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Basis set extrapolations are typically rationalized either from analytical arguments involving the partial-wave or principal expansions of the correlation energy in helium-like systems or from fitting extrapolation parameters to reference energetics for a small(ish) training set. Seeking to avoid both, we explore a third alternative: extracting extrapolation parameters from the requirement that the BSSE (basis set superposition error) should vanish at the complete basis set limit. We find this to be a viable approach provided that the underlying basis sets are not too small and reasonably well balanced. For basis sets not augmented by diffuse functions, BSSE minimization and energy fitting yield quite similar parameters.

High-accuracy composite wave function methods like Weizmann-4 (W4) theory, high-accuracy extrapolated ab initio thermochemistry (HEAT), and the Feller–Peterson–Dixon (FPD) approach enable sub-kJ/mol accuracy in gas-phase thermochemical properties. Their biggest computational bottleneck is the evaluation of the valence post-CCSD(T) correction term. We demonstrate here, for the W4-17 thermochemistry benchmark and subsets thereof, that the Λ coupled-cluster expansion converges more rapidly and smoothly than the regular coupled-cluster series. By means of CCSDT(Q)Λand CCSDTQ(5)Λ, we can considerably (up to an order of magnitude) accelerate W4- and W4.3-type calculations without loss in accuracy, leading to the W4Λ and W4.3Λ computational thermochemistry protocols.

It is well-known that both wave function ab initio and DFT calculations on second-row compounds exhibit anomalously slow basis set convergence unless the basis sets are augmented with additional “tight” (high-exponent) d functions, as in the cc-pV(n+d)Z and aug-cc-pV(n+d)Z basis sets. This has been rationalized as being necessary for a better description of the low-lying 3d orbital, which as the oxidation state increases sinks low enough to act as a back-donation acceptor from chalcogen and halogen lone pairs. This prompts the question whether a similar phenomenon exists for the isovalent compounds of the heavy p-block. We show that for the fourth and fifth row, this is the case, but this time for tight f functions enhancing the description of the low-lying 4f and 5f Rydberg orbitals, respectively. In the third-row heavy p block, the 4f orbitals are too far up, while the 4d orbitals are adequately covered by the basis functions already present to describe the 3d subvalence orbitals.

Double-hybrid density functional theory (DHDFT) offers a pathway to accuracy approaching composite wavefunction approaches such as G4 theory. However, the Görling–Levy second-order perturbation theory (GLPT2) term causes them to partially inherit the slow ∝L–3(with Lthe maximum angular momentum) basis set convergence of correlated wavefunction methods. This could potentially be remedied by introducing F12 explicit correlation: we investigate the basis set convergence of both DHDFT and DHDFT-F12 (where GLPT2 is replaced by GLPT2-F12) for the large and chemically diverse general main-group thermochemistry, kinetics, and noncovalent interactions (GMTKN55) benchmark suite. The B2GP-PLYP-D3(BJ) and revDSD-PBEP86-D4 DHDFs are investigated as test cases, together with orbital basis sets as large as aug-cc-pV5Z and F12 basis sets as large as cc-pVQZ-F12. We show that F12 greatly accelerates basis set convergence of DHDFs, to the point that even the modest cc-pVDZ-F12 basis set is closer to the basis set limit than cc-pV(Q+d)Z or def2-QZVPPD in orbital-based approaches, and in fact comparable in quality to cc-pV(5+d)Z. Somewhat surprisingly, aug-cc-pVDZ-F12 is not required even for the anionic subsets. In conclusion, DHDF-F12/VDZ-F12 eliminates concerns about basis set convergence in both the development and applications of double-hybrid functionals. Mass storage and I/O bottlenecks for larger systems can be circumvented by localized pair natural orbital approximations, which also exhibit much gentler system size scaling.

We have investigated the title question for the W4-08 thermochemical benchmark using l-saturated truncations of a large reference (REF) basis set, as well as for standard F12-optimized basis sets. With the REF basis set, the root-mean-square (RMS) contribution of ifunctions to the MP2-F12 total atomization energies (TAEs) is about 0.01 kcal/mol, the largest individual contributions being 0.04 kcal/mol for P2and P4. However, even for these cases, basis set extrapolation from {g,h} basis sets adequately addresses the problem. Using basis sets insufficiently saturated in the spdfghangular momenta may lead to exaggerated ifunction contributions. For extrapolation from spdfgand spdfghbasis sets, basis set convergence appears to be quite close to the theoretical asymptotic ∝ L–7behavior. We hence conclude that hfunctions are sufficient even for highly demanding F12 applications. With one-parameter extrapolation, spdfand spdfgbasis sets are adequate, aug-cc-pV{T,Q}Z-F12 yielding a RMSD = 0.03 kcal/mol. A limited exploration of CCSD(F12*) and CCSD-F12b suggests our conclusions are applicable to higher-level F12 methods as well.

