NENCI-2021 Part I: A Large Benchmark Database of Non-Equilibrium Non-Covalent Interactions Emphasizing Close Intermolecular Contacts
Zachary M. Sparrow, Brian G. Ernst, Paul T. Joo, Ka Un Lao, Robert A. DiStasio Jr
NNENCI-2021 Part I: A Large Benchmark Database of Non-EquilibriumNon-Covalent Interactions Emphasizing Close Intermolecular Contacts
Zachary M. Sparrow, a) Brian G. Ernst, a) Paul T. Joo, Ka Un Lao, and Robert A. DiStasio Jr. b) Department of Chemistry and Chemical Biology, Cornell University, Ithaca, NY 14853 USA (Dated: 5 February 2021)
In this work, we present nenci- , a benchmark database of approximately , non-equilibrium non-covalent interaction energies for a large and diverse selection of intermolecular complexes of biological andchemical relevance. To meet the growing demand for large and high-quality quantum mechanical data inthe chemical sciences, nenci- starts with the molecular dimers in the widely used S66 and S101databases, and extends the scope of these works by: ( i ) including cation- and anion- π complexes, a fun-damentally important class of non-covalent interactions (NCIs) that are found throughout nature and posea substantial challenge to theory, and ( ii ) systematically sampling all intermolecular potential energysurfaces (PES) by simultaneously varying the intermolecular distance and intermolecular angle in each dimer.Designed with an emphasis on close contacts, the complexes in nenci- were generated by sampling sevenintermolecular distances along each PES (ranging from . × − . × the equilibrium separation) as well asnine intermolecular angles per distance (five for each ion- π complex), yielding an extensive database of , benchmark intermolecular interaction energies ( E int ) obtained at the CCSD(T)/CBS level of theory. The E int values in nenci- span a total of . kcal/mol, ranging from − . kcal/mol to +186 . kcal/mol, witha mean (median) E int value of − . kcal/mol ( − . kcal/mol). In addition, a wide range of intermolecularatom-pair distances are also present in nenci- , where close intermolecular contacts involving atoms thatare located within the so-called van der Waals envelope are prevalent—these interactions in particular posean enormous challenge for molecular modeling and are observed in many important chemical and biologicalsystems. A detailed SAPT-based energy decomposition analysis also confirms the diverse and comprehensivenature of the intermolecular binding motifs present in nenci- , which now includes a significant number ofprimarily induction-bound dimers ( e.g. , cation- π complexes). nenci- thus spans all regions of the SAPTternary diagram, thereby warranting a new four-category classification scheme that includes complexes pri-marily bound by electrostatics ( , ), induction ( ), dispersion ( , ), or mixtures thereof ( , ). Acritical error analysis performed on a representative set of intermolecular complexes in nenci- demon-strates that the E int values provided herein have an average error of ± . kcal/mol, even for complexes withstrongly repulsive E int values, and maximum errors of ± . − . kcal/mol ( i.e. , approximately ± . kJ/mol)for the most challenging cases. For these reasons, we expect that nenci- will play an important role in thetesting, training, and development of next-generation classical and polarizable force fields, density functionaltheory (DFT) approximations, wavefunction theory (WFT) methods, as well as machine learning (ML) basedintra- and inter-molecular potentials. I. INTRODUCTION
With tunable strengths situated between thermal fluc-tuations and covalent bonds, non-covalent interactions(NCIs) are ubiquitous in nature and play a criticalrole in determining the structure, stability, and functionin a number of systems throughout chemistry, biology,physics, and materials science.
One particularly illus-trative example is the famous DNA double helix, whosestructure is stabilized by a complex network of hydro-gen bonds and π - π stacking interactions between con-stituent nucleobases. In organic synthesis and biochem-istry, many catalysts and enzymes function by leveragingNCIs to position/orient substrates for the ensuing reac-tion and/or stabilize critical points along the reactionpathway, e.g. , ion- π interactions can stabilize intermedi- a) These authors contributed equally to this work. b) Electronic mail: [email protected] ates and transition states with excess charge.
Overthe past two decades, NCIs have garnered critical recog-nition throughout the chemical sciences, and have nowbecome an integral part of “chemical intuition” when ra-tionalizing complex chemical structures and/or processesas well as designing molecular systems ( e.g. , catalysts)for optimal performance and/or novel applications. Inthis regard, there are quite a number of NCI-based appli-cations actively under investigation, ranging from crys-tal engineering (where hydrogen and halogen bondingare used to direct molecular assembly) and artificialmolecular machines (where π - π stacking, hydrogen bond-ing, and dispersion/van der Waals (vdW) forces are lever-aged to control complex motion at the nanoscale) todrug discovery (where candidate molecules are selectedand screened based on specific NCIs present in the cor-responding active site). Given their importance and prevalence, it is imper-ative that there exists a suite of computational meth-ods that can provide an accurate, reliable, and compu-tationally efficient description of NCIs for systems rang- a r X i v : . [ phy s i c s . c h e m - ph ] F e b ing from small gas-phase molecular dimers to the com-plex tertiary and quaternary structure of proteins in sol-vent. To meet these goals, a number of computationaltechniques have been developed over the past century,including (but not limited to): model intermolecularpotentials ( e.g. , Lennard-Jones), classical and polar-izable force fields, density functional theory (DFT)approximations with corrections for dispersion/vdW in-teractions, efficient ( i.e. , linear scaling) algorithmsfor highly accurate wavefunction theory (WFT) meth-ods, and more recently, the large and rapidly growingsuite of machine learning (ML) based approaches. During this time, such computational methods have en-joyed tremendous success and made critical contributionsto a number of different fields, e.g. , identifying promisingpharmaceutical molecules, predicting (meta-)stablemolecular crystal polymorphs, and elucidating su-percritical behavior in high-pressure liquid hydrogen, to name a few. However, we would argue that the nextgeneration of theoretical approaches for describing NCIswould tremendously benefit by addressing the followingchallenges in an accurate, reliable, and computationallyefficient manner: ( i ) the need to describe NCIs in largemolecular and condensed-phase systems ( i.e. , collectivemany-body effects, solvation/solvent effects, simultane-ous treatment of short-, intermediate-, and long-rangeNCIs on the same footing); ( ii ) the need to describethe diverse types of NCIs on the same footing ( i.e. , simi-lar performance for hydrogen bonding, π - π stacking, dis-persion, ion- π interactions, etc); and ( iii ) the needto describe NCIs in equilibrium and non-equilibrium sys-tems on the same footing ( i.e. , similar performance acrossentire potential energy surfaces (PES)). An essential part of developing next-generation the-oretical methods for describing NCIs involves testingand/or training new approximations against highly accu-rate benchmark data. For the increasingly popular suiteof ML-based models—which require large amounts ofhigh-quality data to learn the quantum mechanics under-lying NCIs—such reference data is of critical importance.However, such benchmark non-covalent/non-bonded in-teraction (or binding) energies ( E int ) are seldom experi-mentally available, especially for large/complex systemsand non-equilibrium configurations. Instead, one usuallyrelies on quantum chemical and/or quantum Monte Carlomethods ( i.e. , WFT methods) to obtain highly accu-rate and systematically improvable E int values for bench-marking and training purposes. On the WFT side, cou-pled cluster theory including single, double, and pertur-bative triple excitations in conjunction with an extrap-olation to the complete basis set limit (CCSD(T)/CBS)has long been considered the de facto “gold standard”for generating accurate E int data for small- and medium-sized organic molecules, and has therefore been used togenerate a number of seminal benchmark databases forNCIs. One of the first of these databases, the so-called S22 database, includes CCSD(T)-quality E int values for a set of small-/medium-sized biologically- relevant intermolecular complexes (comprised of {C, H,O, N}) in their respective optimized (equilibrium) ge-ometries, and was designed to cover a number of differ-ent intermolecular binding motifs ( i.e. , single and doublehydrogen bonds, dipole-dipole interactions, π - π stack-ing, dispersion, C–H · · · π , etc). Following the successand widespread use of S22 in the testing and param-eterization of many theoretical methods for describingNCIs, the amount of benchmark-quality E int data wassubstantially increased with the introduction of the S66database (which includes equilibrium intermolecularcomplexes of similar size and composition to that foundin S22) as well as extensions thereof to include complexeswith non-equilibrium intermolecular distances (along aseries of dissociation curves in S22x5 and S66x8 )and