Scaled MP3 non-covalent interaction energies agree closely with accurate CCSD(T) benchmark data

Towards tailoring hydrophobic interaction with uranyl(VI) oxygen for C-H activation

Bovine serum albumin (BSA) has a uranyl(VI) binding hotspot where uranium is tightly bound by three carboxylates. Uranyl oxygen is "soaked" into the hydrophobic core of BSA. Isopropyl hydrogen of Val is trapped near UO22+ and upon photoexcitation, C-H bond cleavage is initiated. A unique hydrophobic contact with "yl"-oxygen, as observed here, can be used to induce C-H activation.

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.

An in-silico NMR laboratory for nuclear magnetic shieldings computed via finite fields: Exploring nucleus-specific renormalizations of MP2 and MP3

We developed and implemented a method-independent, fully numerical, finite difference approach to calculating nuclear magnetic resonance shieldings, using gauge-including atomic orbitals. The resulting capability can be used to explore non-standard methods, given only the energy as a function of finite-applied magnetic fields and nuclear spins. For example, standard second-order Møller-Plesset theory (MP2) has well-known efficacy for 1H and 13C shieldings and known limitations for other nuclei such as 15N and 17O. It is, therefore, interesting to seek methods that offer good accuracy for 15N and 17O shieldings without greatly increased compute costs, as well as exploring whether such methods can further improve 1H and 13C shieldings. Using a small molecule test set of 28 species, we assessed two alternatives: κ regularized MP2 (κ-MP2), which provides energy-dependent damping of large amplitudes, and MP2.X, which includes a variable fraction, X, of third-order correlation (MP3). The aug-cc-pVTZ basis was used, and coupled cluster with singles and doubles and perturbative triples [CCSD(T)] results were taken as reference values. Our κ-MP2 results reveal significant improvements over MP2 for 13C and 15N, with the optimal κ value being element-specific. κ-MP2 with κ = 2 offers a 30% rms error reduction over MP2. For 15N, κ-MP2 with κ = 1.1 provides a 90% error reduction vs MP2 and a 60% error reduction vs CCSD. On the other hand, MP2.X with a scaling factor of 0.6 outperformed CCSD for all heavy nuclei. These results can be understood as providing renormalization of doubles amplitudes to partially account for neglected triple and higher substitutions and offer promising opportunities for future applications.

Assessment of three-body dispersion models against coupled-cluster benchmarks for crystalline benzene, carbon dioxide, and triazine

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, R_min, shows a strong correlation with the three-body contribution to the lattice energy, and, here, the largest of the closest-contact distances, R_max, serves as a cutoff criterion to limit the number of trimers to be considered. We considered all trimers up to R_max=15Å. The trimers with R_min<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 R_min>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 R_max=10Å. From these results, it appears that in molecular crystals where dispersion dominates the three-body contribution to the lattice energy, the trimers with R_min>4Å can be computed with the MP2+ATM method to reduce the computational cost, and those with R_max>10Å appear to be basically negligible.

Ionization energies of metallocenes: a coupled cluster study of cobaltocene

Open-shell transition metal chemistry presents challenges to contemporary electronic structure methods, based on either density functional or wavefunction theory. While CCSD(T) is the well-trusted gold standard for maingroup thermochemistry, the accuracy and robustness of the method is less clear for open-shell transition metal chemistry, requiring benchmarking of CCSD(T)-based protocols against either higher-level theory or experiment. Ionization energies (IEs) of metallocenes provide an interesting test case with metallocenes being common redox reagents as well as playing roles as redox mediators and cocatalysts in redox catalysis. Using highly accurate ZEKE-MATI experimental measurements of gas phase adiabatic (5.3275 ± 0.0006 eV) and vertical (5.4424 ± 0.0006 eV) ionization energies of cobaltocene, we systematically assessed the accuracy of the local coupled-cluster method DLPNO-CCSD(T) with respect to geometry, reference determinant, basis set size and extrapolation schemes, PNO cut-off and extrapolation, local triples approximation, relativistic effects and core-valence correlation. We show that PNO errors are controllable via the recently introduced PNO extrapolation schemes and that the expensive iterative triples (T₁) contribution can be made more manageable by calculating it as a smaller-basis/smaller PNO-cutoff correction. The reference determinant turns out to be a critical aspect in these calculations with the HF determinant resulting in large DLPNO-CCSD(T) errors, likely due to the qualitatively flawed molecular orbital spectrum. The BP86 functional on the other hand was found to provide reference orbitals giving small DLPNO-CCSD(T) errors, likely due to more realistic orbitals as suggested by the more consistent MO spectrum compared to HF. A protocol including complete basis set extrapolations with correlation-consistent basis sets, complete PNO space extrapolations, iterative triples- and core-valence correlation corrections was found to give errors of -0.07 eV and -0.03 eV for adiabatic- and vertical-IE of cobaltocene, respectively, giving close to chemical accuracy for both properties. A computationally efficient DLPNO-CCSD(T) protocol was devised and tested against adiabatic ionization energies of 6 different metallocenes (V, Cr, Mn, Fe, Co, Ni). For the other metallocenes, the iterative triples (T₁) and PNO extrapolation contributions turn out to be even more important. The results give errors close to the experimental uncertainty, similar to recent auxiliary-field quantum Monte Carlo results. The quality of the reference determinant orbitals is identified as the main source of uncertainty in CCSD(T) calculations of metallocenes.

Benchmarking two-body contributions to crystal lattice energies and a range-dependent assessment of approximate methods

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%.

