Extended Koopmans' theorem: Approximate ionization energies from MCSCF wave functions

The extended Koopmans' theorem has been implemented using multiconfigurational self-consistent field wave functions calculated with the GAMESS, HONDO, and SIRIUS programs. The results of illustrative calculations are presented for the molecules HF, H₂O, NH₃, CH₄, N₂, CO, HNC, HCN, C₂H₂, H₂CO, and B₂H₆. The lowest extended Koopmans' theorem ionization potentials agree well within the experimental values and the ionization potentials representing excited states of the ions show some improvements over the Koopmans' theorem values in most cases. The extended Koopmans' theorem is easily implemented and the time required to calculate the ionization energies is insignificant compared to the time required to calculate the wave function of the un-ionized molecule. © 1992 by John Wiley & Sons, Inc.     

The extended Koopmans' theorem: vertical ionization potentials from natural orbital functional theory

The Piris natural orbital functional, PNOF5, has been used to predict vertical ionization potentials of a selected set of 30 organic and inorganic spin-compensated molecules by means of the extended Koopmans' theorem. Electron affinities of 10 selected radicals have also been estimated as the inverse of the ionization potentials of the anionic species, calculated at the experimental geometries of the neutral radicals. The basis set limit effects have been assessed by inspecting the data obtained for the Dunning's basis set series cc-pVXZ and aug-cc-pVXZ (X = D, T, Q, 5). The performance of the PNOF5 is established by carrying out a statistical analysis of the mean absolute errors (MAEs) with respect to the experiment values. The calculated PNOF5 ionization potentials and electron affinities agree satisfactorily with the corresponding experimental data, with MAEs smaller than 0.5 eV.     

Calculation of vertical ionization potentials of ketene by perturbation corrections to Koopmans' theorem

The vertical ionization potentials of ketene are calculated by perturbation corrections to Koopmans' theorem. The present results are compared with those from the pseudonatural orbital coupled electron pair approach and the experimental values.     

Extended Koopmans' theorem at the second-order perturbation theory

The extended Koopmans' theorem (EKT), when combined with the second-order Møller−Plesset (MP2) perturbation theory through the relaxed density matrix approach [J. Cioslowski, P. Piskorz, and G. Liu, J. Chem. Phys. 1997, 107, 6,804], provides a straightforward way to calculate the ionization potentials (IPs) as an one electron quantity. However, such an EKT-MP2 method often suffers from the negative occupation problem, failing to provide the complete IP spectra for a system of interest. Here a small positive number scheme is proposed to cure this problem so as to remove the associated unphysical results. In order to obtain an in-depth physical interpretation of the EKT-MP2 method, we introduce a Koopmans-type quantity, named KT-MP2, based on which the respective contribution from the relaxation and the correlation parts in the EKT-MP2 results are recognized. Furthermore, the close relationship between the EKT-MP2 method and the derivative approach of the MP2 energy with respect to the orbital occupation numbers [N. Q. Su and X. Xu, J. Chem. Theory Comput. 2015, 11, 4,677] is revealed. When these MP2-based methods are applied to a set of atoms and molecules, new insights are gained on the role played by the relaxation and the correlation effects in the electron ionization processes.     

Ionization potentials of ferrocene and Koopmans' theorem. Anab initio LCAO-MO-SCF calculation

The first ionization potentials of ferrocene have been computed in the LCAO-MO-SCF scheme as the difference of the total energy for the neutral molecule and the positive ion. The corresponding sequence of ionization potentials is found to be IP.(e _2g ) _1g )1u e _1g ) in good agreement with the experimental assignment. However, this is different from the sequence of orbital energies for the neutral molecule which is found to bea _1g (3d)<e _2g (-Cp)a _2u (-Cp)e _2u (-Cp)<e _2g (3d)1g (-Cp)e _1u (-Cp).It is concluded that Koopmans' theorem is not valid for the ferrocene molecule. This is traced to the different extent of the electronic rearrangement which occurs upon ionization, depending on the nature (ligand or metal) of the orbital involved in the ionization process.

---

Zusammenfassung Das erste Ionisierungspotential von Ferrocen ist im Rahmen des LCAO-MO-SCF-Verfahrens als Differenz der Gesamtenergie von Molekül zu Ion berechnet worden. Dabei ergibt sich in guter Übereinstimmung mit dem Experiment die Folge I.P.(e _2g )1g )e _1u )e _1g ). Sie ist allerdings durchaus von der Orbitalreihenfolge des neutralen Moleküls, diea _1g (3d)<e _2g (-Cp)a _2u (-Cp)e _2u (-Cp)<e _2g (3d)<e _1g (-Cp)e _1u(-Cp). ist, verschieden, woraus sich die Nichtgültigkeit des Koopmans-Theorems für Ferrocen ergibt, und zwar läßt sie sich auf den unterschiedlichen Umfang der Elektronenumordnung infolge Ionisation zurückverfolgen, je nachdem, aus welchem MO die Ionisierung stattfindet.

