Single-determinant-based symmetry-adapted perturbation theory without single-exchange approximation

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.

Benchmarking Swaths of Intermolecular Interaction Components with Symmetry-Adapted Perturbation Theory

A benchmark database for interaction energy components of various noncovalent interactions (NCIs) along their dissociation curve is one of the essential needs in theoretical chemistry, especially for the development of force fields and machine-learning methods. We utilize DFT-SAPT or SAPT(DFT) as one of the most accurate methods to generate an extensive stock of the energy components, including dispersion energies extrapolated to the complete basis set limit (CBS). Precise analyses of the created data, and benchmarking the total interaction energies against the best available CCSD(T)/CBS values, reveal different aspects of the methodology and the nature of NCIs. For example, error cancellation effects between the S2 approximation and nonexact xc-potentials occur, and large charge transfer energies in some systems, including heavy atoms, can explain the lower accuracy of DFT-SAPT. This method is perfect for neutral complexes containing light nonmetals, while other systems with heavier atoms should be treated carefully. In the last part, a representative data set for all NCIs is extracted from the original data.

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

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

Classical Exchange Polarization: An Anisotropic Variable Polarizability Model

Polarizability, or the tendency of the electron distribution to distort under an electric field, often depends on the local chemical environment. For example, the polarizability of a chloride ion is larger in gas phase compared to a chloride ion solvated in water. This effect is due to the restriction the Pauli exclusion principle places on the allowed electron states. Because no two electrons can occupy the same state, when a highly polarizable atom comes in close contact with other atoms or molecules, the space of allowed states can dramatically decrease. This constraint suggests that an accurate molecular mechanics polarizability model should depend on the radial distance between neighboring atoms. This paper introduces a variable polarizability model within the framework of the HIPPO (Hydrogen-like Intermolecular Polarizable Potential) force field, by damping the polarizability as a function of the orbital overlap of two atoms. This effectively captures the quantum mechanical exchange polarization effects, without explicit utilization of antisymmetrized wave functions. We show that the variable polarizability model remarkably improves the two-body polarization energies and three-body energies of ion-ion and ion-water systems. Under this model, no manual tuning of atomic polarizabilities for monatomic ions is required; the gas-phase polarizability can be used because an appropriate damping function is able to correct the polarizability at short range.

Physical mechanisms of intermolecular interactions from symmetry-adapted perturbation theory

Symmetry-adapted perturbation theory (SAPT) is a method for computational studies of noncovalent interactions between molecules. This method will be discussed here from the perspective of establishing the paradigm for understanding mechanisms of intermolecular interactions. SAPT interaction energies are obtained as sums of several contributions. Each contribution possesses a clear physical interpretation as it results from some specific physical process. It also exhibits a specific dependence on the intermolecular separation R. The four major contributions are the electrostatic, induction, dispersion, and exchange energies, each due to a different mechanism, valid at any R. In addition, at large R, SAPT interaction energies are seamlessly connected with the corresponding terms in the asymptotic multipole expansion of interaction energy in inverse powers of R. Since such expansion explicitly depends on monomers' multipole moments and polarizabilities, this connection provides additional insights by rigorously relating interaction energies to monomers' properties.

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.

Efficient Density-Fitted Explicitly Correlated Dispersion and Exchange Dispersion Energies

The leading-order dispersion and exchange-dispersion terms in symmetry-adapted perturbation theory (SAPT), E_disp⁽²⁰⁾ and E_exch-disp⁽²⁰⁾, suffer from slow convergence to the complete basis set limit. To alleviate this problem, explicitly correlated variants of these corrections, E_disp⁽²⁰⁾-F12 and E_exch-disp⁽²⁰⁾-F12, have been proposed recently. However, the original formalism (M., Kodrycka , J. Chem. Theory Comput. 2019, 15, 5965-5986), while highly successful in terms of improving convergence, was not competitive to conventional orbital-based SAPT in terms of computational efficiency due to the need to manipulate several kinds of two-electron integrals. In this work, we eliminate this need by decomposing all types of two-electron integrals using robust density fitting. We demonstrate that the error of the density fitting approximation is negligible when standard auxiliary bases such as aug-cc-pVXZ/MP2FIT are employed. The new implementation allowed us to study all complexes in the A24 database in basis sets up to aug-cc-pV5Z, and the E_disp⁽²⁰⁾-F12 and E_exch-disp⁽²⁰⁾-F12 values exhibit vastly improved basis set convergence over their conventional counterparts. The well-converged E_disp⁽²⁰⁾-F12 and E_exch-disp⁽²⁰⁾-F12 numbers can be substituted for conventional E_disp⁽²⁰⁾ and E_exch-disp⁽²⁰⁾ ones in a calculation of the total SAPT interaction energy at any level (SAPT0, SAPT2+3, ...). We show that the addition of F12 terms does not improve the accuracy of low-level SAPT treatments. However, when the theory errors are minimized in high-level SAPT approaches such as SAPT2+3(CCD)δMP2, the reduction of basis set incompleteness errors thanks to the F12 treatment substantially improves the accuracy of small-basis calculations.

