Distributed molecular polarisabilities and asymptotic intermolecular interaction energies†

Unifying Charge-Flow Polarization Models

We show that several models where electric polarization in molecular systems is modeled by charge-flow between atoms can all be considered as different manifestations of a general underlying mathematical structure. The models can be classified according to whether they employ atomic or bond parameters and whether they employ atom/bond hardness or softness. We show that an ab initio calculated charge response kernel can be considered as the inverse screened Coulombic matrix projected onto the zero-charge subspace, and this may provide a method for deriving charge screening functions to be used in force fields. The analysis suggests that some models contain redundancies, and we argue that a parameterization of charge-flow models in terms of bond softness is preferable as it depends on local quantities and decay to zero upon bond dissociation, while bond hardness depends on global quantities and increases toward infinity upon bond dissociation.

Non-covalent interactions from a Quantum Chemical Topology perspective

About half a century after its little-known beginnings, the quantum topological approach called QTAIM has grown into a widespread, but still not mainstream, methodology of interpretational quantum chemistry. Although often confused in textbooks with yet another population analysis, be it perhaps an elegant but somewhat esoteric one, QTAIM has been enriched with about a dozen other research areas sharing its main mathematical language, such as Interacting Quantum Atoms (IQA) or Electron Localisation Function (ELF), to form an overarching approach called Quantum Chemical Topology (QCT). Instead of reviewing the latter’s role in understanding non-covalent interactions, we propose a number of ideas emerging from the full consequences of the space-filling nature of topological atoms, and discuss how they (will) impact on interatomic interactions, including non-covalent ones. The architecture of a force field called FFLUX, which is based on these ideas, is outlined. A new method called Relative Energy Gradient (REG) is put forward, which is able, by computation, to detect which fragments of a given molecular assembly govern the energetic behaviour of this whole assembly. This method can offer insight into the typical balance of competing atomic energies both in covalent and non-covalent case studies. A brief discussion on so-called bond critical points is given, highlighting concerns about their meaning, mainly in the arena of non-covalent interactions.

Using atomic charges to model molecular polarization

We review different models for introducing electric polarization in force fields, with special focus on methods where polarization is modelled at the atomic charge level. While electric polarization has been included in several force fields, the common approach has been to focus on atomic dipole polarizability. Several approaches allow modelling electric polarization by using charge-flow between charge sites instead, but this has been less exploited, despite that atomic charges and charge-flow is expected to be more important than atomic dipoles and dipole polarizability. A number of challenges are required to be solved for charge-flow models to be incorporated into polarizable force fields, for example how to parameterize the models and how to make them computational efficient.

SAPT codes for calculations of intermolecular interaction energies

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

Automatic Generation of Flexible-Monomer Intermolecular Potential Energy Surfaces

A method is developed for automatic generation of nonreactive intermolecular two-body potential energy surfaces (PESs) including intramonomer degrees of freedom. This method, called flex-autoPES, is an extension of the autoPES method developed earlier, which assumes rigid monomers. In both cases, the whole PES development proceeds without any human intervention. The functional form used is a sum of products of site-site functions (both atomic and off-atomic sites can be used). The leading terms with sites involving different monomers are of physically motivated form. The long-range part of a PES is computed from monomer properties without using any dimer information. The close-range part is fitted to dimer interaction energies computed using electronic structure methods. Virtually any method can be used in such calculations, but the use of symmetry-adapted perturbation theory provides a seamless connection to the long-range part of the PES. The performance of the flex-autoPES code was tested by developing a full-dimensional PES for the water dimer and PESs including only some soft intramonomer degrees of freedom for the ethylene glycol dimer and for the ethylene glycol-water dimer. In the case of the water dimer, the root-mean-square error (RMSE) of the PES from the data points with negative total energies is 0.03 kcal/mol, and we expect this PES to be more accurate than any previously published PES of this type. For the ethylene glycol dimer and the ethylene glycol-water dimers, the analogous RMSEs are 0.25 and 0.1 kcal/mol, respectively.

Molecular dimers of methane clathrates: ab initio potential energy surfaces and variational vibrational states

Motivated by the energetic and environmental relevance of methane clathrates, highly accurate ab initio potential energy surfaces (PESs) have been developed for the three possible dimers of the methane and water molecules: (H₂O)₂, CH₄·H₂O, and (CH₄)₂.

Do Semilocal Density-Functional Approximations Recover Dispersion Energies at Small Intermonomer Separations?

The methods that add dispersion energies to interaction energies computed using density-functional theory (DFT), known as DFT+D methods, taper off the dispersion energies at distances near van der Waals minima and smaller based on an assumption that DFT starts to reproduce the dispersion energies there. We show that this assumption is not correct as the alleged contribution behaves unphysically and originates to a large extent from nonexchange-correlation terms. Thus, dispersion functions correct DFT in this region for deficiencies unrelated to dispersion interactions.

An Estimation of Hybrid Quantum Mechanical Molecular Mechanical Polarization Energies for Small Molecules Using Polarizable Force-Field Approaches

In this work, we report two polarizable molecular mechanics (polMM) force field models for estimating the polarization energy in hybrid quantum mechanical molecular mechanical (QM/MM) calculations. These two models, named the potential of atomic charges (PAC) and potential of atomic dipoles (PAD), are formulated from the ab initio quantum mechanical (QM) response kernels for the prediction of the QM density response to an external molecular mechanical (MM) environment (as described by external point charges). The PAC model is similar to fluctuating charge (FQ) models because the energy depends on external electrostatic potential values at QM atomic sites; the PAD energy depends on external electrostatic field values at QM atomic sites, resembling induced dipole (ID) models. To demonstrate their uses, we apply the PAC and PAD models to 12 small molecules, which are solvated by TIP3P water. The PAC model reproduces the QM/MM polarization energy with a R² value of 0.71 for aniline (in 10,000 TIP3P water configurations) and 0.87 or higher for other 11 solute molecules, while the PAD model has a much better performance with R² values of 0.98 or higher. The PAC model reproduces reference QM/MM hydration free energies for 12 solute molecules with a RMSD of 0.59 kcal/mol. The PAD model is even more accurate, with a much smaller RMSD of 0.12 kcal/mol, with respect to the reference. This suggests that polarization effects, including both local charge distortion and intramolecular charge transfer, can be well captured by induced dipole type models with proper parametrization.

