Orbital Eigenvalues Research Articles

Tuning electronic eigenvalues of benzene via doping

Using variable atomic numbers within molecular grand-canonical ensemble theory, the highest occupied Kohn-Sham eigenvalue of isoelectronic benzene derivatives is tuned. The performed transmutational changes correspond to the iterative doping with boron and nitrogen. The molecular Fukui function proves to be a reliable index in order to predict the changes in the highest occupied molecular orbital eigenvalue due to doping.

Modeling temporary anions in density functional theory: Calculation of the Fukui function

Two approaches are investigated for modeling electron densities of temporary anions in density functional theory (DFT). Both rely on an artificial binding of the excess electron, in one case by a compact basis set and in the other by a potential wall. The key feature of the calculations is that the degree of binding is controlled in both cases by knowledge of the negative electron affinity of the corresponding neutral, approximated in terms of DFT local functional frontier orbital eigenvalues and vertical ionization potential, A=-(epsilon(LUMO)+epsilon(HOMO))-I. To illustrate the two approaches, Fukui functions for nucleophilic attack are determined in four molecules with increasingly negative electron affinities. They yield very similar results, which are notably different to those determined without artificial electron binding. The use of a potential wall has the attractive feature that large, diffuse basis sets can be used, avoiding the need for a compact basis, tailored to a particular molecule.

Modeling the adiabatic connection in H2

Full configuration interaction (FCI) data are used to quantify the accuracy of approximate adiabatic connection (AC) forms in describing the ground state potential energy curve of H2, within spin-restricted density functional theory (DFT). For each internuclear separation R, accurate properties of the AC are determined from large basis set FCI calculations. The parameters in the approximate AC form are then determined so as to reproduce these FCI values exactly, yielding an exchange-correlation energy expressed entirely in terms of FCI-derived quantities. This is combined with other FCI-derived energy components to give the total electronic energy; comparison with the FCI energy quantifies the accuracy of the AC form. Initial calculations focus on a [1/1]-Padé-based form. The potential energy curve determined using the procedure is a notable improvement over those from existing DFT functionals. The accuracy near equilibrium is quantified by calculating the bond length and vibrational wave numbers; errors in the latter are below 0.5%. The molecule dissociates correctly, which can be traced to the use of virtual orbital eigenvalues in the slope in the noninteracting limit, capturing static correlation. At intermediate R, the potential energy curve exhibits an unphysical barrier, similar to that noted previously using the random phase approximation. Alternative forms of the AC are also considered, paying attention to size extensivity and the behavior in the strong-interaction limit; none provide an accurate potential energy curve for all R, although good accuracy can be achieved near equilibrium. The study demonstrates how data from correlated ab initio calculations can provide valuable information about AC forms and highlight areas where further theoretical progress is required.

Comparison of DFT Methods for Molecular Orbital Eigenvalue Calculations

We report how closely the Kohn-Sham highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) eigenvalues of 11 density functional theory (DFT) functionals, respectively, correspond to the negative ionization potentials (-IPs) and electron affinities (EAs) of a test set of molecules. We also report how accurately the HOMO-LUMO gaps of these methods predict the lowest excitation energies using both time-independent and time-dependent DFT (TD-DFT). The 11 DFT functionals include the local spin density approximation (LSDA), five generalized gradient approximation (GGA) functionals, three hybrid GGA functionals, one hybrid functional, and one hybrid meta GGA functional. We find that the HOMO eigenvalues predicted by KMLYP, BH&HLYP, B3LYP, PW91, PBE, and BLYP predict the -IPs with average absolute errors of 0.73, 1.48, 3.10, 4.27, 4.33, and 4.41 eV, respectively. The LUMOs of all functionals fail to accurately predict the EAs. Although the GGA functionals inaccurately predict both the HOMO and LUMO eigenvalues, they predict the HOMO-LUMO gap relatively accurately (approximately 0.73 eV). On the other hand, the LUMO eigenvalues of the hybrid functionals fail to predict the EA to the extent that they include HF exchange, although increasing HF exchange improves the correspondence between the HOMO eigenvalue and -IP so that the HOMO-LUMO gaps are inaccurately predicted by hybrid DFT functionals. We find that TD-DFT with all functionals accurately predicts the HOMO-LUMO gaps. A linear correlation between the calculated HOMO eigenvalue and the experimental -IP and calculated HOMO-LUMO gap and experimental lowest excitation energy enables us to derive a simple correction formula.