A non-empirical intermolecular force-field for trinitrobenzene and its application in crystal structure prediction

An anisotropic atom-atom distributed intermolecular force-field (DIFF) for rigid trinitrobenzene (TNB) is developed using distributed multipole moments, dipolar polarizabilities, and dispersion coefficients derived from the charge density of the isolated molecule. The short-range parameters of the force-field are fitted to first- and second-order symmetry-adapted perturbation theory dimer interaction energy calculations using the distributed density-overlap model to guide the parameterization of the short-range anisotropy. The second-order calculations are used for fitting the damping coefficients of the long-range dispersion and polarization and also for relaxing the isotropic short-range coefficients in the final model, DIFF-srL2(rel). We assess the accuracy of the unrelaxed model, DIFF-srL2(norel), and its equivalent without short-range anisotropy, DIFF-srL0(norel), as these models are easier to derive. The model potentials are contrasted with empirical models for the repulsion-dispersion fitted to organic crystal structures with multipoles of iterated stockholder atoms (ISAs), FIT(ISA,L4), and with Gaussian Distributed Analysis (GDMA) multipoles, FIT(GDMA,L4), commonly used in modeling organic crystals. The potentials are tested for their ability to model the solid state of TNB. The non-empirical models provide more reasonable relative lattice energies of the three polymorphs of TNB and propose more sensible hypothetical structures than the empirical force-field (FIT). The DIFF-srL2(rel) model successfully has the most stable structure as one of the many structures that match the coordination sphere of form III. The neglect of the conformational flexibility of the nitro-groups is a significant approximation. This methodology provides a step toward force-fields capable of representing all phases of a molecule in molecular dynamics simulations.

Nonapproximated third-order exchange induction energy in symmetry-adapted perturbation theory

The exchange terms in symmetry-adapted perturbation theory (SAPT) are normally calculated within the so-called S² or single exchange approximation, which approximates the all-electron antisymmetrizer by interchanges of at most one electron pair between the interacting molecules. This approximation is typically very accurate at the van der Waals minimum separation and at larger intermolecular distances but begins to deteriorate at short range. Nonapproximated expressions for the second-order SAPT exchange corrections have been derived some time ago by Schäffer and Jansen [Mol. Phys. 111, 2570 (2013)]. In this work, we extend Schäffer and Jansen's formalism to derive and implement a nonapproximated expression for the third-order exchange-induction correction. Numerical tests on several representative noncovalent databases show that the S² approximation underestimates the exchange-induction contributions in both second and third orders. This underestimation is very similar in relative terms, but the larger absolute values of the third-order exchange-induction effects, and their near complete cancellation with the corresponding induction energies, make the third-order errors more severe. In the worst-case scenario of interactions involving ions, the breakdown of the S² approximation can result in a qualitatively wrong, attractive character of SAPT total energies at short range {as first observed by Lao and Herbert [J. Phys. Chem. A 116, 3042 (2012)]}. As expected, the inclusion of the full third-order exchange-induction energy in place of its S²-approximated counterpart restores the correct, repulsive short-range behavior of the SAPT potential energy curves computed through the third order.

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.

The Molpro quantum chemistry package

Molpro is a general purpose quantum chemistry software package with a long development history. It was originally focused on accurate wavefunction calculations for small molecules but now has many additional distinctive capabilities that include, inter alia, local correlation approximations combined with explicit correlation, highly efficient implementations of single-reference correlation methods, robust and efficient multireference methods for large molecules, projection embedding, and anharmonic vibrational spectra. In addition to conventional input-file specification of calculations, Molpro calculations can now be specified and analyzed via a new graphical user interface and through a Python framework.

