Correlation energy density from <i>ab</i> <i>initio</i> first- and second-order density matrices: A benchmark for approximate functionals

1999

101

The Kohn-Sham ͑KS͒ solution is constructed from an accurate CI density and the KS exchange and correlation energies E x and E c , as well as the corresponding exchange and exchange-correlation energy densities ⑀ x (r) and ⑀ xc (r), which are obtained for the hydrogen abstraction reaction HϩH 2 and the symmetrical four-center exchange reaction H 2 ϩH 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 ͑TS͒ of the reactions where the left-right correlation is enhanced. Standard GGAs tend to underestimate the barrier height for the reaction HϩH 2 and to overestimate it for the reaction H 2 ϩH 2 . For H 2 ϩH 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 ϩH 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 HϩH 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 with the purpose of reducing the approximate correlation energy for intermediate values, which are expected to characterize the TS's of radical abstraction reactions.

Section: Introductionsupporting

confidence: 66%

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

Schipper

1999

101

“…We have observed in a study of the correlation energy density ⑀ c (r) Ref. 31 for dissociating H 2 that the GGA lacks the left-right correlation. This is also evident from the large error in W c PW as compared to the exact W c in the same case 23 ͑Ϫ0.057 instead of Ϫ0.207 at Rϭ5.0 bohr͒.…”

Section: Ks and Gga Exchange And Correlation Energiesmentioning

confidence: 99%

“…31 33 with eight s-and four p-type functions has been used, which has been augmented with extra p and d polarization functions. For N and F the correlation-consistent polarized core-valence triple zeta added ͑cc-pCVTZ͒ basis sets 34 have been used.…”

Section: Calculation Of the Ks Quantitiesmentioning

confidence: 99%

Exchange and correlation energy in density functional theory: Comparison of accurate density functional theory quantities with traditional Hartree–Fock based ones and generalized gradient approximations for the molecules Li2, N2, F2

Schipper

1997

205

115

The density functional definition of exchange and correlation differs from the traditional one. In order to calculate the density functional theory ͑DFT͒, quantities accurately, molecular Kohn-Sham ͑KS͒ solutions have been obtained from ab initio wave functions for the homonuclear diatomic molecules Li 2 , N 2 , F 2 . These afford the construction of the KS determinant ⌿ s and the calculation of its total electronic energy E KS and the kinetic, nuclear-attraction and Coulomb repulsion components T s , V, W H as well as the ͑DFT͒ exchange energy E x and correlation energy E c . Comparison of these DFT quantities has been made on one hand with the corresponding HartreeFock ͑HF͒ quantities and on the other hand with local density approximation ͑LDA͒ and generalized gradient approximation ͑GGA͒. Comparison with HF shows that the correlation errors in the components T, V, and W H of the total energy are much larger for HF than KS determinantal wave functions. However, the total energies E KS and E HF appear to be close to each other, as well as the exchange energies E x and E x HF and correlation energies E c and E c HF . The KS determinantal wave function and the KS orbitals therefore correspond to much improved kinetic and Coulombic energies, while having only a slightly larger total correlation energy. It is stressed that these properties of the Kohn-Sham orbitals make them very suitable for use in the molecular orbital theories of chemistry. Comparison of the accurate Kohn-Sham exchange and correlation energies with LDA and GGA shows that the GGA exchange energies are consistently too negative, while the GGA correlation energies are not negative enough. It is argued that the GGA exchange functionals represent effectively not only exchange, but also the molecular non-dynamical correlation, while the GGA correlation functionals represent dynamical correlation only. © 1997 American Institute of Physics. ͓S0021-9606͑97͒00337-1͔

“…͑See also the next section for this definition.͒ A procedure to construct ⑀ xc from ab initio firstand second-order density matrices has been proposed in Ref. 17 and examples of the correlation energy density ⑀ c have been obtained for the two-electron systems He and H 2 .…”

Section: ͑5͒mentioning

confidence: 99%

“…To define ⑀ xc and to provide a partitioning for v xc , we use the following expression for E xc ͓ ͔; 12,17 E xc ͓ ͔ϭW xc ͓ ͔ϩT c ͓ ͔, ͑6͒…”

Section: Partitioning Of V Xc and ⑀ Xcmentioning

confidence: 99%

Molecular exchange-correlation Kohn–Sham potential and energy density from ab initio first- and second-order density matrices: Examples for XH (X=Li, B, F)

Leeuwen

1996

Self Cite

The molecular Kohn-Sham exchange-correlation potential v xc and the energy density ⑀ xc have been constructed from ab initio first-and second-order density matrices for the series XH ͑XϭLi, B, F͒. The way various effects of electronic structure and electron correlation manifest themselves in the shape of v xc and ⑀ xc has been analyzed by their decomposition into various components; the potential of the exchange-correlation hole, the kinetic component and ͑in the case of v xc ͒ the ''response'' component. The kinetic energy of noninteracting particles T s , the kinetic part of the exchange-correlation energy T c , and the energy of the highest occupied molecular orbital ⑀ N have been obtained with reasonable accuracy and the effect of bond formation on these functionals has been studied. 1