Assessment of the Second-Order Statically Screened Exchange Correction to the Random Phase Approximation for Correlation Energies

With increasing interelectronic distance, the screening of the electron–electron interaction by the presence of other electrons becomes the dominant source of electron correlation. This effect is described by the random phase approximation (RPA) which is therefore a promising method for the calculation of weak interactions. The success of the RPA relies on the cancellation of errors, which can be traced back to the violation of the crossing symmetry of the 4-point vertex, leading to strongly overestimated total correlation energies. By the addition of second-order screened exchange (SOSEX) to the correlation energy, this issue is substantially reduced. In the adiabatic connection (AC) SOSEX formalism, one of the two electron–electron interaction lines in the second-order exchange term is dynamically screened (SOSEX(W, v_c)). A related SOSEX expression in which both electron–electron interaction lines are statically screened (SOSEX(W(0), W(0))) is obtained from the G3W2 contribution to the electronic self-energy. In contrast to SOSEX(W, v_c), the evaluation of this correlation energy expression does not require an expensive numerical frequency integration and is therefore advantageous from a computational perspective. We compare the accuracy of the statically screened variant to RPA and RPA+SOSEX(W, v_c) for a wide range of chemical reactions. While both methods fail for barrier heights, SOSEX(W(0), W(0)) agrees very well with SOSEX(W, v_c) for charged excitations and noncovalent interactions where they lead to major improvements over RPA.

Accurate Property Prediction by Second Order Perturbation Theory: The REMP and OO-REMP Hybrids

The prediction of molecular properties such as equilibrium structures or vibrationalwavenumbers is a routine task in computational chemistry. If very high accuracy is required, however, the use of computationally demanding ab initio wavefunction methods is mandatory. We present property calculations utilizing the REMP and OO-REMP hybrid perturbation theories showing that with the latter approach, very accurate results are obtained at second order in perturbation theory. Specifically, equilibrium structures and harmonic vibrational wavenumbers as well as dipole moments of closed and open shell molecules were calculated and compared to the best available experimental results or very accurate calculations.OO-REMP is capable of predicting bond lengths of small closed and open shell molecules with an accuracy of 0.2 pm and 0.5 pm, respectively, often within the range of experimental uncertainty. Equilibrium harmonic vibrational wavenumbers are predicted with an accuracy better than 20 cm−1 . Dipole moments of small closed and open shell molecules are reproduced with a relative error of less than 3 %. Across all investigated properties it turns out that a 20 %:80 % MP:RE mixing ratio consistently provides the best results. This is in line with our previous findings featuring closed and open shell reaction energies.

A Minimum Quantum Chemistry CCSD(T)/CBS Data Set of Dimeric Interaction Energies for Small Organic Functional Groups: Heterodimers

We extend our previous quantum chemistry calculations of interaction energies for 31 homodimers of small organic functional groups (the SOFG-31 data set) by including 239 heterodimers with monomers selected within the SOFG-31 data set, thus resulting in the SOFG-31+239 data set. The minimum-level theoretical scheme contains (1) the basis set superposition error corrected supermolecule (BSSE-SM) approach for intermolecular interactions; (2) the second-order Møller-Plesset perturbation theory (MP2) with the Dunning's aug-cc-pVXZ (X = D, T, Q) basis sets for the geometry optimization and correlation energy calculations; and (3) the single-point energy calculations with the coupled cluster with single, double, and perturbative triple excitations method at the complete basis set limit [CCSD(T)/CBS] using the well-tested extrapolation methods for the MP2 energy calibrations. In addition, we have performed a parallel series of energy decomposition calculations based on the symmetry adapted perturbation theory (SAPT) in order to gain chemical insights. That the above procedure cannot be further reduced has been proven to be very crucial for constructing reliable data sets of interaction energies. The calculated CCSD(T)/CBS interaction energy data can serve as a benchmark for testing or training less accurate but more efficient calculation methods, such as the electronic density functional theory. As an application, we employ a segmental SAPT model previously developed for the SOFG-31 data set to predict binding energies of large heterodimer complexes. These model energy "quanta" can be used in coarse-grained molecular dynamics simulations by avoiding large-scale calculations.

Benchmarking CASPT3 Vertical Excitation Energies

Based on 280 reference vertical transition energies of various natures (singlet, triplet, valence, Rydberg, n → π∗, π → π∗, and double excitations) extracted from the QUEST database, we assess the accuracy of third-order multireference perturbation theory, CASPT3, in the context of molecular excited states. When one applies the disputable ionization- potential-electron-affinity (IPEA) shift, we show that CASPT3 provides a similar accuracy as its second-order counterpart, CASPT2, with the same mean absolute error of 0.11 eV. However, as already reported, we also observe that the accuracy of CASPT3 is almost insensitive to the IPEA shift, irrespective of the transition type and system size, with a small reduction of the mean absolute error to 0.09 eV when the IPEA shift is switched off.

Conformational Energy Benchmark for Longer n-Alkane Chains

We present the first benchmark set focusing on the conformational energies of highly flexible, long n-alkane chains, termed ACONFL. Unbranched alkanes are ubiquitous building blocks in nature, so the goal is to be able to calculate their properties most accurately to improve the modeling of, e.g., complex (biological) systems. Very accurate DLPNO-CCSD(T1)/CBS reference values are provided, which allow for a statistical meaningful evaluation of even the best available density functional methods. The performance of established and modern (dispersion corrected) density functionals is comprehensively assessed. The recently introduced r²SCAN-V functional shows excellent performance, similar to efficient composite DFT methods like B97-3c and r²SCAN-3c, which provide an even better cost-accuracy ratio, while almost reaching the accuracy of much more computationally demanding hybrid or double hybrid functionals with large QZ AO basis sets. In addition, we investigated the performance of common wave function methods, where MP2/CBS surprisingly performs worse compared to the simple D4 dispersion corrected Hartree-Fock. Furthermore, we investigate the performance of several semiempirical and force field methods, which are commonly used for the generation of conformational ensembles in multilevel workflows or in large scale molecular dynamics studies. Outstanding performance is obtained by the recently introduced general force field, GFN-FF, while other commonly applied methods like the universal force field yield large errors. We recommend the ACONFL as a helpful benchmark set for parametrization of new semiempirical or force field methods and machine learning potentials as well as a meaningful validation set for newly developed DFT or dispersion methods.