We have evaluated a set of accurate canonical CCSD(T) energies for stationary points on the potential energy surface for Ru(II, III) chloride carbonyl catalysis of two competing reactions between benzene and methyl acrylate (MA), namely, hydroarylation and oxidative coupling. We have then applied this set to evaluate the performance of localized orbital coupled-cluster methods and several new and common density functionals. We find that (a) DLPNO-CCSD(T) with TightPNO cutoffs is an acceptable substitute for full canonical CCSD(T) calculations on this system; (b) for the closed-shell systems where it could be applied, LNO-CCSD(T) with tight convergence criteria is very close to the canonical results; (c) the recent ωB97X-V and ωB97M-V functionals exhibit superior performance to commonly used DFT functionals in both closed- and open-shell calculations; (d) the revDSD-PBEP86 revision of the DSD-PBEP86 double hybrid represents an improvement over the original, even though transition metals were not involved in its parametrization; and (e) DSD-SCAN and DOD-SCAN show comparable efficiency. Most tested (meta)-GGA and hybrid density functionals perform better for open-shell than for closed-shell complexes; this is not the case for the double hybrids considered.

We have explored the use of range separation as a possible avenue for further improvement on our revDSD minimally empirical double hybrid functionals. Such ωDSD functionals encompass the XYG3 type of double hybrid (i.e., xDSD) as a special case for ω → 0. As in our previous studies, the large and chemically diverse GMTKN55 benchmark suite was used for evaluation. Especially when using the D4 rather than D3BJ dispersion model, xDSD has a slight performance advantage in WTMAD2. As in previous studies, PBEP86 is the winning combination for the semilocal parts. xDSDn-PBEP86-D4 marginally outperforms the previous “best in class” ωB97M(2) Berkeley double hybrid but without range separation and using fewer than half the number of empirical parameters. Range separation turns out to offer only marginal further improvements on GMTKN55 itself. While ωB97M(2) still yields better performance for small-molecule thermochemistry, this is compensated in WTMAD2 by the superior performance of the new functionals for conformer equilibria. Results for two external test sets with pronounced static correlation effects may indicate that range-separated double hybrids are more resilient to such effects.

For revDSD double hybrids, the Görling–Levy second-order perturbation theory component is an Achilles’ heel when applied to systems with significant near-degeneracy (“static”) correlation. We have explored its replacement by the direct random phase approximation (dRPA), inspired by the SCS-dRPA75 functional of Kállay and co-workers. The addition to the final energy of both a D4 empirical dispersion correction and of a semilocal correlation component lead to significant improvements, with DSD-PBEdRPA75-D4 approaching the performance of revDSD-PBEP86-D4 and the Berkeley ωB97M(2). This form appears to be fairly insensitive to the choice of the semilocal functional but does exhibit stronger basis set sensitivity than the PT2-based double hybrids (due to much larger prefactors for the nonlocal correlation). As an alternative, we explored adding an MP3-like correction term (in a medium-sized basis set) to a range-separated ωDSD-PBEP86-D4 double hybrid and found it to have significantly lower WTMAD2 (weighted mean absolute deviation) for the large and chemically diverse GMTKN55 benchmark suite; the added computational cost can be mitigated through density fitting techniques.

We present a general purpose double-hybrid DFT parametrization based on the BLYP functional, spin-component scaled (SCS) MP2-like correlation and a dispersion correction, called DSD-BLYP. Six training sets were used, including main group and transition state thermochemistry, kinetics, and dispersion forces. This new parametrization is usually 10−15% more accurate than the already exceptional B2GP-PLYP double hybrid, at the same computational cost. Its principal benefit is greater robustness for systems with significant nondynamical correlation. If a scaling factor is included in the harmonic frequency calculations, B2GP-PLYP was found to give very accurate results for kinetics, thermochemistry, and frequencies.

A hierarchy of wavefunction composite methods (cWFT), based on G4-type cWFT methods available for elements H through Rn, was recently reported by the present authors [J. Chem. Theor. Comput.2020, 16, 4238]. We extend this hierarchy by considering the inner-shell correlation energy in the second-order Møller–Plesset correction and replacing the Weigend–Ahlrichs def2-mZVPP(D) basis sets used with complete basis set extrapolation from augmented correlation-consistent core–valence triple-ζ, aug-cc-pwCVTZ(-PP), and quadruple-ζ, aug-cc-pwCVQZ(-PP), basis sets, thus creating cc-G4-type methods. For the large and chemically diverse GMTKN55 benchmark suite, they represent a substantial further improvement and bring WTMAD2 (weighted mean absolute deviation) down below 1 kcal/mol. Intriguingly, the lion’s share of the improvement comes from better capture of valence correlation; the inclusion of core–valence correlation is almost an order of magnitude less important. These robust correlation-consistent cWFT methods approach the CCSD(T) complete basis limit with just one or a few fitted parameters. Particularly, the DLPNO variants such as cc-G4-T-DLPNO are applicable to fairly large molecules at a modest computational cost, as is (for a reduced range of elements) a different variant using MP2-F12/cc-pVTZ-F12 for the MP2 component.

