A computational study of a catalytic cycle generates state energies (the E-representation), whereas experiments lead to rate constants (the k-representation). Based on transition state theory (TST), these are equivalent representations. Nevertheless, until recently, there has been no simple way to calculate the efficiency of a catalytic cycle, that is, its turnover frequency (TOF), from a theoretically obtained energy profile. In this Account, we introduce the energetic span model that enables one to evaluate TOFs in a straightforward manner and in affinity with the Curtin-Hammett principle. As shown herein, the model implies a change in our kinetic concepts. Analogous to Ohm's law, the catalytic chemical current (the TOF) can be defined by a chemical potential (independent of the mechanism) divided by a chemical resistance (dependent on the mechanism and the nature of the catalyst). This formulation is based on Eyring's TST and corresponds to a steady-state regime. In many catalytic cycles, only one transition state and one intermediate determine the TOF. We call them the TOF-determining transition state (TDTS) and the TOF-determining intermediate (TDI). These key states can be located, from among the many states available to a catalytic cycle, by assessing the degree of TOF control (X(TOF)); this last term resembles the structure-reactivity coefficient in classical physical organic chemistry. The TDTS-TDI energy difference and the reaction driving force define the energetic span (δE) of the cycle. Whenever the TDTS appears after the TDI, δE is the energy difference between these two states; when the opposite is true, we must also add the driving force to this difference. Having δE, the TOF is expressed simply in the Arrhenius-Eyring fashion, wherein δE serves as the apparent activation energy of the cycle. An important lesson from this model is that neither one transition state nor one reaction step possess all the kinetic information that determines the efficiency of a catalyst. Additionally, the TDI and TDTS are not necessarily the highest and lowest states, nor do they have to be adjoined as a single step. As such, we can conclude that a change in the conceptualization of catalytic cycles is in order: in catalysis, there are no rate-determining steps, but rather rate-determining states. We also include a study on the effect of reactant and product concentrations. In the energetic span approximation, only the reactants or products that are located between the TDI and TDTS accelerate or inhibit the reaction. In this manner, the energetic span model creates a direct link between experimental quantities and theoretical results. The versatility of the energetic span model is demonstrated with several catalytic cycles of organometallic reactions.

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").

Spin-component scaled double hybrids including dispersion correction were optimized for many exchange and correlation functionals. Even DSD-LDA performs surprisingly well. DSD-PBEP86 emerged as a very accurate and robust method, approaching the accuracy of composite ab initio methods at a fraction of their computational cost.

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. Recently, the SCS concept was also extended to higher level post HF methods such as SCS-MP3 14,35,36 and SCS-CCSD. 37,38

The efficiency of catalytic cycles is measured by their turnover frequency (TOF). The degree of TOF control determines which states contribute most to the rate of the cycle, and thus indicates the steps that have the highest impact on the cycle. A kinetic model developed by Christiansen (Christiansen, J. A. Adv. Catal. 1953, 5, 311) for catalytic cycles is implemented here in a form that utilizes state energies. This enables one to assess the efficiency of quantum mechanically computed catalytic cycles like the palladium-catalyzed cross-coupling and Heck reactions, to test alternative hypotheses, and to make some predictions. This implementation can also account for effects such as Sabatier's volcano curve for heterogeneous catalysis. The model leads to a dependence of the TOF for any cycle on the "corrected" energy span quantity, deltaE, whose precise expression depends on the location of the summit and trough of the cycle in the step sequence of the cycle. Thus, knowing the highest energy transition state, the most abundant reaction intermediate, and the reaction energy enables one to make quick predictions about relative efficiency of cycles. At the same time, the degree of TOF control determines which states contribute most to the rate of reaction, and thus indicates the values to be included in the calculation of the energetic span and the steps that may be tinkered with to improve the cycle.

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Following up on an earlier preliminary communication (Kozuch and Martin, Phys. Chem. Chem. Phys. 2011, 13, 20104), we report here in detail on an extensive search for the most accurate spin-component-scaled double hybrid functionals [of which conventional double hybrids (DHs) are a special case]. Such fifth-rung functionals approach the performance of composite ab initio methods such as G3 theory at a fraction of their computational cost, and with analytical derivatives available. In this article, we provide a critical analysis of the variables and components that maximize the accuracy of DHs. These include the selection of the exchange and correlation functionals, the coefficients of each component [density functional theory (DFT), exact exchange, and perturbative correlation in both the same spin and opposite spin terms], and the addition of an ad-hoc dispersion correction; we have termed these parametrizations "DSD-DFT" (Dispersion corrected, Spin-component scaled, Double-hybrid DFT). Somewhat surprisingly, the quality of DSD-DFT is only mildly dependent on the underlying DFT exchange and correlation components, with even DSD-LDA yielding respectable performance. Simple, nonempirical GGAs appear to work best, whereas meta-GGAs offer no advantage (with the notable exception of B95c). The best correlation components appear to be, in that order, B95c, P86, and PBEc, while essentially any good GGA exchange yields nearly identical results. On further validation with a wider variety of thermochemical, weak interaction, kinetic, and spectroscopic benchmarks, we find that the best functionals are, roughly in that order, DSD-PBEhB95, DSD-PBEP86, DSD-PBEPW91, and DSD-PBEPBE. In addition, DSD-PBEP86 and DSD-PBEPBE can be used without source code modifications in a wider variety of electronic structure codes. Sample job decks for several commonly used such codes are supplied as electronic Supporting Information.

The S66x8 dataset for noncovalent interactions of biochemical relevance has been re-examined by means of MP2-F12 and CCSD(F12*)(T) methods. We deem our revised benchmark data to be reliable to about 0.05 kcal mol(-1) RMS. Most levels of DFT perform quite poorly in the absence of dispersion corrections: somewhat surprisingly, that is even the case for the double hybrids and for dRPA75. Analysis of optimized D3BJ parameters reveals that the main benefit of dRPA75 and DSD double hybrids alike is the treatment of midrange dispersion. dRPA75-D3BJ is the best performer overall at RMSD = 0.10 kcal mol(-1). The nonlocal VV10 dispersion functional is especially beneficial for the double hybrids, particularly in DSD-PBEP86-NL (RMSD = 0.12 kcal mol(-1)). Other recommended dispersion-corrected functionals with favorable price/performance ratios are ωB97X-V, and, surprisingly, B3LYP-D3BJ and BLYP-D3BJ (RMSDs of 0.23, 0.20 and 0.23 kcal mol(-1), respectively). Without dispersion correction (but parametrized for midrange interactions) M06-2X has the lead (RMSD = 0.45 kcal mol(-1)). A collection of three energy-based diagnostics yields similar information to an SAPT analysis about the nature of the noncovalent interaction. Two of those are the percentages of Hartree-Fock and of post-MP2 correlation effects in the interaction energy; the third, CSPI = [IE - IE]/[IE + IE] or its derived quantity DEBC = CSPI/(1 + CSPI(2))(1/2), describes the character of the MP2 correlation contribution, ranging from 0 (purely dispersion) to 1 (purely other effects). In addition, we propose an improved, parameter-free scaling for the (T) contribution based on the Ecorr[CCSD-F12b]/Ecorr[CCSD] and Ecorr[CCSD(F12*)]/Ecorr[CCSD] ratios. For Hartree-Fock and conventional DFT calculations, full counterpoise generally yields the fastest basis set convergence, while for double hybrids, half-counterpoise yields faster convergence, as previously established for correlated ab initio methods.