Qualitative representation of the Löwdin function f(E) for a FCI problem with K = 8 CSFs and H 11 = E ref , to show the general behaviour of this function. In this example, the horizontal asymptote corresponds to E ref (i.e., single configuration representations: RHF for closed shell systems, ROHF or GVB for open shell singlets and triplets). The points E 1 ≤ E 2 ≤ E 3 ≤ ⋯ ≤ E 8 correspond to the set of eigenvalues of the full H matrix of dimension K = 8 (i.e., those in which f(E) = E). The K − 1 = 7 vertical asymptotes a 1 ≤ a 2 ≤ a 3 ≤ ⋯ ≤ a 7 correspond to the eigenvalues of the K − 1 reduced H matrix, namely, H II,II. See text for more explanation details of this plot.
Relevant values for the Löwdin function (dimension of the FCI space, eigenvalues of the FCI matrix, etc.) and energies obtained using HF and several standard parameterizations of the E XC [ρ] functional for LDA, 17,18 GGA (PW91PW91, 19 BLYP, 20,21 and PBEPBE 22 ), meta-GGA (revTPSS, 23 PKZBPKZB, 24 and VSXC 25 ) and hybrid (B3LYP 21,26 and PBE0 27 ) approximations. In all cases, the basis set is STO-3G and the geometries correspond to CISD/6-31++g** optimised structures. 12 (Boldface italics are used to separate the different blocks: FCI dimensions and eigenvalues (rows 1–7), f (E = 0) (row 8), E(HF/ROHF) = H 11 (row 9), and DFT based calculations (rows 10–18)).
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