Understanding the Many-Body Basis Set Superposition Error: Beyond Boys and Bernardi

Scalable generalized screening for high-order terms in the many-body expansion: Algorithm, open-source implementation, and demonstration

The many-body expansion lies at the heart of numerous fragment-based methods that are intended to sidestep the nonlinear scaling of ab initio quantum chemistry, making electronic structure calculations feasible in large systems. In principle, inclusion of higher-order n-body terms ought to improve the accuracy in a controllable way, but unfavorable combinatorics often defeats this in practice and applications with n ≥ 4 are rare. Here, we outline an algorithm to overcome this combinatorial bottleneck, based on a bottom-up approach to energy-based screening. This is implemented within a new open-source software application ("Fragme∩t"), which is integrated with a lightweight semi-empirical method that is used to cull subsystems, attenuating the combinatorial growth of higher-order terms in the graph that is used to manage the calculations. This facilitates applications of unprecedented size, and we report four-body calculations in (H2O)64 clusters that afford relative energies within 0.1 kcal/mol/monomer of the supersystem result using less than 10% of the unique subsystems. We also report n-body calculations in (H2O)20 clusters up to n = 8, at which point the expansion terminates naturally due to screening. These are the largest n-body calculations reported to date using ab initio electronic structure theory, and they confirm that high-order n-body terms are mostly artifacts of basis-set superposition error.

Nuclear Magnetic Resonance Study of CO2 Capture by Fluoroalkylamines: Ammonium Ion pKa Depression via Fluorine Modification and Thermochemistry of Carbamylation

We are developing energy-efficient and reversible carbon capture and release (CCR) systems that mimic the Lys201 carbamylation reaction in the active site of ribulose-1,5-bisphosphate carboxylase-oxygenase (RuBisCO). The multiequilibria scenario ammonium ion Xa ⇌ amine Xb ⇌ carbamic acid Xc ⇌ carbamate Xd requires the presence of both free amine and CO2 for carbamylation and is affected by the pKa(Xa). Two fluorination strategies aimed at ammonium ion pKa depression and low pH carbamylation were analyzed with (2,2,2-trifluoroethyl)butylamine 2b and 2,2-difluoropropylamine 3b and compared to butylamine 1b. The determination of K1 and ΔG1 of the carbamylation reactions requires the solution of multiequilibria systems of equations based on initial conditions, 1H NMR measurements of carbamylation yields over a wide pH range, and knowledge of K2-K5 values. K2 and K3 describe carbonic acid acidity, and ammonium ion acidities K4 were measured experimentally. We calibrated carbamic acid acidities K5 based on the measured value K6 of aminocarbamic acid using isodesmic reactions. The proton exchange reactions were evaluated with ab initio computations at the APFD/6-311+G* level in combination with continuum solvation models and explicit solvation. The utilities of 1-3 will be discussed as they pertain to the development of fluorine-modified RuBisCO-mimetic reversible CCR systems.

DFT and TD-DFT study of hydrogen bonded complexes of aspartic acid and n water (n = 1 and 2)

CONTEXT

Hydrogen bonds (HB) influence the conformational preferences of biomolecules and their optical and electronic properties. The directional interaction of molecules of water can be a prototype to understand the effects of HBs on biomolecules. Among the neurotransmitters (NT), L-aspartic acid (ASP) stands out due to its importance in health and as a precursor of several biomolecules. As it presents different functional groups and readily forms inter- and intramolecular HBs, ASP can be considered a prototype for understanding the behavior of NTs when interacting by HB with other substances. Although several theoretical studies have been performed in the past on isolated ASP and its formed complexes with water, both in gas and liquid phases, using DFT and TD-DFT formalisms, these works did not perform large basis set calculations or study electronic transitions of ASP-water complexes. We investigated the HB interactions in complexes of ASP and water molecules. The results show that the interactions between the carboxylic groups of ASP with water molecules, forming cyclic structures with two HBs, lead to more stable and less polar complexes than other conformers formed between water and the NH₂ group. It was observed that there is a relationship between the deviation in the UV-Vis absorption band of the ASP and the interactions of water with the HOMO and LUMO orbitals with the stabilization/destabilization of the S₁ state to the S₀ of the complexes. However, in some cases, such as 1:1 complex ASP-W2, this analysis may be inaccurate due to small changes in ΔE.

