Abstract
The Extended Koopman’s Theorem (EKT) provides a straightforward way to compute charged excitations from any level of theory. In this work we make the link with the many-body effective energy theory (MEET) that we derived to calculate the spectral function, which is directly related to photoemission spectra. In particular, we show that at its lowest level of approximation the MEET removal and addition energies correspond to the so-called diagonal approximation of the EKT. Thanks to this link, the EKT and the MEET can benefit from mutual insight. In particular, one can readily extend the EKT to calculate the full spectral function, and choose a more optimal basis set for the MEET by solving the EKT secular equation. We illustrate these findings with the examples of the Hubbard dimer and bulk silicon.
Highlights
Within the lowest level of approximation in terms of one- and twobody density matrices, the manybody effective energy theory (MEET) equations correspond to the so-called diagonal approximation to the Extended Koopman’s Theorem (EKT) (DEKT) equations
Using the test case of the Hubbard dimer with two different on-site interactions U1 and U2 for site 1 and site 2 we showed the effect of the basis set on the MEET energies: in particular HOMO-LUMO gap in the basis sets which solve the EKT secular equations is smaller than the HOMO-LUMO gap obtained using the natural orbital basis set
We showed that using the currently available approximations the DEKT band gap of Si largely deviates from the DEKT value obtained using QMC (12.9 eV vs 4.4 eV at the Γ point) and, there is no effect of the basis set (EKT vs DEKT) on the DEKT energies, contrary to what is observed within QMC, where, small, there is a significant difference
Summary
The Extended Koopman’s Theorem (EKT) (Morrell et al, 1975; Smith and Day, 1975) has been derived in quantum chemistry and used within various frameworks, from functional theories based on reduced quantities, such as reduced-density matrix functional theory (Gilbert, 1975) (e.g., Pernal and Cioslowski, 2005; Leiva and Piris, 2005; Piris et al, 2012; Piris et al, 2013) and many-body perturbation theory based on Green’s functions (Hedin, 1965) (e.g., Dahlen and van Leeuwen, 2005; Stan et al, 2006; Stan et al, 2009), to wavefunction-based methods (e.g., Cioslowski et al, 1997; Kent et al, 1998; Bozkaya, 2013; Zheng, 2016; Bozkaya and Ünal, 2018; Pavlyukh, 2019; Lee et al, 2021). Much effort is devoted to develop better approximations to Σ (Springer et al, 1998; Zhukov et al, 2004; Shishkin et al, 2007; Kuneš et al., 2007; Guzzo et al, 2011; Romaniello et al, 2012; Lischner et al., 2013; Stefanucci et al, 2014) or to develop novel ways to determine G (Lani et al, 2012; Berger et al, 2014) In this spirit in these last years we have developed the many-body effective energy theory (MEET) (Di Sabatino et al, 2016), in which the spectral function is expressed in terms of density matrices, or, alternatively, in terms of moments of G, as reported in Ref.
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