     

Koopmans' theorem ionization potentials of methyl and silyl halides,chalcogenides, amines,and phosphines: An ab InitioSCF study

Koopmans' theorem ionization potentials have been calculated for a series of hydrides, methyls, and silyls H_nX, (CH₃)_nX, and (SiH₃)_nX (X = F, Cl, n = 1; X = O, S, n = 2; X = N, P, n = 3), together with some mixed species (MH₃)_nXH_3-n (X = N, P; M = C, Si) using ab initio SCF methods. The calculated values give excellent agreement with experimental values without the inclusion of d functions. For the chlorides, HCl, CH₃Cl, and SiH₃Cl, the values vary rather little over a wide range of basis sets, and are unaffected by the inclusion of d functions.     

The virial theorem as an entail for Koopmans' theorem in atomic calculations

The frozen approximation in the Koopmans' theorem suggests that the virial theorem is not obeyed. By imposing the virial theorem to Koopmans' theorem, we observe that the ionization potential of an atomic orbital is directly related to the respective kinetic energy and that the virial theorem introduces some reorganization effect on the electronic cloud. The quantity of reorganization introduced is not hazard, depending on the type of atom as well as the atomic orbital.     

Extended Koopmans' theorem and sudden ionization processes

The extended Koopmans' theorem eigenvalues of a correlated ground state are shown to represent the centroids of groups of ionization processes in the sudden limit. Lowest order relaxation effects are absent.     

Koopmans' theorem for large molecular systems within density functional theory

It is shown that in density functional theory (DFT), Koopmans' theorem for a large molecular system can be stated as follows: The ionization energy of the system equals the negative of the highest occupied molecular orbital (HOMO) energy plus the Coulomb electrostatic energy of removing an electron from the system, or equivalently, the ionization energy of an N-electron system is the negative of the arithmetic average of the HOMO energy of this system and the lowest unoccupied molecular orbital (LUMO) energy of the (N - 1)-electron system. Relations between this DFT Koopmans' theorem and its existing counterparts in the literature are discussed. Some of the previous results are generalized and some are simplified. DFT calculation results of a fullerene molecule, a finite single-walled carbon nanotube and a finite boron nitride nanotube are presented, indicating that this Koopmans' theorem approximately holds, even if the orbital relaxation is taken into consideration.     

Koopmans' theorem in the ROHF method: canonical form for the Hartree-Fock Hamiltonian

Since the classic work of Roothaan [Rev. Mod. Phys. 32, 179 (1960)], the one-electron energies of a ROHF method are known as ambiguous quantities having no physical meaning. Together with this, it is often assumed in present-day computational studies that Koopmans' theorem is valid in a ROHF method. In this work we analyze the specific dependence of orbital energies on the choice of the basic equations in a ROHF method which are the Euler equations and different forms of the generalized Hartree-Fock equation. We first prove that the one-electron open-shell energies epsilon(m) derived by the Euler equations can be related to the respective ionization potentials I(m) via the modified Koopmans' formula I(m)= -epsilon(m)f(m) where f(m) is an occupation number. As compared to this, neither the closed-shell orbital energies nor the virtual ones derived by the Euler equations can be related to the respective ionization potentials and electron affinities via Koopmans' theorem. Based on this analysis, we derive the new (canonical) form for the Hamiltonian of the Hartree-Fock equation, the eigenvalues of which obey Koopmans' theorem for the whole energy spectrum. A discussion of new orbital energies is presented on the examples of a free N atom and an endohedral N@C(60) (I(h)). The vertical ionization potentials and electron affinities estimated via Koopmans' theorem are compared with the respective observed data and, for completeness, with the respective estimates derived via a DeltaSCF method. The agreement between observed data and their estimates via Koopmans' theorem is qualitative and, in general, appears to possess the same accuracy level as in the closed-shell SCF.     

Assessment of the extended Koopmans' theorem for the chemical reactivity: Accurate computations of chemical potentials,chemical hardnesses,and electrophilicity indices

The extended Koopmans' theorem (EKT) provides a straightforward way to compute ionization potentials and electron affinities from any level of theory. Although it is widely applied to ionization potentials, the EKT approach has not been applied to evaluation of the chemical reactivity. We present the first benchmarking study to investigate the performance of the EKT methods for predictions of chemical potentials (μ) (hence electronegativities), chemical hardnesses (η), and electrophilicity indices (ω). We assess the performance of the EKT approaches for post‐Hartree–Fock methods, such as Møller–Plesset perturbation theory, the coupled‐electron pair theory, and their orbital‐optimized counterparts for the evaluation of the chemical reactivity. Especially, results of the orbital‐optimized coupled‐electron pair theory method (with the aug‐cc‐pVQZ basis set) for predictions of the chemical reactivity are very promising; the corresponding mean absolute errors are 0.16, 0.28, and 0.09 eV for μ, η, and ω, respectively. © 2015 Wiley Periodicals, Inc.     

