Basis set convergence of CCSD(T) equilibrium geometries using a large and diverse set of molecular structures

Molecular structures with spectroscopic accuracy at DFT cost by the templating synthon approach and the PCS141 database

The computation of accurate geometric parameters at density functional theory cost for large molecules in the gas phase is addressed through a novel strategy that combines quantum chemical models with machine learning techniques. The first key step is the expansion of a database of accurate semi-experimental equilibrium structures with additional molecular geometries optimized by version 2 of the Pisa composite scheme. Then, the templating synthon approach is used to improve the accuracy of structures optimized by a hybrid density functional paired with a double zeta basis set, leveraging chemical similarity to cluster different molecular environments and refine bond lengths and valence angles. A set of prototypical biomolecular building blocks is used to demonstrate that it is possible to achieve spectroscopic accuracy for molecular systems too large to be treated by state-of-the-art composite wavefunction methods. In addition, a freely accessible web-based tool has been developed to facilitate the post-processing of geometries optimized using standard electronic structure codes, thereby providing an accurate and efficient tool for the computational study of medium- to large-sized molecules, also accessible to experiment-oriented researchers.

Post-CCSD(T) Thermochemistry of Chlorine Fluorides as a Challenging Test Case for Evaluating Density Functional Theory and Composite Ab Initio Methods

Quantum chemistry plays a key role in exploring the chemical properties of highly reactive chlorine polyfluoride compounds (ClFn). Here, we investigate the thermochemical properties of ClFn species (n=2-6) by means of high-level thermochemical procedures approximating the CCSDT(Q) and CCSDTQ5 energies at the complete basis set limit. We consider total atomization energies (TAEs), Cl-F bond dissociation energies (BDEs), F2 elimination energies (F2 elim.), ionization potentials (IPs), and electron affinities (EAs). The TAEs have significant contributions from post-CCSD(T) correlation effects. The higher-order triple excitations, CCSDT-CCSD(T), are negative and amount to -0.338 (ClF2), -0.727 (ClF3), -0.903 (ClF4), -1.335 (ClF5), and -1.946 (ClF6) kcal/mol. However, the contributions from quadruple (and, where available, also quintuple) excitations are much larger and positive and amount to +1.335 (ClF2), +1.387 (ClF3), +2.367 (ClF4), +2.399 (ClF5), and +3.432 (ClF6) kcal/mol. Thus, the contributions from post-CCSD(T) excitations exceed the threshold of chemical accuracy in nearly all cases. Due to their increasing hyper-valency and multireference character, the ClFn series provides an interesting and challenging test case for both density functional theory and low-level composite ab initio procedures. Here, we highlight the limitations in achieving overall chemical accuracy across all DFT and most composite ab initio procedures.

Accurate Structure and Spectroscopic Properties of Azulene and Its Derivatives by Means of Pisa Composite Schemes and Vibrational Perturbation Theory to Second Order

The structural and spectroscopic properties in the gas phase of azulene and some of its N-bearing derivatives have been analyzed by a general computational strategy based on the recent Pisa composite schemes (PCSs). First of all, an accurate semiexperimental equilibrium structure has been derived for azulene and employed to validate the geometrical parameters delivered by different quantum chemical methods. Next, different isomerization energies (azulene to naphthalene, 1-aza-azulene to quinoline and to other isomers) have been computed by an explicitly correlated PCS version employing frozen natural orbitals. Accurate geometries have been obtained by a cheaper PCS variant based on a double-hybrid functional improved by one-parameter bond corrections, with the same functional providing also remarkable harmonic frequencies. The corresponding equilibrium rotational constants show average deviations within 0.1% from experimental results when taking into account anharmonic vibrational corrections obtained by a global hybrid functional. Therefore, reliable computational estimates have been produced for the rotational constants of several nitrogen derivatives (isomeric aza-azulenes and guaiazulene), whose non-negligible dipole moments could allow experimental microwave characterizations. An analogous approach delivers infrared spectra in remarkable agreement with their experimental counterparts for naphthalene, quinoline, and azulene, together with reliable predictions for the still-unknown spectrum of 1-aza-azulene. In addition to their intrinsic interest, the results of this paper further confirm that a very accurate yet robust and user-friendly tool is now available for aiding high-resolution spectroscopic studies of quite large systems of current technological and/or biological interest.

Accurate Geometries of Large Molecules at DFT Cost by Semiexperimental and Coupled Cluster Templating Fragments

Accurate geometries of small semirigid molecules in the gas phase are available thanks to high-resolution spectroscopy and accurate quantum chemical approaches. These results can be employed for validating cheaper low-level quantum chemical models or correcting the corresponding structures of large molecules. On these grounds, in this work, a large panel of semiexperimental equilibrium structures already available in the literature is used to confirm the average error (1 mÅ for bond lengths and 2 mrad for valence angles) of a version of the Pisa composite schemes (PCS2), which is applicable to molecules containing up to about 20 atoms. Then, the geometries of 30 additional medium-sized systems were optimized at the PCS2 level to cover a more balanced chemical space containing moieties poorly represented in SE compilations. The final database is available on a public domain Web site (https://www.skies-village.it/databases/) and can be employed for correcting structures of larger molecules obtained by hybrid or double-hybrid density functionals in the framework of the templating molecule approach. Several examples show that corrections based on the structures of building blocks taken from this database reduce the error of the B3LYP geometrical parameters of large molecules by nearly an order of magnitude without increasing the computational cost. Furthermore, the results of different density functional theory (DFT) or wave function (e.g., MP2) models can be improved in the same way by simply computing both the whole molecule and suitable building blocks at the chosen level. Then, whenever reference structures of some building blocks containing up to about 20 atoms are not available, they can be purposely optimized at the PCS2 level by employing reasonable computer resources. Therefore, a new DFT-cost tool is now available for the accurate characterization of large molecules by experiment-oriented scientists.

