Alternative separation of exchange and correlation energies in range-separated density-functional perturbation theory

An alternative separation of short-range exchange and correlation energies is used in the framework of second-order range-separated density-functional perturbation theory. This alternative separation was initially proposed by Toulouse et al. [Theor. Chem. Acc. 114, 305 (2005)] and relies on a long-range-interacting wave function instead of the noninteracting Kohn-Sham one. When second-order corrections to the density are neglected, the energy expression reduces to a range-separated double-hybrid (RSDH) type of functional, RSDHf, where “f” stands for “full-range integrals” as the regular full-range interaction appears explicitly in the energy expression when expanded in perturbation theory. In contrast to the usual RSDH functionals, RSDHf describes the coupling between long- and short-range correlations as an orbital-dependent contribution. Calculations on the first four noble-gas dimers show that this coupling has a significant effect on the potential energy curves in the equilibrium region, improving the accuracy of binding energies and equilibrium bond distances when second-order perturbation theory is appropriate.

Figure 1

Long-range $E^{\left(2\right)\mathrm{lr},\mu }$ (dashed red line) and long-range–short-range $E^{\left(2\right)\mathrm{lr}-\mathrm{sr},\mu }$ (dotted blue line) MP2 correlation energies, as well as the sum of the two contributions (solid green line), computed with the HF-srDFT orbitals and orbital energies for He (top left), Ne (top right), Ar (bottom left), and Kr (bottom right) atoms when varying the $\mu$ parameter.

Figure 2

Interaction energy curves obtained for He${}_2$ (top left), Ne${}_2$ (top right), Ar${}_2$ (bottom left), and Kr${}_2$ (bottom right) with conventional srLDA schemes (thin dotted and double-dot-dashed blue lines) and the alternative range-separated models (thick red lines). See text for further details. Comparison is made with MP2 (thin dashed green line), CCSD(T) (dot-dashed black line), and the experimental (solid black line) results of Ref. [20]. The $\mu$ parameter is set to $0.4a_0^{-1}$.

Figure 3

Decomposition of the $\mu$-dependent RSDHf (thick double-dot-dashed red line) and MP2-srLDA (thin double-dot-dashed blue line) interaction energies for Ar${}_2$ at both $R=7.013a_0$ (left) and $10.0a_0$ (right) bond distances. $E^{\left(2\right)\mathrm{lr}}$, $E^{\left(2\right)\mathrm{lr}-\mathrm{sr}}$, $E_{\mathrm{x}}^{\mathrm{sr}}\left[\Phi \right]=-{\sum }_{ij}{\langle ij|ji\rangle }^{\mathrm{sr},\mu }$, and $n$ denote the purely long-range MP2 interaction energy, the long-range–short-range coupling contribution, the exact short-range exchange energy contribution computed with the HF-srDFT orbitals, and the HF-srDFT density, respectively. See text for further details. Comparison is made with conventional CCSD(T) (dotted-dashed black line) and experimental (black solid line) results.

Figure 4

RSDHf (red double-dot-dashed lines), MP2-srLDA (blue dashed lines), and regular MP2 (green dotted lines) interaction energy curves of Ar${}_2$ with (thick lines) and without (thin lines) BSSE correction. The $\mu$ parameter is set to $0.4a_0^{-1}$.

Figure 5

Basis set (aug-cc-pV$n$Z) convergence for the RSDHf and MP2-srLDA total energies in He${}_2$ (left) and Ne${}_2$ (right). Experimental equilibrium distances [20] and $\mu =0.4a_0^{-1}$ were used. Comparison is made with conventional MP2 and CCSD(T) results.