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The many-body expansion, where one computes the total energy of a supersystem as the sum of the dimer, trimer, tetramer, etc., subsystems, provides a convenient approach to compute the lattice energies of molecular crystals. We investigate approximate methods for computing the non-additive three-body contributions to the crystal lattice energy of the polar molecules acetic acid, imidazole, and formamide, comparing to coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] level benchmarks. Second-order Møller–Plesset perturbation theory (MP2), if combined with a properly damped Axilrod–Teller–Muto dispersion potential, displays excellent agreement with CCSD(T) at a substantially reduced cost. Errors between dispersion-corrected MP2 and CCSD(T) are less than 1 kJ mol−1 for all three crystals. However, the three-body energy requires quite large distance cutoffs to converge, up to 20 Å or more. 

Symmetry-adapted perturbation theory (SAPT) directly computes intermolecular interaction energy in terms of electrostatics, exchange-repulsion, induction/polarization, and London dispersion components. In SAPT based on Hartree–Fock (“SAPT0”) or based on density functional theory, the most time-consuming step is the computation of the dispersion terms. Previous work has explored the replacement of these expensive dispersion terms with simple damped asymptotic models. We recently examined [Schriber et al. J. Chem. Phys. 154, 234107 (2021)] the accuracy of SAPT0 when replacing its dispersion term with Grimme’s popular -D3 correction, reducing the computational cost scaling from O(N5) to O(N3). That work optimized damping function parameters for SAPT0-D3/jun-cc-pVDZ using estimates of the coupled-cluster complete basis set limit [CCSD(T)/CBS] on a 8299 dimer dataset. Here, we explore the accuracy of SAPT0-D3 with additional basis sets, along with an analogous model using -D4. Damping parameters are rather insensitive to basis sets, and the resulting SAPT0-D models are more accurate on average for total interaction energies than SAPT0. Our results are surprising in several respects: (1) improvement of -D4 over -D3 is negligible for these systems, even charged systems where -D4 should, in principle, be more accurate; (2) addition of Axilrod–Teller–Muto terms for three-body dispersion does not improve error statistics for this test set; and (3) SAPT0-D is even more accurate on average for total interaction energies than the much more computationally costly density functional theory based SAPT [SAPT(DFT)] in an aug-cc-pVDZ basis. However, SAPT0 and SAPT0-D3/D4 interaction energies benefit from significant error cancellation between exchange and dispersion terms. 

Symmetry-adapted perturbation theory (SAPT) is an ab initio approach that directly computes noncovalent interaction energies in terms of electrostatics, exchange repulsion, induction/polarization, and London dispersion components. Due to its high computational scaling, routine applications of even the lowest order of SAPT are typically limited to a few hundred atoms. To address this limitation, we report here the addition of electrostatic embedding to the SAPT (EE-SAPT) and ISAPT (EE-ISAPT) methods. We illustrate the embedding scheme using water trimer as a prototype example. Then, we show that EE-SAPT/EE-ISAPT can be applied for efficiently and accurately computing noncovalent interactions in large systems, including solvated dimers and protein–ligand systems. In the latter application, particular care must be taken to properly handle the quantum mechanics/molecular mechanics boundary when it cuts covalent bonds. We investigate various schemes for handling charges near this boundary and demonstrate which are most effective in the context of charge-embedded SAPT. 

We present an efficient, open-source formulation for coupled-cluster theory through perturbative triples with domain-based local pair natural orbitals [DLPNO-CCSD(T)]. Similar to the implementation of the DLPNO-CCSD(T) method found in the ORCA package, the most expensive integral generation and contraction steps associated with the CCSD(T) method are linear-scaling. In this work, we show that the t1-transformed Hamiltonian allows for a less complex algorithm when evaluating the local CCSD(T) energy without compromising efficiency or accuracy. Our algorithm yields sub-kJ mol−1 deviations for relative energies when compared with canonical CCSD(T), with typical errors being on the order of 0.1 kcal mol−1, using our TightPNO parameters. We extensively tested and optimized our algorithm and parameters for non-covalent interactions, which have been the most difficult interaction to model for orbital (PNO)-based methods historically. To highlight the capabilities of our code, we tested it on large water clusters, as well as insulin (787 atoms). 

Quantifying intermolecular interactions with quantum chemistry (QC) is useful for many chemical problems, including understanding the nature of protein–ligand interactions. 

The focal-point approximation can be used to estimate a high-accuracy, slow quantum chemistry computation by combining several lower-accuracy, faster computations. We examine the performance of focal-point methods by combining second-order Møller–Plesset perturbation theory (MP2) with coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] for the calculation of harmonic frequencies and that of fundamental frequencies using second-order vibrational perturbation theory (VPT2). In contrast to standard CCSD(T), the focal-point CCSD(T) method approaches the complete basis set (CBS) limit with only triple-ζ basis sets for the coupled-cluster portion of the computation. The predicted harmonic and fundamental frequencies were compared with the experimental values for a set of 20 molecules containing up to six atoms. The focal-point method combining CCSD(T)/aug-cc-pV(T + d)Z with CBS-extrapolated MP2 has mean absolute errors vs experiment of only 7.3 cm−1 for the fundamental frequencies, which are essentially the same as the mean absolute error for CCSD(T) extrapolated to the CBS limit using the aug-cc-pV(Q + d)Z and aug-cc-pV(5 + d)Z basis sets. However, for H2O, the focal-point procedure requires only 3% of the computation time as the extrapolated CCSD(T) result, and the cost savings will grow for larger molecules. 

