantum chemical methods in the future.

Quantum Tunneling Instability in Pericyclic Reactions

Several cycloreversion reactions of the retro-Diels-Alder type were computationally assessed to understand their quantum tunneling (QT) reactivity. N2, CO, and other leaving groups were considered based on their strong exothermicity, as it reduces their thermodynamic and kinetic stabilities. Our results indicate that for many of these reactions, it is essential to take into account their QT decomposition rate, which can massively weaken their molecular stability and shorten their half-lives even at deep cryogenic temperatures. In practical terms, this indicates that many supposedly stable molecules will actually be unsynthesizable or unisolable, and therefore trying to prepare or detect them would be a futile attempt. In addition, we discuss the importance of tunneling to correctly understand the enthalpy of activation and the collective atomic effect on the tunneling kinetic isotope effects to test if third-row atoms can tunnel in a chemical reaction. This project raises the question of the importance of in silico chemistry to guide in vitro chemistry, especially in cases where the latter cannot solve its own uncertainties.

Approaching the Complete Basis Set Limit for Spin-State Energetics of Mononuclear First-Row Transition Metal Complexes

Convergence to the complete basis set (CBS) limit is analyzed for the problem of spin-state energetics in mononuclear first-row transition metal (TM) complexes by taking under scrutiny a benchmark set of 18 energy differences between spin states for 13 chemically diverse TM complexes. The performance of conventional CCSD(T) and explicitly correlated CCSD(T)-F12a/b calculations in approaching the CCSD(T)/CBS limits is systematically studied. An economic computational protocol is developed based on the CCSD-F12a approximation and (here proposed) modified scaling of the perturbative triples term (T#). This computational protocol recovers the relative spin-state energetics of the benchmark set in excellent agreement with the reference CCSD(T)/CBS limits (mean absolute deviation of 0.4, mean signed deviation of 0.2, and maximum deviation of 0.8 kcal/mol) and enables performing canonical CCSD(T) calculations for mononuclear TM complexes sized up to ca. 50 atoms, which is illustrated by application to heme-related metalloporphyrins. Furthermore, a good transferability of the basis set incompleteness error (BSIE) is demonstrated for spin-state energetics computed using CCSD(T) and other wave function methods (MP2, CASPT2, CASPT2/CC, NEVPT2, and MRCI + Q), which justifies efficient focal-point approximations and simplifies the construction of multimethod benchmark studies.

A Machine Learning Force Field for Bio-Macromolecular Modeling Based on Quantum Chemistry-Calculated Interaction Energy Datasets

Accurate energy data from noncovalent interactions are essential for constructing force fields for molecular dynamics simulations of bio-macromolecular systems. There are two important practical issues in the construction of a reliable force field with the hope of balancing the desired chemical accuracy and working efficiency. One is to determine a suitable quantum chemistry level of theory for calculating interaction energies. The other is to use a suitable continuous energy function to model the quantum chemical energy data. For the first issue, we have recently calculated the intermolecular interaction energies using the SAPT0 level of theory, and we have systematically organized these energies into the ab initio SOFG-31 (homodimer) and SOFG-31-heterodimer datasets. In this work, we re-calculate these interaction energies by using the more advanced SAPT2 level of theory with a wider series of basis sets. Our purpose is to determine the SAPT level of theory proper for interaction energies with respect to the CCSD(T)/CBS benchmark chemical accuracy. Next, to utilize these energy datasets, we employ one of the well-developed machine learning techniques, called the CLIFF scheme, to construct a general-purpose force field for biomolecular dynamics simulations. Here we use the SOFG-31 dataset and the SOFG-31-heterodimer dataset as the training and test sets, respectively. Our results demonstrate that using the CLIFF scheme can reproduce a diverse range of dimeric interaction energy patterns with only a small training set. The overall errors for each SAPT energy component, as well as the SAPT total energy, are all well below the desired chemical accuracy of ~1 kcal/mol.

Reliable Dimerization Energies for Modeling of Supramolecular Junctions

Accurate estimates of intermolecular interaction energy, ΔE, are crucial for modeling the properties of organic electronic materials and many other systems. For a diverse set of 50 dimers comprising up to 50 atoms (Set50-50, with 7 of its members being models of single-stacking junctions), benchmark ΔE data were compiled. They were obtained by the focal-point strategy, which involves computations using the canonical variant of the coupled cluster theory with singles, doubles, and perturbative triples [CCSD(T)] performed while applying a large basis set, along with extrapolations of the respective energy components to the complete basis set (CBS) limit. The resulting ΔE data were used to gauge the performance for the Set50-50 of several density-functional theory (DFT)-based approaches, and of one of the localized variants of the CCSD(T) method. This evaluation revealed that (1) the proposed "silver standard" approach, which employs the localized CCSD(T) method and CBS extrapolations, can be expected to provide accuracy better than two kJ/mol for absolute values of ΔE, and (2) from among the DFT techniques, computationally by far the cheapest approach (termed "ωB97X-3c/vDZP" by its authors) performed remarkably well. These findings are directly applicable in cost-effective yet reliable searches of the potential energy surfaces of noncovalent complexes.

Intermolecular Non-Bonded Interactions from Machine Learning Datasets

Accurate determination of intermolecular non-covalent-bonded or non-bonded interactions is the key to potentially useful molecular dynamics simulations of polymer systems. However, it is challenging to balance both the accuracy and computational cost in force field modelling. One of the main difficulties is properly representing the calculated energy data as a continuous force function. In this paper, we employ well-developed machine learning techniques to construct a general purpose intermolecular non-bonded interaction force field for organic polymers. The original ab initio dataset SOFG-31 was calculated by us and has been well documented, and here we use it as our training set. The CLIFF kernel type machine learning scheme is used for predicting the interaction energies of heterodimers selected from the SOFG-31 dataset. Our test results show that the overall errors are well below the chemical accuracy of about 1 kcal/mol, thus demonstrating the promising feasibility of machine learning techniques in force field modelling.