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.

Physical nature of silane⋯carbene dimers revealed by state-of-the-art ab initio calculations

Using the SAPT2 + 3(CCD)δMP2 method in complete basis set (CBS) limit, it is shown that the interactions in the recently studied silane⋯carbene dimers are mainly dispersive in nature. Consequently, slow convergence of dispersion energy also forces slow convergence of the interaction energy. Therefore, obtaining very accurate values requires extrapolation of the correlation part to the CBS limit. The most accurate values obtained at the CCSD(T)/CBS level of theory show that the studied silane⋯carbene dimers are rather weakly bound, with interaction energies ranging from about -1.9 to -1.3 kcal/mol. Comparing to CCSD(T)/CBS, it will be shown that SCS-MP2 and MP2C methods clearly underestimate and methods based on SAPT2+ and having some third-order corrections, as well as the MP2 method, overestimate values of interaction energies. Popular SAPT(DFT) method performs better than SCS-MP2 and MP2C; nevertheless, underestimation is still considerable. The underestimation is slightly quenched if third-order dispersion energy and its exchange counterpart is added to the SAPT(DFT). The closest value of CCSD(T)/CBS has been given by the SAPT2 + (3)(CCD)δMP2 method in quadruple-ζ basis set. © 2019 Wiley Periodicals, Inc.

Interactions of benzene, naphthalene, and azulene with alkali-metal and alkaline-earth-metal atoms for ultracold studies

We consider collisional properties of polyatomic aromatic hydrocarbon molecules immersed into ultracold atomic gases and investigate intermolecular interactions of exemplary benzene, naphthalene, and azulene with alkali-metal (Li, Na, K, Rb, and Cs) and alkaline-earth-metal (Mg, Ca, Sr, and Ba) atoms. We apply the state-of-the-art ab initio techniques to compute the potential energy surfaces (PESs). We use the coupled cluster method restricted to single, double, and noniterative triple excitations to reproduce the correlation energy and the small-core energy-consistent pseudopotentials to model the scalar relativistic effects in heavier metal atoms. We also report the leading long-range isotropic and anisotropic dispersion and induction interaction coefficients. The PESs are characterized in detail, and the nature of intermolecular interactions is analyzed and benchmarked using symmetry-adapted perturbation theory. The full three-dimensional PESs are provided for the selected systems within the atom-bond pairwise additive representation and can be employed in scattering calculations. The present study of the electronic structure is the first step toward the evaluation of prospects for sympathetic cooling of polyatomic aromatic molecules with ultracold atoms. We suggest azulene, an isomer of naphthalene which possesses a significant permanent electric dipole moment and optical transitions in the visible range, as a promising candidate for electric field manipulation and buffer-gas or sympathetic cooling.

ZMP-SAPT: DFT-SAPT using ab initio densities

Symmetry Adapted Perturbation Theory (SAPT) has become an important tool when predicting and analyzing intermolecular interactions. Unfortunately, Density Functional Theory (DFT)-SAPT, which uses DFT for the underlying monomers, has some arbitrariness concerning the exchange-correlation potential and the exchange-correlation kernel involved. By using ab initio Brueckner Doubles densities and constructing Kohn-Sham orbitals via the Zhao-Morrison-Parr (ZMP) method, we are able to lift the dependence of DFT-SAPT on DFT exchange-correlation potential models in first order. This way, we can compute the monomers at the coupled-cluster level of theory and utilize SAPT for the intermolecular interaction energy. The resulting ZMP-SAPT approach is tested for small dimer systems involving rare gas atoms, cations, and anions and shown to compare well with the Tang-Toennies model and coupled cluster results.

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.

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.

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.

The nature of three-body interactions in DFT: Exchange and polarization effects

We propose a physically motivated decomposition of density functional theory (DFT) 3-body nonadditive interaction energies into the exchange and density-deformation (polarization) components. The exchange component represents the effect of the Pauli exclusion in the wave function of the trimer and is found to be challenging for density functional approximations (DFAs). The remaining density-deformation nonadditivity is less dependent upon the DFAs. Numerical demonstration is carried out for rare gas atom trimers, Ar₂-HX (X = F, Cl) complexes, and small hydrogen-bonded and van der Waals molecular systems. None of the tested semilocal, hybrid, and range-separated DFAs properly accounts for the nonadditive exchange in dispersion-bonded trimers. By contrast, for hydrogen-bonded systems, range-separated DFAs achieve a qualitative agreement to within 20% of the reference exchange energy. A reliable performance for all systems is obtained only when the monomers interact through the Hartree-Fock potential in the dispersion-free Pauli blockade scheme. Additionally, we identify the nonadditive second-order exchange-dispersion energy as an important but overlooked contribution in force-field-like dispersion corrections. Our results suggest that range-separated functionals do not include this component, although semilocal and global hybrid DFAs appear to imitate it in the short range.