The influence of a solvent environment on direct non-covalent interactions between two molecules: A symmetry-adapted perturbation theory study of polarization tuning of π-π interactions by water

High-level quantum chemical computations have provided significant insight into the fundamental physical nature of non-covalent interactions. These studies have focused primarily on gas-phase computations of small van der Waals dimers; however, these interactions frequently take place in complex chemical environments, such as proteins, solutions, or solids. To better understand how the chemical environment affects non-covalent interactions, we have undertaken a quantum chemical study of π-π interactions in an aqueous solution, as exemplified by T-shaped benzene dimers surrounded by 28 or 50 explicit water molecules. We report interaction energies (IEs) using second-order Møller-Plesset perturbation theory, and we apply the intramolecular and functional-group partitioning extensions of symmetry-adapted perturbation theory (ISAPT and F-SAPT, respectively) to analyze how the solvent molecules tune the π-π interactions of the solute. For complexes containing neutral monomers, even 50 explicit waters (constituting a first and partial second solvation shell) change total SAPT IEs between the two solute molecules by only tenths of a kcal mol^-1, while significant changes of up to 3 kcal mol^-1 of the electrostatic component are seen for the cationic pyridinium-benzene dimer. This difference between charged and neutral solutes is attributed to large non-additive three-body interactions within solvated ion-containing complexes. Overall, except for charged solutes, our quantum computations indicate that nearby solvent molecules cause very little "tuning" of the direct solute-solute interactions. This indicates that differences in binding energies between the gas phase and solution phase are primarily indirect effects of the competition between solute-solute and solute-solvent interactions.

Exploring the accuracy and usefulness of semi-empirically scaled ADC schemes by blending second and third order terms

Different approaches to mixed-order algebraic-diagrammatic construction (ADC) schemes are investigated. The performance of two different strategies for scaling third-order contributions to the ADC secular matrix is evaluated. Both considered schemes employ a single tuning parameter and conserve general properties inherent to all ADC methods such as hermiticity and size-consistency.The first approach, scaled-matrix ADC[(2)+x(3)], scales all contributions first occurring in ADC(3) equally and leads to an improvement of the accuracy of excitation energies compared to ADC(3) for x=0.4-0.5. However, with respect to excited state dipole moments this method provides lower accuracy than ADC(3). The second scaling approach, MP[(1)+x(2)]-ISR(3), scales the second order contributions of the ground-state wavefunction and derives a rigorous ADC scheme via the intermediate state representation (ISR) formalism. Although the error in excitation energies is not improved, this method provides insight into the relevance of the individual terms of the ADC(3) matrix and indicates, that the MP(2) wavefunction is indeed the optimal reference wavefunction for deriving a third-order single-reference ADC scheme.

UREMP, RO-REMP, and OO-REMP: Hybrid perturbation theories for open-shell electronic structure calculations

An accurate description of the electron correlation energy in closed- and open-shell molecules is shown to be obtained by a second-order perturbation theory (PT) termed REMP. REMP is a hybrid of the Retaining the Excitation degree (RE) and the Møller–Plesset (MP) PTs. It performs particularly encouragingly in an orbital-optimized variant (OO-REMP) where the reference wavefunction is given by an unrestricted Slater determinant whose spin orbitals are varied such that the total energy becomes a minimum. While the approach generally behaves less satisfactorily with unrestricted Hartree–Fock references, reasonable performance is observed for restricted Hartree–Fock and restricted open-shell Hartree–Fock references. Inclusion of single excitations to OO-REMP is investigated and found—as in similar investigations—to be dissatisfying as it deteriorates performance. For the non-multireference subset of the accurate W4-11 benchmark set of Karton et al. [Chem. Phys. Lett. 510, 165–178 (2011)], OO-REMP predicts most atomization and reaction energies with chemical accuracy (1 kcal mol−1) if complete-basis-set extrapolation with augmented and core-polarized basis sets is used. For the W4-11 related test-sets, the error estimates obtained with the OO-REMP method approach those of coupled-cluster with singles, doubles and perturbative triples [CCSD(T)] within 20%–35%. The best performance of OO-REMP is found for a mixing ratio of 20%:80% MP:RE, which is essentially independent of whether radical stabilization energies, barrier heights, or reaction energies are investigated. Orbital optimization is shown to improve the REMP approach for both closed and open shell cases and outperforms coupled-cluster theory with singles and doubles (CCSD), spin-component scaled Møller-Plesset theory at second order (SCS-MP2), and density functionals, including double hybrids in all the cases considered.

Stable Carbenes as Structural Components of Partially Saturated Sulfur-Containing Heterocycles

Recently, an unusual elongation of the C-S bond was observed experimentally for some sulfur-containing heterocycles. Using a superior ab initio (SCS-MP2/cc-pVTZ) level of theory, we showed that the phenomenon can be explained by a contribution of a donor-acceptor adduct of a carbene with an unsaturated ligand. One may achieve further elongation of the C-S bond, eventually turning it to a coordinate one, by increasing the stability of each part of the system as, e.g., in the utmost case of spiro adducts with Arduengo carbenes. The effect of carbene stability was quantified by employing the isodesmic reactions of carbene exchange.

