Calculation of vertical ionization potentials with the Piris natural orbital functional

The Piris natural orbital functional (PNOF) based on a new approach for the two-electron cummulant has been used to predict vertical ionization potentials of 15 molecules. The ionization energies have been calculated using the extended Koopmans' theorem. The calculated PNOF values are in good agreement with the corresponding experimental data.     

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.     

On Koopmans' theorem in density functional theory

This paper clarifies why long-range corrected (LC) density functional theory gives orbital energies quantitatively. First, the highest occupied molecular orbital and the lowest unoccupied molecular orbital energies of typical molecules are compared with the minus vertical ionization potentials (IPs) and electron affinities (EAs), respectively. Consequently, only LC exchange functionals are found to give the orbital energies close to the minus IPs and EAs, while other functionals considerably underestimate them. The reproducibility of orbital energies is hardly affected by the difference in the short-range part of LC functionals. Fractional occupation calculations are then carried out to clarify the reason for the accurate orbital energies of LC functionals. As a result, only LC functionals are found to keep the orbital energies almost constant for fractional occupied orbitals. The direct orbital energy dependence on the fractional occupation is expressed by the exchange self-interaction (SI) energy through the potential derivative of the exchange functional plus the Coulomb SI energy. On the basis of this, the exchange SI energies through the potential derivatives are compared with the minus Coulomb SI energy. Consequently, these are revealed to be cancelled out only by LC functionals except for H, He, and Ne atoms.     

The extended Koopmans' theorem Fock operator

By expanding the wave function of a system of N particles in terms of products of functions of one and (N-1) particles, the one-particle, nonlocal operator F?_EKT (extended Koopmans' theorem) is determined. It is shown that although this operator is nonhermitian, its eigenvalues and eigenfunctions represent the ionization energies and occupied orbitals, respectively. The eigenfunctions of F?_EKT are the one-particle functions that enter into the expansion of the wave function of the system as partners of the (N-1)-particle wave functions. The eingenvalues are also one-particle energies that, multipled by the orbital occupancy probalities, enter the expression for the total N-particle energy of the system.     

Extended Koopmans' theorem ionization potentials for beryllium atom shake-up transitions

The ionization potentials were calculated for Be using the extended Koopmans' theorem (EKT ) using several full configuration interaction (CI ) and multiconfigurational-self-consistent-field (MCSCF ) wave functions as reference wave functions. The wave functions used account for 89.7–96.7% of the correlation energy. Comparisons are made with experimental values and with δCI values calculated as the difference in energy obtained from CI wave functions for Be and Be⁺. The best EKT IP differed from the δCI value by 0.0003 eV for the lowest IP and by 0.0006 eV for ionization into the lowest ²P state of Be⁺. A calculation of ionization into the second ²P state of Be⁺ requires diffuse orbitals that are unimportant in the wave function for the ground state of Be. This results in small natural orbital occupation numbers for natural orbitals needed in the EKT calculation. © 1994 John Wiley & Sons, Inc.     

Koopmans' theorem and virtual orbital energies in the general SCF theory

The equality of the ionization potential and the orbital energy of the electron being removed is investigated using a general SCF theory for open-shell configurations. The significance of virtual orbital energies is investigated in the same context.

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Zusammenfassung Die Gleichsetzung von Ionisierungsenergie und entsprechender Einelektronenenergie wird für Konfigurationen mit unabgeschlossenen Schalen mit Hilfe einer allgemeinen SCE-Theorie geprüft. In diesem Rahmen wird auch die Bedeutung von virtuellen Einelektronenenergien untersucht.

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Résumé L'équation entre le potentiel d'ionisation et l'énergie orbitale de l'électron correspondant est examinée à l'aide d'une théorie générale SCE pour les configurations à couches ouvertes. Au cadre de cette théorie, la signification des énergies d'orbitales inoccupées est étudiée.

     

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.     

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.     

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.     

The extended Koopmans' theorem Fock operator and the generalized overlap amplitude one-electron operator

The wave function of a system may be expanded in terms of eigenfunctions of the N −1 electron Hamiltonian times one-particle functions known as generalized overlap amplitudes (GOAS). The one-electron operator whose eigenfunctions are the GOAS is presented, without using an energy-dependent term as in the one-particle Green function or propagator approach. It is shown that this operator and the extended Koopmans' theorem (EKT) one-electron operator are of similar form, but perform complementary roles. The GOA operator begins with one-electron densities and total energies of N −1 electron states to generate the two-matrix and total energy of an N-electron state. The EKT operator begins with the two-matrix of an N-electron state to generate one-electron densities and ionization potentials (or approximations thereto) for N −1 electron states. However, whereas the EKT orbitals must be linearly independent, no such restriction applies to the GOAS. © 1996 John Wiley & Sons, Inc.     

Localized orbital corrections for the calculation of ionization potentials and electron affinities in density functional theory

This paper describes the extension of a previously reported empirical localized orbital correction model to the correction of ionization potential energies (IP) and electron affinities (EA) for atoms and molecules of first and second row elements. The B3LYP localized orbital correction version of the model (B3LYP-LOC) uses 22 heuristically determined parameters that improve B3LYP DFT IP and EA energy calculations on the G2 data set of 134 molecules from a mean absolute deviation (MAD) from experiment of 0.137 to 0.039 eV. The method significantly reduces the number of outliers and overall MAD to error levels below that achieved with G2 wave function based theory; furthermore, the new model has zero additional computational cost beyond standard DFT calculations. Although the model is heuristic and is based on a multiple linear regression to experimental errors, each of the parameters is justified on physical grounds, and each provides insight into the fundamental limitations of DFT, most importantly the failure of current DFT methods to accurately account for nondynamical electron correlation.     