Challenging Dogmas: Hydrogen Bond Revisited

Hydrogen bond directionality in the water dimer is explained on the basis of symmetry-adapted intermolecular perturbation theory which directly separates the intermolecular interaction energy into four physically interpretable components: electrostatics, exchange-repulsion, dispersion, and induction. Analysis of these four main contributions to the binding energy allows a deeper understanding of the dominant factors ruling the mutual arrangement of the two monomers. A preference for the linear configuration is shown to be due to a subtle interplay of all four energy components. While the first-order terms, electrostatic and exchange-repulsion, almost perfectly cancel each other near the equilibrium geometry of the dimer, the importance of the second- and higher-order terms, induction and dispersion, becomes evident.

Analysis of transition state stabilization by non-covalent interactions in the Houk-List model of organocatalyzed intermolecular Aldol additions using functional-group symmetry-adapted perturbation theory

Rational design of catalysts would be aided by a better understanding of how non-covalent interactions stabilize transition states. Here, we apply the newly-developed Functional-Group Symmetry-Adapted Perturbation Theory (F-SAPT) to quantify non-covalent interactions in transition states of the proline-catalyzed intermolecular aldol reaction between benzaldehyde and cyclohexanone, according to the Houk-List mechanism [Bahmanyar et al., J. Am. Chem. Soc., 2003, 125, 2475]. A recent re-examination of this organocatalytic reaction by Rzepa and co-workers [Armstrong et al., Chem. Sci., 2014, 5, 2057] used electron density analysis to identify three key non-covalent interactions thought to influence stereoselectivity: (1) a favorable electrostatic interaction (originally identified by Houk and List) between the NCH(δ+) group of the enamine intermediate and the (δ-)O[double bond, length as m-dash]C of benzaldehyde; (2) a C-H/π interaction between the cyclohexene group of the enamine intermediate and the benzaldehyde phenyl ring; (3) a stabilizing contact between an ortho-hydrogen of the phenyl and an oxygen of the carboxylic acid group of the enamine. These three interactions have been directly computed using F-SAPT, which confirms the stabilizing interaction between an ortho-hydrogen and the carboxylic acid in the (S,S) and (R,S) transition state stereoisomers. F-SAPT analysis also finds stabilizing dispersion and electrostatic interactions due to a C-H/π interaction between the cyclohexene and phenyl groups in the (S,S) and (R,R) transition states. However, unfavorable exchange-repulsion cancels the attractive terms that favor these stereoisomers. Surprisingly, the interaction thought to be most important for stereoselectivity, the NCH(δ+)(δ-)O[double bond, length as m-dash]C interaction, is actually found to be repulsive due to the negative charge on the nitrogen. Hence, our results indicate that geometric analysis and/or density-based analysis does not necessarily produce a reliable picture of non-covalent stabilization. As confirmed by high-level coupled-cluster computations, intermolecular interaction energies are strongest for the (R,R) transition states, which are not the experimentally favored products. This suggests that at least for this reaction, stereoselectivity is also strongly dependent on the energy required to distort the reacting molecules into the transition state geometry.

Intermolecular symmetry-adapted perturbation theory study of large organic complexes

Binding energies for the complexes of the S12L database by Grimme [Chem. Eur. J. 18, 9955 (2012)] were calculated using intermolecular symmetry-adapted perturbation theory combined with a density-functional theory description of the interacting molecules. The individual interaction energy decompositions revealed no particular change in the stabilisation pattern as compared to smaller dimer systems at equilibrium structures. This demonstrates that, to some extent, the qualitative description of the interaction of small dimer systems may be extrapolated to larger systems, a method that is widely used in force-fields in which the total interaction energy is decomposed into atom-atom contributions. A comparison of the binding energies with accurate experimental reference values from Grimme, the latter including thermodynamic corrections from semiempirical calculations, has shown a fairly good agreement to within the error range of the reference binding energies.