Optimizing the regularization in size-consistent second-order Brillouin-Wigner perturbation theory

Despite its simplicity and relatively low computational cost, second-order Møller-Plesset perturbation theory (MP2) is well-known to overbind noncovalent interactions between polarizable monomers and some organometallic bonds. In such situations, the pairwise-additive correlation energy expression in MP2 is inadequate. Although energy-gap dependent amplitude regularization can substantially improve the accuracy of conventional MP2 in these regimes, the same regularization parameter worsens the accuracy for small molecule thermochemistry and density-dependent properties. Recently, we proposed a repartitioning of Brillouin-Wigner perturbation theory that is size-consistent to second order (BW-s2), and a free parameter (α) was set to recover the exact dissociation limit of H2 in a minimal basis set. Alternatively α can be viewed as a regularization parameter, where each value of α represents a valid variant of BW-s2, which we denote as BW-s2(α). In this work, we semi-empirically optimize α for noncovalent interactions, thermochemistry, alkane conformational energies, electronic response properties, and transition metal datasets, leading to improvements in accuracy relative to the ab initio parameterization of BW-s2 and MP2. We demonstrate that the optimal α parameter (α = 4) is more transferable across chemical problems than energy-gap-dependent regularization parameters. This is attributable to the fact that the BW-s2(α) regularization strength depends on all of the information encoded in the t amplitudes rather than just orbital energy differences. While the computational scaling of BW-s2(α) is iterative O(N5), this effective and transferable approach to amplitude regularization is a promising route to incorporate higher-order correlation effects at second-order cost.

Corrected density functional theory and the random phase approximation: Improved accuracy at little extra cost

We recently introduced an efficient methodology to perform density-corrected Hartree-Fock density functional theory [DC(HF)-DFT] calculations and an extension to it we called "corrected" HF DFT [C(HF)-DFT] [Graf and Thom, J. Chem. Theory Comput. 19 5427-5438 (2023)]. In this work, we take a further step and combine C(HF)-DFT, augmented with a straightforward orbital energy correction, with the random phase approximation (RPA). We refer to the resulting methodology as corrected HF RPA [C(HF)-RPA]. We evaluate the proposed methodology across various RPA methods: direct RPA (dRPA), RPA with an approximate exchange kernel, and RPA with second-order screened exchange. C(HF)-dRPA demonstrates very promising performance; for RPA with exchange methods, on the other hand, we often find over-corrections.

Competition between Solvent···Solvent and Solvent···Solute Interactions in the Microhydration of the Tetrafluoroborate Anion, BF4-(H2O)n=1,2,3,4

This study systematically examines the interactions of the tetrafluoroborate anion (BF4-) with up to four water molecules (BF4-(H2O)n=1,2,3,4). Full geometry optimizations and subsequent harmonic vibrational frequency computations are performed using a variety of density functional theory (DFT) methods (B3LYP, B3LYP-D3BJ, and M06-2X) and the MP2 ab initio method with a triple-ζ correlation consistent basis set augmented with diffuse functions on all non-hydrogen atoms (cc-pVTZ for H and aug-cc-pVTZ for B, O, and F; denoted as haTZ). Optimized structures and harmonic vibrational frequencies were also obtained with the CCSD(T) ab initio method and the haTZ basis set for the mono- and dihydrate (n = 1, 2) structures. The 2-body:Many-body (2b:Mb) technique, in which CCSD(T) computations capture the 1- and 2-body contributions to the interactions and MP2 computations recover all higher-order contributions, was used to extend these demanding computations to the tri- and tetrahydrate (n = 3, 4) systems. Four, five, and eight new stationary points have been identified for the di-, tri-, and tetrahydrate systems, respectively. The global minimum of the monohydrate adopts a symmetric double ionic hydrogen bond motif with C2v symmetry and an electronic dissociation energy of 13.17 kcal mol-1 at the CCSD(T)/haTZ level of theory. This strong solvent···solute interaction, however, competes with solute···solute interactions in the lowest-energy BF4-(H2O)n=2,3,4 minima that are not seen in the other di-, tri-, or tetrahydrate minima. The latter interactions help increase the 2b:Mb dissociation energies to more than 26, 41, and 51 kcal mol-1 for n = 2, 3, and 4, respectively. Structures that form hydrogen bonds between the solvating water molecules also exhibit the largest shifts in the harmonic OH stretching frequencies for the waters of hydration. These shifts can exceed -280 cm-1 relative to an isolated H2O molecule at the 2b:Mb/haTZ level of theory.

Approximating large-basis coupled-cluster theory vibrational frequencies using focal-point approximations

The focal-point approximation can be used to estimate a high-accuracy, slow quantum chemistry computation by combining several lower-accuracy, faster computations. We examine the performance of focal-point methods by combining second-order Møller-Plesset perturbation theory (MP2) with coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] for the calculation of harmonic frequencies and that of fundamental frequencies using second-order vibrational perturbation theory (VPT2). In contrast to standard CCSD(T), the focal-point CCSD(T) method approaches the complete basis set (CBS) limit with only triple-ζ basis sets for the coupled-cluster portion of the computation. The predicted harmonic and fundamental frequencies were compared with the experimental values for a set of 20 molecules containing up to six atoms. The focal-point method combining CCSD(T)/aug-cc-pV(T + d)Z with CBS-extrapolated MP2 has mean absolute errors vs experiment of only 7.3 cm-1 for the fundamental frequencies, which are essentially the same as the mean absolute error for CCSD(T) extrapolated to the CBS limit using the aug-cc-pV(Q + d)Z and aug-cc-pV(5 + d)Z basis sets. However, for H2O, the focal-point procedure requires only 3% of the computation time as the extrapolated CCSD(T) result, and the cost savings will grow for larger molecules.

