Seeking for reliable double-hybrid density functionals without fitting parameters: The PBE0-2 functional
Graphical abstract
Highlights
► We propose a double-hybrid density functional without fitting to accurate data. ► This new functional is shown to perform well for self-interaction problems. ► This new functional is shown to perform well for noncovalent interactions.
Introduction
Over the past two decades, Kohn–Sham density functional theory (KS-DFT) [1] has become a very popular electronic structure method for molecular and solid-state systems, due to its computational efficiency and reasonable accuracy. Although the exact exchange–correlation (XC) density functional Exc[ρ] in KS-DFT remains unknown, accurate approximations to Exc[ρ] have been continuously developed to extend the applicability of KS-DFT to a wide range of systems [2].
The hierarchy of approximations to Exc[ρ] has been formulated as a Jacob’s ladder connecting the earth (Hartree theory) to the heaven (chemical accuracy) [3]. The first rung of the ladder is the local density approximation (LDA) [4], [5], representing the XC energy density by the local density. Going beyond the LDA, there has been an increasing interest in the ‘parameter-free’ functionals developed by Perdew and co-workers, such as PBE [6] and TPSS [7], demonstrating the usefulness of these functionals. As LDA, PBE, and TPSS belong to the first, second, and third rungs of the ladder, respectively, they systematically improve upon the description of short-range XC effects by climbing up the ladder (with slightly increasing computational costs). However, due to the lack of accurate treatment of nonlocal XC effects, these semilocal functionals (first three rungs) can lead to significant errors, characterized by self-interaction error (SIE), noncovalent interaction error (NCIE), and static correlation error (SCE), in situations where these failures occur [2], [8].
The SIEs of density functional approximations (DFAs) can be reduced by hybrid DFT methods (fourth rung) [9], combining a fraction of the exact Hartree–Fock (HF) exchange with a semilocal functional. As the fraction of HF exchange included in a hybrid functional can be rationalized by perturbation theory arguments [10], the PBE-based hybrid functional PBE0 [11], has gradually gained its popularity. In view of the successive evolution of the PBE-based functionals: LDA (first rung) → PBE (second rung) → TPSS (third rung) → PBE0 (fourth rung) → ? (fifth rung), naturally, the question is, what could be a functional on the fifth rung (highest rung) of the ladder? In this Letter, we propose a PBE-based double-hybrid functional (fifth rung), and demonstrate its superiority to the functionals on the first four rungs of the ladder, for a wide range of applications.
Section snippets
The PBE0-2 functional
The SIEs and NCIEs associated with DFAs can be simultaneously reduced by double-hybrid methods (fifth rung) [12], [13], [14], [15], [16], [17], combining a fraction of HF exchange and a fraction of second-order Møller–Plesset (MP2) correlation [18] with a semilocal functional. Accordingly, a PBE-based double-hybrid functional is given bywhere is the HF exchange, is the PBE exchange, is the PBE correlation, and is the MP2
Results
For a comprehensive comparison of different functionals, we examine the performance of PBE0-2, other PBE-based functionals (LDA [4], [5], PBE [6], TPSS [7], PBE0 [11], and PBE0-DH [20]), three popular functionals (B3LYP (a hybrid functional) [9], [28], B2PLYP (a double-hybrid functional) [14], and B2PLYP-D3 (a double-hybrid functional with empirical dispersion corrections) [17]), and the ab initio MP2 method [18], on various test sets involving the 223 atomization energies (AEs) of the G3/99
Conclusions
In summary, we have developed a PBE-based double-hybrid functional, employing a half-and-half mixing of the PBE and MP2 correlation functionals with the LS1DH approximation in Eq. (2). Owing to its significant improvement over LDA, PBE, TPSS, and PBE0, this functional, denominated PBE0-2, fits well into the fifth rung of the ladder. Our results indicate that PBE0-2 is generally comparable or superior to PBE0-DH in performance. As PBE0-2 contains a very large fraction (≈79%) of HF exchange and a
Acknowledgments
This work was supported by National Science Council of Taiwan (Grant No. NSC98-2112-M-002-023-MY3), National Taiwan University (Grant Nos. 99R70304 and 10R80914-1), and NCTS of Taiwan.
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