Weakly Bonded Complexes of Aliphatic and Aromatic Carbon Compounds Described with Dispersion Corrected Density Functional Theory

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

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

First-Principles Models for van der Waals Interactions in Molecules and Materials: Concepts, Theory, and Applications

Noncovalent van der Waals (vdW) or dispersion forces are ubiquitous in nature and influence the structure, stability, dynamics, and function of molecules and materials throughout chemistry, biology, physics, and materials science. These forces are quantum mechanical in origin and arise from electrostatic interactions between fluctuations in the electronic charge density. Here, we explore the conceptual and mathematical ingredients required for an exact treatment of vdW interactions, and present a systematic and unified framework for classifying the current first-principles vdW methods based on the adiabatic-connection fluctuation-dissipation (ACFD) theorem (namely the Rutgers-Chalmers vdW-DF, Vydrov-Van Voorhis (VV), exchange-hole dipole moment (XDM), Tkatchenko-Scheffler (TS), many-body dispersion (MBD), and random-phase approximation (RPA) approaches). Particular attention is paid to the intriguing nature of many-body vdW interactions, whose fundamental relevance has recently been highlighted in several landmark experiments. The performance of these models in predicting binding energetics as well as structural, electronic, and thermodynamic properties is connected with the theoretical concepts and provides a numerical summary of the state-of-the-art in the field. We conclude with a roadmap of the conceptual, methodological, practical, and numerical challenges that remain in obtaining a universally applicable and truly predictive vdW method for realistic molecular systems and materials.

Transferable Atomic Multipole Machine Learning Models for Small Organic Molecules

Accurate representation of the molecular electrostatic potential, which is often expanded in distributed multipole moments, is crucial for an efficient evaluation of intermolecular interactions. Here we introduce a machine learning model for multipole coefficients of atom types H, C, O, N, S, F, and Cl in any molecular conformation. The model is trained on quantum-chemical results for atoms in varying chemical environments drawn from thousands of organic molecules. Multipoles in systems with neutral, cationic, and anionic molecular charge states are treated with individual models. The models' predictive accuracy and applicability are illustrated by evaluating intermolecular interaction energies of nearly 1,000 dimers and the cohesive energy of the benzene crystal.

Exploring the Limit of Accuracy of the Global Hybrid Meta Density Functional for Main-Group Thermochemistry, Kinetics, and Noncovalent Interactions

The hybrid meta density functionals M05-2X and M06-2X have been shown to provide broad accuracy for main group chemistry. In the present article we make the functional form more flexible and improve the self-interaction term in the correlation functional to improve its self-consistent-field convergence. We also explore the constraint of enforcing the exact forms of the exchange and correlation functionals through second order (SO) in the reduced density gradient. This yields two new functionals called M08-HX and M08-SO, with different exact constraints. The new functionals are optimized against 267 diverse main-group energetic data consisting of atomization energies, ionization potentials, electron affinities, proton affinities, dissociation energies, isomerization energies, barrier heights, noncovalent complexation energies, and atomic energies. Then the M08-HX, M08-SO, M05-2X, and M06-2X functionals and the popular B3LYP functional are tested against 250 data that were not part of the original training data for any of the functionals, in particular 164 main-group energetic data in 7 databases, 39 bond lengths, 38 vibrational frequencies, and 9 multiplicity-changing electronic transition energies. These tests include a variety of new challenges for complex systems, including large-molecule atomization energies, organic isomerization energies, interaction energies in uracil trimers, and bond distances in crowded molecules (in particular, cyclophanes). The M08-HX functional performs slightly better than M08-SO and M06-2X on average, significantly better than M05-2X, and much better than B3LYP for a combination of main-group thermochemistry, kinetics, noncovalent interactions, and electronic spectroscopy. More important than the slight improvement in accuracy afforded by M08-HX is the conformation that the optimization procedure works well for data outside the training set. Problems for which the accuracy is especially improved by the new M08-HX functional include large-molecule atomization energies, noncovalent interaction energies, conformational energies in aromatic peptides, barrier heights, multiplicity-changing excitation energies, and bond lengths in crowded molecules.

Accurate DFT Descriptions for Weak Interactions of Molecules Containing Sulfur

Dispersion corrected atom centered potentials (DCACPs) have been shown to significantly improve the density functional theory (DFT) description of weak interactions. In this work, we have calibrated a DCACP for sulfur in combination with the widely used Generalized Gradient Approximation (GGA) BLYP, thereby augmenting the existing library of DCACPs for the first- and second-row elements H, C, N, O, and rare gases. Three weakly bound complexes as well as elemental (orthorhombic) sulfur are used as test cases to evaluate the transferability of the DCACP to different chemical environments. It is found that the sulfur DCACP systematically improves the agreement of DFT-calculated weak interactions with respect to MP2 and CCSD(T) level results.