Simple and Efficient Route toward Improved Energetics within the Framework of Density-Corrected Density Functional Theory

The crucial step in density-corrected Hartree-Fock density functional theory (DC(HF)-DFT) is to decide whether the density produced by the density functional for a specific calculation is erroneous and, hence, should be replaced by, in this case, the HF density. We introduce an indicator, based on the difference in noninteracting kinetic energies between DFT and HF calculations, to determine when the HF density is the better option. Our kinetic energy indicator directly compares the self-consistent density of the analyzed functional with the HF density, is size-intensive, reliable, and most importantly highly efficient. Moreover, we present a procedure that makes best use of the computed quantities necessary for DC(HF)-DFT by additionally evaluating a related hybrid functional and, in that way, not only "corrects" the density but also the functional itself; we call that procedure corrected Hartree-Fock density functional theory (C(HF)-DFT).

Quantifying the Intrinsic Strength of C-H⋯O Intermolecular Interactions

It has been recognized that the C-H⋯O structural motif can be present in destabilizing as well as highly stabilizing intermolecular environments. Thus, it should be of interest to describe the strength of the C-H⋯O hydrogen bond for constant structural factors so that this intrinsic strength can be quantified and compared to other types of interactions. This description is provided here for C_2h-symmetric dimers of acrylic acid by means of the calculations that employ the coupled-cluster theory with singles, doubles, and perturbative triples [CCSD(T)] together with an extrapolation to the complete basis set (CBS) limit. Dimers featuring the C-H⋯O and O-H⋯O hydrogens bonds are carefully investigated in a wide range of intermolecular separations by the CCSD(T)/CBS approach, and also by the symmetry-adapted perturbation theory (SAPT) method, which is based on the density-functional theory (DFT) treatment of monomers. While the nature of these two types of hydrogen bonding is very similar according to the SAPT-DFT/CBS calculations and on the basis of a comparison of the intermolecular potential curves, the intrinsic strength of the C-H⋯O interaction is found to be about a quarter of its O-H⋯O counterpart that is less than one might anticipate.

Noncovalent Interactions in Hydrated Nitrosonium Ion Clusters Mediated by Hydrogen-Bonded Water Networks

As important species in the D region of the ionosphere, hydrated nitrosonium ion clusters [NO⁺(H₂O)_n] are also archetypal and concise models to illustrate effects of different solvent shells. We have investigated noncovalent interactions in NO⁺(H₂O)₃ and NO⁺(H₂O)₄ isomers with high levels of ab initio and symmetry-adapted perturbation theory (SAPT) methods. On the basis of our computations, the exchange energies become much more repulsive, whereas the induction energies are significantly more attractive for the noncovalent interactions of NO⁺ with hydrogen-bonded water chains. Combined with analyses of the electron densities for the NO⁺(H₂O)₃ and NO⁺(H₂O)₄ isomers, we propose that the counteracting effect of the exchange and induction energies could be deemed as an index for the tendency to form the HO-NO covalent bond. Moreover, we have also found that the third-order induction terms are very important to evaluate reasonable charge transfer energies with the SAPT computations.

Single-Point Extrapolation to the Complete Basis Set Limit through Deep Learning

Machine learning (ML) offers an attractive method for making predictions about molecular systems while circumventing the need to run expensive electronic structure calculations. Once trained on ab initio data, the promise of ML is to deliver accurate predictions of molecular properties that were previously computationally infeasible. In this work, we develop and train a graph neural network model to correct the basis set incompleteness error (BSIE) between a small and large basis set at the RHF and B3LYP levels of theory. Our results show that, when compared to fitting to the total potential, an ML model fitted to correct the BSIE is better at generalizing to systems not seen during training. We test this ability by training on single molecules while evaluating on molecular complexes. We also show that ensemble models yield better behaved potentials in situations where the training data is insufficient. However, even when only fitting to the BSIE, acceptable performance is only achieved when the training data sufficiently resemble the systems one wants to make predictions on. The test error of the final model trained to predict the difference between the cc-pVDZ and cc-pV5Z potential is 0.184 kcal/mol for the B3LYP density functional, and the ensemble model accurately reproduces the large basis set interaction energy curves on the S66x8 dataset.

High-Order Quantum-Mechanical Analysis of Hydrogen Bonding in Hachimoji and Natural DNA Base Pairs