Accurate Vibrational and Ro-Vibrational Contributions to the Properties of Large Molecules by a New Engine Employing Curvilinear Internal Coordinates and Vibrational Perturbation Theory to Second Order

The unbiased comparison between theory and experiment requires approaches more sophisticated than the basic harmonic-oscillator rigid-rotor model, for taking into account vibrational averaging effects and ro-vibrational couplings in molecules of increasing size. Second-order vibrational perturbation theory based on curvilinear internal coordinates (ICs) offers a remarkable compromise between accuracy and computational cost, thanks to the reduction of mode-mode couplings with respect to their counterparts based on Cartesian coordinates. Therefore, we have developed, implemented, and validated a general engine employing ICs, which allows the accurate evaluation of vibrational averages and ro-vibrational couplings for molecules containing up to about 50 atoms beyond the harmonic approximation. After validation of the new tool for relatively small molecules, the effectiveness of ICs has been demonstrated for some flexible and/or quite large molecular bricks of life.

Accurate structures and rotational constants of bicyclic monoterpenes at DFT cost by means of the bond-corrected Pisa composite scheme (BPCS)

The structural, conformational, and spectroscopic properties in the gas phase of 20 bicyclic monoterpenes and monoterpenoids have been analyzed by a new accurate, reduced-cost computational strategy. In detail, the revDSD-PBEP86 double-hybrid functional in conjunction with the D3BJ empirical dispersion corrections and a suitable triple-zeta basis set provides accurate geometrical parameters, whence equilibrium rotational constants, which are further improved by proper account of core-valence correlation. Average deviations within 0.1% between computed and experimental rotational constants are reached when taking into account the vibrational corrections obtained by the B3LYP functional in conjunction with a double-zeta basis set in the framework of second-order vibrational perturbation theory. In addition to their intrinsic interest, the studied terpenes further extend the panel of systems for which the proposed strategy has provided accurate results at density functional theory cost. Therefore, a very accurate yet robust and user-friendly tool is now available for systematic investigations of the role of stereo-electronic effects on the properties of large systems of current technological and/or biological interest by experimentally oriented researchers.

Toward Accurate Quantum Chemical Methods for Molecules of Increasing Dimension: The New Family of Pisa Composite Schemes

The new versions of the Pisa composite scheme introduced in the present paper are based on the careful selection of different quantum chemical models for energies, geometries, and vibrational frequencies, with the aim of maximizing the accuracy of the overall description while retaining a reasonable cost for all the steps. In particular, the computation of accurate electronic energies has been further improved introducing more reliable complete basis set extrapolations and estimation of core-valence correlation, together with improved basis sets for third-row atoms. Furthermore, the reduced-cost frozen natural orbital (FNO) model has been introduced and validated for large molecules. Accurate molecular structures can be obtained avoiding complete basis set extrapolation and evaluating core-valence correlation at the MP2 level. Unfortunately, analytical gradients are not available for the FNO version of the model. Therefore, for large molecules, an accurate reduced-cost alternative is offered by evaluation of valence contributions with a double-hybrid functional in conjunction with the same MP2 contribution for core-valence correlation or by means of a one-parameter approximation. The same double-hybrid functional and basis set are employed to evaluate zero-point energies and partition functions. After the validation of the new models for small systems, a panel of molecular bricks of life has been used to analyze their performances for problems of current fundamental or technological interest. The fully black-box implementation of the computational workflow paves the way toward the accurate yet not prohibitively expensive study of medium- to large-sized molecules also by experimentally oriented researchers.

A Comprehensive Survey on Deep Graph Representation Learning

Graph representation learning aims to effectively encode high-dimensional sparse graph-structured data into low-dimensional dense vectors, which is a fundamental task that has been widely studied in a range of fields, including machine learning and data mining. Classic graph embedding methods follow the basic idea that the embedding vectors of interconnected nodes in the graph can still maintain a relatively close distance, thereby preserving the structural information between the nodes in the graph. However, this is sub-optimal due to: (i) traditional methods have limited model capacity which limits the learning performance; (ii) existing techniques typically rely on unsupervised learning strategies and fail to couple with the latest learning paradigms; (iii) representation learning and downstream tasks are dependent on each other which should be jointly enhanced. With the remarkable success of deep learning, deep graph representation learning has shown great potential and advantages over shallow (traditional) methods, there exist a large number of deep graph representation learning techniques have been proposed in the past decade, especially graph neural networks. In this survey, we conduct a comprehensive survey on current deep graph representation learning algorithms by proposing a new taxonomy of existing state-of-the-art literature. Specifically, we systematically summarize the essential components of graph representation learning and categorize existing approaches by the ways of graph neural network architectures and the most recent advanced learning paradigms. Moreover, this survey also provides the practical and promising applications of deep graph representation learning. Last but not least, we state new perspectives and suggest challenging directions which deserve further investigations in the future.

Unbiased Comparison between Theoretical and Experimental Molecular Structures and Properties: Toward an Accurate Reduced-Cost Evaluation of Vibrational Contributions

The tremendous development of hardware and software is constantly increasing the role of quantum chemical (QC) computations in the assignment and interpretation of experimental results. However, an unbiased comparison between theory and experiment requires the proper account of vibrational averaging effects. In particular, high-resolution spectra in the gas phase are now available for molecules containing up to about 50 atoms, which are too large for a brute-force approach with the available QC methods of sufficient accuracy. In the present paper, we introduce hybrid approaches, which allow the accurate evaluation of vibrational averaging effects for molecules of this size beyond the harmonic approximation, with special attention being devoted to rotational constants. After the validation of new tools for relatively small molecules, the β-estradiol hormone and a prototypical molecular motor have been considered to witness the feasibility of accurate computations for large molecules.

Accurate Structures and Rotational Constants of Steroid Hormones at DFT Cost: Androsterone, Testosterone, Estrone, β-Estradiol, and Estriol

A comprehensive analysis of the structural, conformational, and spectroscopic properties in the gas phase has been performed for five prototypical steroid hormones, namely, androsterone, testosterone, estrone, β-estradiol, and estriol. The revDSD-PBEP86 double-hybrid functional in conjunction with the D3BJ empirical dispersion and a suitable triple-ζ basis set provides accurate conformational energies and equilibrium molecular structures, with the latter being further improved by proper account of core-valence correlation. Average deviations within 0.1% between computed and experimental ground state rotational constants are reached when adding to those equilibrium values vibrational corrections obtained at the cost of standard harmonic frequencies thanks to the use of a new computational tool. Together with the intrinsic interest of the studied hormones, the accuracy of the results obtained at DFT cost for molecules containing about 50 atoms paves the way toward the accurate investigations of other flexible bricks of life.