The Adsorption of Small Molecules on the Copper Paddle-Wheel: Influence of the Multi-Reference Ground State

We report a theoretical study of the adsorption of a set of small molecules (C₂H₂, CO, CO₂, O₂, H₂O, CH₃OH, C₂H₅OH) on the metal centers of the “copper paddle-wheel”—a key structural motif of many MOFs. A systematic comparison between DFT of different rungs, single-reference post-HF methods (MP2, SOS–MP2, MP3, DLPNO–CCSD(T)), and multi-reference approaches (CASSCF, DCD–CAS(2), NEVPT2) is performed in order to find a methodology that correctly describes the complicated electronic structure of paddle-wheel structure together with a reasonable description of non-covalent interactions. Apart from comparison with literature data (experimental values wherever possible), benchmark calculations with DLPNO–MR–CCSD were also performed. Despite tested methods show qualitative agreement in the majority of cases, we showed and discussed reasons for quantitative differences as well as more fundamental problems of specific cases.

State-Of-The-Art Computations of Vertical Electron Affinities with the Extended Koopmans' Theorem Integrated with the CCSD(T) Method

Accurate computation of electron affinities (EAs), within 0.10 eV, is one of the most challenging problems in modern computational quantum chemistry. The extended Koopmans' theorem (EKT) enables direct computations of electron affinities (EAs) from any level of the theory. In this research, the EKT approach based on the coupled-cluster singles and doubles with perturbative triples [CCSD(T)] method is applied to computations of EAs for the first time. For efficiency, the density-fitting (DF) technique is used for two-electron integrals. Further, the EKT-CCSD(T) method is applied to three test sets of atoms and closed- and open-shell molecules, denoted A16, C10, and O33, respectively, for comparison with the experimental electron affinities. For the A16, C10, and O33 sets, the EKT-CCSD(T) approach, along with the aug-cc-pV5Z basis set, provide mean absolute errors (MAEs) of 0.05, 0.08, and 0.09 eV, respectively. Hence, our results demonstrate that high-accuracy computations of EAs can be achieved with the EKT-CCSD(T) approach. Further, when the EKT-CCSD(T) approach is not computationally affordable, the EKT-MP2.5, EKT-LCCD, and EKT-CCSD methods can be considered, and their results are also reasonably accurate. The huge advantage of the EKT method for the computation of IPs is that it comes for free in an analytic gradient computation. Hence, one needs neither separate computations for neutral and ionic species, as in the case of common approaches, nor additional efforts to obtain IPs, as in the case of equation-of-motion approaches. Overall, we believe that the present research may open new avenues in EA computations.

Surprisingly Good Performance of XYG3 Family Functionals Using a Scaled KS-MP3 Correlation

By adding a GLPT3 (third-order Görling-Levy perturbation theory, or KS-MP3) term E₃ to the XYG7 form for a double hybrid, we are able to bring down WTMAD2 (weighted total mean absolute deviation) for the very large and chemically diverse GMTKN55 benchmark to an unprecedented 1.17 kcal/mol, competitive with much costlier composite wave function ab initio approaches. Intriguingly, (a) the introduction of E₃ makes an empirical dispersion correction redundant; (b) generalized gradient approximation (GGA) or meta-GGA semilocal correlation functionals offer no advantage over the local density approximation (LDA) in this framework; (c) if a dispersion correction is retained, then simple Slater exchange leads to no significant loss in accuracy. It is possible to create a six-parameter functional with WTMAD2 = 1.42 that has no post-LDA density functional theory components and no dispersion correction in the final energy.

Exploring the Limits of Second- and Third-Order Møller-Plesset Perturbation Theories for Noncovalent Interactions: Revisiting MP2.5 and Assessing the Importance of Regularization and Reference Orbitals

This work systematically assesses the influence of reference orbitals, regularization, and scaling on the performance of second- and third-order Møller-Plesset perturbation theory wave function methods for noncovalent interactions (NCIs). Testing on 19 data sets (A24, DS14, HB15, HSG, S22, X40, HW30, NC15, S66, AlkBind12, CO2Nitrogen16, HB49, Ionic43, TA13, XB18, Bauza30, CT20, XB51, and Orel26rad) covers a wide range of different NCIs including hydrogen bonding, dispersion, and halogen bonding. Inclusion of potential energy surfaces from different hydrogen bonds and dispersion-bound complexes gauges accuracy for nonequilibrium geometries. Fifteen methods are tested. In notation where nonstandard choices of orbitals are denoted as methods:orbitals, these are MP2, κ-MP2, SCS-MP2, OOMP2, κ-OOMP2, MP3, MP2.5, MP3:OOMP2, MP2.5:OOMP2, MP3:κ-OOMP2, MP2.5:κ-OOMP2, κ-MP3:κ-OOMP2, κ-MP2.5:κ-OOMP2, MP3:ωB97X-V, and MP2.5:ωB97X-V. Furthermore, we compare these methods to the ωB97M-V and B3LYP-D3 density functionals, as well as CCSD. We find that the κ-regularization (κ = 1.45 au was used throughout) improves the energetics in almost all data sets for both MP2 (in 17 out of 19 data sets) and OOMP2 (16 out of 19). The improvement is significant (e.g., the root-mean-square deviation (RMSD) for the S66 data set is 0.29 kcal/mol for κ-OOMP2 versus 0.67 kcal/mol for MP2) and for interactions between stable closed-shell molecules, not strongly dependent on the reference orbitals. Scaled MP3 (with a factor of 0.5) using κ-OOMP2 reference orbitals (MP2.5:κ-OOMP2) provides significantly more accurate results for NCIs across all data sets with noniterative O(N⁶) scaling (S66 data set RMSD: 0.10 kcal/mol). Across the entire data set of 356 points, the improvement over standard MP2.5 is approximately a factor of 2: RMSD for MP3:κ-OOMP2 is 0.25 vs 0.50 kcal/mol for MP2.5. The use of high-quality density functional reference orbitals (ωB97X-V) also significantly improves the results of MP2.5 for NCI over a Hartree-Fock orbital reference. All our assessments and conclusions are based on the use of the medium-sized aug-cc-pVTZ basis to yield results that are directly compared against complete basis set limit reference values.

