Symmetry-adapted perturbation theory based on unrestricted Kohn-Sham orbitals for high-spin open-shell van der Waals complexes

Superhalogens inside fullerenes X@C2n (X = BO2, BeF3; 2n = 60, 70)

The exploration of endohedral fullerenes has garnered significant attention recently due to their distinctive chemical, electrochemical, and optoelectronic properties. Charge transfer, which usually occurs from encapsulated species to fullerenes, importantly affects the structures and properties of endohedral fullerenes. In this study, we theoretically investigated endohedral superhalogen fullerenes X@C2n (X = BO2, BeF3; 2n = 60, 70), in which the charge is reversely transferred from the fullerene to the superhalogen, by using density functional theory calculations and ab initio molecular dynamics simulations. Both natural population analysis and the quantum theory of atoms in molecules confirm about one electron transfer from the fullerene to the superhalogen, resulting in the formal valence state of X-@C2n+. Energy decomposition analysis on the interaction between the superhalogen and fullerene revealed that electrostatic energy contributes predominantly to the total interaction energy. These endohedral superhalogen fullerenes with cationic fullerenes were predicted to be able to serve as building blocks for one dimensional fullerene-based nanowires when combined with endohedral alkali-metallofullerenes with anionic fullerenes.

Second quantization-based symmetry-adapted perturbation theory: Generalizing exchange beyond single electron pair approximation

This paper presents a general second-quantized form of a permutation operator interchanging n pairs of electrons between interacting subsystems in the framework of the symmetry-adapted perturbation theory (SAPT). We detail the procedure for constructing this operator through the consecutive multiplication of single-pair permutation operators. This generalized form of the permutation operator has enabled the derivation of universal formulas for S2n approximations of the exchange energies in the first and second order of the interaction operator. We present expressions for corrections of S4 approximations and assess its efficacy on a selection of systems anticipated to exhibit a slowly converging overlap expansion. Additionally, we outline a method to sum the overlap expansion series to infinity in second-quantization, up to the second order in V. This new approach offers an alternative to the existing formalism based on density-matrix formulations. When combined with a symbolic algebra program for automated derivations, it paves the way for advancements in SAPT theory, particularly for intricate wavefunction theories.

Dispersion Energy from the Time-Independent Coupled-Cluster Polarization Propagator

We present a new method of calculation of the dispersion energy in the second-order symmetry-adapted perturbation theory. Using the Longuet-Higgins integral and time-independent coupled-cluster response theory, one shows that the general expression for the dispersion energy can be written in terms of cluster amplitudes and the excitation operators σ, which can be obtained by solving a linear equation. We introduced an approximate scheme dubbed CCPP2(T) for the dispersion energy accurate to the second order of intramonomer correlation, which includes certain classes to be summed to infinity. Assessment of the accuracy of the CCPP2(T) dispersion energy against the FCI dispersion for He₂ demonstrates its high accuracy. For more complex systems, CCPP2(T) matches the accuracy of the best methods introduced for calculations of dispersion so far. The method can be extended to higher-order levels of excitations, providing a systematically improvable theory of dispersion interaction.

Symmetry-Adapted Perturbation Theory Based on Multiconfigurational Wave Function Description of Monomers

We present a formulation of the multiconfigurational (MC) wave function symmetry-adapted perturbation theory (SAPT). The method is applicable to noncovalent interactions between monomers which require a multiconfigurational description, in particular when the interacting system is strongly correlated or in an electronically excited state. SAPT(MC) is based on one- and two-particle reduced density matrices of the monomers and assumes the single-exchange approximation for the exchange energy contributions. Second-order terms are expressed through response properties from extended random phase approximation (ERPA). The dispersion components of SAPT(MC) have been introduced in our previous works [HapkaM.J. Chem. Theory Comput.2019, 15, 1016−102730525591; HapkaM.J. Chem. Theory Comput.2019, 15, 6712–672331670950]. SAPT(MC) is applied either with generalized valence bond perfect pairing (GVB) or with complete active space self-consistent field (CASSCF) treatment of the monomers. We discuss two model multireference systems: the H₂ ··· H₂ dimer in out-of-equilibrium geometries and interaction between the argon atom and excited state of ethylene. Using the C₂H₄* ··· Ar complex as an example, we examine second-order terms arising from negative transitions in the linear response function of an excited monomer. We demonstrate that the negative-transition terms must be accounted for to ensure qualitative prediction of induction and dispersion energies and develop a procedure allowing for their computation. Factors limiting the accuracy of SAPT(MC) are discussed in comparison with other second-order SAPT schemes on a data set of small single-reference dimers.

