We investigate palladium-induced activation of the C-H, C-C, C-F, and C-Cl bonds in methane, ethane, cyclopropane, fluoromethane, and chloromethane, using relativistic density functional theory (DFT) at ZORA-BLYP/TZ2P. Our purpose is to arrive at a qualitative understanding, based on accurate calculations, of the trends in activation barriers and transition state (TS) geometries (e.g. early or late along the reaction coordinate) in terms of the reactants' properties. To this end, we extend the activation strain model (in which the activation energy Delta E(not equal) is decomposed into the activation strain Delta E(not equal)(strain) of the reactants and the stabilizing TS interaction Delta E(not equal)(int) between the reactants) from a single-point analysis of the TS to an analysis along the reaction coordinate zeta, that is, Delta E(zeta)=Delta E(strain)(zeta)+Delta E(int)(zeta). This extension enables us to understand qualitatively, trends in the position of the TS along zeta and, therefore, the values of the activation strain Delta E(not equal)(strain)=Delta E(strain)(zeta(TS)) and TS interaction Delta E(not equal)(int)=Delta E(int)(zeta(TS)) and trends therein. An interesting insight that emerges is that the much higher barrier of metal-mediated C-C versus C-H activation originates from steric shielding of the C-C bond in ethane by C-H bonds. Thus, before a favorable stabilizing interaction with the C-C bond can occur, the C-H bonds must be bent away, which causes the metal-substrate interaction Delta E(int)(zeta) in C-C activation to lag behind. Such steric shielding is not present in the metal-mediated activation of the C-H bond, which is always accessible from the hydrogen side. Other phenomena that are addressed are anion assistance, competition between direct oxidative insertion (OxIn) versus the alternative S(N)2 pathway, and the effect of ring strain.

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We have computed a state-of-the-art benchmark potential energy surface (PES) for the archetypal oxidative addition of the ethane C-C bond to the palladium atom and have used this to evaluate the performance of 24 popular density functionals, covering LDA, GGA, meta-GGA, and hybrid density functionals, for describing this reaction. The ab initio benchmark is obtained by exploring the PES using a hierarchical series of ab initio methods [HF, MP2, CCSD, CCSD(T)] in combination with a hierarchical series of five Gaussian-type basis sets, up to g polarization. Relativistic effects are taken into account either through a relativistic effective core potential for palladium or through a full four-component all-electron approach. Our best estimate of kinetic and thermodynamic parameters is -10.8 (-11.3) kcal/mol for the formation of the reactant complex, 19.4 (17.1) kcal/mol for the activation energy relative to the separate reactants, and -4.5 (-6.8) kcal/mol for the reaction energy (zero-point vibrational energy-corrected values in parentheses). Our work highlights the importance of sufficient higher angular momentum polarization functions for correctly describing metal-d-electron correlation. Best overall agreement with our ab initio benchmark is obtained by functionals from all three categories, GGA, meta-GGA, and hybrid DFT, with mean absolute errors of 1.5 to 2.5 kcal/mol and errors in activation energies ranging from -0.2 to -3.2 kcal/mol. Interestingly, the well-known BLYP functional compares very reasonably with a slight underestimation of the overall barrier by -0.9 kcal/mol. For comparison, with B3LYP we arrive at an overestimation of the overall barrier by 5.8 kcal/mol. On the other hand, B3LYP performs excellently for the central barrier (i.e., relative to the reactant complex) which it underestimates by only -0.1 kcal/mol.

To obtain a state-of-the-art benchmark potential energy surface ͑PES͒ for the archetypal oxidative addition of the methane C-H bond to the palladium atom, we have explored this PES using a hierarchical series of ab initio methods "Hartree-Fock, second-order Møller-Plesset perturbation theory, fourth-order Møller-Plesset perturbation theory with single, double and quadruple excitations, coupled cluster theory with single and double excitations ͑CCSD͒, and with triple excitations treated perturbatively ͓CCSD͑T͔͒… and hybrid density functional theory using the B3LYP functional, in combination with a hierarchical series of ten Gaussian-type basis sets, up to g polarization. Relativistic effects are taken into account either through a relativistic effective core potential for palladium or through a full four-component all-electron approach. Counterpoise corrected relative energies of stationary points are converged to within 0.1-0.2 kcal/mol as a function of the basis-set size. Our best estimate of kinetic and thermodynamic parameters is Ϫ8.1 ͑Ϫ8.3͒ kcal/mol for the formation of the reactant complex, 5.8 ͑3.1͒ kcal/mol for the activation energy relative to the separate reactants, and 0.8 ͑Ϫ1.2͒ kcal/mol for the reaction energy ͑zero-point vibrational energy-corrected values in parentheses͒. This agrees well with available experimental data. Our work highlights the importance of sufficient higher angular momentum polarization functions, f and g, for correctly describing metal-d-electron correlation and, thus, for obtaining reliable relative energies. We show that standard basis sets, such as LANL2DZϩ1 f for palladium, are not sufficiently polarized for this purpose and lead to erroneous CCSD͑T͒ results. B3LYP is associated with smaller basis set superposition errors and shows faster convergence with basis-set size but yields relative energies ͑in particular, a reaction barrier͒ that are ca. 3.5 kcal/mol higher than the corresponding CCSD͑T͒ values.

We have computed a state-of-the-art benchmark potential energy surface (PES) for the archetypal oxidative addition of the chloromethane C-Cl bond to the palladium atom and have used this to evaluate the performance of 26 popular density functionals, covering LDA, GGA, meta-GGA, and hybrid density functionals, for describing this reaction. The ab initio benchmark is obtained by exploring the PES using a hierarchical series of ab initio methods [HF, MP2, CCSD, and CCSD(T)] in combination with a hierarchical series of seven Gaussian-type basis sets, up to g polarization. Relativistic effects are taken into account through a full four-component all-electron approach. Our best estimate of kinetic and thermodynamic parameters is -11.2 (-10.8) kcal/mol for the formation of the most stable reactant complex, 3.8 (2.7) kcal/mol for the activation energy of direct oxidative insertion (OxIn), and -28.0 (-28.8) kcal/mol for the reaction energy (all energies relative to separate reactants, zero-point vibra-tional energy-corrected values in parentheses). Our work highlights the importance of sufficient higher angular momentum polarization functions for correctly describing metal-d-electron correlation. The best overall agreement with our ab initio benchmark is obtained by functionals from all three categories, GGA, meta-GGA, and hybrid DFT, with mean absolute errors of 0.8-3.0 kcal/mol and errors in activation energies for OxIn ranging from 0.0 to 1.2 kcal/mol. For example, three well-known functionals, BLYP, OLYP, and B3LYP, compare very reasonably with, respectively, an underestimation of the barrier for OxIn of -4.2 kcal/mol and overestimations of 4.2 and 1.6 kcal/mol. Interestingly, all important features of the CCSD(T) benchmark potential energy surfaces for the Pd-induced activation of C-H, C-C, C-F, and C-Cl bonds are reproduced correctly within a few kcal/mol by BLYP, OLYP, and B3LYP, while at the same time, none of these functionals is the "best one" in each individual case. This follows from an overall comparison of the results of the present as well as previous studies.