Optimized scale factors for calculating vibrational harmonic and fundamental frequencies and zero-point energies have been determined for 145 electronic model chemistries, including 119 based on approximate functionals depending on occupied orbitals, 19 based on single-level wave function theory, three based on the neglect-of-diatomic-differential-overlap, two based on doubly hybrid density functional theory, and two based on multicoefficient correlation methods. Forty of the scale factors are obtained from large databases, which are also used to derive two universal scale factor ratios that can be used to interconvert between scale factors optimized for various properties, enabling the derivation of three key scale factors at the effort of optimizing only one of them. A reduced scale factor optimization model is formulated in order to further reduce the cost of optimizing scale factors, and the reduced model is illustrated by using it to obtain 105 additional scale factors. Using root-mean-square errors from the values in the large databases, we find that scaling reduces errors in zero-point energies by a factor of 2.3 and errors in fundamental vibrational frequencies by a factor of 3.0, but it reduces errors in harmonic vibrational frequencies by only a factor of 1.3. It is shown that, upon scaling, the balanced multicoefficient correlation method based on coupled cluster theory with single and double excitations (BMC-CCSD) can lead to very accurate predictions of vibrational frequencies. With a polarized, minimally augmented basis set, the density functionals with zero-point energy scale factors closest to unity are MPWLYP1M (1.009), τHCTHhyb (0.989), BB95 (1.012), BLYP (1.013), BP86 (1.014), B3LYP (0.986), MPW3LYP (0.986), and VSXC (0.986).

The diverse barrier height database DBH24 is updated by using W4 and W3.2 data (Karton, A.; Tarnopolsky, A.; Lamère, J.-F.; Schatz, G. C.; Martin, J. M. L. J. Phys. Chem. A 2008, 112, 12868) to replace previous W1 values; we call the new database DBH24/08. We used the new database to assess 348 model chemistries, each consisting of a combination of a wave function theory level or a density functional approximation with a one-electron basis set. All assessments are made by simultaneous consideration of accuracy and cost. The assessment includes several electronic structure methods and basis sets that have not previously been systematically tested for barrier heights. Some conclusions drawn in our previous work (Zheng, J.; Zhao, Y.; Truhlar, D. G. J. Chem. Theory Comput. 2007, 3, 569) are still valid when using this improved database and including more model chemistries. For example, BMC-CCSD is again found to be the best method whose cost scales as N(6), and its cost is an order of magnitude smaller than the N(7) method with best performance-to-cost ratio, G3SX(MP3), although the mean unsigned error is only marginally higher, namely 0.70 kcal/mol vs 0.57 kcal/mol. Other conclusions are now broader in scope. For example, among single-reference N(5) methods (that is, excluding MRMP2), we now conclude not only that doubly hybrid density functionals and multicoefficient extrapolated density functional methods perform better than second-order Møller-Plesset-type perturbation theory (MP2) but also that they perform better than any correlation-energy-scaled MP2 method. The most recommended hybrid density functionals, if functionals are judged only on the basis of barrier heights, are M08-SO, M06-2X, M08-HX, BB1K, BMK, PWB6K, MPW1K, BHandHLYP, and TPSS25B95. MOHLYP and HCTH are found to be the best performing local density functionals for barrier heights. The basis set cc-pVTZ+ is more efficient than aug-cc-pVTZ with similar accuracy, especially for density functional theory. The basis sets cc-pVDZ+, 6-31+G(d,p), 6-31B(d,p), 6-31B(d), MIDIY+, MIDIX+, and MIDI! are recommended for double-ζ-quality density functional calculations on large systems for their good balance between accuracy and cost, and the basis sets cc-pVTZ+, MG3S, MG3SXP, and aug-cc-pVDZ are recommended for density functional calculations when larger basis sets are affordable. The best performance of any methods tested is attained by CCSD(T)(full)/aug-cc-pCV(T+d)Z with a mean unsigned error of 0.46 kcal/mol; however, this is several orders of magnitude more expensive than M08-SO/cc-pVTZ+, which has a mean unsigned error of only 0.90 kcal/mol.

We present a perspective on the use of diffuse basis functions for electronic structure calculations by density functional theory and wave function theory. We especially emphasize minimally augmented basis sets and calendar basis sets. We base our conclusions on our previous experience with commonly computed quantities, such as bond energies, barrier heights, electron affinities, noncovalent (van der Waals and hydrogen bond) interaction energies, and ionization potentials, on Stephens et al.'s results for optical rotation and on our own new calculations (presented here) of polarizabilities and of potential energy curves of van der Waals complexes. We emphasize the benefits of partial augmentation of the higher-zeta basis sets in preference to full augmentation at a lower ζ level. Benefits and limitations of the use of fully, partially, and minimally augmented basis sets are reviewed for different electronic structure methods and molecular properties. We have found that minimal augmentation is almost always enough for density functional theory (DFT) when applied to ionization potentials, electron affinities, atomization energies, barrier heights, and hydrogen-bond energies. For electric dipole polarizabilities, we find that augmentation beyond minimal has an average effect of 8% at the polarized triple-ζ level and 5% at the polarized quadruple-ζ level. The effects are larger for potential energy curves of van der Waals complexes. The effects are also larger for wave function theory (WFT). Even for WFT though, full augmentation is not needed for most purposes, and a level of augmentation between minimal and full is optimal for most problems. The calendar basis sets named after the months provide a convergent sequence of partially augmented basis sets that can be used for such calculations. The jun-cc-pV(T+d)Z basis set is very useful for MP2-F12 calculations of barrier heights and hydrogen bond strengths.

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