S-Diagnostic─An a Posteriori Error Assessment for Single-Reference Coupled-Cluster Methods

We propose a novel a posteriori error assessment for the single-reference coupled-cluster (SRCC) method called the S-diagnostic. We provide a derivation of the S-diagnostic that is rooted in the mathematical analysis of different SRCC variants. We numerically scrutinized the S-diagnostic, testing its performance for (1) geometry optimizations, (2) electronic correlation simulations of systems with varying numerical difficulty, and (3) the square-planar copper complexes [CuCl4]2-, [Cu(NH3)4]2+, and [Cu(H2O)4]2+. Throughout the numerical investigations, the S-diagnostic is compared to other SRCC diagnostic procedures, that is, the T1, D1, max T2, and D2 diagnostics as well as different indices of multideterminantal and multireference character in coupled-cluster theory. Our numerical investigations show that the S-diagnostic outperforms the T1, D1, max T2 and D2 diagnostics and is comparable to the indices of multideterminantal and multireference character in coupled-cluster theory in their individual fields of applicability. The experiments investigating the performance of the S-diagnostic for geometry optimizations using SRCC reveal that the S-diagnostic correlates well with different error measures at a high level of statistical relevance. The experiments investigating the performance of the S-diagnostic for electronic correlation simulations show that the S-diagnostic correctly predicts strong multireference regimes. The S-diagnostic, moreover, correctly detects the successful SRCC computations for [CuCl4]2-, [Cu(NH3)4]2+, and [Cu(H2O)4]2+, which have been known to be misdiagnosed by T1 and D1 diagnostics in the past. This shows that the S-diagnostic is a promising candidate for an a posteriori diagnostic for SRCC calculations.

A larger basis set describes atomization energy core-valence correction better than a higher-order coupled-cluster method

The accuracy of coupled-cluster methods for the computation of core-valence correction to atomization energy was assessed. Truncation levels up to CCSDTQP were considered together with (aug-)cc-pwCVnZ (n = D, T, Q, 5) basis sets and three different extrapolation techniques (canonical and flexible Helgaker formula and Riemann zeta function extrapolation). With the exception of CCSD, a more accurate correction can be obtained from a larger basis set using a lower-level coupled-cluster method, and not vice versa. For the CCSD(T) level, it also implies faster computations with modern codes. We also discussed the importance of moving to higher-order or all-electron methods for geometry optimizations. The present study provides the general knowledge needed for the most accurate state-of-the-art computations.

CCSD(T) Rotational Constants for Highly Challenging C5H2 Isomers-A Comparison between Theory and Experiment

We evaluate the accuracy of CCSD(T) and density functional theory (DFT) methods for the calculation of equilibrium rotational constants (Ae, Be, and Ce) for four experimentally detected low-lying C5H2 isomers (ethynylcyclopropenylidene (2), pentatetraenylidene (3), ethynylpropadienylidene (5), and 2-cyclopropen-1-ylidenethenylidene (8)). The calculated rotational constants are compared to semi-experimental rotational constants obtained by converting the vibrationally averaged experimental rotational constants (A0, B0, and C0) to equilibrium values by subtracting the vibrational contributions (calculated at the B3LYP/jun-cc-pVTZ level of the theory). The considered isomers are closed-shell carbenes, with cumulene, acetylene, or strained cyclopropene moieties, and are therefore highly challenging from an electronic structure point of view. We consider both frozen-core and all-electron CCSD(T) calculations, as well as a range of DFT methods. We find that calculating the equilibrium rotational constants of these C5H2 isomers is a difficult task, even at the CCSD(T) level. For example, at the all-electron CCSD(T)/cc-pwCVTZ level of the theory, we obtain percentage errors ≤0.4% (Ce of isomer 3, Be and Ce of isomer 5, and Be of isomer 8) and 0.9-1.5% (Be and Ce of isomer 2, Ae of isomer 5, and Ce of isomer 8), whereas for the Ae rotational constant of isomers 2 and 8 and Be rotational constant of isomer 3, high percentage errors above 3% are obtained. These results highlight the challenges associated with calculating accurate rotational constants for isomers with highly challenging electronic structures, which is further complicated by the need to convert vibrationally averaged experimental rotational constants to equilibrium values. We use our best CCSD(T) rotational constants (namely, ae-CCSD(T)/cc-pwCVTZ for isomers 2 and 5, and ae-CCSD(T)/cc-pCVQZ for isomers 3 and 8) to evaluate the performance of DFT methods across the rungs of Jacob's Ladder. We find that the considered pure functionals (BLYP-D3BJ, PBE-D3BJ, and TPSS-D3BJ) perform significantly better than the global and range-separated hybrid functionals. The double-hybrid DSD-PBEP86-D3BJ method shows the best overall performance, with percentage errors below 0.5% in nearly all cases.

From Molecules with a Planar Tetracoordinate Carbon to an Astronomically Known C5H2 Carbene

Ethynylcyclopropenylidene (2), an isomer of C₅H₂, is a known molecule in the laboratory and has recently been identified in Taurus Molecular Cloud-1 (TMC-1). Using high-level coupled-cluster methods up to the CCSDT(Q)/CBS level of theory, it is shown that two isomers of C₅H₂ with a planar tetracoordinate carbon (ptC) atom, (SP-4)-spiro[2.2]pent-1,4-dien-1,4-diyl (11) and (SP-4)-spiro[2.2]pent-1,4-dien-1,5-diyl (13), serve as the reactive intermediates for the formation of 2. Here, a theoretical connection has been established between molecules containing ptC atoms (11 and 13) and a molecule (2) that is present nearly 430 light years away, thus providing evidence for the existence of ptC species in the interstellar medium. The reaction pathways connecting the transition states and the reactants and products have been confirmed by intrinsic reaction coordinate calculations at the CCSDT(Q)/CBS//B3LYP-D3BJ/cc-pVTZ level. While isomer 11 is non-polar (μ = 0), isomers 2 and 13 are polar, with dipole moment values of 3.52 and 5.17 Debye at the CCSD(T)/cc-pVTZ level. Therefore, 13 is also a suitable candidate for both laboratory and radioastronomical studies.