non-equilibrium intermolecular angles (at the equi-librium distance in S66a8). During the same time, otherbenchmark NCI databases were constructed to reflect thediverse number of NCI types (or binding motifs) foundin: halogen-containing systems (X40x10), nucleobasedimers (ACHC), charge transfer complexes (CT), alkane dimers (ADIM6), large molecular dimers (L7), host-guest complexes (S12L), halogen-bonded systems(XB18), sulfur-containing systems (SULFURx8), andmany more. Along similar lines, there are alsoNCI databases based on a symmetry-adapted perturba-tion theory (SAPT) decomposition of E int into compo-nents ( i.e. , electrostatics, exchange, induction, and dis-persion), which have been used to train force fields formolecular dynamics simulations. Of particular in-terest here is the S101x7 database, which starts withthe molecular dimers in S66 and expands this set to in-clude additional biologically-relevant complexes con-taining halogens ( i.e. , F, Cl, Br) and second-row elements( i.e. , S and P), as well as additional intermolecular com-plexes involving charged systems and/or water. Like theS66x8 database, S101x7 also includes complexes withnon-equilibrium intermolecular distances by computingSAPT-based E int values for select points along each in-termolecular PES; in the S101x7 case, these seven pointsranged from . × − . × the equilibrium intermolecu-lar separation in an effort to better capture short-rangecharge penetration effects. While such existing databases are growing in size,most are still relatively small (containing (cid:46) inter-action energies), making them insufficient for the rapidlygrowing field of ML-based intra-/inter-molecular poten-tials. While composite databases can be consider-ably larger, the accuracy and reliability of such compileddata is inconsistent and can potentially be a source ofboth random and systematic error. Due to the high com-putational cost of generating benchmark E int values forlarge systems, most existing databases (with the excep-tion of L7 and S12L ) have been limited to small-to-medium organic/biological molecules (usually containing < atoms); as a consequence, many of these databasesdo not capture the collective nature of NCIs ( i.e. , many-body effects, solvation/solvent effects, NCIs across mul-tiple length scales) present in large/complex moleculesand condensed-phase systems. In addition, most existingdatabases have focused on common intermolecular bind-ing motifs such as hydrogen and halogen bonding, π - π stacking, dipole-dipole interactions, dispersion, and C–H · · · π interactions, while other important binding motifs(like cation- and anion- π interactions) have been largelyunderrepresented. As such, these databases tend to in-clude intermolecular complexes that are primarily boundby electrostatics, dispersion, or a mixture thereof, buthave not included intermolecular complexes that are pri-marily induction-bound. Furthermore, prior databases( e.g. , S22x5, S66x8, S66a8, S101x7) primarily focused onthe equilibrium geometry and a single displacement fromthe equilibrium geometry ( i.e. , scaling the intermolecu-lar distance or rotating one monomer), but very few haveexplored wider swaths of the intermolecular PES. In thisregard, most databases have also only slightly touchedupon close intermolecular contacts ( i.e. , the short-rangeand often repulsive sector of the intermolecular PES), al-though there are some examples where such short-rangeconsiderations have been incorporated ( e.g. , S101x7 and R160x6 ). As a result, the performance of manytheoretical methods for accurately and reliably describ-ing NCIs in large and complex systems, for a diverse arrayof binding motifs, and across significant portions of theintermolecular PES is simply not well known.Accurate and reliable descriptions of non-equilibriumNCIs at reduced intermolecular separations—where sev-eral strong and competing short-range intermolecularforces are at play—are important for a number of reasonsand pose a substantial challenge to theory. For instance,there are numerous examples throughout chemistry andchemical biology where close intermolecular contacts areeither present at equilibrium or force the system to adopta different configuration. A striking example of thiswas recently observed when studying the enantioselec-tivity of sBOX catalysts, where a combination of attrac-tive and repulsive NCIs are responsible for the enantio-determining C – CN bond formation in chiral nitriles. Intermolecular close contacts also play a crucial role inthe study of systems operating under high-pressure con-ditions, ranging from the microscopic structure of super-critical water to the high-pressure synthesis of com-pounds with atypical compositions and novel properties as well as the search for high- T c superconducting materi-als. Theoretically speaking, SAPT decomposition stud-ies have shown that the intermolecular distance canhave a profound influence on the absolute and relativemagnitudes of the underlying E int components ( i.e. , elec-trostatics, exchange, induction, and dispersion), imply-ing that the forces present in short-range non-equilibriumintermolecular complexes of small/simpler molecules canmimic those found in larger/more complicated systemsat equilibrium separations. Interestingly, this also sug-gests that training and/or testing theoretical methods onnon-equilibrium configurations (particularly in the short-range) of small-to-medium molecular dimers can be used as a surrogate for describing the NCIs in a more diverserange of large (and possibly intractable) systems. Sincefinite-temperature molecular dynamics and Monte Carlosimulations require a consistent treatment of the struc-tures and energetics across the entire PES, an accurateand reliable treatment of non-equilibrium NCIs (includ-ing short-range as well as intermediate- and long-ranginteractions) is of enormous importance for these applica-tions as well. However, the difficulties in obtaining suchan accurate and reliable theoretical description of non-equilibrium NCIs across multiple length scales shouldalso be emphasized. For instance, the long-range sec-tor of the intermolecular PES requires a balanced de-scription of both electrostatics and dispersion, and thiscan be particularly challenging when dealing with NCIsthat also include charged species and/or molecules withsubstantial multipole moments. For larger intermolecu-lar separations, intermolecular energies (and forces) tendto be small, which provides additional challenges whentrying to describe points along the PES on the samefooting. At reduced intermolecular separations, the in-creased amount of orbital (or density) overlap betweenmonomers gives rise to a complex interplay betweenstrongly attractive and strongly repulsive intermolecularforces ( e.g. , charge transfer and penetration, Pauli repul-sion, many-body exchange-correlation effects, etc), andan error when describing any one of these componentscan lead to disastrous results. For such short-rangenon-equilibrium NCIs, the performance of the currentsuite of theoretical methods is still an open question, anda number of studies have reported higher errors for re-pulsive intermolecular contacts.
In this regime,even the suitability of high-level WFT-based approachesfor generating benchmark E int data is still largely unre-solved as such approaches suffer from issues related tothe use of incomplete basis sets ( i.e. , basis set incom-pleteness and superposition errors) in conjunction withan approximate treatment of electron correlation effects(including questions regarding the reliability of pertur-bative expansions).In this work, we directly address the aforementionedchallenges needed for training, testing, and develop-ing next-generation theoretical approaches for describingNCIs by introducing nenci- , a benchmark databaseof approximately , N on- E quilibrium N on- C ovalent I nteraction energies for a diverse selection of molec-ular dimers of biological and chemical relevance. Start-ing with the dimers in the S101 (and hence S66 )databases, which contain a diverse set of intermolecularbinding motifs ( i.e. , single and double hydrogen bonds,halogen bonds, ion-dipole and dipole-dipole interactions, π - π stacking, dispersion, X–H · · · π ) as well as a largenumber of molecular dimers involving water (which rep-resents a crucial first step towards generating benchmark E int values in aqueous environments), nenci- ex-tends the scope of these seminal works in two direc-tions. For one, nenci- includes cation- and anion- π complexes, a fundamentally important and particularlystrong class of NCIs that are primarily induction-bound and characterized by equilibrium E int values which aretypically larger in magnitude than hydrogen bonds andsalt bridges. As such, an accurate and reliable descriptionof ion- π interactions poses substantial difficulties for the-ory, and their inclusion in nenci- directly addressesthe challenge of simultaneously describing diverse NCItypes on the same footing ( i.e. , point ( ii ) above). Sec-ondly, nenci- also includes an extensive and system-atic sampling of equilibrium and non-equilibrium config-urations on each of the intermolecular PES by si-multaneously varying the intermolecular distance and in-termolecular angle in each dimer. Designed with an em-phasis on close intermolecular contacts, the complexesin nenci- were generated by sampling seven inter-molecular distances (ranging from . × − . × the equi-librium