Perspective: Advances and challenges in treating van der Waals dispersion forces in density functional theory

Electron dispersion forces play a crucial role in determining the structure and properties of biomolecules, molecular crystals, and many other systems. However, an accurate description of dispersion is highly challenging, with the most widely used electronic structure technique, density functional theory (DFT), failing to describe them with standard approximations. Therefore, applications of DFT to systems where dispersion is important have traditionally been of questionable accuracy. However, the last decade has seen a surge of enthusiasm in the DFT community to tackle this problem and in so-doing to extend the applicability of DFT-based methods. Here we discuss, classify, and evaluate some of the promising schemes to emerge in recent years. A brief perspective on the outstanding issues that remain to be resolved and some directions for future research are also provided.

Liquid Structure and Cluster Formation in Ionic Liquid/Water Mixtures – An Extensive ab initio Molecular Dynamics Study on 1-Ethyl-3-Methylimidazolium Acetate/Water Mixtures – Part

We present an extensive ab initio molecular dynamics study on mixtures of the room temperature ionic liquid 1-ethyl-3-methylimidazolium acetate with water. To show the dependence of the properties on the concentration, we simulated four different systems: Pure IL, pure water, and two binary mixtures with different mixing ratios. We found that the imidazolium rings are stacking on top of each other in the pure ionic liquid, which is a result of the strong dispersion interactions between the cations. With increasing water content, this ordering is disturbed. While in the pure IL the anions are located almost exclusively within the cation's ring plane, they occupy both the in-plane and the on-top position in the mixture. For very high water content, the anions are mainly found on top of the imidazolium ring. The ethyl chains of the cations attract each other via dispersion force and form clusters, indicating the existence of microheterogeneity in our simulations. We also analyzed the topology of the hydrogen bond network and found that the cation is forming hydrogen bonds to multiple anions at the same time, and is therefore a bridging agent. Likewise, the anion is bridging between different cations. At moderate water content, two acetate ions are frequently coordinated to the same water molecule, leading to the formation of acetate/water clusters.

Intricacies of Describing Weak Interactions Involving Halogen Atoms within Density Functional Theory

In this work we assess the performance of different dispersion-corrected density functional theory (DFT) approaches (M06, M06-2X, DFT-D3, and DCACP) in reproducing high-level wave function based benchmark calculations on the weakly bound halogen dimers X2···X2 and X2···Ar (for X = F, Cl, Br, and I), as well as the prototype halogen bonded complexes H3CX···OCH2 (X = Cl, Br, I). In spite of the generally good performance of all tested methods for weakly bound systems, their performance for halogen-containing compounds varies largely. We find maximum errors in the energies with respect to the CCSD(T) reference values of 0.13 kcal/mol for DCACP, 0.22 kcal/mol for M06-2X, 0.47 kcal/mol for BLYP-D3, and 0.77 kcal/mol for M06. The root-mean-square deviations are 0.13 kcal/mol for DCACP and M06-2X, 0.44 kcal/mol for M06, and 0.51 kcal/mol for BLYP-D3.

Real-world predictions from ab initio molecular dynamics simulations

In this review we present the techniques of ab initio molecular dynamics simulation improved to its current stage where the analysis of existing processes and the prediction of further chemical features and real-world processes are feasible. For this reason we describe the relevant developments in ab initio molecular dynamics leading to this stage. Among them, parallel implementations, different basis set functions, density functionals, and van der Waals corrections are reported. The chemical features accessible through AIMD are discussed. These are IR, NMR, as well as EXAFS spectra, sampling methods like metadynamics and others, Wannier functions, dipole moments of molecules in condensed phase, and many other properties. Electrochemical reactions investigated by ab initio molecular dynamics methods in solution, on surfaces as well as complex interfaces, are also presented.

Assessment of the Performance of DFT and DFT-D Methods for Describing Distance Dependence of Hydrogen-Bonded Interactions

