Breakdown of the single-exchange approximation in third-order symmetry-adapted perturbation theory

We report third-order symmetry-adapted perturbation theory (SAPT) calculations for several dimers whose intermolecular interactions are dominated by induction. We demonstrate that the single-exchange approximation (SEA) employed to derive the third-order exchange-induction correction (E(exch-ind)((30))) fails to quench the attractive nature of the third-order induction (E(ind)((30))), leading to one-dimensional potential curves that become attractive rather than repulsive at short intermolecular separations. A scaling equation for (E(exch-ind)((30))), based on an exact formula for the first-order exchange correction, is introduced to approximate exchange effects beyond the SEA, and qualitatively correct potential energy curves that include third-order induction are thereby obtained. For induction-dominated systems, our results indicate that a "hybrid" SAPT approach, in which a dimer Hartree-Fock calculation is performed in order to obtain a correction for higher-order induction, is necessary not only to obtain quantitative binding energies but also to obtain qualitatively correct potential energy surfaces. These results underscore the need to develop higher-order exchange-induction formulas that go beyond the SEA.     

Orbital relaxation and the third-order induction energy in symmetry-adapted perturbation theory

Theoretical investigations of the induction interaction between closed-shell molecules which fully account for the orbital relaxation effects are presented. Explicit expressions for the third-order induction energy in terms of molecular integrals and orbital energies are given and implemented within the sapt2008 program for symmetry-adapted perturbation theory (SAPT) calculations. Numerical investigations for the He–He, He–LiH, Ar–Ar, H₂–CO, H₂O–H₂O, and H₂O–NH₃ model dimers show that the orbital relaxation increases the third-order induction interaction by 15 to 50% at near-equilibrium geometries, with the largest effect observed for complexes involving highly polar monomers. At large intermonomer separations, the relaxed third-order induction energy perfectly recovers the difference $\delta E^{\rm HF}_{\rm int}$ between the Hartree–Fock interaction energy and the sum of the uncorrelated SAPT contributions through second order in the intermolecular interaction operator. At the near-equilibrium geometries, the sum of the relaxed third-order induction and exchange-induction energies reproduces, however, only a small fraction (6 to 15%) of $\delta E^{\rm HF}_{\rm int}$ for the nonpolar systems and about 40 to 60% for the polar ones. A comparison of the complete SAPT calculations with the coupled-cluster treatment with single, double, and noniterative triple excitations [CCSD(T)] suggests that the pure SAPT approach with all the available third-order corrections is more accurate for nonpolar systems while for the polar ones the hybrid approach including $\delta E^{\rm HF}_{\rm int}$ gives better results.     

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.     

Assessing the Accuracy of SAPT(DFT) Interaction Energies by Comparison with Experimentally Derived Noble Gas Potentials and Molecular Crystal Lattice Energies

The density functional version of symmetry‐adapted perturbation theory, SAPT(DFT), is a computationally efficient method for calculating intermolecular interaction energies. We evaluate its accuracy by comparison with experimentally determined noble gas interaction potentials and sublimation enthalpies, most of which have not been previously calculated using this method. In order to compare the results with wavefunction methods, we also calculate these quantities using MP2 and, for noble gas dimers, using CCSD(T). For the crystal lattice energy calculations, we include corrections to the dispersion, electrostatic, and induction energies that account for the finite interaction distance cutoff and higher‐order induction contributions. Overall, the energy values extrapolated to the complete basis set limit show that SAPT(DFT) achieves significantly better agreement with experiment than MP2.     

Intermolecular potentials based on symmetry-adapted perturbation theory with dispersion energies from time-dependent density-functional calculations

Recently, three of us have proposed a method [Phys. Rev. Lett. 91, 33201 (2003)] for an accurate calculation of the dispersion energy utilizing frequency-dependent density susceptibilities of monomers obtained from time-dependent density-functional theory (DFT). In the present paper, we report numerical calculations for the helium, neon, water, and carbon dioxide dimers and show that for a wide range of intermonomer separations, including the van der Waals and short-range repulsion regions, the method provides dispersion energies with accuracies comparable to those that can be achieved using the current most sophisticated wave-function methods. If the dispersion energy is combined with (i) the electrostatic and first-order exchange interaction energies as defined in symmetry-adapted perturbation theory (SAPT) but computed using monomer Kohn-Sham (KS) determinants, and (ii) the induction energy computed using the coupled KS static response theory, (iii) the exchange-induction and exchange-dispersion energies computed using KS orbitals and orbital energies, the resulting method, denoted by SAPT(DFT), produces very accurate total interaction potentials. For the helium dimer, the only system with nearly exact benchmark values, SAPT(DFT) reproduces the interaction energy to within about 2% at the minimum and to a similar accuracy for all other distances ranging from the strongly repulsive to the asymptotic region. For the remaining systems investigated by us, the quality of the SAPT(DFT) interaction energies is so high that these energies may actually be more accurate than the best available results obtained with wave-function techniques. At the same time, SAPT(DFT) is much more computationally efficient than any method previously used for calculating the dispersion and other interaction energy components at this level of accuracy.     