Molecular Tailoring Approach for Estimating Individual Intermolecular Interaction Energies in Benzene Clusters

There is no general method available for the estimation of individual intermolecular interaction energies in weakly bound molecular clusters, and such studies are limited only to the dimer. Recently, we proposed a molecular tailoring approach-based method for the estimation of individual O-H···O hydrogen bond energies in water clusters. In the present work, we extend the applicability of this method for estimating the individual intermolecular interaction energies in benzene clusters, which are expected to be small. The basis set superposition error (BSSE)-corrected individual intermolecular interaction energies in linear (LN) benzene clusters, LN-(Bz)_n n = 3-7, were calculated to be in the range from -1.75 to -2.33 kcal/mol with the cooperativity contribution falling between 0.05 and 0.20 kcal/mol, calculated at the MP2.5/aug-cc-pVDZ level of theory. In the case of non-linear (NLN) benzene clusters, NLN-(Bz)_n n = 3-5, the BSSE-corrected individual intermolecular interaction energies exhibit a wider range from -1.16 to -2.55 kcal/mol with cooperativity contribution in the range from 0.02 to -0.61 kcal/mol. The accuracy of these estimated values was validated by adding the sum of interaction energies to the sum of monomer energies. These estimated molecular energies of clusters were compared with their actual calculated values. The small difference (<0.3 kcal/mol) in these two values suggests that our estimated individual intermolecular interaction energies in benzene clusters are quite reliable.

OO-REMP: Approaching Chemical Accuracy with Second-Order Perturbation Theory

We present a perturbation theory (PT) providing second-order energies that reproduce main group chemistry benchmark sets for reaction energies, barrier heights, and atomization energies with mean absolute deviations below 1 kcal mol^-1. The PT is defined as a constrained mixture of the unperturbed Hamiltonians of the Retaining the Excitation degree (RE) and the Møller-Plesset (MP) PTs. The orbitals of the reference wave function, a single unrestricted Slater determinant, are iteratively optimized to minimize the total energy. For all benchmark sets, good and near optimal performance of OO-REMP was observed for an unperturbed Hamiltonian consisting of 25% MP and 75% RE contributions.

Accurate Electronic Excitation Energies in Full-Valence Active Space via Bootstrap Embedding

Fragment embedding has been widely used to circumvent the high computational scaling of using accurate electron correlation methods to describe the electronic ground states of molecules and materials. However, similar applications that utilize fragment embedding to treat electronic excited states are comparably less reported in the literature. The challenge here is twofold. First, most fragment embedding methods are most effective when the property of interest is local, but the change of the wave function upon excitation is nonlocal in general. Second, even for local excitations, an accurate estimate of, for example, the excitation energy can still be challenging owing to the need for a balanced treatment of both the ground and the excited states. In this work, we show that bootstrap embedding (BE), a fragment embedding method developed recently by our group, is promising toward describing general electronic excitations. Numerical simulations show that the excitation energies in full-valence active space (FVAS) can be well-estimated by BE to an error of ∼0.05 eV using relatively small fragments, for both local excitations and the excitations of some large dye molecules that exhibit strong charge-transfer characters. We hence anticipate BE to be a promising solution to accurately describing the excited states of large chemical systems.

Prototypical π-π dimers re-examined by means of high-level CCSDT(Q) composite ab initio methods

The benzene-ethene and parallel-displaced (PD) benzene-benzene dimers are the most fundamental systems involving π-π stacking interactions. Several high-level ab initio investigations calculated the binding energies of these dimers using the coupled-cluster with singles, doubles, and quasi-perturbative triple excitations [CCSD(T)] method at the complete basis set [CBS] limit using various approaches such as reduced virtual orbital spaces and/or MP2-based basis set corrections. Here, we obtain CCSDT(Q) binding energies using a Weizmann-3-type approach. In particular, we extrapolate the self-consistent field (SCF), CCSD, and (T) components using large heavy-atom augmented Gaussian basis sets [namely, SCF/jul-cc-pV{5,6}Z, CCSD/jul-cc-pV{Q,5}Z, and (T)/jul-cc-pV{T,Q}Z]. We consider post-CCSD(T) contributions up to CCSDT(Q), inner-shell, scalar-relativistic, and Born-Oppenheimer corrections. Overall, our best relativistic, all-electron CCSDT(Q) binding energies are ∆E_e,all,rel = 1.234 (benzene-ethene) and 2.550 (benzene-benzene PD), ∆H₀ = 0.949 (benzene-ethene) and 2.310 (benzene-benzene PD), and ∆H₂₉₈ = 0.130 (benzene-ethene) and 1.461 (benzene-benzene PD) kcal mol^-1. Important conclusions are reached regarding the basis set convergence of the SCF, CCSD, (T), and post-CCSD(T) components. Explicitly correlated calculations are used as a sanity check on the conventional binding energies. Overall, post-CCSD(T) contributions are destabilizing by 0.028 (benzene-ethene) and 0.058 (benzene-benzene) kcal mol^-1, and thus, they cannot be neglected if sub-chemical accuracy is sought (i.e., errors below 0.1 kcal mol^-1). CCSD(T)/aug-cc-pwCVTZ core-valence corrections increase the binding energies by 0.018 (benzene-ethene) and 0.027 (benzene-benzene PD) kcal mol^-1. Scalar-relativistic and diagonal Born-Oppenheimer corrections are negligibly small. We use our best CCSDT(Q) binding energies to evaluate the performance of MP2-based, CCSD-based, and lower-cost composite ab initio procedures for obtaining these challenging π-π stacking binding energies.