The large and chemically diverse GMTKN55 benchmark was used as a training set for parametrizing composite wave function thermochemistry protocols akin to G4(MP2)XK theory (Chan, B.; Karton, A.; Raghavachari, K. J. Chem. Theory Comput. 2019, 15, 4478–4484). On account of their availability for elements H through Rn, Karlsruhe def2 basis sets were employed. Even after reparametrization, the GMTKN55 WTMAD2 (weighted mean absolute deviation, type 2) for G4(MP2)-XK is actually inferior to that of the best rung-4 DFT functional, ωB97M-V. By increasing the basis set for the MP2 part to def2-QZVPPD, we were able to substantially improve performance at modest cost (if an RI-MP2 approximation is made), with WTMAD2 for this G4(MP2)-XK-D method now comparable to the better rung-5 functionals (albeit at greater cost). A three-tier approach with a scaled MP3/def2-TZVPP intermediate step, however, leads to a G4(MP3)-D method that is markedly superior to even the best double hybrids ωB97M(2) and revDSD-PBEP86-D4. Evaluating the CCSD(T) component with a triple-ζ, rather than split-valence, basis set yields only a modest further improvement that is incommensurate with the drastic increase in computational cost. G4(MP3)-D and G4(MP2)-XK-D have about 40% better WTMAD2, at similar or lower computational cost, than their counterparts G4 and G4(MP2), respectively: detailed comparison reveals that the difference lies in larger molecules due to basis set incompleteness error. An E2/{T,Q} extrapolation and a CCSD(T)/def2-TZVP step provided the G4-T method of high accuracy and with just three fitted parameters. Using KS orbitals in MP2 leads to the G4(MP3|KS)-D method, which entirely eliminates the CCSD(T) step and has no steps costlier than scaled MP3; this shows a path forward to further improvements in double-hybrid density functional methods. None of our final selections require an empirical HLC correction; this cuts the number of empirical parameters in half and avoids discontinuities on potential energy surfaces. G4-T-DLPNO, a variant in which post-MP2 corrections are evaluated at the DLPNO-CCSD(T) level, achieves nearly the accuracy of G4-T but is applicable to much larger systems.

Localized orbital coupled cluster theory has recently emerged as a nonempirical alternative to DFT for large systems. Intuitively, one might expect such methods to perform less well for highly delocalized systems. In the present work, we apply both canonical CCSD(T) approximations and a variety of localized approximations to a set of flexible expanded porphyrins—macrocycles that can switch between Hückel, figure-eight, and Möbius topologies under external stimuli. Both minima and isomerization transition states are considered. We find that Möbius(-like) structures have much stronger static correlation character than the remaining structures, and that this causes significant errors in DLPNO-CCSD(T) and even DLPNO-CCSD(T1) approaches, unless TightPNO cutoffs are employed. If sub-kcal mol–1accuracy with respect to canonical relative energies is required even for Möbius-type systems (or other systems plagued by strong static correlation), then Nagy and Kallay’s LNO-CCSD(T) method with “tight” settings is the suitable localized approach. We propose the present POLYPYR21 data set as a benchmark for localized orbital methods or, more broadly, for the ability of lower-level methods to handle energetics with strongly varying degrees of static correlation.

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–1with 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–1for 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.

We present a family of minimally empirical double-hybrid DFT functionals parametrized against the very large and diverse GMTKN55 benchmark. The very recently proposed ωB97M(2) empirical double hybrid (with 16 adjustable parameters) has the lowest WTMAD2 (weighted mean absolute deviation over GMTKN55) ever reported at 2.19 kcal/mol. However, refits of the DSD-BLYP and DSD-PBEP86 spin-component-scaled, dispersion-corrected double hybrids can achieve WTMAD2 values as low as 2.33 with the very recent D4 dispersion correction (2.42 kcal/mol with the D3(BJ) dispersion term) using just a handful of adjustable parameters. If we use full DFT correlation in the initial orbital evaluation, the xrevDSD-PBEP86-D4 functional reaches WTMAD2 = 2.23 kcal/mol, statistically indistinguishable from ωB97M(2) but using just four nonarbitrary adjustable parameters (and three semiarbitrary ones). The changes from the original DSD parametrizations are primarily due to noncovalent interaction energies for large systems, which were undersampled in the original parametrization set. With the new parametrization, same-spin correlation can be eliminated at minimal cost in performance, which permits revDOD-PBEP86-D4 and revDOD-PBE-D4 functionals that scale as N4or even N3with the size of the system. Dependence of WTMAD2 for DSD functionals on the percentage of HF exchange is roughly quadratic; it is sufficiently weak that any reasonable value in the 64% to 72% range can be chosen semiarbitrarily. DSD-SCAN and DOD-SCAN double hybrids involving the SCAN nonempirical meta-GGA as the semilocal component have also been considered and offer a good alternative if one wishes to eliminate either the empirical dispersion correction or the same-spin correlation component. noDispSD-SCAN66 achieves WTMAD2 = 3.0 kcal/mol, compared to 2.7 kcal/mol for DOD-SCAN66-D4. However, the best performance without dispersion corrections (WTMAD2 = 2.8 kcal/mol) is reached by revωB97X-2, a slight reparametrization of the Chai–Head-Gordon range-separated double hybrid. Finally, in the context of double-hybrid functionals, the very recent D4 dispersion correction is clearly superior over D3(BJ).