separation) as well as nine intermolecular anglesper distance (five for each ion- π complex), yielding anextensive database of , benchmark E int values ob-tained at the CCSD(T)/CBS level of theory. In doingso, nenci- directly addresses the challenges of de-scribing the collective nature of NCIs in large/complexsystems ( i.e. , point ( i )) and simultaneously describingNCIs in equilibrium and non-equilibrium systems onthe same footing ( i.e. , point ( iii )). The E int values in nenci- span a total of . kcal/mol, ranging from − . kcal/mol (corresponding to the strongly attrac-tive Li + · · · Benzene ion- π complex) to +186 . kcal/mol(corresponding to a strongly repulsive DMSO · · · DMSOcomplex that has been scaled to . × the equilib-rium intermolecular separation and rotated to a non-equilibrium angle), with a mean (median) E int value of − . kcal/mol ( − . kcal/mol). A detailed SAPT-based energy decomposition analysis demonstrates thediverse and comprehensive nature of nenci- , whichspans all regions of the corresponding ternary diagramand includes intermolecular binding motifs primarilybound by electrostatics ( , ), induction ( ), disper-sion ( , ), or mixtures thereof ( , ). A critical er-ror analysis performed on a representative set of inter-molecular complexes in nenci- demonstrates thatthe E int values provided herein at the CCSD(T)/CBSlevel have an average error of ± . kcal/mol, even forcomplexes with strongly repulsive E int values, and max-imum errors of ± . − . kcal/mol ( i.e. , approximately ± . kJ/mol) for the most challenging cases. Designedto meet the growing demand for large and high-qualityquantum mechanical data in the chemical sciences, weexpect that nenci- will be an important resource fortesting, training, and developing next-generation forcefields, DFT approximations, WFT methods, and ML-based intra-/inter-molecular potentials.The remainder of this manuscript is organized as fol-lows. Section II describes the construction of nenci- ,including the selection of molecular dimers, generationof equilibrium and non-equilibrium intermolecular com-plexes, a detailed description of the employed computa-tional protocol, and a guide to obtaining the database. Section III discusses the properties of nenci- , includ-ing a statistical analysis of the intermolecular interactionenergies and closest intermolecular contacts, an SAPT-based energy decomposition analysis of the intermolec-ular binding motifs, as well as a critical assessment ofthe error in the benchmark E int values provided herein.The manuscript ends with some brief conclusions andfuture directions in Section IV. In a follow-up to thiswork, many popular WFT and DFT methods are ex-plicitly tested on the nenci- database, where it isshown that there is a nearly universal increase in errorwhen describing the repulsive wall of the intermolecularPES and that ion- π complexes can be quite challengingto model in an accurate and reliable fashion. II. CONSTRUCTION OF THE NENCI-2021 DATABASEA. Selection of Molecular Dimers nenci- is a large database of ≈ , benchmarkintermolecular interaction energies ( E int , see Sec. II C)that includes a diverse selection of molecular dimers andbinding motifs of biological and chemical relevance, withan emphasis on non-equilibrium (attractive and repul-sive) configurations and close intermolecular contacts. Asdepicted in the left panel of Fig. 1, the construction of nenci- starts with the molecular dimers in theS101 database (a superset containing the earlier con-structed S66 database), which were carefully chosen tocontain small molecules with the NCIs found in biolog-ical and chemical systems. As such, nenci- inher-its the extensive sampling of molecule types in S66 andS101, which are comprised of the {H, C, N, O, F, P, S,Cl, Br} atom types, range in size from small ( e.g. , H O,ethene, ethyne) to medium ( e.g. , uracil, indole, pentane),and include second- and third-row elements ( e.g. , DMSO,MeCl, BenBr) as well as positively- ( e.g. , MeNH ,Imidazole + , Guanidine + ) and negatively- ( e.g. , AcO – ,H PO , HPO ) charged species. In addition, nenci- also inherits a wide variety of intermolecular bindingmotifs, including dimers with single and double hydro-gen bonds, halogen bonds, and X − H · · · π interactions,as well as intermolecular complexes primarily bound bydispersion, electrostatics ( e.g. , ion-dipole, dipole-dipole,etc), and mixtures thereof. Another salient benefit of us-ing S66 and S101 as the foundation for nenci- is thelarge number of dimers involving water, which providesa crucial first step towards the generation of benchmarkintermolecular interaction energies in an aqueous envi-ronment. nenci- extends these databases in the followingtwo ways: ( i ) it includes new cation- and anion- π complexes for a total of molecular dimers, and( ii ) it systematically samples both equilibrium and non-equilibrium intermolecular distances as well as inter-molecular angles for each dimer (with a particular em-phasis on close intermolecular contacts) for a total of Li + TrifluorotriazineThyminePyridineBenzeneUracil CytosineGuanineAdenineNa + F - Cl - WaterMeOHMeNH PeptideAcOHAcNH Ethene Ethyne PentaneNeopentaneCyclopentane MeSHDMSOHPO - H PO - H PO MeFMeClMeBrBenFBenClBenBrMeNH + Imidazole + Guanidine + AcO - MeSMeImidazolePyrrolidinePhenolIndoleCH F CH Cl CH Br ●● ●● ●● ●● ●● ●● ●● .
70 0 .
80 0 .
90 0 .
95 1 .
00 1 .
05 1 . R0.00E
RA B
FIG. 1. (
Left ) Graphical depiction of the molecular dimers in the nenci- database. nenci- contains all of themolecular dimers in the original S66 database (purple lines), the additional dimers present in the S101 (superset) database (green lines, S66 ⊂ S101), as well as a new set of cation- and anion- π complexes (red lines, S66 ⊂ S101 ⊂ nenci- ).In this graph, each monomer is represented by a vertex, the size of which is proportional to the number of molecular dimersinvolving that monomer; graph edges connecting two vertices indicate a molecular dimer formed from the connected monomers.Bold edges between vertices denote two different molecular dimer orientations involving the connected monomers ( e.g. , forwater–phenol, water is the hydrogen-bond donor in one dimer and the hydrogen-bond acceptor in the other). Chords passingthrough the center of a vertex indicate a molecular dimer formed from a single monomer ( e.g. , there is one water dimer,two uracil dimers, and three pyridine dimers in nenci- ). ( Right ) Overall description of the nenci- database. Foreach of the dimers described above, nenci- generates a series of equilibrium and non-equilibrium configurations by simultaneously sampling seven intermolecular distances and nine intermolecular angles (five for the ion- π complexes due tosymmetry considerations). As such, nenci- includes , benchmark CCSD(T)/CBS intermolecular interaction energies,which correspond to a wide range of equilibrium and non-equilibrium (both repulsive and attractive) geometries and emphasizeclose intermolecular contacts. See Secs. II A–II C for the details regarding the construction of nenci- . , benchmark interaction energies. In particular, nenci- includes ion- π complexes comprised of thesimplest biologically relevant monovalent cations (Li + ,Na + ) and anions (F – , Cl – ) interacting with a representa-tive set of π -systems, which includes the five DNA/RNAnucleobases (adenine, cytosine, guanine, thymine, uracil)as well as benzene, pyridine, and trifluorotriazine. Theinclusion of ion- π complexes in nenci- was primarilydriven by the fact that ion- π interactions are among thestrongest NCIs known (with intermolecular interactionenergies often rivaling that of hydrogen bonds and saltbridges) and have been observed throughout chemistryand biology. This extension was also motivated bysome of our recent work, which used SAPT to demon-strate that cation- π complexes are primarily bound byinduction, while anion- π complexes are bound by a com-plex interplay between induction, dispersion, and electro-statics; as such, their inclusion substantially expands thescope/range of intermolecular binding motifs in nenci- (see Sec. III B). As shown in paper-ii of this se-ries, this complex interplay between intermolecular forces(in addition to the presence of charged atomic species)in ion- π complexes poses a unique challenge when tryingto obtain accurate and reliable intermolecular interactionenergies using both WFT and DFT methods. In addi-tion, the inclusion of promiscuous ion- π binders ( i.e. , π - systems such as the DNA/RNA nucleobases, which canform favorable ion- π complexes with both cations and an-ions ) as well as π -systems that can only form energet-ically favorable ion- π complexes with cations ( e.g. , ben-zene) or anions ( e.g. , trifluorotriazine) is also well-alignedwith one of the fundamental goals of nenci- : to pro-vide a more comprehensive sampling of both attractiveand repulsive non-equilibrium configurations containinga diverse array of NCI types.Motivated by the S22x5, S66x8, and S101x7 databases, in which intermolecular interaction energycurves were constructed for each molecular dimer, aswell as the S66a8 database, in which the inter-molecular angles were sampled, nenci- systemat-ically samples both equilibrium and non-equilibriumintermolecular distances as well as intermolecular an-gles for each of the molecular dimers describedabove. As depicted in the right panel of Fig. 1, nenci- samples seven intermolecular distances( i.e. , . × , . × , . × , . × , . × , . × , . × the equi-librium intermolecular separation) and nine intermolec-ular angles (only five intermolecular angles for the ion- π complexes, vide infra ); for more details, see Sec. II B. nenci- therefore contains benchmark intermolecu-lar interaction energies (see Secs. II C and III C) for × geometries (configurations) for each of the molecular dimers in the S101 database, and × geometries for each of the ion- π complexes, yield-ing a total of ×