Iterative diagonalization for orbital optimization in natural orbital functional theory

A challenging task in natural orbital functional theory is to find an efficient procedure for doing orbital optimization. Procedures based on diagonalization techniques have confirmed its practical value since the resulting orbitals are automatically orthogonal. In this work, a new procedure is introduced, which yields the natural orbitals by iterative diagonalization of a Hermitian matrix F . The off‐diagonal elements of the latter are determined explicitly from the hermiticity of the matrix of the Lagrange multipliers. An expression for diagonal elements is absent so a generalized Fockian is undefined in the conventional sense, nevertheless, they may be determined from an aufbau principle. Thus, the diagonal elements are obtained iteratively considering as starting values those coming from a single diagonalization of the matrix of the Lagrange multipliers calculated with the Hartree‐Fock orbitals after the occupation numbers have been optimized. The method has been tested on the G2/97 set of molecules for the Piris natural orbital functional. To help the convergence, we have implemented a variable scaling factor which avoids large values of the off‐diagonal elements of F . The elapsed times of the computations required by the proposed procedure are compared with a full sequential quadratic programming optimization, so that the efficiency of the method presented here is demonstrated. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009     

Valence orbital ionization potentials from atomic spectral data

Average Energy of Configuration. Valence Orbital Ionization Potentials (VOIPs) are reported for the elements H through Kr in various configurations and for many states of ionization. For the lighter elements the isoelectronic series are fitted to a quadratic equation, VOIP (q)=Aq ² + Bq+C. The significance of the A, B, and C parameters is discussed.

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Zusammenfassung Über die Konfiguration gemittelte Ionisierungsenergien der Valenzorbitale (VOIP's) werden für verschiedene Konfigurationen und Ionisierungsstufen der Elemente H bis Kr angegeben. Bei den leichteren Elementen werden die isoelektronischen Reihen durch Ausgleichsparabeln dargestellt, VOIP (q)=Aq ² + Bq+C (q = Ladung), und die Parameter A, B und C diskutiert.

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Résumé Pour plusieurs configurations et états d'ionisation des éléments H jusqu'à Kr, nous présentons les potentiels d'ionisation, moyennes pour chaque configuration, des orbitales de valence (VOIPs). Pour les éléments légers, les séries isoélectroniques sont représentées par des équations VOIP (q)=Aq ² + Bq + C. Les paramètres A, B et C sont discutés.

     

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.     

Homolytic molecular dissociation in natural orbital functional theory

The dissociation of diatomic molecules of the 14-electron isoelectronic series N(2), O(2)(2+), CO, CN(-) and NO(+) is examined using the Piris natural orbital functional. It is found that the method describes correctly the dissociation limit yielding an integer number of electrons on the dissociated atoms, in contrast to the fractional charges obtained when using the variational two-particle reduced density matrix method under the D, Q and G positivity necessary N-representability conditions. The chemistry of the considered systems is discussed in terms of their dipole moments, natural orbital occupations and bond orders as well as atomic Mulliken populations at the dissociation limit. The values obtained agree well with accurate multiconfigurational wave function based CASSCF results and the available experimental data.     

Core ionization potentials from self-interaction corrected Kohn-Sham orbital energies

We propose a simple self-interaction correction to Kohn-Sham orbital energies in order to apply ground state Kohn-Sham density functional theory to accurate predictions of core electron binding energies and chemical shifts. The proposition is explored through a series of calculations of organic compounds of different sizes and types. Comparison is made versus experiment and the "DeltaKohn-Sham" method employing separate state optimizations of the ground and core hole states, with the use of the B3LYP functional and different basis sets. A parameter alpha is introduced for a best fitting of computed and experimental ionization potentials. It is found that internal parametrizations in terms of basis set expansions can be well controlled. With a unique alpha=0.72 and basis set larger than 6-31G, the core ionization energies (IPs) of the self-interaction corrected Kohn-Sham calculations fit quite well to the experimental values. Hence, self-interaction corrected Kohn-Sham calculations seem to provide a promising tool for core IPs that combines accuracy and efficiency.     

The multiconfigurational spin tensor electron propagator method (MCSTEP): Comparison with extended Koopmans' theorem results

Summary We applied the multiconfigurational spin tensor electron propagator method (MCSTEP) for determining the lowest few (in energy) vertical ionization potentials (IPs) of HF, H₂O, NH₃, CH₄, N₂, CO, HNC, HCN, C₂H₂, H₂CO, and B₂H₆. We chose these molecules so that we could compare MCSTEP IPs with recently reported extended Koopmans' theorem (EKT) IPs on the same molecules. Using standard Dunning core-valence basis sets with relatively small complete active spaces, MCSTEP results are in very good to excellent agreement with experiment. These MCSTEP IPs are obtained using matrices no larger than 400 × 400. EKT matrices are even smaller; however, to obtain similar but generally slightly worse agreement with experiment, fairly large active spaces are required with EKT.     

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.

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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.

     

Analytic second-order energy derivatives in natural orbital functional theory

The analytic energy gradients in the atomic orbital representation have recently been published (Mitxelena and Piris in J Chem Phys 146:014102, 2017) within the framework of the natural orbital functional theory (NOFT). We provide here an alternative expression for them in terms of natural orbitals, and use it to derive the analytic second-order energy derivatives with respect to nuclear displacements in the NOFT. The computational burden is shifted to the calculation of perturbed natural orbitals and occupancies, since a set of linear coupled-perturbed equations obtained from the variational Euler equations must be solved to attain the analytic Hessian at the perturbed geometry. The linear response of both natural orbitals and occupation numbers to nuclear geometry displacements need only specify the reconstruction of the second-order reduced density matrix in terms of occupation numbers.