The electrochemical reduction of CO2has been extensively investigated in recent years, with the expectation that a detailed mechanistic understanding could achieve the goal of finding a stable catalyst with high turnover frequencies and low reduction potentials. In the catalytic cycle of the carbon dioxide hydrogenase enzyme, it has been suggested that the reduced metal center reacts with CO2to form a carboxylate intermediate that is stabilized by hydrogen bonding using a histidine moiety in the second coordination sphere. Using the well-known fac-Re(I)bipyridine(CO)3Cl complex as a starting point, the bipyridine ligand was modified in the second coordination sphere with a thiourea tether that is known to form hydrogen bonds with carbonyl moieties. The resulting Re(I) catalyst was an excellent electrocatalyst for the selective reduction of CO2to CO, with a turnover frequency of 3040 s–1. The binding of CO2to the thiourea tether was observable by 1H NMR, and NOE experiments showed that the hydrogen atoms of the thiourea group were labile. Further experiments indicated that the thiourea moiety is also a local proton source and addition of an external proton source actually inhibits catalysis. The absence of a kinetic isotope effect was explained through DFT calculations that showed that the proton invariably jumps to the nearest CO2oxygen atom to form a metal–carboxylic acid without going through any minimum or transition state. EPR and NMR spectroscopies were used to identify the various reduced intermediates. Thus, the thiourea tether in the second coordination sphere can bind CO2, stabilize carboxylic acid reaction intermediates, and directly act as a local proton source, leading to a significantly more active catalyst.

Benchmark ab initio energies for BEGDB and WATER27 data sets have been re-examined at the MP2 and CCSD(T) levels with both conventional and explicitly correlated (F12) approaches. The basis set convergence of both conventional and explicitly correlated methods has been investigated in detail, both with and without counterpoise corrections. For the MP2 and CCSD–MP2 contributions, rapid basis set convergence observed with explicitly correlated methods is compared to conventional methods. However, conventional, orbital-based calculations are preferred for the calculation of the (T) term, since it does not benefit from F12. CCSD(F12*) converges somewhat faster with the basis set than CCSD-F12b for the CCSD-MP2 term. The performance of various DFT methods is also evaluated for the BEGDB data set, and results show that Head-Gordon’s ωB97X-V and ωB97M-V functionals outperform all other DFT functionals. Counterpoise-corrected DSD-PBEP86 and raw DSD-PBEPBE-NL also perform well and are close to MP2 results. In the WATER27 data set, the anionic (deprotonated) water clusters exhibit unacceptably slow basis set convergence with the regular cc-pVnZ-F12 basis sets, which have only diffuse s and p functions. To overcome this, we have constructed modified basis sets, denoted aug-cc-pVnZ-F12 or aVnZ-F12, which have been augmented with diffuse functions on the higher angular momenta. The calculated final dissociation energies of BEGDB and WATER27 data sets are available in the Supporting Information. Our best calculated dissociation energies can be reproduced through n-body expansion, provided one pushes to the basis set and electron correlation limit for the two-body term; for the three-body term, post-MP2 contributions (particularly CCSD-MP2) are important for capturing the three-body dispersion effects. Terms beyond four-body can be adequately captured at the MP2-F12 level.

We present correlation consistent basis sets for explicitly correlated (F12) calculations, denoted VnZ(-PP)-F12-wis (n= D,T), for the d-block elements. The cc-pVDZ-F12-wis basis set is contracted to [8s7p5d2f] for the 3d-block, while its ECP counterpart for the 4dand 5d-blocks, cc-pVDZ-PP-F12-wis, is contracted to [6s6p5d2f]. The corresponding contracted sizes for cc-pVTZ(-PP)-F12-wis are [9s8p6d3f2g] for the 3d-block elements and [7s7p6d3f2g] for the 4dand 5d-block elements. Our VnZ(-PP)-F12-wis basis sets are evaluated on challenging test sets for metal–organic barrier heights (MOBH35) and group-11 metal clusters (CUAGAU-2). In F12 calculations, they are found to be about as close to the complete basis set limit as the combination of standard cc-pVnZ-F12 on main-group elementswith the standard aug-cc-pV(n+1)Z(-PP) basis sets on the transition metal(s). While our basis sets are somewhat more compact than aug-cc-pV(n+1)Z(-PP), the CPU time benefit is negligible for catalytic complexes that contain only one or two transition metals among dozens of main-group elements; however, it is somewhat more significant for metal clusters.

The relative energies of the YMPJ conformer database of the 20 proteinogenic amino acids, with N- and C-termination, have been re-evaluated using explicitly correlated coupled cluster methods. Lower-cost ab initiomethods such as MP2-F12 and CCSD-F12b actually are outperformed by double-hybrid DFT functionals; in particular, the DSD-PBEP86-NL double hybrid performs well enough to serve as a secondary standard. Among range-separated hybrids, ωB97X-V performs well, while B3LYP-D3BJ does surprisingly well among traditional DFT functionals. Treatment of dispersion is important for the DFT functionals; for the YMPJ set, D3BJ generally works as well as the NL nonlocal dispersion functional. Basis set sensitivity for DFT calculations on these conformers is weak enough that def2-TZVP is generally adequate. For conformer corrections to heats of formation, B3LYP-D3BJ and especially DSD-PBEP86-D3BJ or DSD-PBEP86-NL are adequate for all but the most exacting applications. The revised geometries and energetics for the YMPJ database have been made available as Supporting Information and should be useful in the parametrization and validation of molecular mechanics force fields and other low-cost methods. The very recent dRPA75 method yields good performance, without resorting to an empirical dispersion correction, but is still outperformed by DSD-PBEP86-D3BJ and particularly DSD-PBEP86-NL. Core–valence corrections are comparable in importance to improvements beyond CCSD(T*)/cc-pVDZ-F12 in the valence treatment.