Unravelling the Interactions of Magnetic Ionic Liquids by Energy Decomposition Schemes: Towards a Transferable Polarizable Force Field

This work aims at unravelling the interactions in magnetic ionic liquids (MILs) by applying Symmetry-Adapted Perturbation Theory (SAPT) calculations, as well as based on those to set-up a polarisable force field model for these liquids. The targeted MILs comprise two different cations, namely: 1-butyl-3-methylimidazolium ([Bmim]⁺) and 1-ethyl-3-methylimidazolium ([Emim]⁺), along with several metal halides anions such as [FeCl₄]⁻, [FeBr₄]⁻, [ZnCl₃]⁻ and [SnCl₄]²⁻ To begin with, DFT geometry optimisations of such MILs were performed, which in turn revealed that the metallic anions prefer to stay close to the region of the carbon atom between the nitrogen atoms in the imidazolium fragment. Then, a SAPT study was carried out to find the optimal separation of the monomers and the different contributions for their interaction energy. It was found that the main contribution to the interaction energy is the electrostatic interaction component, followed by the dispersion one in most of the cases. The SAPT results were compared with those obtained by employing the local energy decomposition scheme based on the DLPNO-CCSD(T) method, the latter showing slightly lower values for the interaction energy as well as an increase of the distance between the minima centres of mass. Finally, the calculated SAPT interaction energies were found to correlate well with the melting points experimentally measured for these MILs.

Dataset of noncovalent intermolecular interaction energy curves for 24 small high-spin open-shell dimers

We introduce a dataset of 24 interaction energy curves of open-shell noncovalent dimers, referred to as the O24 × 5 dataset. The dataset consists of high-spin dimers up to 11 atoms selected to assure diversity with respect to interaction types: dispersion, electrostatics, and induction. The benchmark interaction energies are obtained at the restricted open-shell CCSD(T) level of theory with complete basis set extrapolation (from aug-cc-pVQZ to aug-cc-pV5Z). We have analyzed the performance of selected wave function methods MP2, CCSD, and CCSD(T) as well as the F12a and F12b variants of coupled-cluster theory. In addition, we have tested dispersion-corrected density functional theory methods based on the PBE exchange-correlation model. The O24 × 5 dataset is a challenge to approximate methods due to the wide range of interaction energy strengths it spans. For the dispersion-dominated and mixed-type subsets, any tested method that does not include the triples contribution yields errors on the order of tens of percent. The electrostatic subset is less demanding with errors that are typically an order of magnitude smaller than the mixed and dispersion-dominated subsets.

Spectroscopy and Scattering Studies Using Interpolated Ab Initio Potentials

The Born-Oppenheimer potential energy surface (PES) has come a long way since its introduction in the 1920s, both conceptually and in predictive power for practical applications. Nevertheless, nearly 100 years later-despite astonishing advances in computational power-the state-of-the-art first-principles prediction of observables related to spectroscopy and scattering dynamics is surprisingly limited. For example, the water dimer, (H₂O)₂, with only six nuclei and 20 electrons, still presents a formidable challenge for full-dimensional variational calculations of bound states and is considered out of reach for rigorous scattering calculations. The extremely poor scaling of the most rigorous quantum methods is fundamental; however, recent progress in development of approximate methodologies has opened the door to fairly routine high-quality predictions, unthinkable 20 years ago. In this review, in relation to the workflow of spectroscopy and/or scattering studies, we summarize progress and challenges in the component areas of electronic structure calculations, PES fitting, and quantum dynamical calculations.

Probing radical-molecule interactions with a second generation energy decomposition analysis of DFT calculations using absolutely localized molecular orbitals

Intermolecular interactions between radicals and closed-shell molecules are ubiquitous in chemical processes, ranging from the benchtop to the atmosphere and extraterrestrial space. While energy decomposition analysis (EDA) schemes for closed-shell molecules can be generalized for studying radical-molecule interactions, they face challenges arising from the unique characteristics of the electronic structure of open-shell species. In this work, we introduce additional steps that are necessary for the proper treatment of radical-molecule interactions to our previously developed unrestricted Absolutely Localized Molecular Orbital (uALMO)-EDA based on density functional theory calculations. A "polarize-then-depolarize" (PtD) scheme is used to remove arbitrariness in the definition of the frozen wavefunction, rendering the ALMO-EDA results independent of the orientation of the unpaired electron obtained from isolated fragment calculations. The contribution of radical rehybridization to polarization energies is evaluated. It is also valuable to monitor the wavefunction stability of each intermediate state, as well as their associated spin density profiles, to ensure the EDA results correspond to a desired electronic state. These radical extensions are incorporated into the "vertical" and "adiabatic" variants of uALMO-EDA for studies of energy changes and property shifts upon complexation. The EDA is validated on two model complexes, H₂O˙F and FH˙OH. It is then applied to several chemically interesting radical-molecule complexes, including the sandwiched and T-shaped benzene dimer radical cation, complexes of pyridine with benzene and naphthalene radical cations, binary and ternary complexes of the hydroxyl radical with water (˙OH(H₂O) and ˙OH(H₂O)₂), and the pre-reactive complexes and transition states in the ˙OH + HCHO and ˙OH + CH₃CHO reactions. These examples suggest that this second generation uALMO-EDA is a useful tool for furthering one's understanding of both energetic and property changes associated with radical-molecule interactions.