Accurate description of intermolecular interactions involving ions using symmetry-adapted perturbation theory

Three new data sets for intermolecular interactions, AHB21 for anion-neutral dimers, CHB6 for cation-neutral dimers, and IL16 for ion pairs, are assembled here, with complete-basis CCSD(T) results for each. These benchmarks are then used to evaluate the accuracy of the single-exchange approximation that is used for exchange energies in symmetry-adapted perturbation theory (SAPT), and the accuracy of SAPT based on wave function and density-functional descriptions of the monomers is evaluated. High-level SAPT calculations afford poor results for these data sets, and this includes the recently proposed "gold", "silver", and "bronze standards" of SAPT, namely, SAPT2+(3)-δMP2/aug-cc-pVTZ, SAPT2+/aug-cc-pVDZ, and sSAPT0/jun-cc-pVDZ, respectively [ Parker , T. M. , et al. , J. Chem. Phys. 2014 , 140 , 094106 ]. Especially poor results are obtained for symmetric shared-proton systems of the form X(-)···H(+)···X(-), for X = F, Cl, or OH. For the anionic data set, the SAPT2+(CCD)-δMP2/aug-cc-pVTZ method exhibits the best performance, with a mean absolute error (MAE) of 0.3 kcal/mol and a maximum error of 0.7 kcal/mol. For the cationic data set, the highest-level SAPT method, SAPT2+3-δMP2/aug-cc-pVQZ, outperforms the rest of the SAPT methods, with a MAE of 0.2 kcal/mol and a maximum error of 0.4 kcal/mol. For the ion-pair data set, the SAPT2+3-δMP2/aug-cc-pVTZ performs the best among all SAPT methods with a MAE of 0.3 kcal/mol and a maximum error of 0.9 kcal/mol. Overall, SAPT2+3-δMP2/aug-cc-pVTZ affords a small and balanced MAE (<0.5 kcal/mol) for all three data sets, with an overall MAE of 0.4 kcal/mol. Despite the breakdown of perturbation theory for ionic systems at short-range, SAPT can still be saved given two corrections: a "δHF" correction, which requires a supermolecular Hartree-Fock calculation to incorporate polarization effects beyond second order, and a "δMP2" correction, which requires a supermolecular MP2 calculation to account for higher-order induction-dispersion coupling. These corrections serve to remove artifacts introduced by the single exchange approximation in the exchange-induction and exchange-dispersion interactions, and obviate the need for ad hoc scaling of the first- and second-order exchange energies. Finally, some density-functional and MP2-based electronic structure methods are assessed as well, and we find that the best density-functional method for computing binding energies in these data sets is B97M-V/aug-cc-pVTZ, which affords a MAE of 0.4 kcal/mol, whereas complete-basis MP2 affords an MAE of 0.3 kcal/mol.

Tractability gains in symmetry-adapted perturbation theory including coupled double excitations: CCD+ST(CCD) dispersion with natural orbital truncations

This work focuses on efficient and accurate treatment of the intermolecular dispersion interaction using the CCD+ST(CCD) dispersion approach formulated by Williams et al. [J. Chem. Phys. 103, 4586 (1995)]. We apply natural orbital truncation techniques to the solution of the monomer coupled-cluster double (CCD) equations, yielding substantial accelerations in this computationally demanding portion of the SAPT2+(CCD), SAPT2+(3)(CCD), and SAPT2+3(CCD) analyses. It is shown that the wholly rate-limiting dimer-basis particle-particle ladder term can be computed in a reduced natural virtual space which is essentially the same size as the monomer-basis virtual space, with an error on the order of a few thousandths of 1 kcal mol(-1). Coupled with our existing natural orbital techniques for the perturbative triple excitation contributions [E. G. Hohenstein and C. D. Sherrill, J. Chem. Phys. 133, 104107 (2010)], this technique provides speedups of greater than an order of magnitude for the evaluation of the complete SAPT2+3(CCD) decomposition, with a total error of a few hundredths of 1 kcal mol(-1). The combined approach yields tractability gains of almost 2× in the system size, allowing for SAPT2+3(CCD)/aug-cc-pVTZ analysis to be performed for systems such as adenine-thymine for the first time. Natural orbital based SAPT2+3(CCD)/aug-cc-pVTZ results are presented for stacked and hydrogen-bonded configurations of uracil dimer and the adenine-thymine dimer.