Many-Body Dispersion in Molecular Clusters

Role of Exact Exchange and Empirical Dispersion in Density Functional Theory-Based Three-Body Noncovalent Interactions

Total and three-body interaction energies are calculated for a benchmark set of three-body systems using a range of different types of density functional theory (DFT) methods, with the results compared to CCSD(T)/CBS results from the benchmark reference [Phys. Chem. Chem. Phys. 2023, 25, 28621-28637]. Inclusion of Hartree-Fock exchange, via either a global or range-separated hybrid approach or inclusion of empirical dispersion corrections, increases accuracy for total and three-body interactions. Basis set convergence testing shows that the aug-cc-pVTZ basis set is well converged with little to no change seen when using quadruple-ζ basis sets. The accuracy of the DFT methods is similar when calculating interaction energies for both global and local minimum structures. Overall, the CAM-B3LYP-D3BJ, B97D3, and ωB97XD functionals are recommended for calculating three-body interactions.

Stabilization of the Protein Structure by the Many-Body Cooperative Effect in the NH/π Hydrogen-bonding Tryptophan Triad

The indole ring of tryptophan can form NH/π hydrogen bonds, acting both as a hydrogen donor at the NH group in the pyrrole subring and as a hydrogen acceptor at the benzene subring. In the structural core of the trimeric stable protein Pholiota squarrosa lectin (PhoSL), three indoles are symmetrically arranged and form NH/π hydrogen bonds among each other. Here, we conducted quantum chemical calculations on this indole triad by using various methods and basis sets. The analyses revealed cooperativity in triad formation, with the many-body effect contributing approximately -2 kcal mol-1, which significantly stabilizes this protein. Symmetry-adapted perturbation theory ascribed this effect to the induced polarization. The electrostatic potential and atomic charges indeed revealed a charge redistribution through the NH/π hydrogen bond, which was favorable for triad formation.

A neural network-based four-body potential energy surface for parahydrogen

We present an isotropic ab initio (para-H2)4 four-body interaction potential energy surface (PES). The electronic structure calculations are performed at the correlated coupled-cluster theory level, with single, double, and perturbative triple excitations. They use an atom-centered augmented correlation-consistent double zeta basis set, supplemented by a (3s3p2d) midbond function. We use a multilayer perceptron to construct the PES. We apply a rescaling transformation to the output energies during training to improve the prediction of weaker energies in the sample data. At long distances, the interaction energies are adjusted to match the empirically derived four-body dispersion interaction. The four-body interaction energy at short intermolecular separations is net repulsive. The use of this four-body PES, in combination with a first principles pair potential for para-H2 [J. Chem. Phys. 119, 12551 (2015)] and an isotropic ab initio three-body potential for para-H2 [J. Chem. Phys. 156, 044301 (2022)], is expected to provide closer agreement with experimental results.

R-8 Dispersion Interaction: Derivation and Application to the Effective Fragment Potential Method

The anisotropic and isotropic R-8 dispersion contributions (disp8) are derived and implemented within the framework of the effective fragment potential (EFP) method formulated with imaginary frequency-dependent Cartesian polarizability tensors distributed at the centroids of the localized molecular orbitals (LMOs). Two forms of damping functions, intermolecular overlap-based and Tang-Toennies, are extended for disp8. To obtain LMO polarizability tensors centered at LMO centroids, an origin-shifting transformation is derived and implemented for the dipole-octopole polarizability tensor and the quadrupole-quadrupole polarizability tensor. The analytic gradient is derived and implemented for the isotropic disp8 contribution. Relative to the previously implemented empirical EFP disp8 energy, the isotropic disp8 component of the interaction energy improves the overall agreement of the EFP dispersion energies with the symmetry-adapted perturbation theory (SAPT) benchmarks, reducing the mean absolute errors (MAEs) and mean absolute percentage errors for most of the databases examined in this work. While the anisotropic disp8 can further enhance the accuracy of the EFP dispersion energy and yield smaller MAEs, significantly overbound dispersion energies are predicted by the anisotropic disp8 when the maximum element in the intermolecular overlap matrix is greater than 0.1, possibly due to the breakdown of the approximations made in the EFP dispersion derivation at a short range. For potential energy scan databases, the newly developed EFP dispersion model with isotropic disp8 yields the overall correct curvature and good agreement with SAPT benchmarks around equilibrium and longer but overestimates the dispersion interactions at a short range. While the overlap-based dispersion-damping functions produce better MAEs than Tang-Toennies damping functions, further improvement is needed to better screen the large attractive dispersion energies at a short range (overlap >0.1).