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.

Psi4 1.4: Open-source software for high-throughput quantum chemistry

PSI4 is a free and open-source ab initio electronic structure program providing implementations of Hartree-Fock, density functional theory, many-body perturbation theory, configuration interaction, density cumulant theory, symmetry-adapted perturbation theory, and coupled-cluster theory. Most of the methods are quite efficient, thanks to density fitting and multi-core parallelism. The program is a hybrid of C++ and Python, and calculations may be run with very simple text files or using the Python API, facilitating post-processing and complex workflows; method developers also have access to most of PSI4's core functionalities via Python. Job specification may be passed using The Molecular Sciences Software Institute (MolSSI) QCSCHEMA data format, facilitating interoperability. A rewrite of our top-level computation driver, and concomitant adoption of the MolSSI QCARCHIVE INFRASTRUCTURE project, makes the latest version of PSI4 well suited to distributed computation of large numbers of independent tasks. The project has fostered the development of independent software components that may be reused in other quantum chemistry programs.

Evidence of Nonrigidity Effects in the Description of Low-Energy Anisotropic Molecular Collisions of Hydrogen Molecules with Excited Metastable Helium Atoms

Cold collisions serve as a sensitive probe of the interaction potential. In the recent study of Klein et al. (Nature Phys.2017, 13, 35–38), the one-parameter scaling of the interaction potential was necessary to obtain agreement between theoretical and observed patterns of the orbiting resonances for excited metastable helium atoms colliding with hydrogen molecules. Here, we show that the effect of nonrigidity of the H₂ molecule on the resonant structure, absent in the previous study, is critical to predict the correct positions of the resonances in that case. We have complemented the theoretical description of the interaction potential and revised reaction rate coefficients by proper inclusion of the flexibility of the molecule. The calculated reaction rate coefficients are in remarkable agreement with the experimental data without empirical adjustment of the interaction potential. We have shown that even state-of-the-art calculations of the interaction energy cannot ensure agreement with the experiment if such an important physical effect as flexibility of the interacting molecule is neglected. Our findings about the significance of the nonrigidity effects can be especially crucial in cold chemistry, where the quantum nature of molecules is pronounced.

Second-Order Exchange-Dispersion Energy Based on a Multireference Description of Monomers

We present a method for calculation of the second-order exchange-dispersion energy in the framework of the symmetry-adapted perturbation theory (SAPT) for weakly interacting monomers described with multiconfigurational wave functions. The proposed formalism is based on response properties obtained from extended random phase approximation (ERPA) equations and assumes the single-exchange (S²) approximation. The approach is applicable to closed shell systems where static correlation cannot be neglected or to systems in nondegenerate excited states. We examine the new method in combination with either generalized valence bond perfect pairing (GVB) or complete active space self-consistent field (CASSCF) description of the interacting monomers. For model multireference dimers in ground states (H₂···H₂, Be···Be, He···H₂), exchange-dispersion energies are reproduced accurately. For the interaction between the excited hydrogen molecule and the helium atom we found unacceptably large errors which is attributed to the neglect of diagonal double excitations in the employed approximation to the linear response function.

Spin splittings from first-order symmetry-adapted perturbation theory without single-exchange approximation

The recently proposed spin-flip symmetry-adapted perturbation theory (SF-SAPT) first-order exchange energy [Patkowski et al., J. Chem. Phys. 148, 164110 (2018)] enables the standard open-shell SAPT approach to treat arbitrary spin states of the weakly interacting complex. Here, we further extend first-order SF-SAPT beyond the single-exchange approximation to a complete treatment of the exchanges of electrons between monomers. This new form of the exchange correction replaces the single-exchange approximation with a more moderate single-spin-flip approximation. The newly developed expressions are applied to a number of small test systems to elucidate the quality of both approximations. They are also applied to the singlet-triplet splittings in pancake bonded dimers. The accuracy of the single-exchange approximation deteriorates at short intermolecular separations, especially for systems with few electrons and for the high-spin state of the complex. In contrast, the single-spin-flip approximation is exact for interactions involving a doublet molecule and remains highly accurate for any number of unpaired electrons. Because the single-exchange approximation affects the high-spin and low-spin states of pancake bonded complexes evenly, the resulting splitting values are of similar accuracy to those produced by the formally more accurate single-spin-flip approximation.