High-order quantum chemistry is applied to hydrogen-bonded natural DNA nucleobase pairs [adenine:thymine (A:T) and guanine:cytosine (G:C)] and non-natural Hachimoji nucleobase pairs [isoguanine:1-methylcytosine (B:S) and 2-aminoimidazo[1,2a][1,3,5]triazin-4(1H)-one:6-amino-5-nitropyridin-2-one (P:Z)] to see how the intermolecular interaction energies and their energetic components (electrostatics, exchange-repulsion, induction/polarization, and London dispersion interactions) vary among the base pairs. We examined the Hoogsteen (HG) geometries in addition to the traditional Watson-Crick (WC) geometries. Coupled-cluster theory through perturbative triples [CCSD(T)] extrapolated to the complete basis set (CBS) limit and high-order symmetry-adapted perturbation theory (SAPT) at the SAPT2+(3)(CCD)δMP2/aug-cc-pVTZ level are used to estimate highly accurate noncovalent interaction energies. Electrostatic interactions are the most attractive component of the interaction energies, but the sum of induction/polarization and London dispersion is nearly as large, for all base pairs and geometries considered. Interestingly, the non-natural Hachimoji base pairs interact more strongly than the corresponding natural base pairs, by -21.8 (B:S) and -0.3 (P:Z) kcal mol^-1 in the WC geometries, according to CCSD(T)/CBS. This is consistent with the H-bond distances being generally shorter in the non-natural base pairs. The natural base pairs are energetically more stabilized in their Hoogsteen geometries than in their WC geometries. The Hoogsteen geometry makes the A:T base pair slightly more stable, by -0.8 kcal mol^-1, and it greatly stabilizes the G:C⁺ base pair, by -15.3 kcal mol^-1. The G:C⁺ stabilization is mainly due to the fact that C has typically added a proton when found in Hoogsteen geometries. By contrast, Hoogsteen geometries are substantially less favorable than WC geometries for non-natural Hachimoji base pairs, by 17.3 (B:S) and 13.8 (P:Z) kcal mol^-1.

Electronic structure theory on modeling short-range noncovalent interactions between amino acids

While short-range noncovalent interactions (NCIs) are proving to be of importance in many chemical and biological systems, these atypical bindings happen within the so-called van der Waals envelope and pose an enormous challenge for current computational methods. We introduce SNCIAA, a database of 723 benchmark interaction energies of short-range noncovalent interactions between neutral/charged amino acids originated from protein x-ray crystal structures at the "gold standard" coupled-cluster with singles, doubles, and perturbative triples/complete basis set [CCSD(T)/CBS] level of theory with a mean absolute binding uncertainty less than 0.1 kcal/mol. Subsequently, a systematic assessment of commonly used computational methods, such as the second-order Møller-Plesset theory (MP2), density functional theory (DFT), symmetry-adapted perturbation theory (SAPT), composite electronic-structure methods, semiempirical approaches, and the physical-based potentials with machine learning (IPML) on SNCIAA is carried out. It is shown that the inclusion of dispersion corrections is essential even though these dimers are dominated by electrostatics, such as hydrogen bondings and salt bridges. Overall, MP2, ωB97M-V, and B3LYP+D4 turned out to be the most reliable methods for the description of short-range NCIs even in strongly attractive/repulsive complexes. SAPT is also recommended in describing short-range NCIs only if the δMP2 correction has been included. The good performance of IPML for dimers at close-equilibrium and long-range conditions is not transferable to the short-range. We expect that SNCIAA will assist the development/improvement/validation of computational methods, such as DFT, force-fields, and ML models, in describing NCIs across entire potential energy surfaces (short-, intermediate-, and long-range NCIs) on the same footing.

Benchmarking two-body contributions to crystal lattice energies and a range-dependent assessment of approximate methods

Using the many-body expansion to predict crystal lattice energies (CLEs), a pleasantly parallel process, allows for flexibility in the choice of theoretical methods. Benchmark-level two-body contributions to CLEs of 23 molecular crystals have been computed using interaction energies of dimers with minimum inter-monomer separations (i.e., closest contact distances) up to 30 Å. In a search for ways to reduce the computational expense of calculating accurate CLEs, we have computed these two-body contributions with 15 different quantum chemical levels of theory and compared these energies to those computed with coupled-cluster in the complete basis set (CBS) limit. Interaction energies of the more distant dimers are easier to compute accurately and several of the methods tested are suitable as replacements for coupled-cluster through perturbative triples for all but the closest dimers. For our dataset, sub-kJ mol^-1 accuracy can be obtained when calculating two-body interaction energies of dimers with separations shorter than 4 Å with coupled-cluster with single, double, and perturbative triple excitations/CBS and dimers with separations longer than 4 Å with MP2.5/aug-cc-pVDZ, among other schemes, reducing the number of dimers to be computed with coupled-cluster by as much as 98%.

An Imbalance in the Force: The Need for Standardized Benchmarks for Molecular Simulation

Force fields (FFs) for molecular simulation have been under development for more than half a century. As with any predictive model, rigorous testing and comparisons of models critically depends on the availability of standardized data sets and benchmarks. While such benchmarks are rather common in the fields of quantum chemistry, this is not the case for empirical FFs. That is, few benchmarks are reused to evaluate FFs, and development teams rather use their own training and test sets. Here we present an overview of currently available tests and benchmarks for computational chemistry, focusing on organic compounds, including halogens and common ions, as FFs for these are the most common ones. We argue that many of the benchmark data sets from quantum chemistry can in fact be reused for evaluating FFs, but new gas phase data is still needed for compounds containing phosphorus and sulfur in different valence states. In addition, more nonequilibrium interaction energies and forces, as well as molecular properties such as electrostatic potentials around compounds, would be beneficial. For the condensed phases there is a large body of experimental data available, and tools to utilize these data in an automated fashion are under development. If FF developers, as well as researchers in artificial intelligence, would adopt a number of these data sets, it would become easier to compare the relative strengths and weaknesses of different models and to, eventually, restore the balance in the force.