Dimer interaction energies have been well studied in computational chemistry, but they can offer an incomplete understanding of molecular binding depending on the system. In the current study, we present a dataset of focal-point coupled-cluster interaction and deformation energies (summing to binding energies, De) of 28 organic molecular dimers. We use these highly accurate energies to evaluate ten density functional approximations for their accuracy. The best performing method (with a double-ζ basis set), B97M-D3BJ, is then used to calculate the binding energies of 104 organic dimers, and we analyze the influence of the nature and strength of interaction on deformation energies. Deformation energies can be as large as 50% of the dimer interaction energy, especially when hydrogen bonding is present. In most cases, two or more hydrogen bonds present in a dimer correspond to an interaction energy of −10 to −25 kcal mol−1, allowing a deformation energy above 1 kcal mol−1 (and up to 9.5 kcal mol−1). A lack of hydrogen bonding usually restricts the deformation energy to below 1 kcal mol−1 due to the weaker interaction energy. 

To study the contribution of three-body dispersion to crystal lattice energies, we compute the three-body contributions to the lattice energies for crystalline benzene, carbon dioxide, and triazine using various computational methods. We show that these contributions converge quickly as the intermolecular distances between the monomers grow. In particular, the smallest value among the three pairwise intermonomer closest-contact distances, Rmin, shows a strong correlation with the three-body contribution to the lattice energy, and, here, the largest of the closest-contact distances, Rmax, serves as a cutoff criterion to limit the number of trimers to be considered. We considered all trimers up to Rmax=15Å. The trimers with Rmin<4Å contribute 90.4%, 90.6%, and 93.9% of the total three-body contributions for crystalline benzene, carbon dioxide, and triazine, respectively, for the coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] method. For trimers with Rmin>4Å, the second-order Møller–Plesset perturbation theory (MP2) supplemented with the Axilrod–Teller–Muto (ATM) three-body dispersion correction reproduces the CCSD(T) values for the cumulative three-body contributions with errors of less than 0.1 kJ mol−1. Moreover, three-body contributions are converged within 0.15 kJ mol−1 by Rmax=10Å. From these results, it appears that in molecular crystals where dispersion dominates the three-body contribution to the lattice energy, the trimers with Rmin>4Å can be computed with the MP2+ATM method to reduce the computational cost, and those with Rmax>10Å appear to be basically negligible. 

The many-body expansion (MBE) is promising for the efficient, parallel computation of lattice energies in organic crystals. Very high accuracy should be achievable by employing coupled-cluster singles, doubles, and perturbative triples at the complete basis set limit [CCSD(T)/CBS] for the dimers, trimers, and potentially tetramers resulting from the MBE, but such a brute-force approach seems impractical for crystals of all but the smallest molecules. Here, we investigate hybrid or multi-level approaches that employ CCSD(T)/CBS only for the closest dimers and trimers and utilize much faster methods like Møller–Plesset perturbation theory (MP2) for more distant dimers and trimers. For trimers, MP2 is supplemented with the Axilrod–Teller–Muto (ATM) model of three-body dispersion. MP2(+ATM) is shown to be a very effective replacement for CCSD(T)/CBS for all but the closest dimers and trimers. A limited investigation of tetramers using CCSD(T)/CBS suggests that the four-body contribution is entirely negligible. The large set of CCSD(T)/CBS dimer and trimer data should be valuable in benchmarking approximate methods for molecular crystals and allows us to see that a literature estimate of the core-valence contribution of the closest dimers to the lattice energy using just MP2 was overbinding by 0.5 kJ mol−1, and an estimate of the three-body contribution from the closest trimers using the T0 approximation in local CCSD(T) was underbinding by 0.7 kJ mol−1. Our CCSD(T)/CBS best estimate of the 0 K lattice energy is −54.01 kJ mol−1, compared to an estimated experimental value of −55.3 ± 2.2 kJ mol−1. 

Using the many-body expansion to predict crystal lattice energies (CLEs), a pleasantly parallel process, allows for flexibility in the choice of theoretical methods. Benchmark-level two-body contributions to CLEs of 23 molecular crystals have been computed using interaction energies of dimers with minimum inter-monomer separations (i.e., closest contact distances) up to 30 Å. In a search for ways to reduce the computational expense of calculating accurate CLEs, we have computed these two-body contributions with 15 different quantum chemical levels of theory and compared these energies to those computed with coupled-cluster in the complete basis set (CBS) limit. Interaction energies of the more distant dimers are easier to compute accurately and several of the methods tested are suitable as replacements for coupled-cluster through perturbative triples for all but the closest dimers. For our dataset, sub-kJ mol−1 accuracy can be obtained when calculating two-body interaction energies of dimers with separations shorter than 4 Å with coupled-cluster with single, double, and perturbative triple excitations/CBS and dimers with separations longer than 4 Å with MP2.5/aug-cc-pVDZ, among other schemes, reducing the number of dimers to be computed with coupled-cluster by as much as 98%.