METHODS

We studied the landscapes of the ground state surface of different conformers of isolated L-ASP and the L-ASP-(H₂O)_n complexes (n = 1 and 2) using the DFT formalism, with the B3LYP functional, and six different basis sets: 6-31 +  + G(d,p), 6-311 +  + G(d,p), D95 +  + (d,p), D95V +  + (d,p), cc-pVDZ, and, cc-pVTZ basis sets. The cc-pVTZ basis set provides the minimum energy of all conformers, and therefore, we performed the analysis with this basis set. We evaluated the stabilization of the ASP and complexes using the minimum ground state energy, corrected by the zero point energy and the interaction energy between the ASP and the water molecules. We also calculated the vertical electronic transitions S₁ ← S₀, and their properties using the TD-DFT formalism at B3LYP/cc-pVTZ level with the optimized geometries for S₀ state with the same basis set. For the analysis of the vertical transitions of isolated ASP and the ASP-(H₂O)_n complexes, we calculated the electrostatic energy in the S₀ and S₁ states. We performed the calculations with the Gaussian 09 software package. We used the VMD software package to visualize the geometries and shapes of the molecule and complexes.

Extending multi-layer energy-based fragment method for excited-state calculations of large covalently bonded fragment systems

Recently, we developed a low-scaling Multi-Layer Energy-Based Fragment (MLEBF) method for accurate excited-state calculations and nonadiabatic dynamics simulations of nonbonded fragment systems. In this work, we extend the MLEBF method to treat covalently bonded fragment ones. The main idea is cutting a target system into many fragments according to chemical properties. Fragments with dangling bonds are first saturated by chemical groups; then, saturated fragments, together with the original fragments without dangling bonds, are grouped into different layers. The accurate total energy expression is formulated with the many-body energy expansion theory, in combination with the inclusion-exclusion principle that is used to delete the contribution of chemical groups introduced to saturate dangling bonds. Specifically, in a two-layer MLEBF model, the photochemically active and inert layers are calculated with high-level and efficient electronic structure methods, respectively. Intralayer and interlayer energies can be truncated at the two- or three-body interaction level. Subsequently, through several systems, including neutral and charged covalently bonded fragment systems, we demonstrate that MLEBF can provide accurate ground- and excited-state energies and gradients. Finally, we realize the structure, conical intersection, and path optimizations by combining our MLEBF program with commercial and free packages, e.g., ASE and SciPy. These developments make MLEBF a practical and reliable tool for studying complex photochemical and photophysical processes of large nonbonded and bonded fragment systems.

Computational investigation of the increased virulence and pathogenesis of SARS-CoV-2 lineage B.1.1.7

New variants of SARS-CoV-2 are being reported worldwide. The World Health Organization has reported Alpha (B.1.1.7), Beta (B.1.351), Gamma (P.1), Delta (B.1.617.2) and Omicron (B.1.1.529) as the variants of concern. There are speculations that the variants might evade the host immune responses induced by currently available vaccines and develop resistance to drugs under consideration. The first step of viral infection in COVID-19 occurs through the interaction of the spike protein's receptor-binding domain (RBD) with the peptidase domain of the human ACE-2 (hACE-2) receptor. This study aims to get a molecular-level understanding of the mechanism behind the increased infection rate in the alpha variant. We have computationally studied the spike protein interaction in both the wild-type and B.1.1.7 variant with the hACE-2 receptor using molecular dynamics and MM-GBSA based binding free energy calculations. The binding free energy difference shows that the mutant variant of the spike protein has increased binding affinity for the hACE-2 receptor (i.e. ΔG(N501Y,A570D) is in the range -7.2 to -7.6 kcal mol^-1) and the results were validated using Density functional theory. We demonstrate that with the use of state-of-the-art computational approaches, we can, in advance, predict the virulent nature of variants of SARS-CoV-2 and alert the world healthcare system.