Correlated intermolecular interaction components from asymptotically corrected Kohn-Sham orbitals

In these years there was considerable interest inunderstanding of intermolecular forces in energetic(explosive) systems[1—3]. The supermolecular approach(SM) is widely adopted for calculating ab initio in-termolecular interactions. Nevertheless, it is unable toprovide physically meaningful interaction contribu-tions such as electrostatic, induction, repulsion anddispersion energies. In contrast, the symmetry-adaptedperturbation theory (SAPT)[4—8] has the ability to de-rive these correlated…     

Symmetry-adapted perturbation-theory calculations of intermolecular forces employing density-functional description of monomers

A symmetry-adapted perturbation theory based on Kohn-Sham determinants [SAPT(KS)] and utilizing asymptotically corrected exchange-correlation potentials has been applied to the He2, Ne2, (H2O)2, and (CO2)2 dimers. It is shown that SAPT(KS) is able to recover the electrostatic, first-order exchange, second-order induction, and exchange-induction energies with an accuracy approaching and occasionally surpassing that of regular SAPT at the currently programmed theory level. The use of the asymptotic corrections is critical to achieve this accuracy. The SAPT(KS) results can be obtained at a small fraction of the time needed for regular SAPT calculations. The robustness of the SAPT(KS) method with respect to the basis set size is also demonstrated. A theoretical justification for high accuracy of SAPT(KS) predictions for the electrostatic, first-order exchange, and second-order induction energies has been provided.     

Application of symmetry-adapted perturbation theory to small ionic systems

The application of symmetry-adapted perturbation theory (SAPT) to small ionic systems was investigated in the context of the accuracy of calculated interaction energies for alkali halides. Two forms of alkali halides were considered: ion pairs M(+)X(-) (M = Li, Na, K, Rb, and X = F, Cl, Br, I) and dimers (MX)(2). The influence of the order of energy correction terms included in SAPT and the effect of the so-called hybrid approach to SAPT on the accuracy of the calculated energies (such as the interaction energies in the ion pairs and the binding energies in the dimers with respect to two free monomers) were studied. The effects of the size of basis sets, combined with SAPT, on the accuracy were also established.     

Interactions in diatomic dimers involving closed-shell metals

Interaction energies of dimers containing alkaline earth (Be, Mg, and Ca) metals have been investigated using symmetry-adapted perturbation theory (SAPT) and supermolecular (SM) methods. Also, to enable broader comparisons, some calculations have been performed on the Zn dimer and on the He-Mg dimer. Although all of the investigated metallic atoms have closed electronic shells, the quasidegeneracy of the ground states of these atoms with the lowest-lying excited states leads to convergence problems in theories based on a single-determinant reference state. The main goal of the present work was to establish how the quality of the interaction energies computed using various electronic-structure methods changes across the range of atoms. We show that although the convergence problems become somewhat less severe with the increase of the atomic number, single-determinant-based methods do not provide reliable interaction energies for any of the investigated metallic dimers even at the level of the coupled-cluster method with single, double, and noniterative triple excitations [CCSD(T)]. However, interaction energies accurate to within a few percent can be obtained if CCSD(T) calculations in large basis sets are extrapolated to the complete basis set limit and followed by full configuration interaction (FCI) calculations with a frozen-core (FC) approximation. Since the systems considered contain only two valence electrons, FCI/FC calculations have been feasible for all of them except for Zn2, providing the best theoretical estimates of the binding energies to date. We found that a large part of the error of the SAPT results originates from limiting some exchange components to terms proportional to the squares of the intermonomer orbital overlap integrals. When the neglected terms were approximately accounted for, the accuracy improved significantly and became comparable to that of CCSD(T), allowing us to obtain for the first time a physical interpretation of the interaction energies in metallic dimers.     

Estimating and modeling charge transfer from the SAPT induction energy

Recent studies using quantum mechanics energy decomposition methods, for example, SAPT and ALMO, have revealed that the charge transfer energy may play an important role in short ranged inter‐molecular interactions, and have a different distance dependence comparing with the polarization energy. However, the charge transfer energy component has been ignored in most current polarizable or non‐polarizable force fields. In this work, first, we proposed an empirical decomposition of SAPT induction energy into charge transfer and polarization energy that mimics the regularized SAPT method (ED‐SAPT). This empirical decomposition is free of the divergence issue, hence providing a good reference for force field development. Then, we further extended this concept in the context of AMOEBA polarizable force field, proposed a consistent approach to treat the charge transfer phenomenon. Current results show a promising application of this charge transfer model in future force field development. © 2017 Wiley Periodicals, Inc.     