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

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

Appropriate clusterset selection for the prediction of thermodynamic properties of liquid water with QCE theory

Evident in many physical and chemical phenomena, thermodynamics is the study of how energy is stored, transformed and transferred in a molecule or material. However, prediction of these properties with simulation techniques is a non-trivial task as several factors such as composition and intermolecular interactions come into play. While molecular dynamics and ab initio molecular dynamics are the most common techniques for the prediction of thermodynamic properties, there exists many shortcomings associated with their use. Therefore, in this work we instead apply QCE theory to predict the thermodynamic properties of liquid water. This theory assumes that a condensed phase system can be represented as a 'mixture' of varying sized clusters rather than as a continuum. As QCE theory relies on first-principle simulations and statistical thermodynamics to determine the thermodynamic behavior of a system, appropriate selection of clusters is a crucial step towards achieving accurate predictions. In this study, we use molecular dynamics and ab initio calculations to obtain initial configurations of 400 water clusters, Wn where n = 3 to 10 and contrast their stability using two different criteria. The role of entropy towards cluster stabilization is investigated by comparing the binding (ΔE_BIND/mol) and Gibbs free binding energy per molecule (ΔG_BIND/mol) of various Wn at 298.15 K. Initial clustersets are constructed by exploring two-, three-, four and five-combinations of clustersets using the minimum ΔG_BIND/mol structures of Wn. We also expand the ΔG_BIND/mol criteria for Wn of sizes 3 to 7 to include values larger than 0.0 kJ mol^-1 and smaller than 3.0 kJ mol^-1 as a means of improving thermodynamic predictions. 37 of the 459 resulting clustersets predicted the correct boiling point of water and its thermodynamic properties with an accuracy of 95%. A scaled population-weighted infrared spectrum was compared to experimental results to validate the composition of the top 5 clustersets. The predicted spectra showed an exact match within the low frequency range (<1000 cm^-1) with some discrepancy at the high frequency range (>3400 cm^-1). This work highlights that ΔG_BIND/mol is so far the best criteria to apply when determining an appropriate clusterset for QCE theory.

Accurate Prediction of Three-Body Intermolecular Interactions via Electron Deformation Density-Based Machine Learning

This work extends the electron deformation density-based descriptor, originally developed in the electron deformation density-based interaction energy machine learning (EDDIE-ML) algorithm to predict dimer interaction energies, to the prediction of three-body interactions in trimers. Using a sequential learning process to select the training data, the resulting Gaussian process regression (GPR) model predicts the three-body interaction energy within 0.2 kcal mol^-1 of the SRS-MP2/cc-pVTZ reference values for the 3B69 and S22-3 trimer data sets. A hybrid kernel function is introduced, which combines contributions from the average and individual atomic environments, allowing the total trimer interaction energy to be predicted in addition to the three-body contribution using the same descriptor. To extend the range and diversity of trimer interaction energies available in the literature, a new data set based on a protein-ligand crystal structure is introduced, consisting of 509 structures of a central ligand with two protein fragments. Benchmark calculations are provided for the new data set, which contains significantly larger molecular interactions than current databases in the literature in addition to charged fragments. Compared to density funtional theory (DFT)- and wavefunction-based methods for calculating the three-body interaction energy, our model makes predictions in a significantly shorter time frame by reducing the number of required SCF calculations from 7 to 4 performed at the PBE0 level of theory, showcasing the utility and efficiency of our Δ-ML method particularly when applied to larger systems.