Molecular Dynamics Simulations Enforcing Nonperiodic Boundary Conditions: New Developments and Application to the Solvent Shifts of Nitroxide Magnetic Parameters

Multiscale methods combining quantum mechanics and molecular mechanics (QM/MM) have become the most suitable and effective strategies for the investigation of the spectroscopic properties of medium-to-large size chromophores in condensed phases. In this context, we are developing a novel workflow aimed at improving the generality, reliability, and ease of use of the available computational tools. In this paper, we report our latest developments with specific reference to a general protocol based on atomistic simulations, carried out under nonperiodic boundary conditions (NPBC). In particular, we add to our in house MD engine a new efficient treatment of mean field electrostatic contributions to energy and forces, together with the capability of performing the simulations either in the canonical (NVT) or in the isothermal–isobaric (NPT) ensemble. Next, we provide convincing evidence that the NBPC approach enhanced by specific tweaks for rigid body propagation, allows for the simulation of solute–solvent systems with a minimum number of degrees of freedom and large integration time step. After its validation, this new approach is applied to the challenging case of solvatochromic effects on the electron paramagnetic resonance (EPR) spectrum of a prototypical nitroxide radical. To this end, we propose and validate also an automated protocol to extract and weight simulation snapshots, making use of a continuous description of the strength of solute–solvent hydrogen bridges. While further developments are being worked on, the effectiveness of our approach, even in its present form, is demonstrated by the accuracy of the results obtained through an unsupervised approach characterized by a strongly reduced computational cost as compared to that of conventional QM/MM models, without any appreciable deterioration of accuracy.

Quantification of Geometric Errors Made Simple: Application to Main-Group Molecular Structures

Nearly all electronic structure simulations begin with obtaining approximate geometries, making a systematic quantification of errors in approximate molecular structures of key importance. Recently, the geometric energy offset (GEO) framework based on a single and natural measure for quantifying and analyzing these errors has been proposed ( J. Phys. Chem. Lett. 2020, 11, 99579964). An accurate and far less costly approximation to GEO is utilized here to readily quantify errors in main-group structures and analyze them in a chemically intuitive way. The use of semiexperimental geometries as a reference further simplifies the analysis. The analysis reveals new insights into the geometric performance of methods, their rankings, as well as patterns across different classes of methods and basis sets that arise from the analysis.

A DLPNO-CCSD(T) benchmarking study of intermolecular interactions of ionic liquids

The accuracy of correlation energy recovered by coupled cluster single-, double-, and perturbative triple-excitations, CCSD(T), has led to the method being considered the gold standard of computational chemistry. The application of CCSD(T) has been limited to medium-sized molecular systems due to its steep scaling with molecular size. The recent development of alternative domain-based local pair natural orbital coupled-cluster method, DLPNO-CCSD(T), has significantly broadened the range of chemical systems to which CCSD(T) level calculations can be applied. Condensed systems such as ionic liquids (ILs) have a large contribution from London dispersion forces of up to 150 kJ mol^-1 in large-scale clusters. Ionic liquids show appreciable charge transfer effects that result in the increased valence orbital delocalization over the entire ionic network, raising the question whether the application of methods based on localized orbitals is reliable for these semi-Coulombic materials. Here the performance of DLPNO-CCSD(T) is validated for the prediction of correlation interaction energies of two data sets incorporating single-ion pairs of protic and aprotic ILs. DLPNO-CCSD(T) produced results within chemical accuracy with tight parameter settings and a non-iterative treatment of triple excitations. To achieve spectroscopic accuracy of 1 kJ mol^-1 , especially for hydrogen-bonded ILs and those containing halides, the DLPNO settings had to be increased by two orders of magnitude and include the iterative treatment of triple excitations, resulting in a 2.5-fold increase in computational cost. Two new sets of parameters are put forward to produce the performance of DLPNO-CCSD(T) within chemical and spectroscopic accuracy.

Regularized second-order correlation methods for extended systems

Second-order Møller-Plesset perturbation theory (MP2) constitutes the simplest form of many-body wavefunction theory and often provides a good compromise between efficiency and accuracy. There are, however, well-known limitations to this approach. In particular, MP2 is known to fail or diverge for some prototypical condensed matter systems like the homogeneous electron gas (HEG) and to overestimate dispersion-driven interactions in strongly polarizable systems. In this paper, we explore how the issues of MP2 for metallic, polarizable, and strongly correlated periodic systems can be ameliorated through regularization. To this end, two regularized second-order methods (including a new, size-extensive Brillouin-Wigner approach) are applied to the HEG, the one-dimensional Hubbard model, and the graphene-water interaction. We find that regularization consistently leads to improvements over the MP2 baseline and that different regularizers are appropriate for the various systems.

Reliability of computed molecular structures

When the structures of 1342 molecules are optimized by 30 methods and 7 basis sets, there appear 289 (21.54%) problematic molecules and 112 (8.35%) failed ones. When 278 problematic molecules are compared, the best methods are BHandH and LC-wPBE, while B97D, BP86, HFS, VSXC, and HCTH are very unreliable. When 179 problematic molecules are computed with larger basis sets, the smallest mean absolute deviation (MAD) of bond angle (2.3°) is shown by QCISD(T)/cc-pVTZ, while the smallest MAD of bond length (0.021 Å), the best SUM1 (4.9 unit), and the best SUM2 (2.4 unit) are shown by DSDPBEP86(Full), DSDPBEP86, PBE1PBE-D3, MP2, and MP2(Full) in combination with aug-cc-pVQZ, cc-pVQZ, Def2QZVP, Def2TZVPP, and/or 6-311++G(3df,3pd). Very large basis sets, for example, larger than cc-pVTZ usually have to be used to obtain very good structures and the performances of many density-functional theory methods are encouraging. The best results may be the limit of modern computational chemistry.