101 + 35 ×
40 = 7 , equilibriumand non-equilibrium intermolecular complexes. By in-cluding such a systematic sampling of equilibrium andnon-equilibrium structures, nenci- is a relativelylarge database that contains a wide range of attrac-tive and repulsive intermolecular interaction energies (seeSec. III A); as such, we believe that nenci- willbe well-suited for in-depth studies of the NCIs foundthroughout biology and chemistry, as well as trainingand testing next-generation density functional approxi-mations, dispersion corrections, polarizable force fields,and ML-based potentials. By including an extensive setof angularly sampled geometries at . × and . × theequilibrium intermolecular separation, nenci- alsoincludes a wide range of close intermolecular contacts,which are found throughout chemistry and chemical bi-ology, as well as high-pressure systems; here, we stressagain that benchmark intermolecular interaction energiesin this regime not only serve as surrogates for larger/morecomplex systems at equilibrium, but are also importantto ensure similar performance across the entire inter-molecular PES when training, testing, and developingnovel theoretical methods. B. Generation of Equilibrium and Non-EquilibriumIntermolecular Complexes
Unless otherwise specified, all monomer geometrieswere taken from the S66 and S101 databases. For theeight π -systems used to construct the ion- π complexesin nenci- , the monomer geometries for benzene,pyridine, and uracil were taken from the S66 database,while the monomer geometries for the DNA/RNA nucle-obases were taken from our recent work on promiscuousion- π binding. During the construction of the equilib-rium and non-equilibrium molecular dimer geometries,we employed the frozen monomer convention in which allmonomers were kept fixed at their optimized geometries.The molecular dimer geometries in the S66 database were also taken as is and without any changes;for the remaining molecular dimers, equilibrium ge-ometries were optimized (see Sec. II C) along a pre-defined characteristic intermolecular interaction vector.This characteristic intermolecular interaction vector wasbased on the interaction type ( e.g. , hydrogen-bonded,halogen-bonded, dispersion-bound, ion- π , etc) assignedto the molecular dimer via chemical intuition. Dimersthat appear in the S66 database were assigned the sameinteraction type as in the original work, and the re-maining dimers were assigned an interaction type thatwas as consistent as possible with the S66 convention.Given one of the following interaction types, the charac-teristic intermolecular interaction vector was defined as:• For hydrogen- (halogen-) bonded systems, the in-teraction vector points between the hydrogen (halo- gen) bond donor and the hydrogen (halogen) bondacceptor. For double-hydrogen-bonded systems,the interaction vector is defined as the mean ofthe two hydrogen-bond vectors (with both takento originate from the same monomer).• For dispersion-bound systems, the interaction vec-tor points from the center of mass of monomer A to the center of mass of monomer B .• For ion- π complexes, the interaction vector pointsfrom the ion to the nuclear center of charge of the π -system (computed using only the atoms in eachring, i.e. , the five carbons and nitrogen in pyridine).Here, we note in passing that this on-axis placementof the ion does not necessarily correspond to thelowest energy geometry of each ion- π complex. • Finally, there remain a few special cases ( i.e. , theT-shaped benzene dimer), which do not fit well intoany of these categories. Such systems are treatedanalogously with the dispersion-bound complexes,but only a subset of atoms is used in calculatingan effective “molecular center” to ensure that theinteraction vector accurately characterizes the in-teraction. For reference, the atoms used to calcu-late the interaction vector for each such complexare provided in Table S1.All remaining equilibrium molecular dimer geometrieswere obtained by minimizing the intermolecular inter-action energy by rigidly translating monomer A alongthe characteristic intermolecular interaction vector (seeSec. II C), and then used as starting points to generateall non-equilibrium structures.To systematically sample both intermolecular dis-tances and intermolecular angles for the moleculardimers in nenci- , we started with the proceduredevised by Řezáč, Riley, and Hobza when constructingthe S66x8 and S66a8 databases, and extended thisprotocol to accommodate a broader range of intermolec-ular interaction types and orientations. As such, the molecular dimer geometries in the S66a8 database werealso taken as is and without any changes. The procedurefor generating the remaining , non-equilibriumintermolecular complexes in nenci- is outlinedbelow, with STEPS 1–5 graphically illustrated for thewater dimer in Fig. 2: STEP 1.
Starting with an optimized equilibriumintermolecular complex, arbitrarily label each monomeras either A or B (except for the ion- π complexes, inwhich the ion should be labelled as monomer A ). Drawthe characteristic intermolecular interaction vector from B to A (dashed black line) according to the interactiontype assigned to the molecular dimer ( vide supra ).Define the z -axis (solid red arrow) along the interactionvector. STEP 2.
Without loss of generality, assume that A willbe rotated around B (the alternative will be dealt with in FIG. 2. Graphical depiction of STEPS 1–5 in the protocol used for generating four (of eight) non-equilibrium intermolecularangles for the water dimer. As described in the text, a local reference frame ( x -axis: solid blue arrow; y -axis: solid yellowarrow; z -axis: solid red arrow) is defined with respect to the characteristic intermolecular interaction vector (dashed blackline) between monomers A (red) and B (gray), as well as the principal axis on monomer A (solid black line). Preliminarygeometries for the first four non-equilibrium intermolecular angles are then obtained by rotating A around the x - and y -axeson B by θ = ± ◦ . For clarity, the inset to STEP 5 also provides a view down the z -axis of the corresponding non-equilibriumgeometries. To obtain preliminary geometries for the remaining four non-equilibrium intermolecular angles, this procedure isrepeated after swapping the monomer labels. See Sec. II B for more details. STEP 8 below). To determine the axes of rotation, firstfind the principal axis ( C n ) corresponding to monomer A ( i.e. , the molecular axis with the highest degree ( n ) ofrotational symmetry); for the water monomer depictedin Fig. 2, the principal axis is the solid black line labelled C . If no principal axis with n ≥ exists, we follow theconvention used during the construction of the S66a8 database, i.e. , an approximate principal axis is definedby removing all hydrogen atoms from the molecule andreducing the identity of each heavy atom and functionalgroup to identical spheres. STEP 3.
Define the y -axis (solid yellow arrow) tobe perpendicular to the z -axis and the principal axis of A . STEP 4.
Define the x -axis (solid blue arrow) to beperpendicular to the z - and y -axes, thereby completelyspecifying the local reference frame used in this work. STEP 5.
To generate preliminary geometries for thefirst four non-equilibrium intermolecular angles, rotate A about the x - and y -axes passing through the tail ofthe interaction vector ( i.e. , located on monomer B ) by θ = ± ◦ . STEP 6.
For each non-equilibrium intermolecularangle, minimize the intermolecular interaction energy by rigidly translating A along the characteristic intermolec-ular interaction vector (see Sec. II C). For the ion- π complexes that are repulsive along the entire dissociationcurve ( e.g. , Na + · · · trifluorotriazine), the minimum ofthe SAPT exchange + induction + dispersion (EID) en-ergy was used in lieu of the intermolecular interactionenergy (see Sec. II C). Define the intermolecular distance( i.e. , the length of the characteristic intermolecularinteraction vector) in each optimized geometry as theequilibrium ( . × ) intermolecular distance for the givennon-equilibrium intermolecular angle. STEP 7.
For each non-equilibrium intermolecularangle, scale the corresponding (optimized) interactionvector by factors of . × , . × , . × , . × , . × , and . × , and rigidly translate A consistent with each scaledvector. This will provide molecular dimer geometriesalong four separate intermolecular dissociation curvescorresponding to each of the four non-equilibriumintermolecular angles. STEP 8.
Switch the A and B labels, and repeat STEPS1–7. This will provide molecular dimer geometries alongthe intermolecular dissociation curves correspondingto each of the remaining four non-equilibrium inter-molecular angles (for a total of eight non-equilibriumintermolecular angles). N.B.: The ion- π complexesin nenci- only have four unique non-equilibriumintermolecular angles due to the spherical symmetry ofthe ion; as such, STEP 8 is unnecessary and can beskipped for these molecular dimers. STEP 9.