Full Implementation, Optimization, and Evaluation of a Range-Separated Local Hybrid Functional with Wide Accuracy for Ground and Excited States

We report the first full and efficient implementation of range-separated local hybrid functionals (RSLHs) into the TURBOMOLE program package. This enables the computation of ground-state energies and nuclear gradients as well as excitation energies. Regarding the computational effort, RSLHs scale like regular local hybrid functionals (LHs) with system or basis set size and increase timings by a factor of 2-3 in total. An advanced RSLH, ωLH22t, has been optimized for atomization energies and reaction barriers. It is an extension of the recent LH20t local hybrid and is based on short-range PBE and long-range HF exchange-energy densities, a pig2 calibration function to deal with the gauge ambiguity of exchange-energy densities, and reoptimized B95c correlation. ωLH22t has been evaluated for a wide range of ground-state and excited-state quantities. It further improves upon the already successful LH20t functional for the GMTKN55 main-group energetics test suite, and it outperforms any global hybrid while performing close to the top rung-4 functional, ωB97M-V, for these evaluations when augmented by D4 dispersion corrections. ωLH22t performs excellently for transition-metal reactivity and provides good balance between delocalization errors and left-right correlation for mixed-valence systems, with a somewhat larger bias toward localized states compared to LH20t. It approaches the accuracy of the best local hybrids to date for core, valence singlet and triplet, and Rydberg excitation energies while improving strikingly on intra- and intermolecular charge-transfer excitations, comparable to the most successful range-separated hybrids available.

Interexcited State Photophysics I: Benchmarking Density Functionals for Computing Nonadiabatic Couplings and Internal Conversion Rate Constants

We present the first benchmarking study of nonadiabatic matrix coupling elements (NACMEs) calculated using different density functionals. Using the S₁ → S₀ transition in perylene solvated in toluene as a case study, we calculate the photophysical properties and corresponding rate constants for a variety of density functionals from each rung of Jacob's ladder. The singlet photoluminescence quantum yield (sPLQY) is taken as a measure of accuracy, measured experimentally here as 0.955. Important quantum chemical parameters such as geometries, absorption, emission, and adiabatic energies, NACMEs, Hessians, and transition dipole moments were calculated for each density functional basis set combination (data set) using density functional theory based multireference configuration interaction (DFT/MRCI) and compared to experiment where possible. We were able to derive simple relations between the TDDFT and DFT/MRCI photophysical properties; with semiempirical damping factors of ∼0.843 ± 0.017 and ∼0.954 ± 0.064 for TDDFT transition dipole moments and energies to DFT/MRCI level approximations, respectively. NACMEs were dominated by out-of-plane derivative components belonging to the center-most ring atoms with weaker contributions from perturbations along the transverse and longitudinal axes. Calculated theoretical spectra compared well to both experiment and literature, with fluorescence lifetimes between 7.1 and 12.5 ns, agreeing within a factor of 2 with experiment. Internal conversion (IC) rates were then calculated and were found to vary wildly between 10⁶-10¹⁶ s^-1 compared with an experimental rate of the order 10⁷ s^-1. Following further testing by mixing data sets, we found a strong dependence on the method used to obtain the Hessian. The 5 characterized data sets ranked in order of most promising are PBE0/def2-TZVP, ωB97XD/def2-TZVP, HCTH407/TZVP, PBE/TZVP, and PBE/def2-TZVP.

Performance of Localized-Orbital Coupled-Cluster Approaches for the Conformational Energies of Longer n-Alkane Chains

We report an update and enhancement of the ACONFL (conformer energies of large alkanes [J. Phys. Chem. A2022,126, 3521-3535]) dataset. For the ACONF12 (n-dodecane) subset, we report basis set limit canonical coupled-cluster with singles, doubles, and perturbative triples [i.e., CCSD(T)] reference data obtained from the MP2-F12/cc-pV{T,Q}Z-F12 extrapolation, [CCSD(F12*)-MP2-F12]/aug-cc-pVTZ-F12, and a (T) correction from conventional CCSD(T)/aug-cc-pV{D,T}Z calculations. Then, we explored the performance of a variety of single and composite localized-orbital CCSD(T) approximations, ultimately finding an affordable localized natural orbital CCSD(T) [LNO-CCSD(T)]-based post-MP2 correction that agrees to 0.006 kcal/mol mean absolute deviation with the revised canonical reference data. In tandem with canonical MP2-F12 complete basis set extrapolation, this was then used to re-evaluate the ACONF16 and ACONF20 subsets for n-hexadecane and n-icosane, respectively. Combining those with the revised canonical reference data for the dodecane conformers (i.e., ACONF12 subset), a revised ACONFL set was obtained. It was then used to assess the performance of different localized-orbital coupled-cluster approaches, such as pair natural orbital localized CCSD(T) [PNO-LCCSD(T)] as implemented in MOLPRO, DLPNO-CCSD(T0) and DLPNO-CCSD(T1) as implemented in ORCA, and LNO-CCSD(T) as implemented in MRCC, at their respective "Normal", "Tight", "vTight", and "vvTight" accuracy settings. For a given accuracy threshold and basis set, DLPNO-CCSD(T1) and DLPNO-CCSD(T0) perform comparably. With "VeryTightPNO" cutoffs, explicitly correlated DLPNO-CCSD(T1)-F12/VDZ-F12 is the best pick among all the DLPNO-based methods tested. To isolate basis set incompleteness from localized-orbital-related truncation errors (domain, LNOs), we have also compared the localized coupled-cluster approaches with canonical DF-CCSD(T)/aug-cc-pVTZ for the ACONF12 set. We found that gradually tightening the cutoffs improves the performance of LNO-CCSD(T), and using a composite scheme such as vTight + 0.50[vTight - Tight] improves things further. For DLPNO-CCSD(T1), "TightPNO" and "VeryTightPNO" offer a statistically similar accuracy, which gets slightly better when TCutPNO is extrapolated to the complete PNO space limit. Similar to Brauer et al.'s [Phys. Chem. Chem. Phys.2016,18 (31), 20905-20925] previous report for the S66x8 noncovalent interactions, the dispersion-corrected direct random phase approximation (dRPA)-based double hybrids perform remarkably well for the ACONFL set. While the revised reference data do not affect any conclusions on the less accurate methods, they may upend orderings for more accurate methods with error statistics on the same order as the difference between reference datasets.