Noncovalent interactions such as hydrogen bonds, van der Waals forces, and π-π interactions play important roles influencing the structure, stability, and dynamic properties of biomolecules including DNA and RNA base pairs. In an effort to better understand the fundamental physics of hydrogen bonding (H-bonding), we investigate the distance dependence of interaction energies in the prototype bimolecular complexes of formic acid, formamide, and formamidine. Potential energy curves along the H-bonding dissociation coordinate are examined both by establishing reference CCSD(T) interaction energies extrapolated to the complete basis set limit and by assessing the performance of the density functional methods B3LYP, PBE, PBE0, B970, PB86, M05-2X, and M06-2X and empirical dispersion corrected methods B3LYP-D3, PBE-D3, PBE0-D3, B970-D2, BP86-D3, and ωB97X-D, with basis sets 6-311++G(3df,3pd), aug-cc-pVDZ, and aug-cc-pVTZ. Although H-bonding interactions are dominated by electrostatics, it is necessary to properly account for dispersion interactions to obtain accurate energetics. In order to quantitatively probe the nature of hydrogen bonding interactions as a function of distance, we decompose the interaction energy curves into physically meaningful components with symmetry-adapted perturbation theory (SAPT). The SAPT results confirm that the contribution of dispersion and induction are significant at and near equilibrium, although electrostatics dominate. Among the DFT/DFT-D techniques, the best overall results are obtained utilizing counterpoise-corrected ωB97X-D with the aug-cc-pVDZ basis set.

Probing substituent effects in aryl-aryl interactions using stereoselective Diels-Alder cycloadditions

Stereoselective Diels-Alder cycloadditions that probe substituent effects in aryl-aryl sandwich complexes were studied experimentally and theoretically. Computations on model systems demonstrate that the stereoselectivity in these reactions is mediated by differential pi-stacking interactions in competing transition states. This allows relative stacking free energies of substituted and unsubstituted sandwich complexes to be derived from measured product distributions. In contrast to gas-phase computations, dispersion effects do not appear to play a significant role in the substituent effects, in accord with previous experiments. The experimental pi-stacking free energies are shown to correlate well with Hammett sigma(m) constants (r = 0.96). These substituent constants primarily provide a measure of the inductive electron-donating and -withdrawing character of the substituents, not donation into or out of the benzene pi-system. The present experimental results are most readily explained using a recently proposed model of substituent effects in the benzene sandwich dimer in which the pi-system of the substituted benzene is relatively unimportant and substituent effects arise from direct through-space interactions. Specifically, these results are the first experiments to clearly show that OMe enhances these pi-stacking interactions, despite being a pi-electron donor. This is in conflict with popular models in which substituent effects in aryl-aryl interactions are modulated by polarization of the aryl pi-system.

On the Structure and Geometry of Biomolecular Binding Motifs (Hydrogen-Bonding, Stacking, X-H···π): WFT and DFT Calculations

The strengths of noncovalent interactions are generally very sensitive to a number of geometric parameters. Among the most important of these parameters is the separation between the interacting moieties (in the case of an intermolecular interaction, this would be the intermolecular separation). Most works seeking to characterize the properties of intermolecular interactions are mainly concerned with binding energies obtained at the potential energy minimum (as determined at some particular level of theory). In this work, in order to extend our understanding of these types of noncovalent interactions, we investigate the distance dependence of several types of intermolecular interactions, these are hydrogen bonds, stacking interactions, dispersion interactions, and X-H···π interactions. There are several methods that have traditionally been used to treat noncovalent interactions as well as many new methods that have emerged within the past three or four years. Here we obtain reference data using estimated CCSD(T) values at the complete basis set limit (using the CBS(T) method); potential energy curves are also produced using several other methods thought to be accurate for intermolecular interactions, these are MP2/cc-pVTZ, MP2/aug-cc-pVDZ, MP2/6-31G*(0.25), SCS(MI)-MP2/cc-pVTZ, estimated MP2.5/CBS, DFT-SAPT/aug-cc-pVTZ, DFT/M06-2X/6-311+G(2df,2p), and DFT-D/TPSS/6-311++G(3df,3pd). The basis set superposition error is systematically considered throughout the study. It is found that the MP2.5 and DFT-SAPT methods, which are both quite computationally intensive, produce potential energy curves that are in very good agreement to those of the reference method. Among the MP2 techniques, which can be said to be of medium computational expense, the best results are obtained with MP2/cc-pVTZ and SCS(MI)-MP2/cc-pVTZ. DFT-D/TPSS/6-311++G(3df,3pd) is the DFT-based method that can be said to give the most well-balanced description of intermolecular interactions.

Ab initio calculation of van der Waals bonded molecular crystals

Intermolecular interactions in the van der Waals bonded benzene crystal are studied from first principles, by combining exact exchange energies with correlation energies defined by the adiabatic connection fluctuation-dissipation theorem, within the random phase approximation. Correlation energies are evaluated using an iterative procedure to compute the eigenvalues of dielectric matrices, which eliminates the computation of unoccupied electronic states. Our results for the structural and binding properties of solid benzene are in very good agreement with experimental results and show that the framework adopted here is a very promising one to investigate molecular crystals and other condensed systems bound by dispersion forces.