Structural and electronic properties of lithium endohedral doped carbon nanocapsules

Endohedral lithium and lithium ion doped carbon nanocapsule (CNC) systems have been theoretically investigated by performing semi-empirical molecular orbital method at the level of PM3 (UHF and/or RHF) type quantum chemical treatment in order to explore the energetics and electronic structures. The geometries of the studied systems have been optimized and the molecular properties, energies, some selected molecular orbital eigenvalues and dipole moments of the studied capsules are reported. Molecular dynamics simulations have been also performed to study the structural properties and energetics of endohedral lithium and lithium ion doped CNCs.

Poisson's equation solution of Coulomb integrals in atoms and molecules†

The integral bottleneck in evaluating molecular energies arises from the two-electron contributions. These are difficult and time-consuming to evaluate, especially over exponential type orbitals, used here to ensure the correct behaviour of atomic orbitals. In this work, it is shown that the two-centre Coulomb integrals involved can be expressed as one-electron kinetic-energy-like integrals. This is accomplished using the fact that the Coulomb operator is a Green's function of the Laplacian. The ensuing integrals may be further simplified by defining Coulomb forms for the one-electron potential satisfying Poisson's equation therein. A sum of overlap integrals with the atomic orbital energy eigenvalue as a factor is then obtained to give the Coulomb energy. The remaining questions of translating orbitals involved in three and four centre integrals and the evaluation of exchange energy are also briefly discussed. The summation coefficients in Coulomb forms are evaluated using the LU decomposition. This algorithm is highly parallel. The Poisson method may be used to calculate Coulomb energy integrals efficiently. For a single processor, gains of CPU time for a given chemical accuracy exceed a factor of 40. This method lends itself to evaluation on a parallel computer. †Dedicated to Professor Nicholas Handy.

Analytical evaluation of molecular electronic integrals using Poisson's equation: Exponential‐type orbitals and atom pairs

AbstractThe integral bottleneck in evaluating molecular energies arises from the two‐electron contributions. These are difficult and time‐consuming to evaluate, especially over exponential type orbitals, used here to ensure the correct behavior of atomic orbitals. The two‐center two‐electron integrals are essential to describe atom pairs in molecules and distinguish those that are bound. In this work on analytical integration, it is shown that the two‐center Coulomb integrals involved can be expressed as one‐electron kinetic energy‐like integrals. This is accomplished using the fact that the Coulomb operator is a Green's function of the Laplacian. The ensuing integrals may be further simplified by defining spectral forms for the one‐electron potential satisfying Poisson's equation therein. A sum of overlap integrals with the atomic orbital energy eigenvalue as a factor is then obtained to give the Coulomb energy. This is most easily evaluated by direct integration. The orbitals involved in three and four center integrals are translated to two centers. This is discussed very briefly. The evaluation of exchange energy is a straightforward extension of this work. The summation coefficients in spectral forms are evaluated analytically from Gaunt coefficients. The Poisson method may be used to calculate Coulomb energy integrals efficiently. For a single processor, gains of CPU time for a given chemical accuracy exceed a factor of 4. This method lends itself to efficient evaluation on a parallel computer. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem 106, 2006

Ionization potentials and electron affinities in the Perdew–Zunger self-interaction corrected density-functional theory

Using a self-consistent implementation of the Perdew-Zunger self-interaction corrected (PZ-SIC) density-functional theory, we have calculated ionization potentials (IP) and electron affinities (EA) of first- and second-row atoms and a set of small molecules. Several exchange-correlation functionals were tested. IPs and EAs were obtained by two methods: as the difference in self-consistent field (SCF) energies of neutrals and ions (deltaSCF) and as negatives of highest-occupied orbital energies. We found that, except for local spin-density approximation, PZ-SIC worsens DeltaSCF IPs and EAs. On the other hand, PZ-SIC brings orbital eigenvalues into much better agreement with electron removal energies. The Perdew-Zunger SIC seems to over-correct many-electron systems; for molecules it performs worse than for atoms. We also discuss several common approximations to PZ-SIC such as spherical averaging of orbital densities in atoms.