For the equilibrium intermolecular angle, alsoscale the corresponding (optimized) interaction vector byfactors of . × , . × , . × , . × , . × , and . × , andrigidly translate A consistent with each scaled vector.This will provide molecular dimer geometries along theintermolecular dissociation curve corresponding to theequilibrium intermolecular angle. C. Computational Details
Intermolecular interaction energies ( E int ) for each ofthe , intermolecular complexes in nenci- werecomputed via E int = E AB − E A − E B , (1)in which E AB is the total energy of the dimer and E A ( E B ) is the total energy of monomer A ( B ). As men-tioned above, all monomers were kept fixed at their op-timized geometries, and the counterpoise correction ofBoys and Bernardi was applied to minimize basis setsuperposition error (BSSE).Unless otherwise specified, Dunning’s correlation con-sistent basis sets (with and without diffuse functions),namely cc-pVXZ and aug-cc-pVXZ (with X = D, T,Q), along with the frozen core (FC) approximationwere used for all atoms except Li and Na. To provide amore accurate description of the core/valence electronsin the cation- π complexes, the cc-pwCVXZ and aug-cc-pwCVXZ basis sets were used for Li and Na inconjunction with the following modified FC approxima-tion: Li + = 1s (no core) and Na + = [He]2s ([He]core). All calculations employed the resolution-of-the-identity (RI) or density-fitting (DF) approximation dur-ing self-consistent field (SCF) calculations at the mean-field Hartree-Fock (HF) level as well as during post-HF calculations to account for electron correlation ef-fects; the RI/DF approximation has been shown to in-troduce negligible errors when computing intermolecularinteraction energies. Whenever available, the cor-responding JKFIT and RI auxiliary basis sets were usedin conjunction with each primary (atomic orbital) basisset, i.e. , cc-pVXZ-JKFIT/cc-pVXZ-RI were usedwith cc-pVXZ, and aug-cc-pVXZ-JKFIT/aug-cc-pVXZ-RI were used with aug-cc-pVXZ. For the cation- π complexes, the def2-aQZVPP-JKFIT/def2-aQZVPP-RIauxiliary basis sets (which are some of the largestavailable auxiliary basis sets) were taken from the MOL-PRO basis set library and used in conjunctionwith cc-pwCVXZ and aug-cc-pwCVXZ for Li and Na.Throughout this work, we used the abbreviation aXZ todenote the following basis set usage: aug-cc-pVXZ (with aug-cc-pVXZ-JKFIT/aug-cc-pVXZ-RI) for {H, C, N, O,F, S, P, Cl, Br}; aug-cc-pwCVXZ (with def2-aQZVPP-JKFIT/def2-aQZVPP-RI) for {Li, Na}; we also use theabbreviation haXZ ( i.e. , heavy-aug-cc-pVXZ, also knownas jul-cc-pVXZ ) to mean: cc-pVXZ (with cc-pVXZ-JKFIT/cc-pVXZ-RI) for {H}, aug-cc-pVXZ (with aug-cc-pVXZ-JKFIT/aug-cc-pVXZ-RI) for {C, N, O, F, S,P, Cl, Br}, and aug-cc-pwCVXZ (with def2-aQZVPP-JKFIT/def2-aQZVPP-RI) for {Li, Na}.For each molecular dimer and non-equilibrium inter-molecular angle , the corresponding optimal intermolec-ular distance ( . × ) was obtained via a constrained min-imization of E int at the BSSE-corrected MP2/cc-pVTZlevel (see Eq. (1)); for the molecular dimers not includedin the original S66 database, the same procedure wasalso used to obtain the optimal intermolecular distancefor the equilibrium intermolecular angle. In practice, thiswas accomplished by computing E int for a series of dimergeometries in which monomer A (and/or B ) was rigidlytranslated along the characteristic intermolecular inter-action vector (see Sec. II B), and then locating the min-imum value along the corresponding cubic spline inter-polant.Benchmark E int values in nenci- were obtainedusing Eq. (1) with all dimer ( E AB ) and monomer ( E A and E B ) contributions computed using the “gold stan-dard” CCSD(T) method extrapolated to the completebasis set (CBS) limit, i.e. , E CCSD(T) / CBS ≡ E MP2 / CBS + δE CCSD(T) / haTZ . (2)In this expression, the CBS-extrapolated MP2 total en-ergy, E MP2 / CBS ≡ E MP2 / a(TQ)Z = E HF / aQZ + E MP2 / a(TQ)Zcorr , (3)was obtained using the two-point extrapolation proce-dure of Halkier et al. on the MP2 correlation energy,namely, E MP2 / a(XY)Zcorr = X E MP2 / aXZcorr − Y E MP2 / aYZcorr X − Y (4)with X = 3 (aTZ) and Y = 4 (aQZ). The so-called“delta” CCSD(T) correction, δE CCSD(T) / haTZ = E CCSD(T) / haTZ − E MP2 / haTZ , (5)was computed using the haTZ basis set. The accuracyof this scheme for computing E int —in particular for in-termolecular complexes with particularly close contacts( i.e. , . × the equilibrium intermolecular separation)—iscritically assessed below in Sec. III C.The energy decomposition analysis scheme (and clas-sification of intermolecular binding motifs) provided inSec. III B was based on calculations at the SAPT2+/aDZlevel of theory, the so called “silver standard” ofSAPT. FIG. 3. Normalized probability density functions (PDFs) ofthe benchmark E int values in the nenci- database as afunction of the intermolecular distance (with the . × and . × scaled intermolecular distances omitted for clarity).The E int values in nenci- range from − . kcal/mol(most attractive) to +186 . kcal/mol (most repulsive), witha mean (median) interaction energy of − . kcal/mol( − . kcal/mol). Insets display the peaks of the . × , . × ,and . × PDFs as well as the (positive E int ) tails of the . × and . × PDFs.
All calculations in this work were performed usingthe
Psi4 ( v1.2 ) software program. During all HFcalculations, the SCF convergence parameters were setto . × − in the total energy ( e_convergence =1E-8 ) and . × − in the root-mean-square DIIS er-ror ( d_convergence = 1E-8 ). For all CCSD(T) cal-culations, the CCSD convergence parameters were setto . × − in the total energy ( e_convergence =1E-6 ) and . × − in the residual of the t -amplitudes( r_convergence = 1E-5 ). D. Obtaining the NENCI-2021 Database
A single zip file containing the Cartesian coordinatesof the , intermolecular complexes in nenci- (in xyz format) is provided in the Supplementary Material.The properties of each monomer ( i.e. , charge, multiplic-ity, number of atoms), the corresponding benchmark E int value, as well as the CCSD(T)/CBS and SAPT energeticcomponents can be found in the comment line of each xyz file (see README file for additional details).
III. PROPERTIES OF THE NENCI-2021 DATABASEA. Statistical Analysis of Intermolecular Interaction Energiesand Closest Intermolecular Contacts
A well-balanced database of intermolecular interac-tions should have a wide range of E int values, and this isindeed the case for nenci- , as evidenced by the nor-malized E int distributions provided in Fig. 3. With E int values ranging from − . kcal/mol to +186 . kcal/mol,the benchmark intermolecular interaction energies in nenci- span . kcal/mol. In general, the mostattractive (most negative) E int values in nenci- cor-respond to charged intermolecular complexes that tendto be at (or close) to their equilibrium geometries. Forinstance, the single most attractive intermolecular com-plex in nenci- (with E int = − . kcal/mol) corre-sponds to the Li + · · · benzene ion- π system at its equi-librium geometry ( i.e. , with Li + located above the cen-ter of the benzene ring, see Sec. II B). In fact, the topten most attractive intermolecular interactions in nenci- correspond to the various Li + · · · π complexes atslightly different (but close to equilibrium) intermolec-ular distances and angles; these are followed by theionic H O · · · HPO hydrogen-bonded complexes (witha minimum E int = − . kcal/mol) and the Na + · · · π complexes (with a minimum E int = − . kcal/mol). Ingeneral, the most repulsive (most positive) E int valuesin nenci- correspond to intermolecular complexes inwhich the monomers are separated by the shortest dis-tance ( . × ) and rotated away from their equilibriumintermolecular angle, as both of these geometric pertur-bations lead to a rapid increase in the exponentially re-pulsive steric contribution to the interaction energy. Forinstance, the single most repulsive intermolecular com-plex in nenci- (with E int = +186 . kcal/mol) cor-responds to the dimethyl sulfoxide (DMSO) dimer sep-arated by . × the equilibrium intermolecular distanceand rotated to a non-equilibrium angle; in fact, thisdimer has the closest intermolecular contact in the entiredatabase (with d H ··· H = 0 . Å, vide infra ). Other sub-stantially repulsive intermolecular complexes in nenci- include the Na + · · · trifluorotriazine ion- π system(with a maximum E int = +118 . kcal/mol) and anotherDMSO dimer (with E int = +112 . kcal/mol), both ofwhich were characterized by a . × intermolecular sepa-ration and a non-equilibrium intermolecular angle.The mean and median E int values in nenci- are − . kcal/mol and − . kcal/mol, respectively, whichcorrespond to typical interaction energies found in weaklybound molecular dimers. These statistical measuresare primarily governed by the (relatively) large num-ber of intermolecular complexes in nenci- that con-tain monomers in non-equilibrium (angular) orientations.Such geometric perturbations tend to nullify the ener-getic stabilization provided by directional intermolecularbinding motifs ( e.g. , single- and double-hydrogen bonds,dipole-dipole interactions, etc), and often result in com-plexes with weakly attractive E int values. Broken downby the scaled intermolecular distance, the mean (median) E int values are: +21 . ( +11 . ) kcal/mol for . × , +0 . ( +0 . ) kcal/mol for . × , − . ( − . ) kcal/mol for . × , − . ( − . ) kcal/mol for . × , − . ( − . ) kcal/molfor . × , − . ( − . ) kcal/mol for . × , and − . ( − . ) kcal/mol for . × . In total, nenci- con-tains , attractive ( E int < ) and , repulsive( E int > ) intermolecular complexes, and the crossover0from attractive to repulsive E int values typically occursaround . × the equilibrium intermolecular distance. Asone might expect, the proportion of attractive inter-molecular interactions in nenci- quickly diminishesas the distance between monomers decreases; brokendown again by the scaled intermolecular distance, wefind the percentage of attractive (repulsive) E int valuesare: . ( . ) for . × , . ( . ) for . × , . ( . ) for . × , . ( . ) for . × , . ( . ) for . × , . ( . ) for . × , and . ( . ) for . × . Quite interestingly, there are still anumber ( N = 38 ) of attractive intermolecular complexesat the . × scaled intermolecular distance, which gener-ally correspond to strongly favorable dimers such as theLi + · · · π complexes discussed above. In the same breath,there are also quite a few ( N = 19 ) repulsive complexesat the equilibrium ( . × ) distance—some of which evenoccur at the corresponding equilibrium angle, e.g. , thecation- and anion- π complexes involving trifluorotriazineand benzene, respectively.A well-balanced database of intermolecular interac-tions should also sample a wide range of intermolecularatom-pair distances ( i.e. , interatomic distances betweenthe atoms on molecule A and the atoms on molecule B ). Again, this is indeed the case for nenci- , andis demonstrated by the series of normalized PDFs inFig. 