Exact ionization potentials from wavefunction asymptotics: The extended Koopmans’ theorem, revisited

Abstract

A simple explanation is given for the exactness of the extended Koopmans' theorem, (EKT) for computing the removal energy of any many-electron system to the lowest-energy ground state ion of a given symmetry. In particular, by removing the electron from a "removal orbital" of appropriate symmetry that is concentrated in the asymptotic region, one obtains the exact ionization potential and the exact Dyson orbital for the corresponding state of the ion. It is argued that the EKT is not restricted to many-electron systems but holds for any finite many-body system, provided that the interaction vanishes for increasing interparticle distance. A necessary and sufficient condition for the validity of the EKT for any state (not just the lowest-energy states of a given symmetry) in terms of the third-order reduced density matrix is stated and derived.

Electron momentum spectroscopy of metal carbonyls: a reinvestigation of the role of nuclear dynamics

“…Deviations from basis set completeness (BSC) are checked by comparing results employing the aug-cc-pVDZ-PP and aug-cc-pVTZ-PP basis sets. It is worth reminding that Kohn-Sham orbitals and their energies are known to represent valuable approximations to the corresponding Dyson orbitals and (electronically relaxed) ionization energies produced in CI (Configuration Interaction) calculations [80,81], and that, by virtue of Janak's theorem [82] or the extended Koopmans' theorem [83], the approximation becomes exact when considering specifically the HOMO and an (hypothetical) exact exchange-correlation functional.…”

The extended Koopmans' theorem for orbital-optimized methods: Accurate computation of ionization potentials

“…In previous studies 31,33,34 it was discussed that it is very important to employ diffuse basis sets in the EKT computations as in case of anions. 92,93 However, tight diffuse functions (such as DZP++ [94][95][96] ) may destroy the quality of the computed electron affinities (EAs), as it was observed for anions, but some basis sets (such as TZVP and TZVPP 97, 98 ) may provide better EAs even though they do not contain diffuse functions per se.…”

“…[25][26][27][28][29][30][31][32][33][34] In 1993, Sundholm and Olsen 30 examined the exactness of the EKT computing IPs of Be ( 1 S) with the EKT based on the multiconfiguration Hartree-Fock (MCHF) method, and comparing it with the configuration interaction (CI) calculations on Be + ( 2 S). They showed that the difference between the lowest IPs obtained from two methods approaches to zero as the size of the basis set increases.…”

“…Olsen and Sundholm 31 also noted that the missing second-order terms for many-electron systems identified by Pickup and Snijders 28 simply vanish in the complete basis set limit. In 2009, Vanfleteren et al 33 discussed the exactness of the results from the EKT by analyzing the wave function asymptotics and they concluded that one obtains the exact IP when an electron is removed from an orbital, "removal orbital," which is concentrated in the asymptotic region. In another 2009 study, Ernzerhof 34 discussed that the lowest eigenvalue of the EKT procedure in general does not exist.…”

Time-independent density-functional theory for excited states of Coulomb systems

“…This happens, for example, when the spinmultiplicity of the excited state and the ground-state ion differ by more than one. [41][42][43]46 In all cases, a large manifold of bound excited states can be obtained from this functional; in the absence of spatial symmetry it seems likely that the only missing excited states are those associated with an extra spin-flip, relative to the ground-state ion. The key relation, Eq.…”