Geometry-dependent distributed polarizability models for the water molecule

Geometry-dependent distributed polarizability models have been constructed by fits to ab initio calculations at the coupled cluster level of theory with up to noniterative triple excitations in an augmented triple-zeta quality basis set for the water molecule in the field of a point charge. The investigated models include (i) charge-flow polarizabilities between chemically bonded atoms, (ii) isotropic or anisotropic dipolar polarizabilities on oxygen atom or on all atoms, and (iii) combinations of models (i) and (ii). For each model, the polarizability parameters have been optimized to reproduce the induction energy of a water molecule polarized by a point charge successively occupying a grid of points surrounding the molecule. The quality of the models is ascertained by examining their ability to reproduce these induction energies as well as the molecular dipolar and quadrupolar polarizabilities. The geometry dependence of the distributed polarizability models has been explored by changing bond lengths and HOH angle to generate 125 molecular structures (reduced to 75 symmetry-unique ones). For each considered model, the distributed polarizability components have been fitted as a function of the geometry by a Taylor expansion in monomer coordinate displacements up to the sum of powers equal to 4.

Determination of structure and properties of molecular crystals from first principles

CONSPECTUS: Until recently, it had been impossible to predict structures of molecular crystals just from the knowledge of the chemical formula for the constituent molecule(s). A solution of this problem has been achieved using intermolecular force fields computed from first principles. These fields were developed by calculating interaction energies of molecular dimers and trimers using an ab initio method called symmetry-adapted perturbation theory (SAPT) based on density-functional theory (DFT) description of monomers [SAPT(DFT)]. For clusters containing up to a dozen or so atoms, interaction energies computed using SAPT(DFT) are comparable in accuracy to the results of the best wave function-based methods, whereas the former approach can be applied to systems an order of magnitude larger than the latter. In fact, for monomers with a couple dozen atoms, SAPT(DFT) is about equally time-consuming as the supermolecular DFT approach. To develop a force field, SAPT(DFT) calculations are performed for a large number of dimer and possibly also trimer configurations (grid points in intermolecular coordinates), and the interaction energies are then fitted by analytic functions. The resulting force fields can be used to determine crystal structures and properties by applying them in molecular packing, lattice energy minimization, and molecular dynamics calculations. In this way, some of the first successful determinations of crystal structures were achieved from first principles, with crystal densities and lattice parameters agreeing with experimental values to within about 1%. Crystal properties obtained using similar procedures but empirical force fields fitted to crystal data have typical errors of several percent due to low sensitivity of empirical fits to interactions beyond those of the nearest neighbors. The first-principles approach has additional advantages over the empirical approach for notional crystals and cocrystals since empirical force fields can only be extrapolated to such cases. As an alternative to applying SAPT(DFT) in crystal structure calculations, one can use supermolecular DFT interaction energies combined with scaled dispersion energies computed from simple atom-atom functions, that is, use the so-called DFT+D approach. Whereas the standard DFT methods fail for intermolecular interactions, DFT+D performs reasonably well since the dispersion correction is used not only to provide the missing dispersion contribution but also to fix other deficiencies of DFT. The latter cancellation of errors is unphysical and can be avoided by applying the so-called dispersionless density functional, dlDF. In this case, the dispersion energies are added without any scaling. The dlDF+D method is also one of the best performing DFT+D methods. The SAPT(DFT)-based approach has been applied so far only to crystals with rigid monomers. It can be extended to partly flexible monomers, that is, to monomers with only a few internal coordinates allowed to vary. However, the costs will increase relative to rigid monomer cases since the number of grid points increases exponentially with the number of dimensions. One way around this problem is to construct force fields with approximate couplings between inter- and intramonomer degrees of freedom. Another way is to calculate interaction energies (and possibly forces) "on the fly", i.e., in each step of lattice energy minimization procedure. Such an approach would be prohibitively expensive if it replaced analytic force fields at all stages of the crystal predictions procedure, but it can be used to optimize a few dozen candidate structures determined by other methods.

Distributed polarizability models for imidazolium-based ionic liquids

Quantum chemical calculations are used to derive distributed polarizability models sufficiently accurate and compact to be used in classical molecular dynamics simulations of imidazolium-based room temperature ionic liquids. Two distributed polarizability models are fitted to reproduce the induction energy of three imidazolium cations (1,3-dimethyl-, 1-ethyl-3-methyl-, and 1-butyl-3-methylimidazolium) and four anions (tetrafluoroborate, hexafluorophosphate, nitrate, and thiocyanate) polarized by a point charge located successively on a grid of surrounding points. The first model includes charge-flow polarizabilities between first-neighbor atoms and isotropic dipolar polarizability on all atoms (except H), while the second model includes anisotropic dipolar polarizabilities on all atoms (except H). For the imidazolium cations, particular attention is given to the transferability of the distributed polarizability sets. The molecular polarizability and its anisotropy rebuilt by the distributed models are found to be in good agreement with the exact ab initio values for the three cations and 23 additional conformers of 1-ethyl-3-methyl-, 1-butyl-3-methyl-, 1-pentyl-3-methyl-, and 1-hexyl-3-methylimidazolium cations.