4, which quantify a representative set of atom-pairdistances ( i.e. , O · · · H , N · · · H , H · · · H , and C · · · H ) asa function of intermolecular separation. In this figure,we chose to focus on the O · · · H , N · · · H , H · · · H , and C · · · H intermolecular atom-pair distances, as the firsttwo are representative of hydrogen-bonded systems andthe last two are the relevant interatomic distances fornon-bonded complexes in general. Since nenci- wasdesigned with a particular emphasis on close intermolecu-lar contacts, we focus our discussion on the short-distancesectors in these PDFs. As discussed above in the Intro-duction, such close intermolecular contacts are impor-tant in a number of applications, and pose signif-icant difficulty for both WFT and DFT methods alike(see paper-ii in this series), as both strongly attrac-tive and strongly repulsive intermolecular forces must beaccurately described to obtain a quantitatively correct E int value. As the intermolecular distance is reducedfrom . × to . × , the complexes in nenci- sam-ple increasingly closer interatomic distances and beginto more appreciably populate the region inside the cor-responding vdW envelope. In other words, a number ofintermolecular atom-pair distances ( R AB ) are less thanthe sum of the corresponding vdW radii, i.e. , R AB A well-balanced database of intermolecular interac-tions should also sample a wide variety of different bind-ing motifs. Here, we would again argue that this isthe case for nenci- , and demonstrate this pointby the extensively populated ternary diagrams depictedin Fig. 5. Introduced by Kim et al. in the late2000s, these ternary diagrams were constructed usinga SAPT decomposition of E int into the following fourcomponents for each intermolecular complex in nenci- : ε Elst (electrostatics, Elst), ε Exch (exchange, Exch), ε Ind (induction, Ind), and ε Disp (dispersion, Disp), i.e. , E int ≈ ε SAPT = ε Elst + ε Exch + ε Ind + ε Disp . In particular,we performed this decomposition at the SAPT2+/aDZlevel of theory, the so-called “silver standard”of SAPT, which has been shown to have an overallmean absolute error (MAE) of . kcal/mol across theS22, HBC6, NBC10, and HSG databases. Unlike the “bronze standard” sSAPT0/jun-cc-pVDZ, which can underestimate the dispersion component inanion- π complexes by more than , the more so-phisticated SAPT2+/aDZ method employed herein isexpected to more accurately describe ε Disp in the anion- π complexes present in nenci- . As such, thisSAPT level should be well-suited to provide a physicallysound and semi-quantitative characterization of the bind-ing motifs included in nenci- .In previously constructed databases of non-covalentinteractions ( e.g. , S66 and S101 ), each intermolecularcomplex was typically classified into one of three cate-gories, based on whether E int ≈ ε SAPT was dominated by the ε Elst component (Elst-bound), the ε Disp component(Disp-bound), or a mixture (Mix) thereof (Mix-bound).Since the ε Ind component tended to be small in thesecomplexes, the analogous and fourth Ind-bound categorywas deemed to be largely unnecessary. With the additionof , ion- π complexes (in particular, the cation- π systems), the scope of the SAPT decomposition analysisis substantially wider in nenci- , and now encom-passes the Ind-bound regime. As such, we proposea natural extension of the traditional three-categoryclassification scheme made popular by Hobza et al. andSherrill et al. to include the Ind-bound category.To do so, we construct a three-dimensional feature spacedefined by the ε Disp /ε Elst , ε Ind /ε Disp , and ε Elst /ε Ind ratios as follows: STEP 1. To start, a single dimension of the featurespace is chosen as the basis for constructing an initialsub-classification scheme. Although this choice isarbitrary, we will start with the ε Disp /ε Elst ratio, asthis selection is tantamount to constructing the afore-mentioned three-category classification scheme ( i.e. ,Elst-bound, Disp-bound, or Mix-bound). For illustrativepurposes, a ternary diagram ( T ED ) depicting this initialsub-classification scheme is plotted in the left panel ofFig. 5. STEP 2. Intermolecular complexes with | ε Disp /ε Elst | > η are sub-classified as Disp-bound(shaded blue regions in T ED ), while intermolecularcomplexes with | ε Elst /ε Disp | > η are sub-classified asElst-bound (shaded green regions in T ED ). If onestopped at this point, set η = 2 , and classified allother cases as Mix-bound, this initial sub-classificationscheme (based on the single ε Disp /ε Elst feature) wouldbe equivalent to the three-category classification schemedescribed above. Since the value of η is somewhatarbitrary, we have chosen to employ a slightly smallervalue ( η = 3 / ) in the classification scheme introducedin this work; with this choice for η , less intermolecu-lar complexes will be classified as Mix-bound ( vide infra ). STEP 3. To go beyond this three-category classificationscheme, STEP 2 is repeated for the two remaining di-mensions of the feature space. Selection of the ε Ind /ε Disp feature generates the T ID ternary diagram in Fig. 5 andthe analogous sub-classification of intermolecular com-plexes as: Disp-bound (if ε Disp /ε Ind > η ; shaded blueregions in T ID ) or Ind-bound (if ε Ind /ε Disp > η ; shadedred regions). Similarly, the ε Elst /ε Ind feature yieldsthe final required sub-classification scheme: Elst-bound(if | ε Elst /ε Ind | > η ; shaded green regions in T EI ) orInd-bound (if | ε Ind /ε Elst | > η ; shaded red regions). Here,we note in passing that the absolute value (magnitude)must be used for all sub-classifications based on ε Elst ,as the sign of the Elst component can be positive ornegative. STEP 4. To arrive at our extended ( i.e. , four-category)classification scheme, each intermolecular complex that2 Disp (-) Elst (-) Ind (-) Elst (+) Elst Ind Disp Disp (-) Elst (-) Ind (-) Elst (+) FIG. 5. ( Left ) Geometric depiction of the extended four-category classification scheme (based on a SAPT decompositionof E int ) used to classify each intermolecular complex in nenci- as: Elst-bound (E, green), Ind-bound (I, red), Disp-bound (D, blue), or Mix-bound. As described in the main text, this classification scheme can be represented by a fusedternary diagram ( T EID ) which has been colored according to the following rule: each intermolecular complex that has beenassigned the same category (color) in any two of the three sub-classification schemes ( T ED , T ID , T EI ) retains that color in T EID ; otherwise, the complex is classified as Mix-bound (white). ( Middle/Right ) Ternary diagrams depicting the break-down of the SAPT2+/aDZ intermolecular interaction energies of each complex in nenci- according to the contribu-tions from electrostatics ( ε Elst ), induction ( ε Ind ), and dispersion ( ε Disp ). Since ε Elst can be positive (Elst(+)) or negative(Elst(-)), these plots are comprised of two ternary diagrams (one for Elst(+) and one for Elst(-)) that have been fused to-gether. In these ternary diagrams, the shaded polygons are used to reflect the four-category classification scheme describedabove, i.e. , Elst-bound (green), Ind-bound (red), and Disp-bound (blue); complexes that are not located in any one ofthese regions are Mix-bound. In the Middle panel, each point has been colored using an RGB scheme with values givenby: {| ε Ind | / ( | ε Elst | + | ε Ind | + | ε Disp | ) , | ε Elst | / ( | ε Elst | + | ε Ind | + | ε Disp | ) , | ε Disp | / ( | ε Elst | + | ε Ind | + | ε Disp | ) } . In the Right panel,each point is colored according to the scaled intermolecular distance. has been sub-classified (in STEP 2 and STEP 3) withthe same label twice retains that label; otherwise, theintermolecular complex is classified as Mix-bound. Thisfinal classification scheme is graphically depicted in thecolored T EID ternary diagram in Fig. 5, which is assem-bled as an “outer sum” over the colored ternary diagramscorresponding to the sub-classification schemes, i.e. , T EID = T ED ⊕ T ID ⊕ T EI , in which the colors of T EID aredetermined according to the rules described above.Based on this extended four-category classificationscheme, the equilibrium intermolecular complexesin nenci- are comprised of ( . ) Elst-bound, ( . ) Ind-bound, ( . ) Disp-bound, and ( . ) Mix-bound