A Task-Based Approach to Parallel Restricted Hartree-Fock Calculations

In recent years, parallelism via multithreading has become extremely important to the optimization of high-performance electronic structure theory codes. Such multithreading is generally achieved via OpenMP constructs, using a fork-join threading model to enable thread-level data parallelism within the code. An alternative approach to multithreading is task-based parallelism, which displays multiple benefits relative to fork-join thread parallelism. A novel Restricted Hartree-Fock (RHF) algorithm, utilizing task-based parallelism to achieve optimal performance, was developed and implemented into the JuliaChem electronic structure theory software package. The new RHF algorithm utilizes a unique method of shell quartet batch creation, enabling construction and distribution of fine-grained shell quartet batches in a load-balanced manner using the Julia task construct. These shell quartet batches are then distributed statically across message-passing interface (MPI) ranks and dynamically across threads within an MPI rank, requiring no explicit inter-rank or interthread synchronization to do so. Compared to the hybrid MPI/OpenMP RHF algorithm present in the GAMESS software package, the task-based algorithm demonstrates speedups of up to ∼40% for systems in the S22(3) test set of molecules, with system sizes up to ∼1000 basis functions. The JuliaChem algorithm demonstrates the viability of both the task-based parallelism model and the Julia programming language for construction of performant electronic structure theory codes targeting systems of a size of chemical interest.

O(N) Stochastic Evaluation of Many-Body van der Waals Energies in Large Complex Systems

We propose a new strategy to solve the key equations of the many-body dispersion (MBD) model by Tkatchenko, DiStasio Jr., and Ambrosetti. Our approach overcomes the original O(N3) computational complexity that limits its applicability to large molecular systems within the context of O(N) density functional theory. First, to generate the required frequency-dependent screened polarizabilities, we introduce an efficient solution to the Dyson-like self-consistent screening equations. The scheme reduces the number of variables and, coupled to a direct inversion of the iterative subspace extrapolation, exhibits linear-scaling performances. Second, we apply a stochastic Lanczos trace estimator resolution to the equations evaluating the many-body interaction energy of coupled quantum harmonic oscillators. While scaling linearly, it also enables communication-free pleasingly parallel implementations. As the resulting O(N) stochastic massively parallel MBD approach is found to exhibit minimal memory requirements, it opens up the possibility of computing accurate many-body van der Waals interactions of millions-atoms' complex materials and solvated biosystems with computational times in the range of minutes.

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.

The Role of Non-Covalent Interactions on Cluster Formation: Pentamer, Hexamers and Heptamer of Difluoromethane

The role of non-covalent interactions (NCIs) has broadened with the inclusion of new types of interactions and a plethora of weak donor/acceptor partners. This work illustrates the potential of chirped-pulse Fourier transform microwave technique, which has revolutionized the field of rotational spectroscopy. In particular, it has been exploited to reveal the role of NCIs' in the molecular self-aggregation of difluoromethane where a pentamer, two hexamers and a heptamer were detected. The development of a new automated assignment program and a sophisticated computational screening protocol was essential for identifying the homoclusters in conditions of spectral congestion. The major role of dispersion forces leads to less directional interactions and more distorted structures than those found in polar clusters, although a detailed analysis demonstrates that the dominant interaction energy is the pairwise interaction. The tetramer cluster is identified as a structural unit in larger clusters, representing the maximum expression of bond between dimers.

Requirements for an accurate dispersion-corrected density functional

Post-self-consistent dispersion corrections are now the norm when applying density-functional theory to systems where non-covalent interactions play an important role. However, there is a wide range of base functionals and dispersion corrections available from which to choose. In this work, we opine on the most desirable requirements to ensure that both the base functional and dispersion correction, individually, are as accurate as possible for non-bonded repulsion and dispersion attraction. The base functional should be dispersionless, numerically stable, and involve minimal delocalization error. Simultaneously, the dispersion correction should include finite damping, higher-order pairwise dispersion terms, and electronic many-body effects. These criteria are essential for avoiding reliance on error cancellation and obtaining correct results from correct physics.

