Highly accurate first-principles benchmark data sets for the parametrization and validation of density functional and other approximate methods. Derivation of a robust, generally applicable, double-hybrid functional for thermochemistry and thermochemical kinetics

Obtaining Good Performance With Triple-ζ-Type Basis Sets in Double-Hybrid Density Functional Theory Procedures

A variety of combinations of B-LYP-based double-hybrid density functional theory (DHDFT) procedures and basis sets have been examined. A general observation is that the optimal combination of exchange contributions is in the proximity of 30% Becke 1988 (B88) exchange and 70% Hartree-Fock (HF) exchange, while for the correlation contributions, the use of independently optimized spin-component-scaled Møller-Plesset second-order perturbation theory (SCS-MP2) parameters (MP2OS and MP2SS) is beneficial. The triple-ζ Dunning aug'-cc-pVTZ+d and Pople 6-311+G(3df,2p)+d basis sets are found to be cost-effective for DHDFT methods. As a result, we have formulated the DuT-D3 DHDFT procedure, which employs the aug'-cc-pVTZ+d basis set and includes 30% B88 and 70% HF exchange energies, 59% LYP, 47% MP2OS, and 36% MP2SS correlation energies, and a D3 dispersion correction with the parameters s6 = 0.5, sr,6 = 1.569, and s8 = 0.35. Likewise, the PoT-D3 DHDFT procedure was formulated with the 6-311+G(3df,2p)+d basis set and has 32% B88 and 68% HF exchange energies, 63% LYP, 46% MP2OS, and 27% MP2SS correlation energies, and the D3 parameters s6 = 0.5, sr,6 = 1.569, and s8 = 0.30. Testing using the large E3 set of 740 energies demonstrates the robustness of these methods. Further comparisons show that the performance of these methods, particularly DuT-D3, compares favorably with the previously reported DSD-B-LYP and DSD-B-LYP-D3 methods used in conjunction with quadruple-ζ aug'-pc3+d and aug'-def2-QZVP basis sets but at lower computational expense. The previously reported ωB97X-(LP)/6-311++G(3df,3pd) procedure also performs very well. Our findings highlight the cost-effectiveness of appropriate- and moderate-sized triple-ζ basis sets in the application of DHDFT procedures.

The Accuracy of Density Functional Theory in the Description of Cation-π and π-Hydrogen Bond Interactions

Cation-π and π-hydrogen bond interactions are ubiquitous in protein folding, molecular recognition, and ligand-receptor associations. As such systems are routinely studied at the DFT level, it becomes essential to understand the underlying accuracy of the plethora of density functionals currently available for the description of these interactions. For that purpose, we carried out theoretical calculations on two small model systems (benzene-Na(+) and benzene-H2O) that represent a paradigm for those intermolecular interactions and systematically tested 46 density functionals against the results of high-level post-HF methods, ranging from MP2 to extrapolated CCSD(T)/CBS. A total of 13 basis sets were also tested to examine the convergence of the interaction energy with basis set size. The convergence was surprisingly fast, with deviations below 0.2 kcal/mol for double-ζ polarized basis sets with diffuse functions. Concerning functional benchmarking, the Truhlar group functionals were particularly well suited for the description of the π-hydrogen bond interactions. In the case of cation-π interactions, there was not a clear correlation between accuracy and functional sophistication. Despite the large number of functionals predicting interaction energies within chemical accuracy (five for π-hydrogen bond and 20 for cation-π interactions), not a single functional has shown chemical accuracy in both cases. Moreover, if we calculate the average error for these two interactions, only two density functionals resulted in an average error below 1.0 kcal/mol (M06 and HCTH, with average errors of 0.6 and 0.8 kcal/mol). The obtained results serve as a guide for future computer simulations on this kind of system.

Harmonic vibrational frequencies: scale factors for pure, hybrid, hybrid meta, and double-hybrid functionals in conjunction with correlation consistent basis sets

Scale factors for (a) low (<1000 cm(-1)) and high harmonic vibrational frequencies, (b) thermal contributions to enthalpy and entropy, and (c) zero-point vibrational energies have been determined for five hybrid functionals (B3P86, B3PW91, PBE1PBE, BH&HLYP, MPW1K), five pure functionals (BLYP, BPW91, PBEPBE, HCTH93, and BP86), four hybrid meta functionals (M05, M05-2X, M06, and M06-2X) and one double-hybrid functional (B2GP-PLYP) in combination with the correlation consistent basis sets [cc-pVnZ and aug-cc-pVnZ, n = D(2),T(3),Q(4)]. Calculations for vibrational frequencies were carried out on 41 organic molecules and an additional set of 22 small molecules was used for the zero-point vibrational energy scale factors. Before scaling, approximately 25% of the calculated frequencies were within 3% of experimental frequencies. Upon application of the derived scale factors, nearly 90% of the calculated frequencies deviated less than 3% from the experimental frequencies for all of the functionals when the augmented correlation consistent basis sets were used.