dimers. When including all non-equilibrium intermolecular distances and angles, the en-tire nenci- database contains , ( . ) Elst-bound, ( . ) Ind-bound, , ( . ) Disp-bound, and , ( . ) Mix-bound intermolecularcomplexes. Here, we note in passing that this observeddecrease in the percentage of Ind-bound complexes is par-tially due to the inclusion of five (instead of nine) inter-molecular angles for each ion- π complex due to symmetryconsiderations (see Sec. II B). As such, the intermolecularcomplexes in nenci- largely span the entire ternarydiagram in Fig. 5 and therefore contain a diverse array ofbinding motifs; as such, we hope that nenci- will be used to critically examine (and potentially improve) theperformance of theoretical models when faced with thechallenge of simultaneously describing diverse NCI typeson the same footing ( i.e. , point ( ii ) in the Introduction).Here, we note that the apparent bias towards Elst-bound complexes in nenci- is an unavoidable con-sequence of sampling short-range intermolecular sepa-rations; at such distances, there is often a substantialamount of orbital/density overlap between monomers,and charge penetration effects (in ε Elst )tend to be the dominant contribution (over ε Ind and ε Disp ) to ε SAPT . For instance, a significant majority( . ) of the intermolecular complexes at . × areclassified as Elst-bound while approximately half that( . ) of the equilibrium dimers share this label.This increase in the relative number of Elst-bound com-plexes at shorter intermolecular separations is clearlyreflected in the ternary diagram in the right panel ofFig. 5 as well as the percentage of Elst-bound com-plexes when broken down by the scaled intermoleculardistance, i.e. , . ( . × ), . ( . × ), . ( . × ), . ( . × ), . ( . × ), . ( . × ), and . ( . × ). In general, many complexes that are Disp-boundat larger intermolecular distances become Elst-bound orMix-bound at reduced separations where short-range ef-fects ( e.g. , charge penetration) become more significant.On the other hand, the Ind-bound complexes (which are3primarily comprised of cation- π interactions) tend to re-main Ind-bound even at reduced intermolecular separa-tions since charge penetration effects are substantially re-duced when one of the monomers is a monovalent cation( e.g. , Li + or Na + ). For reference, the respective per-centages of Ind-bound, Disp-bound, or Mix-bound com-plexes as a function of the scaled intermolecular distanceare: . , . , . for . × , . , . , . for . × , . , . , . for . × , . , . , . for . × , . , . , . for . × , . , . , . for . × , and . , . , . for . × .Before moving on to consider the error/uncertainty inthe E int values in nenci- , we note in passing that thepositive electrostatics (Elst(+)) region of the ternary di-agram in Fig. 5 is not sampled as well as the (Elst(-)) re-gion. However, nenci- does contain a non-negligible( ) number of intermolecular complexes with ε Elst > .As mentioned above, such complexes are primarily foundamong the cation- π complexes, where the degree of or-bital overlap in the dimer (and hence the energetic stabi-lization due to charge penetration effects) is largely sup-pressed; hence, intermolecular complexes with repul-sive ε Elst values are quite rare and may be adequatelyaccounted for in nenci- . C. Error Analysis and Critical Assessment of the BenchmarkIntermolecular Interaction Energies In addition to being extensive in size and scope, wewould also argue that a well-balanced database of inter-molecular interactions should contain a reliable estimateof the error/uncertainty present in the computed E int values. For ab initio WFT methods, the two primarysources of error when computing E int are: ( i ) incomplete-ness in the one-particle basis set ( i.e. , basis set incom-pleteness error (BSIE)) and ( ii ) the approximate treat-ment of the electron correlation energy ( E corr ). Sincethe mean-field HF contribution to E int converges quicklywith respect to the underlying basis set, we expectthat the BSIE at the E HF / aQZint level will be negligiblewhen compared to the BSIE in the post-HF correlationenergy contributions in Eq. (2). As depicted in Eq. (3),the BSIE in the MP2 correlation energy is largely mit-igated using the two-point extrapolation scheme forapproximating the MP2/CBS limit provided in Eq. (4).Although the δE CCSD(T) correction converges with re-spect to the basis set significantly faster than E MP2corr or E CCSD(T)corr alone, the BSIE in this term isgenerally the largest remaining source of error for ex-trapolation schemes such as that outlined in Eqs. (2)–(5). To mitigate this error (and still remain compu-tationally feasible when generating such a large numberof intermolecular interaction energies), this contributionwas computed using an augmented Dunning-style triple- ζ (haTZ) basis set in nenci- ( cf . Eq. (5)).As such, we will primarily focus on the remaining BSIEin the δE CCSD(T) / haTZ contribution to E int when criti- cally assessing the accuracy of the intermolecular inter-action energies in nenci- . To do so, we will compareour E int values against two different references. As a firstreference value, we computed the δE CCSD(T) correctionin Eq. (5) using a larger (and substantially more expen-sive) augmented quadruple- ζ (aQZ) basis set, i.e. , E REF1 = E MP2 / CBS + δE CCSD(T) / aQZ = E HF / aQZ + E MP2 / a(TQ)Zcorr + δE CCSD(T) / aQZ . (6)in which E MP2 / CBS was computed using Eqs. (3)–(4). Asa second and alternative reference, we simply replacedthe δE CCSD(T) / haTZ correction with a direct two-pointextrapolation of E CCSD(T) using the aTZ and aQZbasis sets, i.e. , E REF2 = E CCSD(T) / a(TQ)Z = E HF / aQZ + E CCSD(T) / a(TQ)Zcorr . (7)By including CCSD(T) calculations in the much largeraQZ basis set, both of these reference values directlyprobe the BSIE in the CCSD(T) contribution, and areexpected to be more reliable than the E int values in the nenci- database.The error of the CCSD(T)/CBS scheme outlined inEqs. (2)–(5) with respect to both E REF1 and E REF2 isshown in Fig. 6 for a select subset of intermolecular com-plexes in nenci- . Plotted as a function of the scaledintermolecular distance (at the equilibrium angle, un-less otherwise noted), this subset of intermolecular com-plexes was chosen to cover the wide array of binding mo-tifs in nenci- , and includes examples of Elst-, Ind-,Disp-, and Mix-bound systems, i.e. , single (H O · · · H O,MeNH · · · MeNH ) and double (AcOH · · · AcOH) hydro-gen bonds, dipole-dipole (MeF · · · MeF), π - π stacking(BZ · · · BZ PD), CH- π (BZ · · · BZ TS), as well as cation- π (Na + · · · BZ) and anion- π (F – · · · BZ) interactions. Asseen in Fig. 6, the E int values in nenci- are gen-erally within ± . kcal/mol of both E REF1 and E REF2 ,and the errors with respect to these references tend to in-crease in magnitude at reduced intermolecular distances.The worst-case scenarios among this subset include theacetic acid dimer (AcOH · · · AcOH, double hydrogen-bonded) and the C parallel-displaced (PD) benzenedimer (BZ · · · BZ PD, π - π stacking), with errors in bothsteadily increasing in magnitude as the intermolecularseparation is decreased; at . × , we report errors of +0 . kcal/mol (AcOH · · · AcOH) and − . kcal/mol(BZ · · · BZ PD) with respect to E REF1 ( +0 . kcal/moland − . kcal/mol when compared to E REF2 , vide in-fra ). In these cases, the increased error is most likely dueto the relatively larger amount of orbital overlap betweenthese monomers at reduced intermolecular separations,where the interplay between short-range intermolecularinteractions ( i.e. , charge penetration, Pauli repulsion,many-body exchange-correlation effects, etc) becomes in-creasingly more challenging to describe in an accurate4 . . . . . Scaled Intermolecular Distance − . − . − . . . . . E rr o r( i n k c a l / m o l ) REF1 . . . . . Scaled Intermolecular Distance REF2 AcOH · · · AcOHH O · · · H O MeF · · · MeFMeNH · · · MeNH BZ · · · BZ PDBZ · · · BZ TS Na + · · · BZF − · · · BZ ± / molDMSO · · · DMSO FIG. 6. Errors (in kcal/mol) in the nenci- E int values (computed using the E CCSD(T) / CBS extrapolation scheme inEqs. (2)–(5)) with respect to E REF1 (Eq. (6); left ) and E REF2 (Eq. (7); right ) for a representative set of intermolecularcomplexes. Plotted as a function of the scaled intermolecular distance (with the . × and . × distances omitted for clarity),all intermolecular complexes (with the exception of DMSO · · · DMSO, black stars) were kept at their equilibrium angle. Errorswith respect to E REF1 and E REF2 were computed as E CCSD(T) / CBS − E REF1 and E CCSD(T) / CBS − E REF2 , respectively. Basedon this error profile (see main text), the errors in the nenci- E int values are ± . kcal/mol on average, but can be as largeas ± . − . kcal/mol ( i.e. , ± kJ/mol, dashed green lines) for some complexes at reduced ( i.e. , . × − . × ) intermolecularseparations. and reliable fashion. This trend is also reflected in theerror profiles corresponding to the two different BZ · · · BZdimers in Fig. 6, where one can see that the error in thePD dimer (more orbital overlap) is noticeably larger inmagnitude than the error in the C T-shaped (TS) dimer(less orbital overlap) at all intermolecular separations.In the same breath, we also note that the error withrespect to E REF1 (or E REF2 ) is non-trivial in general,and does not necessarily follow a direct/straightforwardcorrelation with closest intermolecular contacts and/orthe sign/magnitude of E int . For example, the errors forAcOH · · · AcOH ( E int = +12 . kcal/mol) and BZ · · · BZPD ( E int = +51 . kcal/mol) at . × are both largerthan that found in the