Renormalized second-order perturbation theory for the electron correlation energy: Concept, implementation, and benchmarks

We present a renormalized second-order perturbation theory (rPT2), based on a Kohn-Sham (KS) reference state, for the electron correlation energy that includes the random-phase approximation (RPA), second-order screened exchange (SOSEX), and renormalized single excitations (rSE). These three terms all involve a summation of certain types of diagrams to infinite order, and can be viewed as ``renormalization'' of the second-order direct, exchange, and single-excitation (SE) terms of Rayleigh-Schrödinger perturbation theory based on a KS reference. In this work, we establish the concept of rPT2 and present the numerical details of our SOSEX and rSE implementations. A preliminary version of rPT2, in which the renormalized SE (rSE) contribution was treated approximately, has already been benchmarked for molecular atomization energies and chemical reaction barrier heights and shows a well-balanced performance [J. Paier et al., New J. Phys. 14, 043002 (2012)]. In this work, we present a refined version of rPT2, in which we evaluate the rSE series of diagrams rigorously. We then extend the benchmark studies to noncovalent interactions, including the rare-gas dimers, and the S22 and S66 test sets, as well as the cohesive energy of small copper clusters, and the equilibrium geometry of 10 diatomic molecules. Despite some remaining shortcomings, we conclude that rPT2 gives an overall satisfactory performance across different electronic situations, and is a promising step towards a generally applicable electronic-structure approach.

Figure 1

Goldstone diagrams for RPA (first row) and SOSEX (second row) contributions. Dashed lines represent bare Coulomb interactions, and full lines correspond to KS electrons (arrow up) and holes (arrow down). Third row: RPA+SOSEX energy in the coupled-cluster context. The wiggly together with the arrowed solid lines represent the direct ring-CCD amplitudes $T_{ia,jb}$ [see Eq. (9)]. The contraction between the direct ring-CCD amplitudes and the bare Coulomb interactions (dashed lines) yields the RPA+SOSEX correlation energy.

Figure 2

Goldstone diagrams for a sequence of correlation-energy terms arising from single excitations. Summing these up to infinite order yields the renormalized single-excitation (rSE) contribution. Here, $\Delta v_{pq}=\langle {\psi }_p|\widehat{f}-{\widehat{h}}^0|{\psi }_q\rangle$, and note $\Delta v_{ia}=f_{ia}$.

Figure 3

rPT2 represented in terms of Goldstone diagrams. The three rows of (infinitely summed) diagrams represent the three components of rPT2: RPA, SOSEX, and rSE. The first column shows the (only) three terms in normal (bare) second-order Rayleigh-Schrödinger perturbation theory based on a KS reference.

Figure 4

Binding-energy curves for Ar${}_2$ computed with PBE and RPA-based approaches (standard RPA, RPA+SE, RPA+rSE, and RPA+rSE-diag based on PBE), in comparison with the accurate Tang-Toennies (experimentally derived) curve. The results are obtained using the Gaussian ``aug-cc-pV6Z'' (Ref. 73) basis set. The basis set superposition error (BSSE) is corrected here and in all following calculations using the counterpoise correction scheme (Ref. 74).

Figure 5

Binding-energy curves for rare-gas dimers computed with RPA-based approaches, in comparison with PBE, MP2, and the accurate Tang-Toennies reference curves. He${}_2$, Ne${}_2$, and Ar${}_2$ results are obtained using the aug-cc-pV6Z basis set, and Kr${}_2$ using the aug-cc-pV5Z basis set. All RPA-type calculations are based on the PBE reference.

Figure 6

The percentage errors for the S22 test set for RPA-derived computational schemes (based on PBE reference orbitals), in comparison to PBE and MP2. The CCSD(T)/CBS results of Takatani et al. (Ref. 75) are used as reference. Lines are guides to the eye.

Figure 7

MAEs (in both meV and kcal/mol) for the S66 test set given by RPA, rPT2, and related schemes (based on PBE reference orbitals), in addition to PBE and MP2. The CCSD(T) results of Rezac et al. (Ref. 38) at the CBS limit are used here as reference.

Figure 8

Interaction energies of four molecular dimers with distorted geometries. The data points correspond to shortening or elongating of the dimer along the main noncovalent interaction coordinate (at 0.9, 1.0, 1.2, 1.5, and 2.0 of the equilibrium distance). Geometries and reference CCSD(T) data are taken from the S22×5 test set (Ref. 78).

Figure 9

The MAEs (in both meV and kcal/mol) of the G2-I atomization energies (Ref. 39) obtained with PBE, MP2, RPA, rPT2, and related methods. The Gaussian cc-pV6Z basis set (Ref. 73) was used in all calculations. Reference data are from Ref. 79.

Figure 10

Upper panel: structure of the Cu clusters cut from the Cu bulk. The number in parentheses corresponds to the effective coordination number $C_{\mathrm{eff}}$ of that cluster. Lower panel: effective cohesive energy for the Cu clusters obtained using different approaches. The dashed line marks the experimental cohesive energy for copper bulk (Ref. 81). The fhi-aims tier-4 basis set was used. All RPA-based calculations were done with a PBE reference.

Figure 11

The MAEs (in both meV and kcal/mol) of the HTBH38 and NHTBH38 test sets for barrier heights, obtained with PBE, MP2, RPA, rPT2, and related methods (based on PBE). Reference data are from Refs. 40 and 41. Gaussian cc-pV6Z basis sets were used in the calculations.

Figure 12

The rPT2@PBE binding energy of the methane dimer in its equilibrium geometry as a function of the basis set size. ``XZ'' and ``aXZ'' (X $=$ D,T,Q,5) denote the Dunning ``cc-pVXZ'' and ``aug-cc-pVXZ'' basis respectively, whereas ``tN'' denotes the FHI-aims ``tier N'' basis. ``t3/4'' here means tier 4 basis for C and tier 3 basis for H (note that a tier 4 basis for H is not available). ``t3/4+a5Z-d'' corresponds to the NAO ``tier 3/4'' plus diffuse functions from aug-cc-pV5Z. The BSSE is corrected.