W4Λ: Leveraging Λ Coupled-Cluster for Accurate Computational Thermochemistry Approaches.

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)Λ and 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.

Correlation Consistent Basis Sets for Explicitly Correlated Theory: The Transition Metals.

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 4d and 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 4d and 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 elements with 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.

S66x8 noncovalent interactions revisited: new benchmark and performance of composite localized coupled-cluster methods.

The S66x8 noncovalent interactions benchmark has been re-evaluated at the "sterling silver" level, using explicitly correlated MP2-F12 near the complete basis set limit, CCSD(F12*)/aug-cc-pVTZ-F12, and a (T) correction from conventional CCSD(T)/sano-V{D,T}Z+ calculations. The revised reference values differ by 0.1 kcal mol-1 RMS from the original Hobza benchmark and its revision by Brauer et al., but by only 0.04 kcal mol-1 RMS from the "bronze" level data in Kesharwani et al., Aust. J. Chem., 2018, 71, 238-248. We then used these to assess the performance of localized-orbital coupled cluster approaches with and without counterpoise corrections, such as PNO-LCCSD(T) as implemented in MOLPRO, DLPNO-CCSD(T1) as implemented in ORCA, and LNO-CCSD(T) as implemented in MRCC, for their respective "Normal", "Tight", and "very Tight" settings. We also considered composite approaches combining different basis sets and cutoffs. Furthermore, in order to isolate basis set convergence from domain truncation error, for the aug-cc-pVTZ basis set we compared PNO, DLPNO, and LNO approaches with canonical CCSD(T). We conclude that LNO-CCSD(T) with veryTight criteria performs very well for "raw" (CP-uncorrected), but struggles to reproduce counterpoise-corrected numbers even for veryveryTight criteria: this means that accurate results can be obtained using either extrapolation from basis sets large enough to quench basis set superposition error (BSSE) such as aug-cc-pV{Q,5}Z, or using a composite scheme such as Tight{T,Q} + 1.11[vvTight(T) - Tight(T)]. In contrast, PNO-LCCSD(T) works best with counterpoise, while performance with and without counterpoise is comparable for DLPNO-CCSD(T1). Among more economical methods, the highest accuracies are seen for dRPA75-D3BJ, ωB97M-V, ωB97M(2), revDSD-PBEP86-D4, and DFT(SAPT) with a TDEXX or ATDEXX kernel.

Automatic generation of complementary auxiliary basis sets for explicitly correlated methods.

Explicitly correlated calculations, aside from the orbital basis set, typically require three auxiliary basis sets: Coulomb-exchange fitting (JK), resolution of the identity MP2 (RI-MP2), and complementary auxiliary basis set (CABS). If unavailable for the orbital basis set and chemical elements of interest, the first two can be auto-generated on the fly using existing algorithms, but not the third. In this paper, we present a quite simple algorithm named autoCABS; a Python implementation under a free software license is offered at Github. For the cc-pVnZ-F12 (n = D,T,Q,5), the W4-08 thermochemical benchmark, and the HFREQ2014 set of harmonic frequencies, we demonstrate that autoCABS-generated CABS basis sets are comparable in quality to purpose-optimized OptRI basis sets from the literature, and that the quality difference becomes entirely negligible as n increases.

The MOBH35 Metal-Organic Barrier Heights Reconsidered: Performance of Local-Orbital Coupled Cluster Approaches in Different Static Correlation Regimes.

We have revisited the MOBH35 (Metal-Organic Barrier Heights, 35 reactions) benchmark [Iron; , Janes, J. Phys. Chem. A, 2019, 123 (17), 3761-3781; ibid. 2019, 123, 6379-6380] for realistic organometallic catalytic reactions, using both canonical CCSD(T) and localized orbital approximations to it. For low levels of static correlation, all of DLPNO-CCSD(T), PNO-LCCSD(T), and LNO-CCSD(T) perform well; for moderately strong levels of static correlation, DLPNO-CCSD(T) and (T1) may break down catastrophically, and PNO-LCCSD(T) is vulnerable as well. In contrast, LNO-CCSD(T) converges smoothly to the canonical CCSD(T) answer with increasingly tight convergence settings. The only two reactions for which our revised MOBH35 reference values differ substantially from the original ones are reaction 9 and to a lesser extent 8, both involving iron. For the purpose of evaluating density functional theory (DFT) methods for MOBH35, it would be best to remove reaction 9 entirely as its severe level of static correlation makes it just too demanding for a test. The magnitude of the difference between DLPNO-CCSD(T) and DLPNO-CCSD(T1) is a reasonably good predictor for errors in DLPNO-CCSD(T1) compared to canonical CCSD(T); otherwise, monitoring all of T1, D1, max|tiA|, and 1/(εLUMO - εHOMO) should provide adequate warning for potential problems. Our conclusions are not specific to the def2-SVP basis set but are largely conserved for the larger def2-TZVPP, as they are for the smaller def2-SV(P): the latter may be an economical choice for calibrating against canonical CCSD(T). Finally, diagnostics for static correlation are statistically clustered into groups corresponding to (1) importance of single excitations in the wavefunction; (2a) the small band gap, weakly separated from (2b) correlation entropy; and (3) thermochemical importance of correlation energy, as well as the slope of the DFT reaction energy with respect to the percentage of HF exchange. Finally, a variable reduction analysis reveals that much information on the multireference character is provided by T1, IND/Itot, and the exchange-based diagnostic A100[TPSS].

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

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

Exploring Avenues beyond Revised DSD Functionals: II. Random-Phase Approximation and Scaled MP3 Corrections.

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.

Canonical and DLPNO-Based Composite Wavefunction Methods Parametrized against Large and Chemically Diverse Training Sets. 2: Correlation-Consistent Basis Sets, Core-Valence Correlation, and F12 Alternatives.

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.

Canonical and DLPNO-Based G4(MP2)XK-Inspired Composite Wave Function Methods Parametrized against Large and Chemically Diverse Training Sets: Are They More Accurate and/or Robust than Double-Hybrid DFT?

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.

Argo: a data analysis program for quantum chemical calculations.

The ARGO program is designed for the analysis and display of results obtained from quantum chemical calculations. ARGO currently extracts values of geometrical parameters and energy of each conformer obtained from potential energy surface (PES) scans. It automatically arranges the data into columns and prepares input files for the Gnuplot graphing utility for two- and three-dimensional PES graphs. In addition, it transforms Raman activities into Raman intensities and calculates the IR or Raman intensity ratios. This software, which was written in Python, is freely available, open-source, easy to modify, and pleasant to use. Extensive documentation is also provided with examples and suggestions for code modifications. We plan to create a collaborative program package aimed at delivering endless variety and possibilities for fast and accurate data analysis of quantum chemical calculations.