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.

Explicitly Correlated Double-Hybrid DFT: A Comprehensive Analysis of the Basis Set Convergence on the GMTKN55 Database

Double-hybrid density functional theory (DHDFT) offers a pathway to accuracy approaching composite wavefunction approaches such as G4 theory. However, the Görling-Levy second-order perturbation theory (GLPT2) term causes them to partially inherit the slow ∝L-3 (with L the maximum angular momentum) basis set convergence of correlated wavefunction methods. This could potentially be remedied by introducing F12 explicit correlation: we investigate the basis set convergence of both DHDFT and DHDFT-F12 (where GLPT2 is replaced by GLPT2-F12) for the large and chemically diverse general main-group thermochemistry, kinetics, and noncovalent interactions (GMTKN55) benchmark suite. The B2GP-PLYP-D3(BJ) and revDSD-PBEP86-D4 DHDFs are investigated as test cases, together with orbital basis sets as large as aug-cc-pV5Z and F12 basis sets as large as cc-pVQZ-F12. We show that F12 greatly accelerates basis set convergence of DHDFs, to the point that even the modest cc-pVDZ-F12 basis set is closer to the basis set limit than cc-pV(Q+d)Z or def2-QZVPPD in orbital-based approaches, and in fact comparable in quality to cc-pV(5+d)Z. Somewhat surprisingly, aug-cc-pVDZ-F12 is not required even for the anionic subsets. In conclusion, DHDF-F12/VDZ-F12 eliminates concerns about basis set convergence in both the development and applications of double-hybrid functionals. Mass storage and I/O bottlenecks for larger systems can be circumvented by localized pair natural orbital approximations, which also exhibit much gentler system size scaling.

Origin of the Microsolvation Effect on the Central Barriers of SN2 Reactions

We have quantitatively analyzed the microsolvation effect on the central barriers of microsolvated bimolecular nucleophilic substitution (S_N2) reactions by means of a two-step energy decomposition procedure. According to the first energy decompositions, an obvious increase in the central barrier for a microsolvated S_N2 reaction against its unsolvated counterpart can be mainly ascribed to the fact that the interaction between the solute and the conjunct solvent becomes less attractive from the reactant complex to the transition state. On the basis of the second energy decompositions with symmetry-adapted perturbation theory, this less attractive interaction in the transition state is primarily due to the interplay of the changes in the electrostatic, exchange, and induction components. However, the contribution of the change for the dispersion component is relatively small. A distinct linear correlation has also been observed between the changes of the total interaction energies and those of the corresponding electrostatic components for the microsolvated S_N2 reactions studied in this work. Moreover, the two-step energy decomposition procedure employed in this work is expected to be extensively applied to the gas phase reactions mediated by molecules or clusters.

Non-covalent interactions atlas benchmark data sets 4: σ-hole interactions

The SH250×10 dataset presented here extends the Non-Covalent Interactions Atlas database (https://www.nciatlas.org) to complexes bound by σ-hole interactions - halogen, chalcogen and pnictogen bonds. It comprises 250 complexes where Cl, Br, I, S, Se, P and As interact with diverse electron donors. An accurate CCSD(T)/CBS benchmark is provided for ten points along a dissociation curve of each complex. The SH250×10 set is used in testing a wide variety of DFT functionals and semiempirical quantum-mechanical methods. In DFT calculations, the new data set exposes large errors of some functionals related to exaggerated charge transfer. The size and diversity of the data set have also been exploited in the reparametrization of a halogen-bond correction for the PM6 semiempirical method.

A Minimum Quantum Chemistry CCSD(T)/CBS Data Set of Dimeric Interaction Energies for Small Organic Functional Groups: Heterodimers

We extend our previous quantum chemistry calculations of interaction energies for 31 homodimers of small organic functional groups (the SOFG-31 data set) by including 239 heterodimers with monomers selected within the SOFG-31 data set, thus resulting in the SOFG-31+239 data set. The minimum-level theoretical scheme contains (1) the basis set superposition error corrected supermolecule (BSSE-SM) approach for intermolecular interactions; (2) the second-order Møller-Plesset perturbation theory (MP2) with the Dunning's aug-cc-pVXZ (X = D, T, Q) basis sets for the geometry optimization and correlation energy calculations; and (3) the single-point energy calculations with the coupled cluster with single, double, and perturbative triple excitations method at the complete basis set limit [CCSD(T)/CBS] using the well-tested extrapolation methods for the MP2 energy calibrations. In addition, we have performed a parallel series of energy decomposition calculations based on the symmetry adapted perturbation theory (SAPT) in order to gain chemical insights. That the above procedure cannot be further reduced has been proven to be very crucial for constructing reliable data sets of interaction energies. The calculated CCSD(T)/CBS interaction energy data can serve as a benchmark for testing or training less accurate but more efficient calculation methods, such as the electronic density functional theory. As an application, we employ a segmental SAPT model previously developed for the SOFG-31 data set to predict binding energies of large heterodimer complexes. These model energy "quanta" can be used in coarse-grained molecular dynamics simulations by avoiding large-scale calculations.

