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In this work, two approaches for simulating X-ray absorption (XA) spectra with the complete active space self-consistent field (CASSCF) linear response (LR) method are introduced. The first approach employs the well-known core-valence separation (CVS) approximation, which is predominantly used by many other electronic structure methods for simulating X-ray spectra. The second ansatz uses the harmonic Davidson algorithm (DA) for finding interior eigenvalues that lie close to a target excitation energy shift and virtually solves a shifted-and-inverted (S&I) generalized eigenvalue problem. Our implementations of these Davidson-type algorithms for core spectroscopies converge as rapidly as the standard DA for valence excitations. It is shown in a proof-of-principle application to the Manganese atom that the additional errors for excitations energies introduced by the CVS approximation compared to the exact S&I approach are negligible for K-edges, but become larger than the methodological error of the CASSCF LR method when computing M-edges. We could also demonstrate that, in case of large active-space K-edge calculations, the computational savings of the CVS approximation are huge as the configuration part of the response vectors is completely neglected. When simulating the oxygen K-edge XA spectrum of the permanganate ion, CASSCF LR showed a better agreement with the experimental spectrum than the CAS-CI and NEVPT2 methods, however, the order of the two close-together pre-edge peaks seems to be inverted.