The domain based local pair natural orbital coupled cluster method with single-, double-, and perturbative triple excitations (DLPNO–CCSD(T)) is an efficient quantum chemical method that allows for coupled cluster calculations on molecules with hundreds of atoms. Because coupled-cluster theory is the method of choice if high-accuracy is needed, DLPNO–CCSD(T) is very promising for large-scale chemical application. However, the various approximations that have to be introduced in order to reach near linear scaling also introduce limited deviations from the canonical results. In the present work, we investigate how far the accuracy of the DLPNO–CCSD(T) method can be pushed for chemical applications. We also address the question at which additional computational cost improvements, relative to the previously established default scheme, come. To answer these questions, a series of benchmark sets covering a broad range of quantum chemical applications including reaction energies, hydrogen bonds, and other noncovalent interactions, conformer energies, and a prototype organometallic problem were selected. An accuracy of 1 kcal/mol or better can readily be obtained for all data sets using the default truncation scheme, which corresponds to the stated goal of the original implementation. Tightening of the three thresholds that control DLPNO leads to mean absolute errors and standard deviations from the canonical results of less than 0.25 kcal/mol (<1 kJ/mol). The price one has then to pay is an increased computational time by a factor close to 3. The applicability of the method is shown to be independent of the nature of the reaction. On the basis of the careful analysis of the results, three different sets of truncation thresholds (termed “LoosePNO”, “NormalPNO”, and “TightPNO”) have been chosen for “black box” use of DLPNO–CCSD(T). This will allow users of the method to optimally balance performance and accuracy.

The basis set convergence of explicitly correlated ab initiomethods, when applied to noncovalent interactions, has been considered in the presence (and absence) of Boys–Bernardi counterpoise corrections, as well as using “half-counterpoise” (the average of raw and counterpoise-corrected values) as recently advocated in this journal [Burns, L. A.; Marshall, M. S.; Sherrill, C. D. J. Chem. Theory Comput.2014, 10, 49–57]. Reference results were obtained using basis sets so large that BSSE (basis set superposition error) can be shown to be negligible. For the HF+CABS component, full counterpoise unequivocally exhibits the fastest basis set convergence. However, at the MP2-F12 and CCSD(T*)-F12b levels, surprisingly good uncorrectedresults can be obtained with small basis sets like cc-pVDZ-F12, owing to error compensation between basis set superposition error (which overbinds) and intrinsic basis set insufficiency (which underbinds). For intermediate sets like cc-pVTZ-F12, “half–half” averages work best, while for large basis sets like cc-pVQZ-F12, full counterpoise may be preferred but BSSE in uncorrected values is tolerably small for most purposes. A composite scheme in which CCSD(T)–MP2 “high level corrections” obtained at the CCSD(T*)-F12b/cc-pVDZ-F12 level are combined with “half-counterpoise” MP2-F12/cc-pVTZ-F12 interaction energies yields surprisingly good performance for standard benchmark sets like S22 and S66.

We consider basis set convergence and the effect of various approximations to CCSD(T)-F12 for a representative sample of harmonic frequencies (the HFREQ2014 set). CCSD(T*)(F12*)/cc-pVDZ-F12 offers a particularly favorable compromise between accuracy and computational cost: its RMSD <3 cm–1from the valence CCSD(T) limit is actually less than the remaining discrepancy with the experimental value at the valence CCSD(T) limit (about 5 cm–1RMSD). CCSD(T)-F12a and CCSD(T)-F12b appear to benefit from error compensation between CCSD and (T).

Recent quantum chemical computations demonstrated the electron-acceptance behavior of this highly reactive cyclo[18]carbon (C18) ring with piperidine (pip). The C18–pip complexation exhibited a double-well potential along the N–C reaction coordinate, forming a van der Waals (vdW) adduct and a more stable, strong covalent/dative bond (DB) complex by overcoming a low activation barrier. By means of direct dynamical computations using canonical variational transition state theory (CVT), including the small-curvature tunneling (SCT), we show the conspicuous role of heavy atom quantum mechanical tunneling (QMT) in the transformation of vdW to DB complex in the solvent phase near absolute zero. Below 50 K, the reaction is entirely driven by QMT, while at 30 K, the QMT rate is too rapid (kT∼ 0.02 s–1), corresponding to a half-life time of 38 s, indicating that the vdW adduct will have a fleeting existence. We also explored the QMT rates of other cyclo[n]carbon–pip systems. This study sheds light on the decisive role of QMT in the covalent/DB formation of the C18–pip complex at cryogenic temperatures.