X‐ray Absorption Spectra of CuII and CuIII Complexes of N,N′‐1,2‐Phenylenebis(2‐mercapto‐2‐methylpropionamide)

AbstractThe X‐ray absorption spectra of CuII and CuIII complexes of N,N′‐1,2‐phenylenebis(2‐mercapto‐2‐methylpropionamide) were recorded and analysed in solid phase. The EXAFS spectra of the two samples were refined with full multiple scattering path. Geometry optimisations on the CuII and CuIII complexes were performed by the B3LYP density functional method, with the 6‐31G(d,p) basis set, considering different spin multiplicities. The singlet state of the CuIII complex was shown to be more stable than the triplet state, and a good agreement between the calculated and the corresponding experimental structure was found. Further single‐point calculations on the optimised geometry were carried out at the Hartree−Fock level for obtaining molecular orbital eigenvectors and eigenvalues. The latter were employed as parameters in a new fitting approach to rationalise the shape of the K‐edge of the XAS spectra. (© Wiley‐VCH Verlag GmbH & Co. KGaA, 69451 Weinheim, Germany, 2005)

Solution of atomic orbitals in an interpolating wavelet basis

The Schrödinger equation of hydrogenic atoms and the Hartree-Fock equations of some many-electron atoms are solved using interpolating wavelets as basis functions. The nonstandard operator form is used to compute operators in basis sets including multiple resolution levels. We introduce an algorithm for converting matrices from nonstandard operator form to standard operator form. We also consider the different components of the Hamiltonian and Fock operators separately and derive analytic formulas for their evaluation. Extension to many-electron atoms is done within the Hartree-Fock formalism. Convergence of atomic parameters such as orbital eigenvalues with respect to the number of resolution levels is inspected numerically for hydrogenlike atoms (ions) and some light many-electron atoms (helium, lithium, beryllium, neon, sodium, magnesium, and argon).

Differentiability in density-functional theory: Further study of the locality theorem

The locality theorem in density-functional theory (DFT) states that the functional derivative of the Hohenberg-Kohn universal functional can be expressed as a local multiplicative potential function, and this is the basis of DFT and of the successful Kohn-Sham model. Nesbet has in several papers [Phys. Rev. A 58, R12 (1998); Phys. Rev. A65, 010502 (2001); Adv. Quant. Chem, 43, 1 (2003)] claimed that this theorem is in conflict with fundamental quantum physics, and as a consequence that the Hohenberg-Kohn theory cannot be generally valid. We have commented upon these works [Comment, Phys. Rev. A 67, 056501 (2003)] and recently extended the arguments [Adv. Quantum Chem. 43, 95 (2003)]. We have shown that there is no such conflict and that the locality theorem is inherently exact. In the present work we have furthermore verified this numerically by constructing a local Kohn-Sham potential for the $1s2s^{3}S$ state of helium that generates the many-body electron density and shown that the corresponding $2s$ Kohn-Sham orbital eigenvalue agrees with the ionization energy to nine digits. Similar result is obtained with the Hartree-Fock density. Therefore, in addition to verifying the locality theorem, this result also confirms the so-called ionization-potential theorem.

Self-interaction error in density functional theory: a mean-field correction for molecules and large systems

Corrections to the self-interaction error which is rooted in all standard exchange-correlation functionals in the density functional theory (DFT) have become the object of an increasing interest. After an introduction reminding the origin of the self-interaction error in the DFT formalism, and a brief review of the self-interaction free approximations, we present a simple, yet effective, self-consistent method to correct this error. The model is based on an average density self-interaction correction (ADSIC), where both exchange and Coulomb contributions are screened by a fraction of the electron density. The ansatz on which the method is built makes it particularly appealing, due to its simplicity and its favorable scaling with the size of the system. We have tested the ADSIC approach on one of the classical pathological problem for density functional theory: the direct estimation of the ionization potential from orbital eigenvalues. A large set of different chemical systems, ranging from simple atoms to large fullerenes, has been considered as test cases. Our results show that the ADSIC approach provides good numerical values for all the molecular systems, the agreement with the experimental values increasing, due to its average ansatz, with the size (conjugation) of the systems.

A computational study of the structure‐activity relationships of some p‐hydroxybenzoic acid antioxidants

AbstractEquilibrium structures of all derivatized systems of p‐hydroxybenzoic acid, 3,4‐dihydroxybenzoic acid, 3‐methoxy‐4‐hydroxybenzoic acid, and 3,5‐dimethoxy‐4‐hydroxybenzoic acid and calculated structural and energetic molecular descriptors were determined at the B3LYP/6‐31+G(d) density functional theoretical level in an attempt to study their structure‐activity relationships (SAR). The theoretical antioxidant activity trend, derived in terms of hydrogen‐donating capacity against radicals in lipid systems, is in excellent agreement with the experimental one. The lower antioxidant activity of benzoates, experimentally found relative to the homologous cinnamates, could be due to (i) their lower spin delocalization, (ii) their higher calculated heats of formation values in forming radicals (ΔHOF), and (iii) the much stronger electron‐withdrawing effect of the‐COOH group than‐CH=CHCOOH. The low calculated dipole moment values of the global minimum structures of the antioxidants could facilitate their solubilities in nonpolar solvents, hence the ease of hydrogen abstraction. However, highest occupied molecular orbital (HOMO) eigenvalues can neither predict antioxidant activity nor differentiate the same activity between two series of structurally related compounds. Again, density functional theory calculations provide a good molecular descriptor, ΔHOF, to correlate with the antioxidant activity in molecules showing similar structural characteristics.

