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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.
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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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