Quantum chemical benchmark databases of gold-standard dimer interaction energies

Advances in computational chemistry create an ongoing need for larger and higher-quality datasets that characterize noncovalent molecular interactions. We present three benchmark collections of quantum mechanical data, covering approximately 3,700 distinct types of interacting molecule pairs. The first collection, which we refer to as DES370K, contains interaction energies for more than 370,000 dimer geometries. These were computed using the coupled-cluster method with single, double, and perturbative triple excitations [CCSD(T)], which is widely regarded as the gold-standard method in electronic structure theory. Our second benchmark collection, a core representative subset of DES370K called DES15K, is intended for more computationally demanding applications of the data. Finally, DES5M, our third collection, comprises interaction energies for nearly 5,000,000 dimer geometries; these were calculated using SNS-MP2, a machine learning approach that provides results with accuracy comparable to that of our coupled-cluster training data. These datasets may prove useful in the development of density functionals, empirically corrected wavefunction-based approaches, semi-empirical methods, force fields, and models trained using machine learning methods.

Random-Phase Approximation in Many-Body Noncovalent Systems: Methane in a Dodecahedral Water Cage

The many-body expansion (MBE) of energies of molecular clusters or solids offers a way to detect and analyze errors of theoretical methods that could go unnoticed if only the total energy of the system was considered. In this regard, the interaction between the methane molecule and its enclosing dodecahedral water cage, CH₄···(H₂O)₂₀, is a stringent test for approximate methods, including density functional theory (DFT) approximations. Hybrid and semilocal DFT approximations behave erratically for this system, with three- and four-body nonadditive terms having neither the correct sign nor magnitude. Here, we analyze to what extent these qualitative errors in different MBE contributions are conveyed to post-Kohn-Sham random-phase approximation (RPA), which uses approximate Kohn-Sham orbitals as its input. The results reveal a correlation between the quality of the DFT input states and the RPA results. Moreover, the renormalized singles energy (RSE) corrections play a crucial role in all orders of the many-body expansion. For dimers, RSE corrects the RPA underbinding for every tested Kohn-Sham model: generalized-gradient approximation (GGA), meta-GGA, (meta-)GGA hybrids, as well as the optimized effective potential at the correlated level. Remarkably, the inclusion of singles in RPA can also correct the wrong signs of three- and four-body nonadditive energies as well as mitigate the excessive higher-order contributions to the many-body expansion. The RPA errors are dominated by the contributions of compact clusters. As a workable method for large systems, we propose to replace those compact contributions with CCSD(T) energies and to sum up the remaining many-body contributions up to infinity with supermolecular or periodic RPA. As a demonstration of this approach, we show that for RPA(PBE0)+RSE it suffices to apply CCSD(T) to dimers and 30 compact, hydrogen-bonded trimers to get the methane-water cage interaction energy to within 1.6% of the reference value.

Interaction analyses of SARS-CoV-2 spike protein based on fragment molecular orbital calculations

At the stage of SARS-CoV-2 infection in human cells, the spike protein consisting of three chains, A, B, and C, with a total of 3300 residues plays a key role, and thus its structural properties and the binding nature of receptor proteins to host human cells or neutralizing antibodies has attracted considerable interest. Here, we report on interaction analyses of the spike protein in both closed (PDB-ID: 6VXX) and open (6VYB) structures, based on large-scale fragment molecular orbital (FMO) calculations at the level of up to the fourth-order Møller–Plesset perturbation with singles, doubles, and quadruples (MP4(SDQ)). Inter-chain interaction energies were evaluated for both structures, and a mutual comparison indicated considerable losses of stabilization energies in the open structure, especially in the receptor binding domain (RBD) of chain-B. The role of charged residues in inter-chain interactions was illuminated as well. By two separate calculations for the RBD complexes with angiotensin-converting enzyme 2 (ACE2) (6M0J) and B38 Fab antibody (7BZ5), it was found that the binding with ACE2 or antibody partially compensated for this stabilization loss of RBD.

Visualized IFIE results seen from chain-B of spike protein.

Third-Order Møller-Plesset Theory Made More Useful? The Role of Density Functional Theory Orbitals

The practical utility of Møller-Plesset (MP) perturbation theory is severely constrained by the use of Hartree-Fock (HF) orbitals. It has recently been shown that the use of regularized orbital-optimized MP2 orbitals and scaling of MP3 energy could lead to a significant reduction in MP3 error [Bertels, L. W.; J. Phys. Chem. Lett. 2019, 10, 4170 4176]. In this work, we examine whether density functional theory (DFT)-optimized orbitals can be similarly employed to improve the performance of MP theory at both the MP2 and MP3 levels. We find that the use of DFT orbitals leads to significantly improved performance for prediction of thermochemistry, barrier heights, noncovalent interactions, and dipole moments relative to the standard HF-based MP theory. Indeed, MP3 (with or without scaling) with DFT orbitals is found to surpass the accuracy of coupled-cluster singles and doubles (CCSD) for several data sets. We also found that the results are not particularly functional sensitive in most cases (although range-separated hybrid functionals with low delocalization error perform the best). MP3 based on DFT orbitals thus appears to be an efficient, noniterative O(N⁶) scaling wave-function approach for single-reference electronic structure computations. Scaled MP2 with DFT orbitals is also found to be quite accurate in many cases, although modern double hybrid functionals are likely to be considerably more accurate.