The calculation of heavy atom ionization potentials by the relativistic transition state approximation

The relativistic statistical exchange (Xα) model has been applied to the calculation of the ionization potentials of uranium and thorium. The results are comparable in accuracy to those of the relativistic Hartree-Fock method, if the transition state approximation is used, rather than interpreting the eigenvalues by Koopmans' theorem.     

On the quantum chemical origin for the nonvalidity of Koopmans' theorem in transitionmetal compounds

The quantum chemical origin for the nonvalidity of Koopmans' theorem in transitionmetal compounds of the 3d series is analyzed by means of the Green's function formalism applied in the framework of a semiempirical INDO Hamiltonian. In the case of ferrocene (1), cyclobutadiene iron tricarbonyl (2) and irontetracarbonyl dihydride (3) the self-energy part of a geometric approximation has been partitioned into relaxation and correlation (pair removal, pair relaxation) increments. The breakdown of Koopmans' theorem for strongly localized MOs with large Fe 3d amplitudes is predominantly the result of electronic relaxation lowering the calculated ionization potentials. On the other hand the variation of the pair correlation energy in the cationic hole-state is by no means negligible and acts into the opposite direction as the relaxation increment. These significant pair relaxation contributions explain the wellknown failtures of the ΔSCF approach in combination with large scaleab initio bases. The loss of ground state pair correlation in the outer valence region is small in comparison to relaxation and pair relaxation. The magnitude of the aforementioned reorganization increments has been studied as a function of the localization properties of the MOs and as a function of the one-electron energies of the available particle- and hole-states. The computational findings derived with the INDO model are compared with recentab initio studies.     

Extended Hückel calculations of the ionization potentials of some conjugated hydrocarbons

The prediction of ionization potentials of conjugated hydrocarbons using extended Hückel theory is reevaluated. Consequently, two major modifications to the theory are proposed and then tested on five sample molecules. Allowance for distinct sigma and pi orbital parameters as well as for diagonal and off-diagonal parameters constitute the changes examined and are shown to arise naturally from the theory. Subsequent testing of the reproduction of photoelectron spectra indicates significant improvements over previous usage of extended Hückel theory.     

Accurate electron density and one-electron properties for the beryllium atom

An accurate analytical electron density for the beryllium atom is obtained by using a fast and systematic method recently developed and tested for the neon atom. Asymptotic conditions both at the nucleus and at large distances are obeyed. A point-by-point comparison between our density and the one obtained from an almost “exact” configuration interaction wave function shows that differences are less than 0.5% for r between 0 and 5 bohrs and less than 1 % up to 9 bohrs. The accuracy of the density is also assessed by comparing results of density moments and x-ray scattering factors.     

A method for molecular ionization potentials

A new method for one-electron propagator calculations of molecular inization potentials is proposed, using a large matrix technique The results of some trial calculations on molecular nitrogen are given.     

Calculation of vertical ionization potentials of difluoramine by perturbation corrections to koopmans' theorem

The vertical ionization potentials of difluoramine are calculated by perturbation corrections to Koopmans' theorem. The calculation shows that difluoraimine has three overlapping bands between 15 and 16 eV. The calculated results compare well with the experimental values. The photoelectron spectrum of difluoramine is compared with that of OF₂ and CH₂F₂.     

Extended wedge covariance for wetting and filling transitions

Fluid adsorption on nonplanar and heterogeneous substrates is studied using a simple interfacial model. For systems with short-ranged forces, we find that, by tuning the local strength of the substrate potential, it is possible to find the exact equilibrium interfacial profile as a functional of the wall shape psi x. The tuning of the local substrate potential takes the form of a gauge condition theta x=+/-psi x, where theta x can be interpreted as a local effective contact angle. For wedgelike geometries with asymptotic tilt angle alpha, the midpoint interfacial height and roughness satisfy the same covariance relations previously found for simple linear wedges. For troughlike geometries satisfying the gauge condition, covariance is also found for the two-point correlation function. Predictions for more microscopic Landau and Ising models are also discussed.     

Ligand atom partial charges assignment for complementary electrostatic potentials

Summary The design of molecules to fit into the active site of receptors is a rapidly developing area of pharmacology and medicinal chemistry. A good ligand needs a suitable geometry and also appropriate electrostatic properties. The electrostatic properties of the ligand should complement those of the receptor. We present a method for the assignment of atom-centred point charges for a ligand, based on the electrostatic potential of the receptor. These point charges are chosen to give the best possible complementarity to the receptor electrostatic potential over the van der Waals surface of the ligand. We demonstrate that point charges can be chosen to give good electrostatic complementarity, and suggest that a molecule with similar electrostatic properties should bind well to the receptor.