To better understand the thermochemical kinetics and mechanism of a specific chemical reaction, an accurate estimation of barrier heights (forward and reverse) and reaction energies is vital. Because of the large size of reactants and transition state structures involved in real-life mechanistic studies (e.g., enzymatically catalyzed reactions), density functional theory remains the workhorse for such calculations. In this paper, we have assessed the performance of 91 density functionals for modeling the reaction energies and barrier heights on a large and chemically diverse data set (BH9) composed of 449 organic chemistry reactions. We have shown that range-separated hybrid functionals perform better than the global hybrids for BH9 barrier heights and reaction energies. Except for the PBE-based range-separated nonempirical double hybrids, range separation of the exchange term helps improve the performance for barrier heights and reaction energies. The 16-parameter Berkeley double hybrid, ωB97M(2), performs remarkably well for both properties. However, our minimally empirical range-separated double hybrid functionals offer marginally better accuracy than ωB97M(2) for BH9 barrier heights and reaction energies.

We carried out an extensive survey of wave function and DFT methods to test their accuracy on geometries and dissociation energies of halogen bonds (XB). For that purpose, we built two benchmark sets (XB18 and XB51). Between the DFT methods, it was found that functionals with high exact exchange or long-range corrections were suitable for these dimers, especially M06-2X, ωB97XD, and double hybrids. Dispersion corrections tend to be detrimental, in spite of the fact that XB is considered a noncovalent interaction. Wave function techniques require heavy correlated methods (i.e., CCSD(T)) or parametrized ones (SCS-MP2 or SCS(MI)MP2). Heavy basis sets are needed to obtain high accuracy, such as aVQZ or aVTZ+CP, and ideally a CBS extrapolation. Relativistic ECPs are also important, even for the bromine based dimers. In addition, we explored some XB with new theoretical tools, the NCI (“Non-Covalent Interactions”) method and the NOFF (“Natural Orbital Fukui Functions”).

Radical stabilization energies (RSE)s have been calculated for a variety of boryl radicals complexed to Lewis bases at the G3(MP2)‐RAD level of theory. These are referenced to the BH bond dissociation energy (BDE) in BH3determined at W4.3 level. High RSE values (and thus low BDE(BH) values) have been found for borane complexes of a variety of five‐ and six‐membered ring heterocycles. Variations of RSE values have been correlated with the strength of Lewis acid–Lewis base complex formation at the boryl radical stage. The analysis of charge‐ and spin‐density distributions shows that spin delocalization in the boryl radical complexes constitutes one of the mechanisms of radical stabilization.

A polyoxometalate of the Keggin structure substituted with RuIII, 6Q5[RuIII(H2O)SiW11O39] in which 6Q=(C6H13)4N+, catalyzed the photoreduction of CO2to CO with tertiary amines, preferentially Et3N, as reducing agents. A study of the coordination of CO2to 6Q5[RuIII(H2O)SiW11O39] showed that 1) upon addition of CO2the UV/Vis spectrum changed, 2) a rhombic signal was obtained in the EPR spectrum (gx=2.146, gy=2.100, and gz=1.935), and 3) the 13C NMR spectrum had a broadened peak of bound CO2at 105.78 ppm (Δ1/2=122 Hz). It was concluded that CO2coordinates to the RuIIIactive site in both the presence and absence of Et3N to yield 6Q5[RuIII(CO2)SiW11O39]. Electrochemical measurements showed the reduction of RuIIIto RuIIin 6Q5[RuIII(CO2)SiW11O39] at −0.31 V versus SCE, but no such reduction was observed for 6Q5[RuIII(H2O)SiW11O39]. DFT‐calculated geometries optimized at the M06/PC1//PBE/AUG‐PC1//PBE/PC1‐DF level of theory showed that CO2is preferably coordinated in a side‐on manner to RuIIIin the polyoxometalate through formation of a RuO bond, further stabilized by the interaction of the electrophilic carbon atom of CO2to an oxygen atom of the polyoxometalate. The end‐on CO2bonding to RuIIIis energetically less favorable but CO2is considerably bent, thus favoring nucleophilic attack at the carbon atom and thereby stabilizing the carbon sp2hybridization state. Formation of a O2C–NMe3zwitterion, in turn, causes bending of CO2and enhances the carbon sp2hybridization. The synergetic effect of these two interactions stabilizes both Ru–O and C–N interactions and probably determines the promotional effect of an amine on the activation of CO2by [RuIII(H2O)SiW11O39]5−. Electronic structure analysis showed that the polyoxometalate takes part in the activation of both CO2and Et3N. A mechanistic pathway for photoreduction of CO2is suggested based on the experimental and computed results.

