Benchmark calculations of chemical reactions in density functional theory: Comparison of the accurate Kohn–Sham solution with generalized gradient approximations for the H2+H and H2+H2 reactions

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Abstract
The Kohn–Sham (KS) solution is constructed from an accurate CI density and the KS exchange and correlationenergies E x and E c , as well as the corresponding exchange and exchange-correlation energy densities ε x ( r ) and ε xc ( r ), which are obtained for the hydrogen abstraction reaction H+H 2 and the symmetrical four-center exchange reaction H 2 +H 2 . The KS quantities are compared with those of the standard GGAs. Comparison shows that the GGA exchange functional represents both exchange and molecular nondynamical left–right correlation, while the GGA correlation functional represents only the dynamical part of the correlation. This role of the GGA exchange functional is especially important for the transition states (TS) of the reactions where the left–right correlation is enhanced. Standard GGAs tend to underestimate the barrier height for the reaction H+H 2 and to overestimate it for the reaction H 2 +H 2 . For H 2 +H 2 the Kohn–Sham orbital degeneracy in the square TS is represented with an equi-ensemble KS solution for both accurate KS/CI and GGA, while near the TS ensemble solutions with unequal occupations of the degenerate highest occupied orbitals are obtained. For the GGA ensemble solution a special ensemble formula for the GGA exchange functional is proposed. Application of this formula to the H 2 +H 2 reaction reduces appreciably the reaction barriers calculated with GGAs and leads to much better agreement with the accurate value. The too low GGA barriers for the H+H 2 reaction are attributed to overestimation of the dynamical correlation in the TS by the GGA correlation functionals. In order to correct this error, it is recommended to modify the dependence of the approximate correlation functionals on the local polarization ζ with the purpose of reducing the approximate correlationenergy for intermediate ζ values, which are expected to characterize the TS’s of radical abstraction reactions.