Exchange polarization effects in the interaction of closed-shell systems

Explicit formulae for the calculation of the exchange polarization energy in the interaction of closed-shell atoms or molecules have been derived by assuming neglect of the electron correlation within the noninteracting systems. The dispersion part of the exchange polarization energy has been represented as a sum of contributions arising from the interaction of two, three or four orbitals at a time. Each of these contributions is given by an integral involving the orbitals engaged in the interaction and the pair functions describing the dispersion interaction between these orbitals. The numerical calculations for the interaction of two ground-state beryllium atoms show that the exchange dispersion energy is positive and quenches about 5 to 10 per cent of the dispersion term. This results in a decrease of the interaction energy, computed as a sum of the SCF and dispersion components, by 6 to 30 per cent for interatomic distances ranging from 10 to 7 bohrs.Simplified formulae for estimating the exchange dispersion energy in the interaction of larger systems are also proposed and their accuracy is discussed.     

Symmetry-adapted perturbation theory analysis of the N...HX hydrogen bonds

The main aim of the study was the detailed investigation of the interaction energy decomposition in dimers and trimers containing N...HX bonds of different types. The study of angular dependence of interaction energy terms partitioned according to the symmetry-adapted perturbation theory (SAPT) was performed for the dimers containing N...HX bonds as mentioned above: ammonia-HX (X = F, Cl, Br) and pyridine-HF complexes. It was found that the electrostatic and induction terms exhibit strong angular dependence, while the exchange contributions are less affected. The dispersion terms are virtually nondirectional. In addition, the three-body SAPT interaction energy analysis for the mixed acid-base NH3...(HF)2 and (NH3)2...HF trimers revealed strong differences between interactions of similar strength but different types (i.e., hydrogen bond and general electrostatic interaction). The importance of the induction terms for the nonadditivity of the interaction energy in strongly polar systems was confirmed.     

Toward a consistent treatment of polarization in model QM/MM calculations

The concept of model chemistries within hybrid QM/MM calculations has been addressed through analysis of the polarization energy determined by two distinct approaches based on (i) induced charges and (ii) induced dipoles. The quantum mechanical polarization energy for four configurations of the water dimer has been determined for a range of basis sets using Morokuma energy decomposition analysis. This benchmark value has been compared to the fully classical polarization energy determined using the induced dipole approach, and the molecular mechanics polarization energy calculated using induced charges within the MM region of hybrid QM/MM calculations. From the water dimer calculations, it is concluded that the induced charge approach is consistent with medium sized basis set calculations whereas the induced dipole approach is consistent with large basis set calculations. This result is highly relevant to the concept of QM/MM model chemistries.     

Frozen core and effective core potentials in symmetry-adapted perturbation theory

The application of the frozen-core approximation (FCA) and effective core potentials (ECPs) within symmetry-adapted perturbation theory (SAPT) has been investigated and implemented. Unlike in the case of conventional electronic-structure theories, the development of a frozen-core version of SAPT is not straightforward. In particular, the FCA realizations neglecting excitations from core orbitals and restricting all summation indices to valence orbitals only are no longer equivalent. It is shown that it is necessary in SAPT to keep some terms containing products of the valence orbitals of one monomer and the core orbitals of the other one in the exchange-energy components. When these terms are included or, equivalently, the "infinite-excitation-energy" approximation omitting only the excitations from the core orbitals is used, the accuracy of the frozen-core approximation in SAPT matches that obtained in supermolecular perturbational and coupled-cluster methods. If these terms are neglected, i.e., within the "index-range-restriction" approximation, several exchange corrections are significantly underestimated. When ECPs are used in SAPT, the accuracy of the interaction energies is as good as in conventional supermolecular methods, provided that the residual supermolecular Hartree-Fock term is included. We have found that only some types of ECPs can be reliably used for calculations of interaction energies both in SAPT and in supermolecular approaches. For systems containing heavy atoms, both FCA and the use of ECPs lead to very significant savings of computer time.     

Origins of the stability of imidazole-imidazole, benzene-imidazole, and benzene-indole dimers: CCSD(T)/CBS and SAPT calculations

The respective structures and stabilities of imidazole-imidazole, benzene-imidazole, and benzene-indole dimers have been investigated using different DFT-D functional, MP2, CCSD(T), and SAPT levels of theory with a medium basis set. Comparative analysis of binding energies and structural parameters of the dimers points to a preference for stacking contact or hydrogen bond in an imidazole-imidazole dimer. In contrast, a T-shaped configuration with H-π interaction is maximally advantageous for benzene-imidazole and benzene-indole dimers. High-level ab initio calculations at the CCSD(T)/CBS and DFT-SAPT levels show that classical hydrogen-bonded tilted imidazole-imidazole dimer is a global minimum structure and that it has high electrostatic energy. However, for benzene-imidazole and benzene-indole dimers, the global minimum (N-H···π) structure has high electrostatic energy as well as dispersion energy.     