Sequential addition of CO molecules to cationic aryl–hydrido RhIIIcomplexes of phosphine‐based (PCP) pincer ligands was found to lead first to CH reductive elimination and then to CH oxidative addition, thereby demonstrating a dual role of CO. DFT calculations indicate that the oxidative addition reaction is directly promoted by CO, in contrast to the commonly accepted view that CO hinders such reactions. This intriguing effect was traced to repulsive π interactions along the aryl‐Rh‐CO axis, which are augmented by the initially added CO ligand (due to antibonding interactions between occupied Rh dπorbitals and occupied π orbitals of both CO and the arene moiety), but counteracted by the second CO ligand (due to significant π back‐donation). These repulsive interactions were themselves linked to significant weakening of the π‐acceptor character of CO in the positively charged rhodium complexes, which is concurrent with an enhanced σ‐donating capability. Replacement of the phosphine ligands by an analogous phosphinite‐based (POCOP) pincer ligand led to significant changes in reactivity, whereby addition of CO did not result in CH reductive elimination, but yielded relatively stable mono‐ and dicarbonyl aryl–hydrido POCOP–RhIIIcomplexes. DFT calculations showed that the stability of these complexes arises from the higher electrophilicity of the POCOP ligand, relative to PCP, which leads to partial reduction of the excessive π‐electron density along the aryl‐Rh‐CO axis. Finally, comparison between the effects of CO and acetonitrile on CH oxidative addition revealed that they exhibit similar reactivity, despite their markedly different electronic properties. However, DFT calculations indicate that the two ligands operate by different mechanisms.

Unrestricted coupled cluster spin contamination corrected [UCCSD(T)] and unrestricted Brueckner doubles [UBD(T)] variations of the Weizmann-1 theory (W1), denoted as W1U, W1Usc, and W1BD, respectively, are compared with the restricted open-shell W1 theory [W1(RO)]. The performances of the four W1 variants are assessed with 220 total atomization energies, electron affinities, ionization potentials, and proton affinities in the G2/97 test set, for consistency with the error analysis of the original W1(RO) study. The root-mean-square deviations from the experiment of W1U (0.65 ± 0.48 kcal/mol), W1Usc (0.57 ± 0.48 kcal/mol), W1BD (0.62 ± 0.48 kcal/mol), and W1(RO) (0.57 ± 0.48 kcal/mol) show that the four methods are virtually indistinguishable. This error analysis excludes the “singlet biradicals,” C2and O3, since single determinantal methods are not really adequate for these strongly multireference systems. The unrestricted W1 variants perform poorly for such highly spin-contaminated and multireference species (the largest deviation from experiment for W1Usc is −4.2 ± 0.1 kcal/mol for the O3EA). W1(RO) performs much better than its unrestricted counterparts for these pathological cases (the deviation from experiment is reduced to −1.5 ± 0.1 kcal/mol for the O3EA), though the errors are significantly larger than those for the overall test set. The examples of C2, O3, and the F2potential energy curve indicate that an advantage to using W1BD is that the error in ⟨S2⟩ correlates with the magnitude of the error in energy, whereas W1(RO) loses accuracy without such a warning.

Mission control: η2‐Coordination of zero‐valent platinum is followed by metal insertion into a carbonhalide bond of a stilbene‐based substrate. The overall reactivity trend is opposite from those of other aryl halides and is controlled by the coordination chemistry.

The crystal structure of the new cationic RhIcomplex trans‐[Rh(CO)2(L)2]BF4(L=α2‐(diisopropylphosphino)isodurene) was found to exhibit a nonlinear OCRhCO fragment and weak intramolecular CH⋅⋅⋅Rh interactions. These interactions, which have also been shown to occur in solution, have been examined by density functional theory calculations and found to be inextricably linked to the presence of the distorted OCRhCO fragment. This linkage has also been demonstrated by comparison with a highly similar RhIcomplex, in which these CH⋅⋅⋅Rh interactions are absent. Furthermore, the presence of these weak interactions has been shown to have a significant effect on the reactivity of the metal center.

Both experimental and theoretical evidence suggest that the proton exchange between water and the methyl group in [TpPt(CO)CH3] (1, Tp=hydridotripyrazolylborate) involves the formation and deprotonation of a “sticky” σ‐methane ligand. The efficiency of this nontrivial process has been attributed to the spatial orientation of functional groups that operate in concert to activate a water molecule and then achieve a multistep proton walk from water to an uncoordinated pyrazolyl nitrogen atom, to the methyl ligand, and then back to the nitrogen atom and water. The overall proton‐exchange process has been proposed to involve an initial attack of water at the CO ligand in 1with concerted deprotonation by the uncoordinated pyrazolyl nitrogen atom. The pyrazolium proton is then transferred to the PtCH3bond, leading to a σ‐methane intermediate. Subsequent rotation and deprotonation of the σ‐methane ligand, followed by reformation of 1and water, result in scrambling of the methyl protons with the hydrogen atoms of water. An alternative two‐step process that involves oxidative addition and reductive elimination has also been considered. The two competing mechanistic routes from 1into [D3]‐1, as well as the conversion of 1into [TpPt(CH3)H2] (2), have been examined by density functional theory (DFT) using a variety of exchange‐correlation methods, primarily PW6B95, which was recently shown to be highly accurate for evaluating reactions of late‐transition‐metal complexes. The key role played by the free pyrazolyl nitrogen atom, acting as a proton carrier that abstracts a proton from water and transfers the proton to the PtCH3bond, is reminiscent of the dual functionality of histidine in the catalytic triad of natural serine proteases.