A Computational Journey across Nitroxide Radicals: From Structure to Spectroscopic Properties and Beyond

Nitroxide radicals are characterized by a long-lived open-shell electronic ground state and are strongly sensitive to the chemical environment, thus representing ideal spin probes and spin labels for paramagnetic biomolecules and materials. However, the interpretation of spectroscopic parameters in structural and dynamic terms requires the aid of accurate quantum chemical computations. In this paper we validate a computational model rooted into double-hybrid functionals and second order vibrational perturbation theory. Then, we provide reference quantum chemical results for the structures, vibrational frequencies and other spectroscopic features of a large panel of nitroxides of current biological and/or technological interest.

junChS and junChS-F12 Models: Parameter-free Efficient yet Accurate Composite Schemes for Energies and Structures of Noncovalent Complexes

A recently developed model chemistry (denoted as junChS [Alessandrini, S.; et al. J. Chem. Theory Comput. 2020, 16, 988-1006]) has been extended to the employment of explicitly correlated (F12) methods. This led us to propose a family of effective, reliable, and parameter-free schemes for the computation of accurate interaction energies of molecular complexes ruled by noncovalent interactions. A thorough benchmark based on a wide range of interactions showed that the so-called junChS-F12 model, which employs cost-effective revDSD-PBEP86-D3(BJ) reference geometries, has an improved performance with respect to its conventional counterpart and outperforms well-known model chemistries. Without employing any empirical parameter and at an affordable computational cost, junChS-F12 reaches subchemical accuracy. Accurate characterizations of molecular complexes are usually limited to energetics. To take a step forward, the conventional and F12 composite schemes developed for interaction energies have been extended to structural determinations. A benchmark study demonstrated that the most effective option is to add MP2-F12 core-valence correlation corrections to fc-CCSD(T)-F12/jun-cc-pVTZ geometries without the need of recovering the basis set superposition error and the extrapolation to the complete basis set.

Density Functional Geometries and Zero-Point Energies in Ab Initio Thermochemical Treatments of Compounds with First-Row Atoms (H, C, N, O, F)

Density functionals are often used in ab initio thermochemistry to provide optimized geometries for single-point evaluations at a high level and to supply estimates of anharmonic zero-point energies (ZPEs). Their use is motivated by relatively high accuracy at a modest computational expense, but a thorough assessment of geometry-related error seems to be lacking. We have benchmarked 53 density functionals, focusing on approximations of the first four rungs and on relatively small basis sets for computational efficiency. Optimized geometries of 279 neutral first-row molecules (H, C, N, O, F) are judged by energy penalties relative to the best available geometries, using the composite model ATOMIC/B5 as energy probe. Only hybrid functionals provide good accuracy with root-mean-square errors around 0.1 kcal/mol and maximum errors below 1.0 kcal/mol, but not all of them do. Conspicuously, first-generation hybrids with few or no empirical parameters tend to perform better than highly parameterized ones. A number of them show good accuracy already with small basis sets (6-31G(d), 6-311G(d)). As is standard practice, anharmonic ZPEs are estimated from scaled harmonic values. Statistics of the latter show less performance variation among functionals than observed for geometry-related error, but they also indicate that ZPE error will generally dominate. We have selected PBE0-D3/6-311G(d) for the next version of the ATOMIC protocol (ATOMIC-2) and studied it in more detail. Empirical expressions have been calibrated to estimate bias corrections and 95% uncertainty intervals for both geometry-related error and scaled ZPEs.

Looking for the Elusive Imine Tautomer of Creatinine: Different States of Aggregation Studied by Quantum Chemistry and Molecular Spectroscopy

New spectroscopic experiments and state-of-the-art quantum-chemical computations of creatinine in different aggregation states unequivocally unveiled a significant tuning of tautomeric equilibrium by the environment: from the exclusive presence of the amine tautomer in the solid state and aqueous solution to a mixture of amine and imine tautomers in the gas phase. Quantum-chemical calculations predict the amine species as the most stable tautomer by about 30 kJ mol^-1 in condensed phases. On the contrary, moving to the isolated forms, both Z and E imine isomers become more stable by about 7 kJ mol^-1 . Since the imine isomers and one amine tautomer are separated by significant energy barriers, all of them should be present in the gas phase. This prediction has indeed been confirmed by high-resolution rotational spectroscopy, which provides the first experimental characterization of the elusive imine tautomer. The interpretation of the complicated hyperfine structure of the rotational spectrum, originated by three ¹⁴ N nuclei, makes it possible to use the spectral signatures as a sort of fingerprint for each individual tautomer in the complex sample.

Kinetic Stability of Si2C5H2 Isomer with a Planar Tetracoordinate Carbon Atom

Dissociation pathways of the global minimum geometry of Si2C5H2 with a planar tetracoordinate carbon (ptC) atom, 2,7-disilatricyclo[4.1.0.01,3]hept-2,4,6-trien-2,7-diyl (1), have been theoretically investigated using density functional theory and coupled-cluster (CC) methods. Dissociation of Si-C bond connected to the ptC atom leads to the formation of 4,7-disilabicyclo[4.1.0]hept-1(6),4(5)-dien-2-yn-7-ylidene (4) through a single transition state. Dissociation of C-C bond connected to the ptC atom leads to an intermediate with two identical transition states and leads back to 1 itself. Simultaneous breaking of both Si-C and C-C bonds leads to an acyclic transition state, which forms an acyclic product, cis-1,7-disilahept-1,2,3,5,6-pentaen-1,7-diylidene (19). Overall, two different products, four transition states, and an intermediate have been identified at the B3LYP/6-311++G(2d,2p) level of theory. Intrinsic reaction coordinate calculations have also been done at the latter level to confirm the isomerization pathways. CC calculations have been done at the CCSD(T)/cc-pVTZ level of theory for all minima. Importantly, all reaction profiles for 1 are found be endothermic in Si2C5H2. These results are in stark contrast compared to the structurally similar and isovalent lowest-energy isomer of C7H2 with a ptC atom as the overall reaction profiles there have been found to be exothermic. The activation energies for Si-C, C-C, and Si-C/C-C breaking are found to be 30.51, 64.05, and 61.85 kcal mol−1, respectively. Thus, it is emphasized here that 1 is a kinetically stable molecule. However, it remains elusive in the laboratory to date. Therefore, energetic and spectroscopic parameters have been documented here, which may be of relevance to molecular spectroscopists in identifying this key anti-van’t-Hoff-Le Bel molecule.