Conformational Energy Benchmark for Longer n-Alkane Chains

We present the first benchmark set focusing on the conformational energies of highly flexible, long n-alkane chains, termed ACONFL. Unbranched alkanes are ubiquitous building blocks in nature, so the goal is to be able to calculate their properties most accurately to improve the modeling of, e.g., complex (biological) systems. Very accurate DLPNO-CCSD(T1)/CBS reference values are provided, which allow for a statistical meaningful evaluation of even the best available density functional methods. The performance of established and modern (dispersion corrected) density functionals is comprehensively assessed. The recently introduced r²SCAN-V functional shows excellent performance, similar to efficient composite DFT methods like B97-3c and r²SCAN-3c, which provide an even better cost-accuracy ratio, while almost reaching the accuracy of much more computationally demanding hybrid or double hybrid functionals with large QZ AO basis sets. In addition, we investigated the performance of common wave function methods, where MP2/CBS surprisingly performs worse compared to the simple D4 dispersion corrected Hartree-Fock. Furthermore, we investigate the performance of several semiempirical and force field methods, which are commonly used for the generation of conformational ensembles in multilevel workflows or in large scale molecular dynamics studies. Outstanding performance is obtained by the recently introduced general force field, GFN-FF, while other commonly applied methods like the universal force field yield large errors. We recommend the ACONFL as a helpful benchmark set for parametrization of new semiempirical or force field methods and machine learning potentials as well as a meaningful validation set for newly developed DFT or dispersion methods.

The influence of a solvent environment on direct non-covalent interactions between two molecules: A symmetry-adapted perturbation theory study of polarization tuning of π-π interactions by water

High-level quantum chemical computations have provided significant insight into the fundamental physical nature of non-covalent interactions. These studies have focused primarily on gas-phase computations of small van der Waals dimers; however, these interactions frequently take place in complex chemical environments, such as proteins, solutions, or solids. To better understand how the chemical environment affects non-covalent interactions, we have undertaken a quantum chemical study of π-π interactions in an aqueous solution, as exemplified by T-shaped benzene dimers surrounded by 28 or 50 explicit water molecules. We report interaction energies (IEs) using second-order Møller-Plesset perturbation theory, and we apply the intramolecular and functional-group partitioning extensions of symmetry-adapted perturbation theory (ISAPT and F-SAPT, respectively) to analyze how the solvent molecules tune the π-π interactions of the solute. For complexes containing neutral monomers, even 50 explicit waters (constituting a first and partial second solvation shell) change total SAPT IEs between the two solute molecules by only tenths of a kcal mol^-1, while significant changes of up to 3 kcal mol^-1 of the electrostatic component are seen for the cationic pyridinium-benzene dimer. This difference between charged and neutral solutes is attributed to large non-additive three-body interactions within solvated ion-containing complexes. Overall, except for charged solutes, our quantum computations indicate that nearby solvent molecules cause very little "tuning" of the direct solute-solute interactions. This indicates that differences in binding energies between the gas phase and solution phase are primarily indirect effects of the competition between solute-solute and solute-solvent interactions.

Open-Shell Variant of the London Dispersion-Corrected Hartree-Fock Method (HFLD) for the Quantification and Analysis of Noncovalent Interaction Energies

The London dispersion (LD)-corrected Hartree–Fock (HF) method (HFLD) is an ab initio approach for the quantification and analysis of noncovalent interactions (NCIs) in large systems that is based on the domain-based local pair natural orbital coupled-cluster (DLPNO-CC) theory. In the original HFLD paper, we discussed the implementation, accuracy, and efficiency of its closed-shell variant. Herein, an extension of this method to open-shell molecular systems is presented. Its accuracy is tested on challenging benchmark sets for NCIs, using CCSD(T) energies at the estimated complete basis set limit as reference. The HFLD scheme was found to be as accurate as the best-performing dispersion-corrected exchange-correlation functionals, while being nonempirical and equally efficient. In addition, it can be combined with the well-established local energy decomposition (LED) for the analysis of NCIs, thus yielding additional physical insights.

Small-Basis Set Density-Functional Theory Methods Corrected with Atom-Centered Potentials

Density functional theory (DFT) is currently the most popular method for modeling noncovalent interactions and thermochemistry. The accurate calculation of noncovalent interaction energies, reaction energies, and barrier heights requires choosing an appropriate functional and, typically, a relatively large basis set. Deficiencies of the density-functional approximation and the use of a limited basis set are the leading sources of error in the calculation of noncovalent and thermochemical properties in molecular systems. In this article, we present three new DFT methods based on the BLYP, M06-2X, and CAM-B3LYP functionals in combination with the 6-31G* basis set and corrected with atom-centered potentials (ACPs). ACPs are one-electron potentials that have the same form as effective-core potentials, except they do not replace any electrons. The ACPs developed in this work are used to generate energy corrections to the underlying DFT/basis-set method such that the errors in predicted chemical properties are minimized while maintaining the low computational cost of the parent methods. ACPs were developed for the elements H, B, C, N, O, F, Si, P, S, and Cl. The ACP parameters were determined using an extensive training set of 118655 data points, mostly of complete basis set coupled-cluster level quality. The target molecular properties for the ACP-corrected methods include noncovalent interaction energies, molecular conformational energies, reaction energies, barrier heights, and bond separation energies. The ACPs were tested first on the training set and then on a validation set of 42567 additional data points. We show that the ACP-corrected methods can predict the target molecular properties with accuracy close to complete basis set wavefunction theory methods, but at a computational cost of double-ζ DFT methods. This makes the new BLYP/6-31G*-ACP, M06-2X/6-31G*-ACP, and CAM-B3LYP/6-31G*-ACP methods uniquely suited to the calculation of noncovalent, thermochemical, and kinetic properties in large molecular systems.

