The question of the completeness of the natural orbitals with nonzero occupation numbers for atoms and 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.

Natural amplitudes of the ground state of the helium atom: Benchmark calculations and their relevance to the issue of unoccupied natural orbitals in the H2 molecule

Employment of exact numerical quadratures in the evaluation of matrix elements involving highly accurate wavefunctions of helium (and its isoelectronic congeners) generated with the help of the regularized Krylov sequences of Nakatsuji results in an efficient algorithm for the calculation of natural orbitals and the corresponding natural amplitudes {λ_nl}. The results of such calculations are presented for the total of 600 natural orbitals pertaining to the ground state of the helium atom. The benchmark-quality values of {λ_nl} computed for 1 ≤ n ≤ 100 and 0 ≤ l ≤ 5 reveal gross inaccuracies in the previously published data. In particular, the dependence of λ_nl on n is found to follow very closely a simple power-scaling law λ_nl≈-A_l (n+B_l)^-4 with A_l that, contrary to previous claims, varies only weakly with l. Even more importantly, the numerical trends observed in the present calculations strongly suggest that in the case of the ground state of the helium atom, the only positive-valued natural amplitude is that pertaining to the strongly occupied orbital, i.e., λ₁₀. The relevance of this finding to the existence of unoccupied natural orbitals pertaining to the ground state wavefunction of the H₂ molecule is discussed.

A natural orbital analysis of the long range behavior of chemical bonding and van der Waals interaction in singlet H2: the issue of zero natural orbital occupation numbers

This paper gives a natural orbital (NO) based analysis of the van der Waals interaction in (singlet) H2 at long distance. The van der Waals interaction, even if not leading to a distinct van der Waals well, affects the shape of the interaction potential in the van der Waals distance range of 5-9 bohrs and can be clearly distinguished from chemical bonding effects. In the NO basis the van der Waals interaction can be quantitatively covered with, apart from the ground state configurations (1σ(g))(2) and (1σ(u))(2), just the 4 configurations (2σ(g))(2) and (2σ(u))(2), and (1π(u))(2) and (1π(g))(2). The physics of the dispersion interaction requires and explains the peculiar relatively large positive CI coefficients of the doubly excited electron configurations (2σ(u))(2) and (1π(g))(2) (the occupancy amplitudes of the 2σ(u) and 1π(gx, y) NOs) in the distance range 5-9 bohrs, which have been observed before by Cioslowski and Pernal [Chem. Phys. Lett. 430, 188 (2006)]. We show that such positive occupancy amplitudes do not necessarily lead to the existence of zero occupation numbers at some H-H distances.