Local Energy Decomposition of Open-Shell Molecular Systems in the Domain-Based Local Pair Natural Orbital Coupled Cluster Framework

Local energy decomposition (LED) analysis decomposes the interaction energy between two fragments calculated at the domain-based local pair natural orbital CCSD(T) (DLPNO-CCSD(T)) level of theory into a series of chemically meaningful contributions and has found widespread applications in the study of noncovalent interactions. Herein, an extension of this scheme that allows for the analysis of interaction energies of open-shell molecular systems calculated at the UHF-DLPNO-CCSD(T) level is presented. The new scheme is illustrated through applications to the CH₂···X (X = He, Ne, Ar, Kr, and water) and heme···CO interactions in the low-lying singlet and triplet spin states. The results are used to discuss the mechanism that governs the change in the singlet–triplet energy gap of methylene and heme upon adduct formation.

First-order symmetry-adapted perturbation theory for multiplet splittings

We present a symmetry-adapted perturbation theory (SAPT) for the interaction of two high-spin open-shell molecules (described by their restricted open-shell Hartree-Fock determinants) resulting in low-spin states of the complex. The previously available SAPT formalisms, except for some system-specific studies for few-electron complexes, were restricted to the high-spin state of the interacting system. Thus, the new approach provides, for the first time, a SAPT-based estimate of the splittings between different spin states of the complex. We have derived and implemented the lowest-order SAPT term responsible for these splittings, that is, the first-order exchange energy. We show that within the so-called S² approximation commonly used in SAPT (neglecting effects that vanish as fourth or higher powers of intermolecular overlap integrals), the first-order exchange energies for all multiplets are linear combinations of two matrix elements: a diagonal exchange term that determines the spin-averaged effect and a spin-flip term responsible for the splittings between the states. The numerical factors in this linear combination are determined solely by the Clebsch-Gordan coefficients: accordingly, the S² approximation implies a Heisenberg Hamiltonian picture with a single coupling strength parameter determining all the splittings. The new approach is cast into both molecular-orbital and atomic-orbital expressions: the latter enable an efficient density-fitted implementation. We test the newly developed formalism on several open-shell complexes ranging from diatomic systems (Li⋯H, Mn⋯Mn, …) to the phenalenyl dimer.

Perspective: Found in translation: Quantum chemical tools for grasping non-covalent interactions

Today's quantum chemistry methods are extremely powerful but rely upon complex quantities such as the massively multidimensional wavefunction or even the simpler electron density. Consequently, chemical insight and a chemist's intuition are often lost in this complexity leaving the results obtained difficult to rationalize. To handle this overabundance of information, computational chemists have developed tools and methodologies that assist in composing a more intuitive picture that permits better understanding of the intricacies of chemical behavior. In particular, the fundamental comprehension of phenomena governed by non-covalent interactions is not easily achieved in terms of either the total wavefunction or the total electron density, but can be accomplished using more informative quantities. This perspective provides an overview of these tools and methods that have been specifically developed or used to analyze, identify, quantify, and visualize non-covalent interactions. These include the quantitative energy decomposition analysis schemes and the more qualitative class of approaches such as the Non-covalent Interaction index, the Density Overlap Region Indicator, or quantum theory of atoms in molecules. Aside from the enhanced knowledge gained from these schemes, their strengths, limitations, as well as a roadmap for expanding their capabilities are emphasized.

Psi4 1.1: An Open-Source Electronic Structure Program Emphasizing Automation, Advanced Libraries, and Interoperability

Psi4 is an ab initio electronic structure program providing methods such as Hartree-Fock, density functional theory, configuration interaction, and coupled-cluster theory. The 1.1 release represents a major update meant to automate complex tasks, such as geometry optimization using complete-basis-set extrapolation or focal-point methods. Conversion of the top-level code to a Python module means that Psi4 can now be used in complex workflows alongside other Python tools. Several new features have been added with the aid of libraries providing easy access to techniques such as density fitting, Cholesky decomposition, and Laplace denominators. The build system has been completely rewritten to simplify interoperability with independent, reusable software components for quantum chemistry. Finally, a wide range of new theoretical methods and analyses have been added to the code base, including functional-group and open-shell symmetry adapted perturbation theory, density-fitted coupled cluster with frozen natural orbitals, orbital-optimized perturbation and coupled-cluster methods (e.g., OO-MP2 and OO-LCCD), density-fitted multiconfigurational self-consistent field, density cumulant functional theory, algebraic-diagrammatic construction excited states, improvements to the geometry optimizer, and the "X2C" approach to relativistic corrections, among many other improvements.