Fast and Accurate Quantum Mechanical Modeling of Large Molecular Systems Using Small Basis Set Hartree-Fock Methods Corrected with Atom-Centered Potentials

There has been significant interest in developing fast and accurate quantum mechanical methods for modeling large molecular systems. In this work, by utilizing a machine learning regression technique, we have developed new low-cost quantum mechanical approaches to model large molecular systems. The developed approaches rely on using one-electron Gaussian-type functions called atom-centered potentials (ACPs) to correct for the basis set incompleteness and the lack of correlation effects in the underlying minimal or small basis set Hartree-Fock (HF) methods. In particular, ACPs are proposed for ten elements common in organic and bioorganic chemistry (H, B, C, N, O, F, Si, P, S, and Cl) and four different base methods: two minimal basis sets (MINIs and MINIX) plus a double-ζ basis set (6-31G*) in combination with dispersion-corrected HF (HF-D3/MINIs, HF-D3/MINIX, HF-D3/6-31G*) and the HF-3c method. The new ACPs are trained on a very large set (73 832 data points) of noncovalent properties (interaction and conformational energies) and validated additionally on a set of 32 048 data points. All reference data are of complete basis set coupled-cluster quality, mostly CCSD(T)/CBS. The proposed ACP-corrected methods are shown to give errors in the tenths of a kcal/mol range for noncovalent interaction energies and up to 2 kcal/mol for molecular conformational energies. More importantly, the average errors are similar in the training and validation sets, confirming the robustness and applicability of these methods outside the boundaries of the training set. In addition, the performance of the new ACP-corrected methods is similar to complete basis set density functional theory (DFT) but at a cost that is orders of magnitude lower, and the proposed ACPs can be used in any computational chemistry program that supports effective-core potentials without modification. It is also shown that ACPs improve the description of covalent and noncovalent bond geometries of the underlying methods and that the improvement brought about by the application of the ACPs is directly related to the number of atoms to which they are applied, allowing the treatment of systems containing some atoms for which ACPs are not available. Overall, the ACP-corrected methods proposed in this work constitute an alternative accurate, economical, and reliable quantum mechanical approach to describe the geometries, interaction energies, and conformational energies of systems with hundreds to thousands of atoms.

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.

Reliable prediction of association (free) energies of supramolecular complexes with heavy main group elements – the HS13L benchmark set

We introduce a set of 13 supramolecular complexes featuring diverse non-covalent interactions with heavy main group elements (Zn, As, Se, Te, Br, I), high charges (−2 up to +4), and large systems with up to 266 atoms (HS13L).

Spin-component-scaled and dispersion-corrected second-order Møller-Plesset perturbation theory: A path toward chemical accuracy

Second-order Moller-Plesset perturbation theory (MP2) provides a valuable alternative to density functional theory for modeing problems in organic and biological chemistry. However, MP2 suffers from known limitations in the description...

NENCI-2021. I. A large benchmark database of non-equilibrium non-covalent interactions emphasizing close intermolecular contacts

In this work, we present NENCI-2021, a benchmark database of ∼8000 Non-Equilibirum Non-Covalent Interaction energies for a large and diverse selection of intermolecular complexes of biological and chemical relevance. To meet the growing demand for large and high-quality quantum mechanical data in the chemical sciences, NENCI-2021 starts with the 101 molecular dimers in the widely used S66 and S101 databases and extends the scope of these works by (i) including 40 cation-π and anion-π complexes, a fundamentally important class of non-covalent interactions that are found throughout nature and pose a substantial challenge to theory, and (ii) systematically sampling all 141 intermolecular potential energy surfaces (PESs) by simultaneously varying the intermolecular distance and intermolecular angle in each dimer. Designed with an emphasis on close contacts, the complexes in NENCI-2021 were generated by sampling seven intermolecular distances along each PES (ranging from 0.7× to 1.1× the equilibrium separation) and nine intermolecular angles per distance (five for each ion-π complex), yielding an extensive database of 7763 benchmark intermolecular interaction energies (E_int) obtained at the coupled-cluster with singles, doubles, and perturbative triples/complete basis set [CCSD(T)/CBS] level of theory. The E_int values in NENCI-2021 span a total of 225.3 kcal/mol, ranging from -38.5 to +186.8 kcal/mol, with a mean (median) E_int value of -1.06 kcal/mol (-2.39 kcal/mol). In addition, a wide range of intermolecular atom-pair distances are also present in NENCI-2021, where close intermolecular contacts involving atoms that are located within the so-called van der Waals envelope are prevalent-these interactions, in particular, pose an enormous challenge for molecular modeling and are observed in many important chemical and biological systems. A detailed symmetry-adapted perturbation theory (SAPT)-based energy decomposition analysis also confirms the diverse and comprehensive nature of the intermolecular binding motifs present in NENCI-2021, which now includes a significant number of primarily induction-bound dimers (e.g., cation-π complexes). NENCI-2021 thus spans all regions of the SAPT ternary diagram, thereby warranting a new four-category classification scheme that includes complexes primarily bound by electrostatics (3499), induction (700), dispersion (1372), or mixtures thereof (2192). A critical error analysis performed on a representative set of intermolecular complexes in NENCI-2021 demonstrates that the E_int values provided herein have an average error of ±0.1 kcal/mol, even for complexes with strongly repulsive E_int values, and maximum errors of ±0.2-0.3 kcal/mol (i.e., ∼±1.0 kJ/mol) for the most challenging cases. For these reasons, we expect that NENCI-2021 will play an important role in the testing, training, and development of next-generation classical and polarizable force fields, density functional theory approximations, wavefunction theory methods, and machine learning based intra- and inter-molecular potentials.

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

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

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.