Spin Splitting Energy of Transition Metals: A New, More Affordable Wave Function Benchmark Method and Its Use to Test Density Functional Theory

Accurately predicting the spin splitting energy of chemical species is important for understanding their reactivity and magnetic properties, but it is very challenging, especially for molecules containing transition metals. One impediment to progress is the scarcity of accurate benchmark data. Here we report a set of calculations designed to yield reliable benchmarks for simple transition-metal complexes that can be used to test density functional methods that are affordable for large systems of more practical interest. Various wave function methods are tested against experiment for Fe²⁺, Fe³⁺, and Co³⁺, including CASSCF, CASPT2, CASPT3, MRCISD, MRCISD+Q, ACPF, AQCC, CCSD(T), and CASPT2/CCSD(T) and also a new method called CASPT2.5, which is performed by taking the average of the CASPT2 and CASPT3 energies. We find that MRCISD+Q, ACPF, and AQCC require smaller active spaces for good accuracy than are required by CASPT2 and CASPT3, and this aspect may be important for calculations on larger molecules; here we find that CASPT2.5 extrapolated to a complete basis set is the most suitable method-in terms of computational cost and in terms of accuracy on monatomic systems-and therefore we chose this method for molecular benchmarks. Then Kohn-Sham density functional calculations with 60 exchange-correlation functionals are tested for FeF₂, FeCl₂, and CoF₂. We find that MN15-L, M06-SX, and revM06 have very good agreement with CASPT2.5 benchmarks in terms of both the spin splitting energy and the optimized geometry for each spin state. In addition, we recommend def2-TZVP as the most suitable basis set to perform density functional calculations for molecular spin splitting energies; extra polarization functions in the basis set do not help to increase the accuracy of the spin splitting energy in KS calculations.

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.

How accurate are approximate quantum chemical methods at modelling solute-solvent interactions in solvated clusters?

In this paper, the performance of a wide range of DFT methods is assessed for the calculation of interaction energies of thermal clusters of a solute in water. Three different charge states (neutral, proton transfer transition state and zwitterion) of glycine were solvated by 1 to 40 water molecules as sampled from molecular dynamics simulations. While some ab initio composite methods that employ insufficiently large basis sets incurred significant errors even for a cluster containing only 5 water molecules relative to the W1X-2 benchmark, the DLPNO-CCSD(T)/CBS and DSD-PBEP86 (triple zeta basis set) levels of theory predicted very accurate interaction energies. These levels of theory were used to benchmark the performance of 16 density functionals from different rungs of Jacob's Ladder. Of the Rung 4 functionals examined, the ωB97M-V and ωB97X-V functionals stood out for predicting absolute interaction energies in 40-water clusters with mean absolute deviations (MAD) ∼4 kJ mol-1. The B3LYP-D3(BJ) functional performed exceptionally well with a MAD ∼1.7 kJ mol-1 and is the overall best performing method. Calculations of relative interaction energies allow for cancellation of systematic errors, including basis set truncation and superposition errors, and the ωB97M-V and B3LYP-D3(BJ) double zeta basis set calculations yielded relative interaction energies that are within ∼3 kJ mol-1 of the benchmark. The ONIOM approximation provides another strategy for accelerating the calculation of accurate absolute interaction energies provided that the calculations have converged with respect to the size of the "high-level-layer".

Molecular Design of Aromatic Polythionoesters

As an example of molecular design of new polymers, structures and properties of poly(ethylene thionoterephthalate) (PET[S₂]) and the related polymers have been predicted from calculations of ab initio molecular orbital (MO) theory, rotational isomeric state (RIS) scheme, and periodic density functional theory (DFT). The MO calculations were confirmed by NMR experiments and introduced to the RIS scheme for PET[S₂] to yield its configurational properties, which are compared herein with those of analogous polyester, polythioester, and polydithioester. Configurational properties of randomly thiono-substituted poly(ethylene terephthalate) (PET), PET[S _z O_1-z ], were also evaluated as a function of sulfidity (z). On the assumption that the crystal of PET[S₂] can be expressed as an isomorphic replacement of the PET crystal, the crystal structure was optimized by a periodic DFT simulation and its Young's moduli in the a-, b-, and c-axis directions were, respectively, evaluated to be E_a = 0.94(7.20) GPa, E _b = 19.58(22.26) GPa, and E _c = 142.1(182.4) GPa, where the parenthetic values are those of the PET crystal. There is a possibility that properties of PET[S _z O_1-z ] will be controlled between those of PET and PET[S₂] by adjusting the sulfidity. The potential practical applications of the polythionoesters are also discussed herein. By purely theoretical computations, the structures and properties of the not-yet synthesized polymers were predicted quantitatively; that is, the theoretical molecular design of new polymers has been achieved.

Is ADC(3) as Accurate as CC3 for Valence and Rydberg Transition Energies?

The search for new models rapidly delivering accurate excited-state energies and properties is one of the most active research lines of theoretical chemistry. Along with these developments, the performance of known methods is constantly reassessed on the basis of new benchmark values. In this Letter, we show that the third-order algebraic diagrammatic construction, ADC(3), does not yield transition energies of the same quality as the third-order coupled cluster method, CC3. This is demonstrated by extensive comparisons with several hundred high-quality vertical transition energies obtained with FCI, CCSDTQ, and CCSDT. Direct comparisons with experimental 0-0 energies of small- and medium-size molecules support the same conclusion, which holds for both valence and Rydberg transitions. Considering these results, we introduce a composite approach, ADC(2.5), which consists of averaging the ADC(2) and ADC(3) excitation energies. Although ADC(2.5) does not match the CC3 accuracy, it significantly improves the ADC(3) results, especially for vertical energies.