Fragmentation Method for Computing Quantum Mechanics and Molecular Mechanics Gradients for Force Matching: Validation with Hydration Free Energy Predictions Using Adaptive Force Matching

A fragmentation approach referred to as a simple overlapping region method for force matching (SORForM) is presented. SORForM is designed to enable efficient computation of quantum mechanical (QM) forces for large molecules and is validated in the framework of adaptive force matching (AFM) to develop solute models in water. The SORForM method divides a molecule into overlapping QM regions with each region containing a gradient zone and a buffer zone. The buffer zone ensures that the atoms in the gradient zone have their surroundings unchanged with fragmentation. The performance of the method is validated with mefenamic acid and linalyl acetate by comparing the hydration free energies of AFM models developed with and without SORForM. The AFM hydration free energies are also compared with that of the experiments. The models developed with B3LYP-D3(BJ) and def2-TZVP are in excellent agreement with experiments. Our work shows that PBE-D3(BJ) provides less satisfactory results when compared to B3LYP-D3(BJ). The def2-TZVP basis set is found to greatly improve the agreement with experiments when compared to a double-zeta quality basis set.

Behavior of counterpoise correction in many-body molecular clusters of organic compounds: Hartree-Fock interaction energy perspective

The counterpoise (CP) correction by Boys and Bernardi has been well accepted as a reliable strategy to account for basis set superposition error (BSSE) in intermolecular complexes. The behavior of the CP correction was thoroughly studied in individual molecules of molecular complexes. This work studies the performance of the CP correction in many‐body clusters including three‐body clusters of organic compounds in the 3B‐69 dataset. Additionally, we used crystal structures of polymorphs of benzene, aspirin, and oxalyl dihydrazide (ODH) to construct a many‐body cluster dataset, abbreviated as the MBC‐36 dataset, consisting of two, four and eight molecules, and 16 molecules in the case of benzene. A series of Dunning's basis sets—cc‐pXZ and aug‐cc‐pXZ (X = D and T)—were used to predict CP‐corrected Hartree–Fock (HF) interaction energies of the 3B‐69 and MBC‐36 datasets. The CP‐corrected interaction energies were found to be basis‐set independent, whereas the non‐CP corrected interaction energies were found not to a follow a smooth exponential fitting as previously found for electronic energies of individual molecules. This observation was attributed to the presence of non‐additive induction forces in some clusters. Two 2 × 2 × 2 supercells of benzene polymorphs were constructed to explore the local nature of BSSE effects. A cut‐off radius of 10 Å was demonstrated to be sufficient to fully recover these effects. Although the behavior of CP correction was found to be non‐conventional in many‐body clusters of organic compounds, the use of a small basis set such as cc‐pVDZ showed excellent performance in the prediction of HF interaction energies.

Theoretical investigation of CH-bond activation by photocatalytic excited SO2 and the effects of C-, N-, S-, and Se-doped TiO2