Unified treatment of chemical and van der Waals forces via symmetry-adapted perturbation expansion

We propose a symmetry-adapted perturbation theory (SAPT) expansion of the intermolecular interaction energy which in a finite order provides the correct values of the constants determining the asymptotics of the interaction energy (the van der Waals constants) and is convergent when the energy of the interacting system is submerged in the continuum of Pauli-forbidden states-the situation common when at least one of the monomers has more than two electrons. These desirable features are achieved by splitting the intermolecular electron-nucleus attraction terms of the Hamiltonian into regular (long-range) and singular (short-range) parts. In the perturbation theory development, the regular part is treated as in the conventional polarization theory, which guarantees the correct asymptotics of the interaction energy, while the singular part is weakened sufficiently by an application of permutational symmetry projectors so that a convergent perturbation series is obtained. The convergence is demonstrated numerically, for both the chemical and van der Waals minima, by performing high-order calculations of the interaction energy of the ground-state lithium and hydrogen atoms-the simplest system for which the physical ground state is submerged in the Pauli-forbidden continuum. The obtained expansion enables a systematic extension of SAPT calculations beyond second order with respect to the intermolecular interaction operator.     

Energy components of aqueous solution: Insight from hybrid QM/MM simulations using a polarizable solvent model

An energy decomposition method is present in statistical Monte Carlo simulations of aqueous solutions of a series of organic solutes, making use of a hybrid quantum mechanical and polarizable molecular mechanical (QM/MM-PIPF) approach. In the hybrid QM/MM-PIPF method, the mutual solute–solvent polarization effect is specifically considered through a coupled iterative procedure that ensures the convergence of solvent induced dipoles and the solute wave function. It should be noted that the method is an approximate approach without specifically considering the electronic correlation effect between solute and solvent electrons, and energetic results have not been verified by free energy calculations. Nevertheless, the energy decomposition analysis provides insight into the details of the molecular polarization effect. Qualitative trends of the energy components from such analyses provide guidance in the understanding of the nature of intermolecular interactions in biomolecular systems, whereas quantitative results on specific terms may be utilized to derive empirical, yet computationally more efficient, force fields. Polarization effects are found to be significant, which contribute 10% to 23% to the total solute–solvent interaction energies. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 :1061–1071, 1997     

Interactions of transition metal atoms in high-spin states: Cr2, Sc-Cr, and Sc-Kr

The high-spin van der Waals states are examined for the following dimers: Cr(2) ((13)Sigma(g)(+)), Sc-Cr ((8)Sigma(+), (8)Pi, (8)Delta), and Sc-Kr ((2)Sigma(+), (2)Pi, (2)Delta). These three systems offer a wide range of van der Waals interactions: anomalously strong, intermediate, and typically weak. The single-reference [coupled cluster with single, double, and noniterative triple excitations, RCCSD(T)] method is used in the calculations for all three systems. In addition, a range of configuration-interaction based methods is applied in Cr(2) and Sc-Cr. The three dimers are shown to be bound by the dispersion interaction of varying strength. In a related effort, the dispersion energy and its exchange counterpart are calculated using the newly developed open-shell variant of the symmetry-adapted perturbation theory (SAPT). The restricted open-shell time-dependent Hartree-Fock linear response function is used in the calculations of the dispersion energy in Sc-Cr and Sc-Kr calculations, while the restricted open-shell time-dependent density functional linear response function is used for Cr(2). A hybrid method combining the repulsive restricted open-shell Hartree-Fock (or complete active space self-consistent field) interaction energy with the dispersion and exchange-dispersion terms is tested against the RCCSD(T) results for the three complexes. The Cr(2) ((13)Sigma(g)(+)) complex has the well depth of 807.8 cm(-1) at the equilibrium distance of 6.18a(0) and the dissociation energy of 776.8 cm(-1). The octet-state Sc-Cr is about four times more strongly bound with the order of well depths of (8)Delta>(8)Pi>(8)Sigma(+) and a considerable anisotropy. The enhanced bonding is attributed to the unusually strong dispersion interaction. Sc-Kr ((2)Sigma(+), (2)Pi, (2)Delta) is a typical van der Waals dimer with well depths in the range of 81 cm(-1) ((2)Delta), 84 cm(-1) ((2)Sigma(+)), and 86 cm(-1) ((2)Pi). The hybrid model based on SAPT leads to results which are in excellent qualitative agreement with RCCSD(T) for all three interactions.