Importance of van der Waals interactions in liquid water

We present ab initio molecular dynamics studies on liquid water using density functional theory in conjunction with either dispersion-corrected atom-centered potentials or empirical van der Waals corrections. Our results show that improving the description of van der Waals interactions in DFT-GGA leads to a softening of liquid water's structure with higher mobility. The results obtained with dispersion-corrected atom-centered potentials are especially encouraging. In particular, the radial distribution functions are in better agreement with experiment, and the self-diffusion coefficient increases by more than three-fold compared with the one predicted by the BLYP functional. This work demonstrates that van der Waals interactions are essential in fine-tuning both structural and dynamical properties of liquid water.

Modeling the interplay of inter- and intramolecular hydrogen bonding in conformational polymorphs

The predicted stability differences of the conformational polymorphs of oxalyl dihydrazide and ortho-acetamidobenzamide are unrealistically large when the modeling of intermolecular energies is solely based on the isolated-molecule charge density, neglecting charge density polarization. Ab initio calculated crystal electron densities showed qualitative differences depending on the spatial arrangement of molecules in the lattice with the greatest variations observed for polymorphs that differ in the extent of inter- and intramolecular hydrogen bonding. We show that accounting for induction dramatically alters the calculated stability order of the polymorphs and reduces their predicted stability differences to be in better agreement with experiment. Given the challenges in modeling conformational polymorphs with marked differences in hydrogen bonding geometries, we performed an extensive periodic density functional study with a range of exchange-correlation functionals using both atomic and plane wave basis sets. Although such electronic structure methods model the electrostatic and polarization contributions well, the underestimation of dispersion interactions by current exchange-correlation functionals limits their applicability. The use of an empirical dispersion-corrected density functional method consistently reduces the structural deviations between the experimental and energy minimized crystal structures and achieves plausible stability differences. Thus, we have established which types of models may give worthwhile relative energies for crystal structures and other condensed phases of flexible molecules with intra- and intermolecular hydrogen bonding capabilities, advancing the possibility of simulation studies on polymorphic pharmaceuticals.

Validation of dispersion-corrected density functional theory approaches for ionic liquid systems

The performance of several general gradient approximation, meta general gradient approximation, and hybrid functionals is tested against Møller-Plesset perturbation theory second-order for ionic liquid systems. Additionally, two dispersion-corrected approaches (addition of van der Waals forces by a 1/r(6) term and employing a dispersion-corrected atom-center dispersion pseudopotential) were studied. For the 1-butyl-3-methylimidazolium cation neglecting dispersion results in different trends for structural stabilities. The two applied correction schemes for density functional theory improve the results tremendously. Investigating several 1-butyl-3-methylimidazolium dicianamide ion pairs shows a mean absolute deviation from Møller-Plesset perturbation theory of 35.7 kJ/mol for Hartree-Fock and up to 33.2 kJ/mol for the density functional theory methods. The dispersion-corrected methods reduce the mean absolute deviation to less than 10 kJ/mol. Comparing adducts of the 1-ethyl-3-methylimidazolium dicianamide ion pair with Diels-Alder educts (cyclopentadiene and methylacrylate) shows similar energetic differences as for the ion pairs. Furthermore large deviations in geometries for the intermolecular distances were found for the Hartree-Fock approach (mean absolute deviation: 190 pm) and density functional theory (mean absolute deviation up to 178 pm) while for the dispersion-corrected methods the mean absolute deviation is less than 50 pm.

Predicting noncovalent interactions between aromatic biomolecules with London-dispersion-corrected DFT

Within the framework of Kohn-Sham density functional theory, interaction energies of hydrogen bonded and pi-pi stacked supramolecular complexes of aromatic heterocycles, nucleobase pairs, and complexes of nucleobases with the anti-cancer agent ellipticine as well as its derivatives are evaluated. Dispersion-corrected atom-centered potentials (DCACPs) are employed together with a generalized gradient approximation to the exchange correlation functional. For all systems presented, the DCACP calculations are in very good agreement with available post Hartree-Fock quantum chemical results. Estimates of 3-body contributions (<15% of the respective interaction energy) and deformation energies (5-15% of the interaction energy) are given. Based on our results, we predict a strongly bound interaction energy profile for the ellipticine intercalation process with a stabilization of nearly 40 kcal/mol (deformation energy not taken into account) when fully intercalated. The frontier orbitals of the intercalator-nucleobase complex and the corresponding non-intercalated nucleobases are investigated and show significant changes upon intercalation. The results not only offer some insights into the systems investigated but also suggest that DCACPs can serve as an effective way to achieve higher accuracy in density functional theory without incurring an unaffordable computational overhead, paving ways for more realistic studies on biomolecular complexes in the condensed phase.