A mean-field self-interaction correction in density functional theory: implementation and validation for molecules

The implementation and the validation of a simple, self-consistent method to correct the self-interaction error in density functional theory approaches is presented. This model is based on an average density self-interaction correction. The main advantage of the method rests on its simplicity and favorable scaling with the size of the system. We have tested this method on one of the classical pathological problem for density functional theory: the direct estimation of the ionization potential from orbital eigenvalues. The proposed method provides good values for delocalized molecular systems, while large deviations, are obtained for atomic or strongly localized molecular systems.

Ligand field and density functional descriptions of the d-states and bonding in transition metal complexes

The d-orbital energy sequences for low symmetry transition metal complexes derived from Kohn-Sham density functional theory and ligand field theory are different due to each model's treatment of interelectron repulsion. The implications for providing a unified description of the underlying metal-ligand bonding are analysed and illustrated using conventional and time-dependent DFT. Previous detailed spectroscopic studies have established the d orbital sequence in planar coordination complexes containing pi-donor halide ligands as dx2-y2 >> dxy > dxz, dyz > dz2 while for a sigma-only system like [Pd(NH3)4]2+, ligand field approaches like the angular overlap model (AOM) or cellular ligand field (CLF) model predict dxz, dyz, and dxy, should be degenerate. However, the energies of the Kohn Sham (KS) 'd' orbitals of [PdCl4]2- and [Pd(NH3)4]2+ place dxy below the dxz/dyz pair. Direct use of the KS orbital eigenvalues in AOM or CLF analyses would imply both ligands are pi acceptors. This result is independent of the choice of functional or whether the calculation is carried out in vacuo or in a polarised continuum representing solvation by water. The origin of the difference between the KS and LFT d orbital sequences derives from their treatments of d-d interelectron repulsion. KS orbitals include interelectron repulsion contributions while LFT d orbitals do not. For a low-spin d8 complex, DFT gives less d-d interelectron repulsion in the xy plane leading to a lowering of dxy relative to dxy/dyz. This differential effect can be reversed qualitatively by progressively removing electron density from dz2 and placing it in dx2-y2. When about 0.6 electrons is rearranged, E(dxy) > E(dxz/dyz) for [PdCl4]2- and the dpi orbitals for [Pd(NH3)4]2+ are virtually degenerate. The wider ramifications of these interelectron repulsion effects are discussed for other symmetries. Excited d state energies for [PdCl4]2- are computed using both time dependent density functional theory (TDDFT) and determinant energies. The latter give the experimental sequence 1A2g < 1Eg < 1B1g while TDDFT gives 1Eg < 1A2g < 1B1g. Both give transition energies up to 30% lower than observed. A DFT analysis of the bonding energies in [PdCl4]2- indicates that the Pd-Cl pi bonding in the molecular plane is about 33% weaker than the out-of-plane pi interaction due to non-zero overlap between ligand orbitals. The normal LFT assumption of linear ligation may not always be valid.

Varying the fraction of orbital exchange in density functional theory: Influence on nuclear magnetic resonance shielding constants

A series of hybrid exchange-correlation functionals containing varying fractions of orbital exchange ξ=0.0, 0.1, 0.2, … 1.0, are determined using conventional molecular thermochemical fits. The functionals are used to determine Kohn–Sham nuclear magnetic resonance shielding tensors for a series of small molecules involving first- and second-row atoms; results are compared with experimental values. On average, isotropic and anisotropic shieldings determined using the conventional coupled approach become progressively less accurate as ξ increases from 0.0 to 1.0. By contrast, isotropic and anisotropic shieldings determined from the hybrid Kohn–Sham densities using the uncoupled multiplicative Kohn–Sham (MKS) approach [Chem. Phys. Lett. 337, 341 (2001)] improve significantly as ξ increases from 0.0 to 0.2–0.3; optimal results are more than three times as accurate as the corresponding coupled results. As ξ is further increased, the MKS results degrade. The quality of the Kohn–Sham highest occupied molecular orbital–lowest unoccupied molecular orbital (HOMO–LUMO) eigenvalue difference in the MKS calculations is investigated by comparing it with values determined from coupled cluster Brueckner doubles densities. In line with the shielding observations, optimal HOMO–LUMO differences are obtained near ξ=0.3.