Hindered rotor benchmarks for the transition states of free radical additions to unsaturated hydrocarbons

The hindered internal rotors of 32 transition states (TSs) formed through four free radicals, namely methyl, vinyl, ethyl, methoxy (ĊH3, Ċ2H3, Ċ2H5, CH3) additions to acetylene, ethylene, allene, propyne, and propene (C2H2/C2H4/C3H4-a/C3H4-p/C3H6) are studied. To validate the uncertainties of rate constants that stem from the use of different electronic structure methods to treat hindered rotors, the rotations of the newly formed C-C and/or C-O rotors in the transition states are calculated using commonly used DFT methods (B3LYP, M06-2X, ωB97X-D and B2PLYP-D3 with two Pople basis sets (6-31+G(d,p), 6-311++G(d,p)) and cc-pVTZ). The hindrance potential energies V(χ) calculated using the M06-2X/6-311++G(d,p) method are benchmarked at the CCSD(T), CCSD(T)-F12, DLPNO-CCSD(T) levels of theory with cc-pVTZ-F12 and cc-pVXZ (X = T, Q) basis sets and are extrapolated to the complete basis set (CBS) limit. The DLPNO-CCSD(T)/CBS method is proven to reproduce the CCSD(T)/CBS energies within 0.5 kJ mol-1 and this method is selected as the benchmark for all of the rotors in this study. Rotational constants B(χ) are computed for each method based on the optimized geometries for the hindrance potential via the I(2,3) approximation. Thereafter, the V(χ) and B(χ) values are used to compute hindered internal rotation partition functions, QHR, as a function of temperature. The uncertainties in the V(χ), B(χ) and QHR calculations stem from the use of different DFT methods for the internal rotor treatment are discussed for these newly formed rotors. For rotors formed by + C2 alkenes/alkynes, the V(χ) and QHR values calculated using DFT methods are compared with the DLPNO-CCSD(T)/CBS results and analysed according to reaction types. Based on comparisons of the DFT methods with the benchmarking method, reliable DFT methods are recommended for the treatment of internal rotors for different reaction types considering both accuracy and computational cost. This work, to the authors' knowledge, is the first to systematically benchmark hindrance potentials which can be used to estimate uncertainties in theoretically derived rate constants arising from the choice of different electronic structure methods.

Quantifying and Understanding Errors in Molecular Geometries

Electronic structure calculations are ubiquitous in most branches of chemistry, but all have errors in both energies and equilibrium geometries. Quantifying errors in possibly dozens of bond angles and bond lengths is a Herculean task. A single natural measure of geometric error is introduced, the geometry energy offset (GEO). GEO links many disparate aspects of geometry errors: a new ranking of different methods, quantitative insight into errors in specific geometric parameters, and insight into trends with different methods. GEO can also reduce the cost of high-level geometry optimizations and shows when geometric errors distort the overall error of a method. Results, including some surprises, are given for both covalent and weak interactions.

Calculations of quantum tunnelling rates for muonium reactions with methane, ethane and propane

Thermal rate constants for Mu + CH₄, Mu + C₂H₆ and Mu + C₃H₈ and their equivalent reactions with H were evaluated with ab initio instanton rate theory. The potential-energy surfaces are fitted using Gaussian process regression to high-level electronic-structure calculations evaluated around the tunnelling pathway. This method was able to successfully reproduce various experimental measurements for the rate constant of these reactions. However, it was not able to reproduce the faster-than-expected rate of Mu + C₃H₈ at 300 K reported by Fleming et al. [Phys. Chem. Chem. Phys., 2015, 17, 19901 and Phys. Chem. Chem. Phys., 2020, 22, 6326]. Analysis of our results indicates that the kinetic isotope effect at this temperature is not significantly influenced by quantum tunnelling. We consider many possible factors for the discrepancy between theory and experiment but conclude that in each case, the instanton approximation is unlikely to be the cause of the error. This is in part based on the good agreement we find between the instanton predictions and new multiconfigurational time-dependent Hartree (MCTDH) calculations for Mu + CH₄ using the same potential-energy surface. Further experiments will therefore be needed to resolve this issue.

Effective basis set extrapolations for CCSDT, CCSDT(Q), and CCSDTQ correlation energies

It is well established that extrapolating the coupled-cluster single double triple [CCSD and (T)] correlation energies using empirically motivated extrapolation exponents can accelerate the basis set convergence. Here, we consider the extrapolation of coupled-cluster expansion terms beyond the CCSD(T) level to the complete basis set (CBS) limit. We obtain reference CCSDT-CCSD(T) [T₃-(T)], CCSDT(Q)-CCSDT [(Q)], and CCSDTQ-CCSDT(Q) [T₄-(Q)] contributions from cc-pV{5,6}Z extrapolations for a diverse set of 16 first- and second-row systems. We use these basis-set limit results to fit extrapolation exponents in conjunction with the cc-pV{D,T}Z, cc-pV{T,Q}Z, and cc-pV{Q,5}Z basis set pairs. The optimal extrapolation exponents result in noticeable improvements in performance (relative to α = 3.0) in conjunction with the cc-pV{T,Q}Z basis set pair; however, smaller improvements are obtained for the other basis sets. These results confirm that the basis sets and basis set extrapolations used for obtaining post-CCSD(T) components in composite thermochemical theories such as Weizmann-4 and HEAT are sufficiently close to the CBS limit for attaining sub-kJ/mole accuracy. The fitted extrapolation exponents demonstrate that the T₃-(T) correlation component converges more slowly to the CBS limit than the (Q) and T₄ terms. A systematic investigation of the effect of diffuse functions shows that it diminishes (i) in the order T₃-(T) > (Q) > T₄-(Q) and (ii) with the size of the basis set. Importantly, we find that diffuse functions tend to systematically reduce the T₃-(T) contribution but systematically increases the (Q) contribution. Thus, the use of the cc-pVnZ basis sets benefits from a certain degree of error cancellation between these two components.