The photocatalytic sulfoxidation on TiO₂ discovered by Parrino et al. represents a new, interesting and lower energy route for the synthesis of sulfonic acids. Sulfonic acids are important precursors for dyes, detergents and drugs. In the commonly known industrial process, SO₂ and a specific hydrocarbon are converted into sulfonic acids using high-energy UV light. In this reaction, SO₂ is excited into a metastable triplet state (³SO₂), which has the potential to activate a CH-bond of hydrocarbons and start a radical reaction cycle. By introducing TiO₂ as a photocatalyst, it has been shown that visible light can be used for the synthesis. This offers the potential to be a cost-effective reaction approach for industrial use. However, experimental studies indicate that the initial excitation mechanism of SO₂ on TiO₂ is significantly different from the catalyst-free mechanism. Parrino et al. were able to reveal first evidence for the existence of a charge-transfer process from SO₂ to the TiO₂ surface by means of electrochemical experiments. First theoretical investigations from first principles were able to further substantiate the existence of a charge-transfer. However, to fully understand this mechanism, more accurate methods such as Time Dependent Density Functional Theory (TD-DFT) or ab initio multireference methods such as the Complete Active Space Self Consistent Field (CASSCF) method are required. Furthermore, after understanding the charge-transfer mechanism, the introduction of dopants into TiO₂ can be investigated in order to possibly redshift the excitation energy. This might open the route to using lower energy light for the sulfoxidation of hydrocarbons on TiO₂ as a new potential industrial reaction for the synthesis of sulfonic acids. In this work, we will study the initial step of the photocatalytic sulfoxidation of hydrocarbons using the TD-DFT and CASSCF methods by using a combined approach consisting of calculations with periodic boundary conditions and a newly constructed embedded cluster model. Furthermore, we will explore the effects of doping by introducing four heteroatoms (C, N, S, and Se) into the TiO₂ surfaces anatase[101] and rutile[110] to find a possible enhancement of the photocatalytic reactivity by lowering the electronic excitation energy.

Supramolecular Approach to Tuning the Photophysical Properties of Quadrupolar Squaraines

In the present study, the influence of the hydrogen bonding for the one- and two-photon absorption of the prototypical squaraine dye is investigated with quantum chemistry tools. The central squaraine unit is bound by strong hydrogen bonds with 4-substituted N,N'-diphenylurea and, alternatively, N,N'-diphenylthiourea molecules, which affects to a high extend the properties of the squaraine electron accepting moiety, thus shifting its maximum absorption wavelength and enhancing the TPA cross section. The replacement of oxygen by sulfur atoms in the squaraine central ring, known to affect its photophysical behavior, is considered here as the way of modifying the strength and nature of the intermolecular contacts. Additionally, the influence of the oxygen-by-sulfur replacement is also considered in the N,N'-diphenylurea moiety, as the factor affecting the acidity of the N-H protons. The introduction of the sequence of the substituents of varying electron-donating or electron-withdrawing characters in the position 4 of N,N'-diphenyl(thio)urea subsystems allows to finely tune the hydrogen bonding with the central squaraine unit by further modification of the N-H bond characteristics. All of these structural modifications lead to the controlled adjustment of the electron density distribution, and thus, the properties affected such as transition moments and absorption intensity. Ab initio calculations provide strong support for this way of tailoring of one- or two-photon absorption due to the obtained strong hypsochromic shift of the maximum one-photon absorption wavelength observed particularly for thiosquaraine complexes and an increase in the TPA wavelength together with the increase in the TPA cross section. Moreover, the source of the strong modification of the thiosquaraine OPA in contrast to the pristine oxosquaraine upon N,N'-diphenyl(thio)urea substitution is determined. Furthermore, for the first time, the linear dependence of the non-additivity in the interaction energy on the Hammett substituent constant is reported. The stronger the electron-donating character of the substituent, the larger the three-body non-additive components and the larger their percentage to the total interaction energy.

A unified and flexible formulation of molecular fragmentation schemes

We present a flexible formulation for energy-based molecular fragmentation schemes. This framework does not only incorporate the majority of existing fragmentation expansions but also allows for flexible formulation of novel schemes. We further illustrate its application in multi-level approaches and for electronic interaction energies. For the examples of small water clusters, a small protein, and protein-protein interaction energies, we show how this flexible setup can be exploited to generate a well-suited multi-level fragmentation expansion for the given case. With such a setup, we reproduce the electronic protein-protein interaction energy of ten different structures of a neurotensin and an extracellular loop of its receptor with a mean absolute deviation to the respective super-system calculations below 1 kJ/mol.