intermolecular complex with thelargest (most repulsive) E int value and closest O · · · H dis-tance in nenci- —a non-equilibrium configuration ofDMSO · · · DMSO with E int = +186 . kcal/mol (whoseerrors with respect to E REF1 and E REF2 are depicted bystars in Fig. 6).From this analysis, we believe that the errors in the nenci- E int values are mostly within ± . kcal/mol,but can be as large as . − . kcal/mol ( i.e. , ≈ kJ/mol)for certain systems at reduced intermolecular separa-tions. Here, we note in passing that the δE CCSD(T) / haTZ correction used in nenci- provides a significant im-provement over δE CCSD(T) / aDZ , and yields nearly iden-tical E int values when compared to the more expensive δE CCSD(T) / aTZ approach; this is shown in Fig. S1 andagain emphasizes the need for triple- ζ basis sets when em-ploying the δE CCSD(T) correction scheme. When con-sidering the largest errors in Fig. 6, i.e. , AcOH · · · AcOHand BZ · · · BZ PD, one can see that the errors with re-spect to E REF1 and E REF2 differ by ≈ . kcal/mol;as such, the estimated average error in nenci- ( ± . kcal/mol) is comparable to the difference betweenusing E REF1 or E REF2 as the reference for E int . Gener-ally speaking, it is not clear which of these two quantitiessupplies the more accurate reference for E int ; however, ithas been pointed out by Sherrill and co-workers thatthe δE CCSD(T) correction does not converge monotoni-cally towards the CBS limit, which implies that E REF1 might in fact be a slightly better reference value than E REF2 .As mentioned above, the other primary source of errorwhen computing E int using approximate ab initio WFTmethods is the necessarily incomplete treatment of theelectron correlation energy; while post-CCSD(T) correc-tions tend to be small for equilibrium intermolecular in-teraction energies ( i.e. , < . kcal/mol), whether or notsuch corrections become more substantial at reduced in-termolecular separations still remains unanswered. Withincreasingly unfavorable scaling with both system andbasis set size, such post-CCSD(T) calculations ( i.e. ,CCSDT, CCSDT(Q), CCSDTQ, etc) are computation-5ally prohibitive and could have only been performedon: ( i ) the smaller/smallest systems in nenci- , butwith sufficiently large basis sets (of at least triple- ζ orquadruple- ζ quality) or ( ii ) the larger/largest systems in nenci- , but with reduced and insufficiently large ba-sis sets ( i.e. , double- ζ at best). Since neither of these ap-proaches would have provided an accurate and reliable es-timate of the post-CCSD(T) contributions to E int for thewide range of intermolecular complexes in nenci- , we chose to focus our efforts above on critically assessingthe CCSD(T)/CBS scheme outlined in Eqs. (2)–(5) basedon a quantitative estimate of the remaining BSIE at theCCSD(T) level. Since an accurate and reliable predictionof E int for intermolecular complexes in the repulsive wall( i.e. , inside the vdW envelope) poses a substantive chal-lenge to state-of-the-art DFT and WFT methods (see paper-ii in this series), further benchmarking of thestandard CCSD(T)/CBS approach (possibly via stochas-tic CC or FCI methods) in this regime is anopen challenge for the community and will be of criticalimportance for the development of next-generation DFTfunctionals and ML-based intra-/inter-molecular interac-tion potentials. IV. CONCLUSIONS AND FUTURE DIRECTIONS In this work, we present nenci- : a large and com-prehensive database of approximately , benchmarknon-equilibrium non-covalent interaction energies for adiverse selection of intermolecular complexes of biolog-ical and chemical relevance with a particular empha-sis on close intermolecular contacts. Designed to ad-dress the growing need for extensive high-quality quan-tum mechanical data in the chemical sciences, nenci- starts with the molecular dimers in the widelyused S66, S66x8, S66a8, and S101x7 databases, and extends the scope of these popular works in two di-rections. For one, nenci- includes cation- andanion- π complexes, a fundamentally important class ofNCIs that are found throughout nature and among thestrongest NCIs known. Secondly, nenci- system-atically samples both equilibrium and non-equilibriumconfigurations on all intermolecular PES by si-multaneously varying the intermolecular distance (from . × − . × the equilibrium separation) and intermolec-ular angle (including either five or nine angles for eachdistance, depending on symmetry considerations). Assuch, a wide range of intermolecular atom-pair distancesare present in nenci- , including a large number ofclose intermolecular contacts with atom pairs located in-side their respective vdW envelope; these intermolecu-lar complexes probe a number of different short-rangedNCIs ( e.g. , charge transfer and penetration, Pauli repul-sion, many-body exchange-correlation effects, etc), whichare observed in many important chemical and biologi-cal systems, and pose an enormous challenge for molec-ular modeling. Computed at the CCSD(T)/CBS level of theory, the , benchmark E int values in nenci- range from − . kcal/mol (most attractive) to +186 . kcal/mol (most repulsive), with a total spanof . kcal/mol and a mean (median) E int value of − . kcal/mol ( − . kcal/mol). A detailed SAPT-based analysis was used to confirm the diverse and com-prehensive nature of the intermolecular binding motifspresent in nenci- , which includes a significant num-ber of primarily induction-bound dimers and now spansall regions of the SAPT ternary diagram; this warranteda new four-category classification scheme that includescomplexes primarily bound by electrostatics ( , ), in-duction ( ), dispersion ( , ), or mixtures thereof( , ). Finally, a critical error analysis was performedon a representative set of intermolecular complexes, fromwhich we estimate that the E int values in nenci- have a mean error of ± . kcal/mol and a maximum er-ror of ± . − . kcal/mol for the most challenging cases.For all of these reasons, we believe that the nenci- database is timely and well-suited for testing, train-ing, and developing next-generation force fields, DFT andWFT methods, as well as ML based potentials. An order-of-magnitude larger than any database of non-covalentinteractions currently available, nenci- can be usedfor a variety of different purposes. For one, nenci- could be employed as a single database and used in its en-tirety. Alternatively, nenci- can be split into multi-ple different training and testing data sets—each contain-ing a diverse sample of intermolecular binding motifs—and used for cross-validation studies and statistical errorassessment. When used for such purposes, we note inpassing that strong correlations will likely exist betweendifferent points on a given intermolecular PES; as such,we caution against separating such points between train-ing and testing data sets to avoid issues associated withoverfitting.We end this manuscript with a brief discussion of sev-eral future research directions that could build off thiswork and potentially have an immediate impact in thefield. For one, paper-ii in this series (in prepara-tion) will critically assess the accuracy and reliabilityof a large number of popular DFT and WFT meth-ods when describing the diverse array of non-equilibriumnon-covalent interactions in nenci- , thereby identi-fying the strengths and weaknesses of established first-principles methods. A simple and straightforward exten-sion of nenci- would target dimers with increased in-termolecular distances ( e.g. , beyond . × the equilibriumseparation), as benchmark E int values for such complexescould play an important role in testing and training MLmethods for predicting molecular multipoles and po-larizabilities , as well as addressing important un-resolved questions regarding the treatment of long-rangeelectrostatics in ML-based potentials. Other impor-tant research thrusts would focus on expanding nenci- to further address the three challenges introducedabove: ( i ) the need to describe NCIs in large molecu-lar and condensed-phase systems can be addressed with6extensions that focus on large/complex systems and po-tentially include explicit solvent molecules; ( ii ) the needto describe the diverse types of NCIs on the same footingcan be addressed by including NCI binding motifs thatare either not found or underrepresented in nenci- ( e.g. , triple hydrogen bonds, quadrupole-quadrupole in-teractions, ionic bonds, etc); ( iii ) the need to describeNCIs in equilibrium and non-equilibrium systems on thesame footing can be addressed by including complexesat more extreme (reduced and increased) intermolecu-lar separations and angles as well as complexes betweenmonomers in non-equilibrium configurations. ACKNOWLEDGMENTS All authors thank David Sherrill for helpful scientificdiscussions and Destiny Malloy for creating an early pro-totype of Fig. 1. All authors acknowledge partial finan-cial support from Cornell University through start-upfunding. This material is based upon work supportedby the National Science Foundation under Grant No.CHE-1945676. RAD also gratefully acknowledges finan-cial support from an Alfred P. Sloan Research Fellowship.This research used resources of the National Energy Re-search Scientific Computing Center, which is supportedby the Office of Science of the U.S. Department of Energyunder Contract No. DE-AC02-05CH11231. DATA AVAILABILITY STATEMENT The data that supports the findings of this study areavailable within the article and its supplementary mate-rial. REFERENCES D. Langbein, Theory of van der Waals Attraction (Springer:Berlin, 1974). V. 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