Surprisingly broad applicability of the cc-pVnZ-F12 basis set for ground and excited states

Excellent convergence properties for the (aug-)cc-pVnZ-F12 basis set family, purpose-made for explicitly correlated calculations, are demonstrated with conventional wave function methods and Kohn-Sham density functional theory for various ground and excited-state calculations. Among the ground-state properties studied are dipole moments, covalent bond lengths, and interaction and reaction energies. For excited states, we looked at vertical excitation energies, UV absorption, and excited-state absorption spectra. Convergence is compared against the basis sets cc-pVnZ, def2-nVD, aug-pcseg-n, and nZaPa-NR. It is established that the cc-pVnZ-F12 family consistently yields results of n + 1 quality and better. Especially, the cc-pVDZ-F12 basis set is found to be a basis set of good cost vs performance trade-off.

Performance of Electronic Structure Methods for the Description of Hückel-Möbius Interconversions in Extended π-Systems

Expanded porphyrins provide a versatile route to molecular switching devices due to their ability to shift between several π-conjugation topologies encoding distinct properties. DFT remains the workhorse for modeling such extended macrocycles, when taking into account their size and huge conformational flexibility. Nevertheless, the stability of Hückel and Möbius conformers depends on a complex interplay of different factors, such as hydrogen bonding, π···π stacking, steric effects, ring strain, and electron delocalization. As a consequence, the selection of an exchange-correlation functional for describing the energy profile of topological switches is very difficult. For these reasons, we have examined the performance of a variety of wave function methods and density functionals for describing the thermochemistry and kinetics of topology interconversions across a wide range of macrocycles. Especially for hexa- and heptaphyrins, the Möbius structures have a stronger degree of static correlation than the Hückel and twisted-Hückel structures, and as a result the relative energies of singly twisted structures are a challenging test for electronic structure methods. Comparison of limited orbital space full CI calculations with CCSD(T) calculations within the same active spaces shows that post-CCSD(T) correlation contributions to relative energies are very minor. At the same time, relative energies are weakly sensitive to further basis set expansion, as proven by the minor energy differences between the extrapolated MP2/CBS energies estimated from cc-pV{T,Q}Z, diffuse-augmented heavy-aug-cc-pV{T,Q}Z and explicitly correlated MP2-F12/cc-pVDZ-F12 calculations. Hence, our CCSD(T) reference values are reasonably well-converged in both 1-particle and n-particle spaces. While conventional MP2 and MP3 yield very poor results, SCS-MP2 and particularly SOS-MP2 and SCS-MP3 agree to better than 1 kcal mol-1 with the CCSD(T) relative energies. Regarding DFT methods, the range-separated double hybrids, such as ωB97M(2) and B2GP-PLYP, outperform other functionals with RMSDs of 0.6 and 0.8 kcal mol-1, respectively. While the original DSD-PBEP86 double hybrid performs fairly poorly for these extended π-systems, the errors drop down to 1.9 kcal mol-1 for the revised revDOD-PBEP86-NL, which eliminates the same-spin correlation energy. Minnesota meta-GGA functionals with high fractions of exact exchange (M06-2X and M08-HX) also perform reasonably well, outperforming more robust and significantly less empirically parametrized functionals like SCAN0-D3.

Theoretical Studies of SiC4H2 Isomers Delineate Three Low-Lying Silylidenes Are Missing in the Laboratory

Eleven isomers of SiC₄H₂ lying within 50 kcal mol^-1 have been theoretically investigated using density functional theory and high-level coupled-cluster methods. Among them, four isomers, 1-ethynyl-3-silacycloprop-1(2)-en-3-ylidene (1), diethynylsilylidene (2), 1-sila-1,2,3,4-pentatetraenylidene (4), and 1,3-butadiynylsilylidene (5), have already been identified in the laboratory. The current investigation reports three low-lying (<1 eV) silylidenes [2-methylenesilabicyclo[1.1.0]but-1(3)-en-4-ylidene (3), 4-sila-2-methylenebicyclo[1.1.0]but-1(3)-en-4-ylidene (6), and 3-ethynyl-1-silapropadienylidene (7)] and three high-lying (>1 eV) silylidenes [2-sila-(didehydrovinylidene)cyclopropene (8), an isomer with a planar tetracoordinate carbon (ptC) atom (10), and 1-ethynyl-1-silapropadienylidene (11)], which remain elusive in the laboratory to date. Isomer 9 also contains a ptC atom, which turned out to be a transition state at all levels. Though all isomers are polar (μ ≠ 0), rotational spectrum is available only for 4. Using matrix isolation, three isomers (1, 2, and 5) have been trapped in the laboratory at 10 K. Considering the astrochemical relevance of silicon-carbide clusters in the interstellar medium, the current theoretical data demand new molecular spectroscopic studies on SiC₄H₂. Surprisingly, unlike the isovalent C₅H₂ isomers, where the bent carbenes are yet to be identified in the laboratory, the bent silylidenes (2 and 5) have been trapped in the case of SiC₄H₂. In both the cases, molecules with transannular C-C and/or Si-C bonds remain elusive, though they lie in the low-lying region. Using suitable precursors, whether these peculiar geometries (especially 3 and 6) would be identified or not in the laboratory needs to be addressed by molecular spectroscopists. The present investigation documents structural and spectroscopic information of SiC₄H₂ isomers, which may compliment future molecular spectroscopic observations including radioastronomical searches.