Balanced lipase interactions for degradation-controlled paclitaxel release from lipid cubic phase formulations

Lipid cubic phase (LCP) formulations enhance the intestinal solubility and bioavailability of hydrophobic drugs by reducing precipitation and facilitating their mass transport to the intestinal surface for absorption. LCPs with an ester linkage connecting the acyl chain to the glycerol backbone (monoacylglycerols), are susceptible to chemical digestion by several lipolytic enzymes including lipases, accelerating the release of hydrophobic agents from the lipid bilayers of the matrix. Unlike regular enzymes that transform soluble substrates, lipolytic enzymes act at the interface of water and insoluble lipid. Therefore, compounds that bind to this interface can enhance or inhibit the activity of enzymes to varying extent. Here, we explore how the lipolysis rate can be tuned by the interfacial interaction of porcine pancreatic lipase with monoolein LCPs containing a known lipase inhibitor, tetrahydrolipstatin. Release of the Biopharmaceutical Classification System (BCS) class IV drug, paclitaxel, from the inhibitor-modified LCP was examined in the presence of lipase and its effectors colipase and calcium. By combining experimental dynamic digestion studies, thermodynamic measurements and molecular dynamics simulations of the competitive inhibition of lipase by tetrahydrolipstatin, we reveal the role and mode of action of lipase effectors in creating a precisely-balanced degradation-controlled LCP release system for the poorly soluble paclitaxel drug.

The Many-Body Expansion for Aqueous Systems Revisited: II. Alkali Metal and Halide Ion-Water Interactions

We present a detailed study of the many-body expansion (MBE) for alkali metal and halide ion-water interactions and quantify the effect of these ions on the strength of the surrounding aqueous hydrogen bonding environment. Building on our previous work on neutral water clusters [J. P. Heindel and S. S. Xantheas, J. Chem. Theor. Comput. 16 (11), 6843-6855 (2020)], we carry out the MBE for the alkali metal and halide ion-water clusters, Z^+/-(H₂O)₉, where Z = Li⁺, Na⁺, K⁺, Rb⁺, Cs⁺, F^-, Cl^-, Br^-, and I^- and compare them with the results for a pure water cluster with the same number of "bodies", viz., (H₂O)₁₀. The 2-B ion-water (I-W) interaction accounts for a larger percentage of the total cluster binding energy compared to a pure water cluster of the same size, with the total 3-B term being smaller and of opposite sign (repulsive), whereas higher order terms are essentially negligible. The same oscillating behavior around zero for the MBE terms higher than the 5-B with a basis set that was reported for water clusters is also observed for the ion-water clusters considered here, with the basis set superposition error (BSSE) corrections amending this as in the water cluster case. A remarkable, linear anticorrelation between the total 2-B (I-W), the total 2-B (W-W), and also the 3-B (W-W-W) interactions is found, quantifying the effect of the different ions in disrupting and altering (weakening) the neighboring hydrogen bonded water network: stronger (I-W) interactions result in weaker (W-W) interactions. Additional linear correlations across the two series of alkali metals and halide ions were found between the 3-B (I-W-W) and the 2-B (I-W) as well as between the 3-B (I-W-W) and the 3-B (W-W-W) interactions, suggesting the existence of previously unrealized underlying physics governing these 2-B intermolecular and 3-B collective interactions. Our results further suggest a universal behavior of the two different families of ions (alkali metals and halides) for both the correlations of the various components of the total binding energies and the estimate of the 2-B BSSE correction, which is reported to follow a common profile for ion-water and water-water interactions when cast in terms of reduced distances and energies of the respective dimers. We expect the current results that quantify the interplay between ion-water and water-water interactions in aqueous clusters to impact the development of classical, ab initio-based force fields for monatomic ion solvation, whereas the insights into the nature of the BSSE to be critical in future ab initio-based, many-body molecular dynamics studies.

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.