Theoretical Description of R-X⋯NH3 Halogen Bond Complexes: Effect of the R Group on the Complex Stability and Sigma-Hole Electron Depletion

In the present work, a number of R–X⋯NH₃ (X = Cl, Br, and I) halogen bonded systems were theoretical studied by means of DFT calculations performed at the ωB97XD/6-31+G(d,p) level of theory in order to get insights on the effect of the electron-donating or electron-withdrawing character of the different R substituent groups (R = halogen, methyl, partially fluorinated methyl, perfluoro-methyl, ethyl, vinyl, and acetyl) on the stability of the halogen bond. The results indicate that the relative stability of the halogen bond follows the Cl < Br < I trend considering the same R substituent whereas the more electron-withdrawing character of the R substituent the more stable the halogen bond. Refinement of the latter results, performed at the MP2/6-31+G(d,p) level showed that the DFT and the MP2 binding energies correlate remarkably well, suggesting that the Grimme’s type dispersion-corrected functional produces reasonable structural and energetic features of halogen bond systems. DFT results were also observed to agree with more refined calculations performed at the CCSD(T) level. In a further stage, a more thorough analysis of the R–Br⋯NH₃ complexes was performed by means of a novel electron localization/delocalization tool, defined in terms of an Information Theory, IT, based quantity obtained from the conditional pair density. For the latter, our in-house developed C++/CUDA program, called KLD (acronym of Kullback–Leibler divergence), was employed. KLD results mapped onto the one-electron density plotted at a 0.04 a.u. isovalue, showed that (i) as expected, the localized electron depletion of the Br sigma-hole is largely affected by the electron-withdrawing character of the R substituent group and (ii) the R–X bond is significantly polarized due to the presence of the NH₃ molecule in the complexes. The afore-mentioned constitutes a clear indication of the dominant character of electrostatics on the stabilization of halogen bonds in agreement with a number of studies reported in the main literature. Finally, the cooperative effects on the [Br—CN]_n system (n = 1–8) was evaluated at the MP2/6-31+G(d,p) level, where it was observed that an increase of about ~14.2% on the complex stability is obtained when going from n = 2 to n = 8. The latter results were corroborated by the analysis of the changes on the Fermi-hole localization pattern on the halogen bond zones, which suggests an also important contribution of the electron correlation in the stabilization of these systems.

Curvature and size effects hinder halogen bonds with extended π systems

The curvature of aromatic systems strengthens the interaction by the concave face while it weakens by the convex one. Parallel structures are favoured over halogen bonded ones.

Third-Order Møller-Plesset Perturbation Theory Made Useful? Choice of Orbitals and Scaling Greatly Improves Accuracy for Thermochemistry, Kinetics, and Intermolecular Interactions

We develop and test methods that include second- and third-order perturbation theory (MP3) using orbitals obtained from regularized orbital-optimized second-order perturbation theory, κ-OOMP2, denoted as MP3:κ-OOMP2. Testing MP3:κ-OOMP2 shows RMS errors that are 1.7-5 times smaller than those of MP3 across 7 data sets. To do still better, empirical training of the scaling factors for the second- and third-order correlation energies and the regularization parameter on one of those data sets led to an unregularized scaled (c₂ = 1.0; c₃ = 0.8) denoted as MP2.8:κ-OOMP2. MP2.8:κ-OOMP2 yields significant additional improvement over MP3:κ-OOMP2 in 4 of 6 test data sets on thermochemistry, kinetics, and noncovalent interactions. Remarkably, these two methods outperform coupled cluster with singles and doubles in 5 of the 7 data sets considered, at greatly reduced cost (no O(N6) iterations).

Reassessment of the Mechanisms of Thermal C-H Bond Activation of Methane by Cationic Magnesium Oxides: A Critical Evaluation of the Suitability of Different Density Functionals

The mechanisms of the thermal reactions of the two iconic magnesium oxide cations MgO^.+ and Mg₂ O₂ ^.+ with methane have been re-evaluated at the CCSD(T)/CBS//CCSD/def2-TZVP level of theory. For the reaction of MgO^.+ with CH₄ , only the classical hydrogen-atom transfer (HAT) was found; in contrast, for the Mg₂ O₂ ^.+ /CH₄ couple, both HAT and proton-coupled electron-transfer (PCET) exist as mechanistic variants. In order to evaluate the suitability of density functional theory (DFT) methods, the reactions were computed by using 27 density functionals. The results obtained demonstrate that the various DFT methods often deliver rather different results for both geometric and energetic features. As to the prediction of the apparent barriers, pure functionals give the largest mean absolute errors. BMK, ωB97XD, and the double-hybrid functional mPW2PLYP were confirmed to come closest to the results provided by CCSD(T)/CBS. Thus, mechanistic conclusions based on a single DFT method should be viewed with great caution. In summary, this study may assist in the selection of a suitable quantum chemical method to unravel the mechanistic details of C-H bond activation by charged metal oxides.

Dihalogen and Pnictogen Bonding in Crystalline Icosahedral Phosphaboranes

Noncovalent interactions in the single crystal of 3,6-Cl2-closo-1,2-P2B10H8 and in the crystal of closo-1,7-P2B10Cl10•toluene were analyzed by means of quantum chemical computations. The crystal packing in the second crystal was dominated by numerous B-Cl···Cl-B dihalogen and strong B-P···π pnictogen bonds, the latter of which were characterized by a small length of 3.08 Å and a large interaction energy value, exceeding −10 kcal mol−1.

RI-MP3 calculations of biomolecules based on the fragment molecular orbital method

In this study, the third-order Møller-Plesset perturbation (MP3) theory using the resolution of the identity (RI) approximation was combined with the fragment molecular orbital (FMO) method to efficiently calculate a high-order electron correlation energy of biomolecular systems. We developed a new algorithm for the RI-MP3 calculation, which can be used with the FMO scheme. After test calculations using a small molecule, the FMO-RI-MP3 calculations were performed for two biomolecular systems comprising a protein and a ligand. The computational cost of these calculations was only around 5 and 4 times higher than those of the FMO-RHF calculations. The error associated with the RI approximation was around 2.0% of the third-order correlation contribution to the total energy. However, the RI approximation error in the interaction energy between the protein and ligand molecule was insignificantly small, which reflected the negligible error in the inter fragment interaction energy. © 2018 Wiley Periodicals, Inc.