From short conjugated oligomers to conjugated polymers. Lessons from studies on long conjugated oligomers

Given their utility in a variety of electronic devices, conjugated oligomers and polymers have attracted considerable research interest in recent years. Because polymeric materials consist of very large molecules with a range of molecular weights (that is, they are polydisperse), predicting their electronic properties is a complicated task. Accordingly, their properties are typically estimated by extrapolation of oligomeric properties to infinite chain lengths. In this Account, we discuss the convergence behavior of various electronic properties of conjugated oligomers, often using thiophene oligomers as a representative example. We have observed some general trends in our studies, which we briefly summarize below for five properties. Most of the calculated values are method dependent: the absolute values can be strongly dependent on the computational level used. Band Gap. The generally accepted approximation used to estimate polymer band gap, whereby a plot of HOMO-LUMO gap versus 1/n (where n is the number of monomer units) is extrapolated to infinite n, fails for long oligomers, because convergence behavior is observed for band gaps. At the B3LYP/6-31G(d) level, it is possible to extrapolate oligomer HOMO-LUMO gaps with a second-order polynomial equation. Alternatively, PBC/B3LYP/6-31G(d) is a very good method to reliably predict the band gap of conjugated polymers. Reorganization Energy. Values of the internal reorganization energy (λ) do not scale linearly with 1/n, instead exhibiting an inverse correlation with the square-root of the number of monomer units for n = 2-12. For larger n (10-50), a linear relationship is observed between reorganization energy and the reciprocal chain length, and the extrapolation approaches λ ≈ 0 for infinite numbers of oligomer rings. Ionization Potential. The relationship between the first adiabatic ionization potential IP(1a) of oligothiophenes and oligoselenophenes and chain length linearly correlates with an empirically obtained value of 1/(n(0.75)). The first vertical ionization potential (IP(1v)) linearly correlates with a similarly empirically obtained value of 1/(n(0.70)). Polaron-Bipolaron Balance. The contribution of a polaron pair to the electronic structure of the short oligothiophene dication is small; for medium-length oligothiophene chains, the contribution from the polaron pair state begins to become significant. For longer (above 20-mer) oligothiophenes, the polaron pair state dominates. A similar picture was observed for multications as well as doped oligomers and polymers. The qualitative polaron-bipolaron picture does not change when a dopant is introduced; however, quantitatively, the bipolaron-polaron pair equilibrium shifts toward the bipolaron state. Disproportionation Energy. The stability of a single oligothiophene dication versus two cation radical oligothiophene molecules increases with increasing chain length, and there is an excellent correlation between the relative disproportionation energy and the inverse of chain length. A similar trend is observed in the disproportionation energies of oligothiophene polycations as well as doped oligomer and polymers. We also examine doped oligothiophenes (with explicitly included counterions) and polymers with a repeating polar unit. From our experience, it is clear that different properties converge in different ways, and long oligomers (having about 50 double bonds in the backbone) must often be used to correctly extrapolate polymer properties.

O - H Bond Dissociation Energies

High-level composite, ab initio and density functional theory (DFT) procedures have been employed to study O–H bond dissociation energies (BDEs), as well as radical stabilization energies (RSEs) in the oxygen-centred radicals that are formed in the dissociation of the O–H bonds. Benchmark values are provided by Wn results up to W3.2 and W4.x. We are able to recommend revised BDE values for FO–H (415.6 ± 3 kJ mol–1), MeC(O)O–H (459.8 ± 6 kJ mol–1) and CF3CH2O–H (461.9 ± 6 kJ mol–1) on the basis of high-level calculations. We find that Gn-type procedures are generally reliable and cost-effective, and that some contemporary functionals and double-hybrid DFT procedures also provide adequate O–H BDEs/RSEs. We note that the variations in the O–H BDEs are associated with variations in the stabilities of not only the radicals but also the closed-shell precursor molecules. Most substituents destabilize both species, with σ-electron-withdrawing groups having larger destabilizing effects, while π-electron acceptors are stabilizing. Although there is little correlation between the stabilizing/destabilizing effects of the substituents and the RSEs, we present some general patterns in the RSEs that emerge from the present study.