Density-fitted open-shell symmetry-adapted perturbation theory and application to π-stacking in benzene dimer cation and ionized DNA base pair steps

Symmetry-Adapted Perturbation Theory (SAPT) is one of the most popular approaches to energy component analysis of non-covalent interactions between closed-shell systems, yielding both accurate interaction energies and meaningful interaction energy components. In recent years, the full open-shell equations for SAPT up to second-order in the intermolecular interaction and zeroth-order in the intramolecular correlation (SAPT0) were published [P. S. Zuchowski et al., J. Chem. Phys. 129, 084101 (2008); M. Hapka et al., ibid. 137, 164104 (2012)]. Here, we utilize density-fitted electron repulsion integrals to produce an efficient computational implementation. This approach is used to examine the effect of ionization on π-π interactions. For the benzene dimer radical cation, comparison against reference values indicates a good performance for open-shell SAPT0, except in cases with substantial charge transfer. For π stacking between hydrogen-bonded pairs of nucleobases, dispersion interactions still dominate binding, in spite of the creation of a positive charge.

Energy Decomposition Analysis with a Stable Charge-Transfer Term for Interpreting Intermolecular Interactions

Many schemes for decomposing quantum-chemical calculations of intermolecular interaction energies into physically meaningful components can be found in the literature, but the definition of the charge-transfer (CT) contribution has proven particularly vexing to define in a satisfactory way and typically depends strongly on the choice of basis set. This is problematic, especially in cases of dative bonding and for open-shell complexes involving cation radicals, for which one might expect significant CT. Here, we analyze CT interactions predicted by several popular energy decomposition analyses and ultimately recommend the definition afforded by constrained density functional theory (cDFT), as it is scarcely dependent on basis set and provides results that are in accord with chemical intuition in simple cases, and in quantitative agreement with experimental estimates of the CT energy, where available. For open-shell complexes, the cDFT approach affords CT energies that are in line with trends expected based on ionization potentials and electron affinities whereas some other definitions afford unreasonably large CT energies in large-gap systems, which are sometimes artificially offset by underestimation of van der Waals interactions by density functional theory. Our recommended energy decomposition analysis is a composite approach, in which cDFT is used to define the CT component of the interaction energy and symmetry-adapted perturbation theory (SAPT) defines the electrostatic, polarization, Pauli repulsion, and van der Waals contributions. SAPT/cDFT provides a stable and physically motivated energy decomposition that, when combined with a new implementation of open-shell SAPT, can be applied to supramolecular complexes involving molecules, ions, and/or radicals.

Spatial assignment of symmetry adapted perturbation theory interaction energy components: The atomic SAPT partition

We develop a physically-motivated assignment of symmetry adapted perturbation theory for intermolecular interactions (SAPT) into atom-pairwise contributions (the A-SAPT partition). The basic precept of A-SAPT is that the many-body interaction energy components are computed normally under the formalism of SAPT, following which a spatially-localized two-body quasiparticle interaction is extracted from the many-body interaction terms. For electrostatics and induction source terms, the relevant quasiparticles are atoms, which are obtained in this work through the iterative stockholder analysis (ISA) procedure. For the exchange, induction response, and dispersion terms, the relevant quasiparticles are local occupied orbitals, which are obtained in this work through the Pipek-Mezey procedure. The local orbital atomic charges obtained from ISA additionally allow the terms involving local orbitals to be assigned in an atom-pairwise manner. Further summation over the atoms of one or the other monomer allows for a chemically intuitive visualization of the contribution of each atom and interaction component to the overall noncovalent interaction strength. Herein, we present the intuitive development and mathematical form for A-SAPT applied in the SAPT0 approximation (the A-SAPT0 partition). We also provide an efficient series of algorithms for the computation of the A-SAPT0 partition with essentially the same computational cost as the corresponding SAPT0 decomposition. We probe the sensitivity of the A-SAPT0 partition to the ISA grid and convergence parameter, orbital localization metric, and induction coupling treatment, and recommend a set of practical choices which closes the definition of the A-SAPT0 partition. We demonstrate the utility and computational tractability of the A-SAPT0 partition in the context of side-on cation-π interactions and the intercalation of DNA by proflavine. A-SAPT0 clearly shows the key processes in these complicated noncovalent interactions, in systems with up to 220 atoms and 2845 basis functions.