A scattered wave model of electronic structure of Eu 2+ in BaMgAl 10O 17 and associated excitation processes

The excitation spectrum of a commercial blue phosphor, BaMgAl 10O 17:Eu 2+ (BAM) has been investigated assuming a distribution of the activator ions at three different sites: Beevers–Ross, anti-Beevers–Ross and mid-oxygen sites in the intermediate plane. The energy levels and wavefunctions of the Eu 2+ ions located at these sites have been calculated using the self-consistent field Xα scattered wave cluster molecular orbital approach (SCF-Xα-SW-MO). The molecular orbital eigenvalues are then utilized to develop an energy level diagram and the associated excitation spectrum. A comparison of the calculated excitation peaks with those from experiment provides support for distribution of Eu 2+ ions at multiple sites in BAM.

Simplified implementation of self-interaction correction in sodium clusters

In density-functional theory, the standard Perdew-Zunger method of self-interaction correction leads to exchange-correlation (xc) potentials that are orbital-dependent. We study various simplified implementations of self-interaction correction\char22{}derived in the framework of the so-called optimized effective potential method\char22{}that lead to a single, orbital-independent xc potential. The performance of the various self-interaction corrected functionals is tested for sodium clusters: we first consider a variety of ground-state properties (total energies, Kohn-Sham orbital eigenvalues, and ionic potential curves) and then study the dynamical regime of weak and strong electronic excitations. It turns out that for sodium clusters the simplest, global-averaging scheme gives a good description of static and dynamic properties, even in the regime of strong excitations.

Low-energy electron attachment to fused 1,4-cyclohexadiene rings by means of electron transmission spectroscopy and exponent stabilization calculations

Electron transmission spectroscopy is used for determining the energies of vertical electron attachment to the empty π* orbitals of ethene (1), 1,4-cyclohexadiene (2), 1,4,5,8-tetrahydronaphthalene (3), and 1,4,5,8,9,10-hexahydroanthracene (4), where the number of ethene double bonds, which interact through space and through the CH2 bridges, increases along the series. In contrast with the expectations based on a simple perturbational model, the energy of the first anion state is nearly constant on going from 1 to 4. Moreover, the energy splitting between the lowest and the highest anion states in the larger molecular systems 3 and 4 is smaller than in 1,4-cyclohexadiene. The experimental data are compared with the empty orbital energies of the neutral states supplied by HF calculations using both a standard basis set and one augmented with diffuse functions, using the exponent stabilization method for distinguishing the virtual orbitals which give rise to temporary anion states. The graphs of virtual orbital eigenvalues versus the exponent scaling factor display avoided crossings and regions where the π* molecular orbital energies are relatively stable. The orbital energies determined in correspondence with the avoided crossings do not reproduce the trends of the resonances observed in the spectra. A better match with experiment (although not completely satisfactory) is obtained by determining the energies in the region of stability of the graphs. This set of results also predicts smaller through-space and through-bond interactions.

Density-functional-theory calculations of the total energies, ionization potentials, and optical response properties with the van Leeuwen–Baerends potential

van Leeuwen and Baerends proposed a Becke-like nonlocal correction to the local-density-approximation (LDA) exchange-correlation potential so that its asymptotic structure becomes exact i.e., $\ensuremath{-}1/r$ [Phys. Rev. A 49, 2421 (1994)]. They showed that it significantly improves the value of the highest occupied orbital eigenvalue of atoms and molecules. However, the correction is exchangelike in nature. With this in mind, in this paper we investigate how this correction affects the total energies and highest eigenvalues within the exchange-only approximation. We show that the potential also corrects the LDA errors substantially within this approximation, and leads to total energies and high eigenvalues which compare well with their Hartree-Fock counterparts. Improvement in the asymptotic behavior of the potential should also result in better values of the response properties of these systems. We show that with this correction one obtains better estimates, both within the exchange-only approximation and with correlation included, of the linear and nonlinear polarizabilities of inert gas atoms. This is quite significant, since the LDA is known to overestimate the nonlinear polarizabilities of these atoms by roughly 100%. On the other hand, for alkaline-earth atoms the values of polarizabilities obtained with this correction are not satisfactory. Nonetheless, hyperpolarizabilities show a marked improvement over the LDA results.