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Analytical double-hybrid density functional based on the polynomial series expansion of adiabatic connection: a quadratic approximation.

We present a systematic derivation of double-hybrid density functional (DHDF) based on the polynomial series expansion of adiabatic connection formula in the closed interval λ = [0,1] without a loss of generality. Because of the tendency of Wλ having a small (but not negligible) curvature at equilibrium, we first evaluate the chemical validity of quadratic approximation for Wλ using the large GMTKN30 benchmark database. The resulting functional, obtained analytically and denoted by quadratic adiabatic connection functional-PT2 (QACF-2), is found to be robust and accurate (2.35 kcal/mol of weighted total mean absolute deviation error, WTMAD), comparable or slightly improved compared to other flavors of existing parameter-free DHDFs (2.45 or 3.29 kcal/mol for PBE0-2 or PBE0-DH, respectively). The nonlocal expansion coefficients obtained for the current QACF-2 (aHF = 2/3, aPT2 = 1/3) also offer some interesting observation, in that the latter analytical coefficients are very similar to the empirically optimized coefficients in some of the best DHDFs today with high accuracy (1.5 kcal/mol). Effects of quadratic truncation in QACF-2 have been further assessed and justified by estimating the higher-order corrections to be as much as 0.54 kcal/mol. The present derivation and numerical experiments suggest that the quadratic λ dependence, despite its simplicity, is a surprisingly good approximation to the adiabatic connection that can serve as a good starting point for further development of accurate parameter-free density functionals.