Comment on ‘‘The exactness of the extended Koopmans’ theorem: A numerical study’’ [J. Chem. Phys. 98, 3999 (1993)]

Numerical analysis of the complete active-space extended Koopmans's theorem

We investigate the numerical accuracy of the extended Koopmans's theorem (EKT) in reproducing the full configuration interaction (FCI) and complete active-space configuration interaction (CAS-CI) ionization energies (IEs) of atomic and molecular systems calculated as the difference between the energies of N and (N - 1) electron states. In particular, we study the convergence of the EKT IEs to their exact values as the basis set and the active space sizes vary. We find that the first FCI EKT IEs approach their exact counterparts as the basis set size increases. However, increasing the basis set or the active space sizes does not always lead to more accurate CAS-CI EKT IEs. Our investigation supports the observation of Davidson et al. [J. Chem. Phys. 155, 051102 (2021)] that the FCI EKT IEs can be systematically improved with arbitrary numerical accuracy by supplementing the basis set with diffuse functions of appropriate symmetry, which allow the detached electron to travel far away from the reference system. By changing the exponent and the center of the diffuse functions, our results delineate a complex pattern for the CAS-CI EKT IE of LiH, which can be important for the spectroscopic studies of small molecules.

ERRORS IN APPROXIMATE IONIZATION ENERGIES DUE TO THE ONE-ELECTRON SPACE TRUNCATION OF THE EKT EIGENPROBLEM

Unless the approximate wavefunction of the parent system is expressed in terms of explicitly correlated basis functions, the finite size of the generalized Fock matrix is unlikely to be the leading source of the truncation error in the ionization energy E produced by the EKT (extended Koopmans' theorem) formalism. This conclusion is drawn from a rigorous analysis that involves error partitioning into the parent- and ionized-system contributions, the former being governed by asymptotic power laws when the underlying wavefunction is assembled from a large number of spinorbitals and the latter arising from the truncation of the infinite-dimensional matrix V whose elements involve the 1-, 2- and 3-matrices of the parent system. Quite surprisingly, the decay of the second contribution with the number $n$ of the natural spinorbitals (NOs) employed in the construction of the truncated V turns out to be strongly system-dependent even in the simplest case of the 1S states of two-electron systems, following the n-5 power law for the helium atom while exhibiting an erratic behavior for the H- anion. This phenomenon, which stems from the presence of the so-called solitonic natural spinorbitals among the NOs, renders the extrapolation of the EKT approximates of E to the complete-basis-set limit generally unfeasible. However, attaining that limit is not contingent upon attempted reproduction of the ill-defined one-electron function known as 'the removal orbital', which does not have to be invoked in the derivation of EKT and whose expansion in terms of the NOs diverges.

Complete-active-space extended Koopmans theorem method

The complete-active-space (CAS) extended Koopmans theorem (EKT) method is defined as a special case of the EKT in which the reference state is a CAS configuration interaction (CI) expansion and the electron removal operator acts only on the active orbitals. With these restrictions, the EKT is equivalent to the CI procedure involving all hole-state configurations derived from the active space of the reference wavefunction and has properties analogous to those of the original Koopmans theorem. The equivalence is used to demonstrate in a transparent manner that the first ionization energy predicted by the EKT is in general not exact, i.e., not equal to the difference between the full CI energies of the neutral and the ion, but can approach the full CI result with arbitrary precision even within a finite basis set. The findings also reconcile various statements about the EKT found in the literature.

Exploring the Effect of the Irradiation Time on Photosensitized Dendrimer-Based Nanoaggregates for Potential Applications in Light-Driven Water Photoreduction

Fourth generation polyamidoamine dendrimer (PAMAM, G4) modified with fluorescein units (F) at the periphery and Pt nanoparticles stabilized by L-ascorbate were prepared. These dendrimers modified with hydrophobic fluorescein were used to achieve self-assembling structures, giving rise to the formation of nanoaggregates in water. The photoactive fluorescein units were mainly used as photosensitizer units in the process of the catalytic photoreduction of water propitiated by light. Complementarily, Pt-ascorbate nanoparticles acted as the active sites to generate H2. Importantly, the study of the functional, optical, surface potential and morphological properties of the photosensitized dendrimer aggregates at different irradiation times allowed for insights to be gained into the behavior of these systems. Thus, the resultant photosensitized PAMAM-fluorescein (G4-F) nanoaggregates (NG) were conveniently applied to light-driven water photoreduction along with sodium L-ascorbate and methyl viologen as the sacrificial reagent and electron relay agent, respectively. Notably, these aggregates exhibited appropriate stability and catalytic activity over time for hydrogen production. Additionally, in order to propose a potential use of these types of systems, the in situ generated H2 was able to reduce a certain amount of methylene blue (MB). Finally, theoretical electronic analyses provided insights into the possible excited states of the fluorescein molecules that could intervene in the global mechanism of H2 generation.