Kinetics and Thermodynamics of Reactions Involving Criegee Intermediates: An Assessment of Density Functional Theory and Ab Initio Methods Through Comparison with CCSDT(Q)/CBS Data

Reactions involving Criegee intermediates (CIs, R₁ R₂ COO) are important in atmospheric ozonolysis models. In recent years, density functional theory (DFT) and CCSD(T)-based ab initio methods are increasingly being used for modeling reaction profiles involving CIs. We obtain highly accurate CCSDT(Q)/CBS reaction energies and barrier heights for ring-closing reactions involving atmospherically important CIs (R₁ /R₂ = H, Me, OH, OMe, F, CN, cyclopropene, ethylene, acetaldehyde, and acrolein). We use this benchmark data to evaluate the performance of DFT, double-hybrid DFT (DHDFT), and ab initio methods for the kinetics and thermodynamics of these reactions. We find that reaction energies are more challenging for approximate theoretical procedures than barrier heights. Overall, taking both reaction energies and barrier heights into account, only one of the 58 considered DFT methods (the meta-GGA MN12-L) attains near chemical accuracy, with root-mean-square deviations (RMSDs) of 3.5 (barrier heights) and 4.7 (reaction energies) kJ mol^-1 . Therefore, MN12-L is recommended for investigations where CCSD(T)-based methods are not computationally feasible. For reaction barrier heights performance does not strictly follow Jacob's Ladder, for example, DHDFT methods do not perform better than conventional DFT methods. Of the ab initio methods, the cost-effective CCSD(T)/CBS(MP2) approach gives the best performance for both reaction energies and barrier heights, with RMSDs of 1.7 and 1.4 kJ mol^-1 , respectively. All the considered Gaussian-n methods show good performance with RMSDs below the threshold of chemical accuracy for both reaction energies and barrier heights, where G4(MP2) shows the best overall performance with RMSDs of 2.9 and 1.5 kJ mol^-1 , respectively. © 2019 Wiley Periodicals, Inc.

Combined Electron-Diffraction, Spectroscopic, and Theoretical Determination of the Structure of N-Deuterio- trans-Methyldiazene, CH3N═ND. Conformational Effects of the N═N Double Bond

Gas phase electron-diffraction (GED) data obtained at a nozzle-tip temperature of 273 K have been combined with spectroscopic vibrational-rotational constants to determine the structure of trans-methyldiazene, an important prototype for the N═N double bond. The N-deuterio form CH₃N═ND was used in the study since it is appreciably more stable than CH₃N═NH. Both the theoretical and experimental results are consistent with a planar C _s trans-CNND framework. The experimental results ( r_α⁰/ r_g²⁷³) are 1.465(2)/1.467(2) Å for the CN bond, 1.248(1)/1.251(1) Å for the N═N double bond, and 1.037(17)/1.048(17) Å for the ND bond. The NND angle is 105.9(20)/105.6(20)° and the CNN angle is 112.4(5)/112.2(5)°, where the uncertainties in parentheses are twice the standard deviation from a simultaneous least-squares fit of the GED and microwave data. For the methyl group, both theory and experiment indicate that two CH bonds are symmetrically arranged out of the molecular plane while the third CH' lies in the plane in an eclipsed (not staggered) cis-H'CNN arrangement. Theoretical calculations (B3LYP/cc-PVnZ and CCSD(T)/cc-PVnZ) suggest a slight distortion of the methyl group, with a tilt of the methyl top axis about 5° away from the N═N bond. The experimental data are consistent with this picture but are equally consistent with an undistorted methyl group. Inclusion of distortions predicted by theory in a complete basis set limit (CBS) lead to a preferred analysis with average values of 1.086(5)/1.106(5) Å for the CH bond length and an average HCH angle of 108.3(8)/107.8(8)°. Features of the structure of methyldiazene and related compounds are discussed. It is found that the short N═N bond length in the diazenes produces much greater steric repulsion than in analogous ethylene compounds and this effect leads to some interesting conformational and distortion differences for attached CH₃ groups.

Ab initio instanton rate theory made efficient using Gaussian process regression

Ab initio instanton rate theory is a computational method for rigorously including tunnelling effects into the calculations of chemical reaction rates based on a potential-energy surface computed on the fly from electronic-structure theory. This approach is necessary to extend conventional transition-state theory into the deep-tunnelling regime, but it is also more computationally expensive as it requires many more ab initio calculations. We propose an approach which uses Gaussian process regression to fit the potential-energy surface locally around the dominant tunnelling pathway. The method can be converged to give the same result as from an on-the-fly ab initio instanton calculation but it requires far fewer electronic-structure calculations. This makes it a practical approach for obtaining accurate rate constants based on high-level electronic-structure methods. We show fast convergence to reproduce benchmark H + CH4 results and evaluate new low-temperature rates of H + C2H6 in full dimensionality at a UCCSD(T)-F12b/cc-pVTZ-F12 level.

Energetic and spectroscopic properties of the low-lying C7H2 isomers: a high-level ab initio perspective

We use high-level ab initio CCSD(T) and CCSDT(Q) methods to investigate the energetic and spectroscopic properties of nine low-lying isomers of C₇H₂, which lie within 1 eV. Among these, heptatriynylidene (1), 1-(buta-1,3-diynyl)cyclopropenylidene (2) and heptahexaenylidene (9) have been detected experimentally. The other six isomers, 1,2-(diethynyl)cyclopropenylidene (3), bicyclo[4.1.0]hepta-1,2,4,5-tetraene-7-ylidene (4), cyclohepta-1,2,3,4-tetraen-6-yne (5), bicyclo[4.1.0]hepta-4,6-diene-2-yne-7-ylidene (6), bicyclo[4.1.0]hepta-1,5-diene-3-yne-7-ylidene (7) and 1-(buta-1,3-diynyl)propadienylidene (8), remain hypothetical to date. Except for 1, all of the isomers are associated with a non-zero dipole moment (μ≠ 0). Although Fourier-transform microwave spectroscopy had detected 2 and 9, our study reveals that six hypothetical isomers (3-8) are thermodynamically sandwiched between the experimentally known and astronomically relevant isomers 2 and 9. The structural parameters, dipole moments, rotational and centrifugal distortion constants, harmonic vibrational frequencies, and infra-red intensities presented here may be useful for the laboratory detection of these previously unidentified isomers (3-8) and also all others (2-9) in astronomical sources.