Efficient and Accurate Double-Hybrid-Meta-GGA Density Functionals-Evaluation with the Extended GMTKN30 Database for General Main Group Thermochemistry, Kinetics, and Noncovalent Interactions

We present an extended and improved version of our recently published database for general main group thermochemistry, kinetics, and noncovalent interactions [J. Chem. Theory Comput. 2010, 6, 107], which is dubbed GMTKN30. Furthermore, we suggest and investigate two new double-hybrid-meta-GGA density functionals called PTPSS-D3 and PWPB95-D3. PTPSS-D3 is based on reparameterized TPSS exchange and correlation contributions; PWPB95-D3 contains reparameterized PW exchange and B95 parts. Both functionals contain fixed amounts of 50% Fock-exchange. Furthermore, they include a spin-opposite scaled perturbative contribution and are combined with our latest atom-pairwise London-dispersion correction [J. Chem. Phys. 2010, 132, 154104]. When evaluated with the help of the Laplace transformation algorithm, both methods scale as N(4) with system size. The functionals are compared with the double hybrids B2PLYP-D3, B2GPPLYP-D3, DSD-BLYP-D3, and XYG3 for GMTKN30 with a quadruple-ζ basis set. PWPB95-D3 and DSD-BLYP-D3 are the best functionals in our study and turned out to be more robust than B2PLYP-D3 and XYG3. Furthermore, PWPB95-D3 is the least basis set dependent and the best functional at the triple-ζ level. For the example of transition metal carbonyls, it is shown that, mainly due to the lower amount of Fock-exchange, PWPB95-D3 and PTPSS-D3 are better applicable than the other double hybrids. Finally, we discuss in some detail the XYG3 functional [Proc. Nat. Acad. Sci. U.S.A. 2009, 106, 4963], which makes use of B3LYP orbitals and electron densities. We show that it is basically a highly nonlocal variant of B2PLYP and that its partially good performance is mainly due to a larger effective amount of perturbative correlation compared to other double hybrids. We finally recommend the PWPB95-D3 functional in general chemistry applications.

Carbondioxide rare-gas systems: sensitivity of basis sets and double-hybrid density functionals

This study emphasizes on the performance of six newly developed double-hybrid density functionals (DHDF) in explaining the potential energy curves of different carbondioxide rare-gas systems. The basis set sensitivity has also been explored with the use of three basis sets. Our results suggest that for lighter He/Ne-CO(2) complexes, proper choice of DHDF and basis set lead to results those matches exactly with earlier calculations and also with the experiment. On the other hand, for heavier Ar/Kr-CO(2) complexes although the equilibrium separation distance matches exactly with earlier observations, the interaction energy values lie far apart. The overall investigation emphasizes on the fact that one has to tune the methods and basis sets properly to achieve good and satisfactory results.

Probing substituent effects in aryl-aryl interactions using stereoselective Diels-Alder cycloadditions

Stereoselective Diels-Alder cycloadditions that probe substituent effects in aryl-aryl sandwich complexes were studied experimentally and theoretically. Computations on model systems demonstrate that the stereoselectivity in these reactions is mediated by differential pi-stacking interactions in competing transition states. This allows relative stacking free energies of substituted and unsubstituted sandwich complexes to be derived from measured product distributions. In contrast to gas-phase computations, dispersion effects do not appear to play a significant role in the substituent effects, in accord with previous experiments. The experimental pi-stacking free energies are shown to correlate well with Hammett sigma(m) constants (r = 0.96). These substituent constants primarily provide a measure of the inductive electron-donating and -withdrawing character of the substituents, not donation into or out of the benzene pi-system. The present experimental results are most readily explained using a recently proposed model of substituent effects in the benzene sandwich dimer in which the pi-system of the substituted benzene is relatively unimportant and substituent effects arise from direct through-space interactions. Specifically, these results are the first experiments to clearly show that OMe enhances these pi-stacking interactions, despite being a pi-electron donor. This is in conflict with popular models in which substituent effects in aryl-aryl interactions are modulated by polarization of the aryl pi-system.

Can DFT methods correctly and efficiently predict the coordination number of copper(I) complexes? A case study

The coordination number of various experimentally known Cu(I) compounds is studied using density functional theory. Various basis sets are tested, aiming to establish a reliable level for prediction of the coordination number of these and other Cu(I) complexes. It is found that most levels exhibit correct trends, namely, the bulkier ligands demonstrate larger preference for coordination of two ligands. Proper absolute values are obtained when dispersion corrections are also included in the calculations. It is concluded that the fairly small modified 6-31+G* basis set due to Pulay represents a good compromise between accuracy and efficiency, followed by Balabanov and Peterson's all-electron aug-cc-pVDZ basis set. The overall energy is decomposed into various components whose relative contribution to the overall tendency of forming a complex with a particular coordination is examined. It is shown that two opposing contributions play a major role: the interaction energy of the ligand being added and the deformation energy of the copper's coordination sphere prior to the ligand addition. The former being a stabilizing contribution, leads to higher coordination numbers while the later, a destabilizing contribution, is shown to favor lower coordination numbers.