Fantasy versus reality in fragment-based quantum chemistry

Since the introduction of the fragment molecular orbital method 20 years ago, fragment-based approaches have occupied a small but growing niche in quantum chemistry. These methods decompose a large molecular system into subsystems small enough to be amenable to electronic structure calculations, following which the subsystem information is reassembled in order to approximate an otherwise intractable supersystem calculation. Fragmentation sidesteps the steep rise (with respect to system size) in the cost of ab initio calculations, replacing it with a distributed cost across numerous computer processors. Such methods are attractive, in part, because they are easily parallelizable and therefore readily amenable to exascale computing. As such, there has been hope that distributed computing might offer the proverbial "free lunch" in quantum chemistry, with the entrée being high-level calculations on very large systems. While fragment-based quantum chemistry can count many success stories, there also exists a seedy underbelly of rarely acknowledged problems. As these methods begin to mature, it is time to have a serious conversation about what they can and cannot be expected to accomplish in the near future. Both successes and challenges are highlighted in this Perspective.

Assessing Electronic Structure Methods for Long-Range Three-Body Dispersion Interactions: Analysis and Calculations on Well-Separated Metal Atom Trimers

Three-body dispersion interactions are much weaker than their two-body counterpart. However, their importance grows quickly as the number of interacting monomers rises. To explore the numerical performance of correlation methods for long-range three-body dispersion, we performed calculations on eight very simple dispersion-dominated model metal trimers: Na₃, Mg₃, Zn₃, Cd₃, Hg₃, Cu₃, Ag₃, and Au₃. One encouraging aspect is that relatively small basis sets of augmented triple-ζ size appear to be adequate for three-body dispersion in the long-range. Coupled cluster calculations were performed at high levels to assess MP3, CCSD, CCSD(T), empirical density functional theory dispersion (D3), and the many-body dispersion (MBD) approach. We found that the accuracy of CCSD(T) was generally significantly lower than for two-body interactions, with errors sometimes reaching 20% in the investigated systems, while CCSD and particularly MP3 were generally more erratic. MBD is found to perform better than D3 at large distances, whereas the opposite is true at shorter distances. When computing reference numbers for three-body dispersion, care should be taken to appropriately represent the effect of the connected triple excitations, which are significant in most cases and incompletely approximated by CCSD(T).

Understanding non-covalent interactions in larger molecular complexes from first principles

Non-covalent interactions pervade all matter and play a fundamental role in layered materials, biological systems, and large molecular complexes. Despite this, our accumulated understanding of non-covalent interactions to date has been mainly developed in the tens-of-atoms molecular regime. This falls considerably short of the scales at which we would like to understand energy trends, structural properties, and temperature dependencies in materials where non-covalent interactions have an appreciable role. However, as more reference information is obtained beyond moderately sized molecular systems, our understanding is improving and we stand to gain pertinent insights by tackling more complex systems, such as supramolecular complexes, molecular crystals, and other soft materials. In addition, accurate reference information is needed to provide the drive for extending the predictive power of more efficient workhorse methods, such as density functional approximations that also approximate van der Waals dispersion interactions. In this perspective, we discuss the first-principles approaches that have been used to obtain reference interaction energies for beyond modestly sized molecular complexes. The methods include quantum Monte Carlo, symmetry-adapted perturbation theory, non-canonical coupled cluster theory, and approaches based on the random-phase approximation. By considering the approximations that underpin each method, the most accurate theoretical references for supramolecular complexes and molecular crystals to date are ascertained. With these, we also assess a handful of widely used exchange-correlation functionals in density functional theory. The discussion culminates in a framework for putting into perspective the accuracy of high-level wavefunction-based methods and identifying future challenges.

Understanding the molecular orientation growth on a nanometer scale and adjustable electron transition performance of a terpyridyl derivative under different external environments

TD-DFT calculation is used to forecast the aggregation style of PYTPY under different external conditions and predict the related morphology and electron transition performance.