Fragment-Based Local Coupled Cluster Embedding Approach for the Quantification and Analysis of Noncovalent Interactions: Exploring the Many-Body Expansion of the Local Coupled Cluster Energy

Herein, we introduce a fragment-based local coupled cluster embedding approach for the accurate quantification and analysis of noncovalent interactions in molecular aggregates. Our scheme combines two different expansions of the domain-based local pair natural orbital coupled cluster (DLPNO-CCSD(T)) energy: the many-body expansion (MBE) and the local energy decomposition (LED). The low-order terms in the MBE are initially computed in the presence of an environment that is treated at a low level of theory. Then, LED is used to decompose the energy of each term in the embedded MBE into additive fragment and fragment-pairwise contributions. This information is used to quantify the total energy of the system while providing at the same time in-depth insights into the nature and cooperativity of noncovalent interactions. Two different approaches are introduced and tested, in which the environment is treated at different levels of theory: the local coupled cluster in the Hartree-Fock (LCC-in-HF) method, in which the environment is treated at the HF level; and the electrostatically embedded local coupled cluster method (LCC-in-EE), in which the environment is replaced by point charges. Both schemes are designed to preserve as much as possible the accuracy of the parent local coupled cluster method for total energies, while being embarrassingly parallel and less memory intensive. These schemes appear to be particularly promising for the study of large and complex molecular aggregates at the coupled cluster level, such as condensed phase systems and protein-ligand interactions.

Effective Fragment Potentials for Flexible Molecules: Transferability of Parameters and Amino Acid Database

An accurate but efficient description of noncovalent interactions is a key to predictive modeling of biological and materials systems. The effective fragment potential (EFP) is an ab initio-based force field that provides a physically meaningful decomposition of noncovalent interactions of a molecular system into Coulomb, polarization, dispersion, and exchange-repulsion components. An EFP simulation protocol consists of two steps, preparing parameters for molecular fragments by a series of ab initio calculations on each individual fragment, and calculation of interaction energy and properties of a total molecular system based on the prepared parameters. As the fragment parameters (distributed multipoles, polarizabilities, localized wave function, etc.) depend on a fragment geometry, straightforward application of the EFP method requires recomputing parameters of each fragment if its geometry changes, for example, during thermal fluctuations of a molecular system. Thus, recomputing fragment parameters can easily become both computational and human bottlenecks and lead to a loss of efficiency of a simulation protocol. An alternative approach, in which fragment parameters are adjusted to different fragment geometries, referred to as "flexible EFP", is explored here. The parameter adjustment is based on translations and rotations of local coordinate frames associated with fragment atoms. The protocol is validated on extensive benchmark of amino acid dimers extracted from molecular dynamics snapshots of a cryptochrome protein. A parameter database for standard amino acids is developed to automate flexible EFP simulations in proteins. To demonstrate applicability of flexible EFP in large-scale protein simulations, binding energies and vertical electron ionization and electron attachment energies of a lumiflavin chromophore of the cryptochrome protein are computed. The results obtained with flexible EFP are in a close agreement with the standard EFP procedure but provide a significant reduction in computational cost.

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.

Exciton energy transfer reveals spectral signatures of excited states in clusters

Electronic excitation and concomitant energy transfer leading to Penning ionization in argon-acetylene clusters generated in a supersonic expansion are investigated with synchrotron-based photoionization mass spectrometry and electronic structure calculations. Spectral features in the photoionization efficiency of the mixed argon-acetylene clusters reveal a blue shift from the 2P1/2 and 2P3/2 excited states of atomic argon. Analysis of this feature suggests that excited states of argon clusters transfer energy to acetylene, resulting in its ionization and successive evaporation of argon. Theoretically calculated Arn (n = 2-6) cluster spectra are in excellent agreement with experimental observations, and provide insight into the structure and ionization dynamics of the clusters. A comparison between argon-acetylene and argon-water clusters reveals that argon solvates water better, allowing for higher-order excitons and Rydberg states to be populated. These results are explained by theoretical calculations of respective binding energies and structures.

What is “many-body” dispersion and should I worry about it?

“Many-body” dispersion can refer to two distinct phenomena, here termed electronic and atomic many-body effects, both of which cause the dispersion energy to be non-additive.

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.