The novel π‐accepting, pincer‐type ligand, dipyrrolylphoshinoxylene (DPyPX), is introduced. This ligand has the strongest π‐accepting phosphines used so far in the PCP family of ligands and this results in some unusual coordination chemistry. The rhodium(I) complex, [(DPyPX)Rh(CO)(PR3)] (4, R=Ph, Et, pyrrolyl) is prepared by treating the relevant [(DPyPX)Rh(PR3)] (3) complex with CO and is remarkably resistant to loss of either ligand. X‐ray crystallographic analysis of complex 4 b(R=Et) reveals an unusual cisoidcoordination of the PCP phosphine ligands. These observations are supported by density functional theory (DFT) calculations.

The Lewis acid-base reaction between Sn(CH3)3X and Y⁻ (with X, Y = F, Cl, Br and I) has been studied using quantum chemical calculations. Complexation energies were calculated at the Density Functional Theory (DFT) level and rationalized on the basis of a local application of the hard and soft acids and bases principle. It was observed that smaller differences in the local softness of the interacting sites in the Lewis acid and base correspond to stronger interactions. Moreover, the calculated sequences in complexation energies can be reproduced using equations containing chemical concepts introduced within the framework of conceptual density functional theory and rooted in the hard and soft acids and bases principle and referring only to the reactants. A method of treating the electronegativity and softness of the halide anions is presented based on a Taylor expansion of the electronegativity of the neutral halogens and the softness-polarizability proportionality. Experimental evidence for the calculated sequences was gathered from measured ¹¹⁷Sn chemical shifts and ¹J (¹³C-^119/117Sn) coupling constant changes upon complexation. (© Wiley-VCH Verlag GmbH & Co. KGaA, 69451 Weinheim, Germany, 2003)

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A long‐standing controversy concerning the heat of formation of methylenimine has been addressed by means of the W2 (Weizmann‐2) thermochemical approach. Our best calculated values, ΔH°f,298(CH2NH) = 21.1±0.5 kcal/mol and ΔH°f,298(CH2NH2+) = 179.4±0.5 kcal/mol, are in good agreement with the most recent measurements but carry a much smaller uncertainty. As a byproduct, we obtain the first‐ever accurate anharmonic force field for methylenimine: upon consideration of the appropriate resonances, the experimental gas‐phase band origins are all reproduced to better than 10 cm−1. Consideration of the difference between a fully anharmonic zero‐point vibrational energy and B3LYP/cc‐pVTZ harmonic frequencies scaled by 0.985 suggests that the calculation of anharmonic zero‐point vibrational energies can generally be dispensed with, even in benchmark work, for rigid molecules. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1297–1305, 2001

In this theoretical study on the Heck reaction we explore the feasibility of an alternative pathway that involves a PdII/PdIVredox system. Usually, the catalytic cycle is formulated based on a Pd0/PdIImechanism. We performed quantum chemical calculations using density functional theory on a model system that consisted of diphosphinoethane (DPE) as a bidentate ligand and the substrates ethylene and phenyl iodide to compare both mechanisms. Accordingly, the PdII/PdIVmechanism is most likely to occur in the equatorial plane of an octahedral PdIVcomplex. The energy profiles of both reaction pathways under consideration are largely parallel. A major difference is found for the oxidative addition of the C−I bond to the palladium centre. This is a rate‐determining step of the PdII/PdIVmechanism, while it is facile for a Pd0catalyst. The calculations show that intermediate ligand detachment and re‐attachment is necessary in the course of the oxidative addition to PdII. Therefore, we expect the PdII/PdIVmechanism to be only feasible if a weakly coordinating ligand is present.

The energetics of isomerization between the azulene and naphthalene radical cations have been investigated using the hybrid density functional method B3LYP with the cc-pVDZ basis sets. CCSD/cc-pVDZ energy calculations were also carried out for selected points along the reaction coordinate. The transition state barrier energies for isomerization are lower than the dissociation limit of C₈H₆^&z.rad;+ (benzocyclobutadiene^&z.rad;+) + C₂H₂ deduced earlier. A key intermediate is a hydrogen shifted naphthalene isomer analogous to the intermediate suggested in the Dewar–Becker isomerization mechanism for neutral azulene. The norcaradiene isomer of the Dewar–Becker mechanism was found to be a transition structure in the ionic system. Results of the present density functional theory (DFT) and coupled cluster calculations are discussed in the light of recent experimental evidence.

Abstract.: The coupling between improvement of the basis set and the valence electron correlation method has been studied quantitatively for the total atomization energies (TAEs) of a number of small molecules, using basis sets of up to [7s6p5d4f3g2h/5s4p3d2f1g] quality. Very significant coupling is found to exist. Using a scaled basis set extrapolation beyond [6s5p4d3f2g/ 4s3p2d1f] at the MP2 or CCSD level, mean absolute errors of 0.18 and 0.15 kcal/mol, respectively, can be obtained for the TAEs of a number of small polyatomic molecules, compared to 0.12 kcal/mol using CCSD(T) throughout.

DFT calculations were used to reveal the unexpected reactivity and mechanism of the addition of acetone to metallabenzene, metallapyrylium and metallathiabenzene complexes of iridium.

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