A density-functional study of the structures, binding energies and total spins of Ni-Fe clusters using nonlocal norm-conserving pseudopotentials and the generalized gradient approximation

We report ab initio calculations of the structures, binding energies, and total spins of the clusters Ni(13), Ni(19), Ni(23), Ni(26), Ni(12)Fe, Ni(11)Fe(2), Ni(18)Fe, Ni(17)Fe(2), Ni(22)Fe, Ni(20)Fe(3), and Ni(25)Fe using a density-functional method that employs linear combination of atomic orbitals as basis sets, nonlocal norm-conserving pseudopotentials, and the generalized gradient approximation for exchange and correlation. Our results show that the Fe-doped Ni clusters, which have icosahedral or polyicosahedral ground-state structures similar to those of the corresponding pure Ni clusters, are most stable with the Fe atoms occupying internal positions, as has also been inferred from experimental results on the adsorption of molecular nitrogen on the cluster surfaces. We also rule out the possibility that the experimentally observed difference between the (nonpolyicosahedral) configurations of N(2)-saturated Ni(26) and N(2)-saturated Ni(25)Fe be due to the influence of the Fe atom on the energy of the underlying metal cluster.     

A generalized gradient approximation for exchange derived from the model potential of van Leeuwen and Baerends

The common way to obtain energies from Kohn-Sham exchange potentials is by using the Levy-Perdew virial relation. For potentials that are not functional derivatives (i.e., nearly all model exchange potentials in existence), this approach leads to energy expressions that lack translational and rotational invariance. We propose a method for constructing potential-based energy functionals that are free from these artifacts. It relies on the same line-integration technique that gives rise to the Levy-Perdew relation, but uses density scaling instead of coordinate scaling. The method is applicable to any exchange or correlation potential that depends on the density explicitly, and correctly recovers the parent energy functional from a functional derivative. To illustrate our approach we develop a properly invariant generalized gradient approximation for exchange starting from the model potential of van Leeuwen and Baerends.     

Long-range and short-range Coulomb correlation effects as simulated by Hartree–Fock,local density approximation,and generalized gradient approximation exchange functionals

 Exchange functionals used in density functional theory (DFT) are generally considered to simulate long-range electron correlation effects. It is shown that these effects can be traced back to the self-interaction error (SIE) of approximate exchange functionals. An analysis of the SIE with the help of the exchange hole reveals that both short-range (dynamic) and long-range (nondynamic) electron correlation effects are simulated by DFT exchange where the local density approximation (LDA) accounts for stronger effects than the generalized gradient expansion (GGA). This is a result of the fact that the GGA exchange hole describes the exact exchange hole close to the reference electron more accurately than the LDA hole does. The LDA hole is more diffuse, thus leading to an underestimation of exchange and stronger SIE effects, where the magnitude of the SIE energy is primarily due to the contribution of the core orbitals. The GGA exchange hole is more compact, which leads to an exaggeration of exchange in the bond and the nonbonding region and negative SIE contributions. Partitioning of the SIE into intra-/interelectronic and individual orbital contributions makes it possible to test the performance of a given exchange functional in different regions of the molecule. It is shown that Hartree–Fock exchange always covers some long-range effects via interelectronic exchange while self-interaction-corrected DFT is lacking these effects. Received: 25 May 2002 / Accepted: 7 October 2002 / Published online: 21 January 2003 Correspondence to: E. Kraka e-mail: kraka@theoc.gn.se Acknowledgements. This work was supported financially by the Swedish Natural Science Research Council (NFR). Calculations were done on the supercomputers of Nationellt Superdatorcentrum (NSC), Link?ping, Sweden. The authors thank the NSC for a generous allotment of computer time.     

Avoiding singularity problems associated with meta-GGA (generalized gradient approximation) exchange and correlation functionals containing the kinetic energy density

Convergence problems of meta-GGA (generalized gradient approximation) XC (exchange and correlation) functionals containing a self-interaction correction term are traced back to a singularity of the latter that occurs at critical points of the electron density. This is demonstrated for the bond critical point of equilibrium and stretched H2. A simple remedy is suggested that cures meta-XC functionals such as VSXC, TPSS, M05, M06, and their derivatives without extra cost.     

First-principles local density approximation (generalized gradient approximation) +U study of catalytic CenOm clusters: U value differs from bulk

Ceria possesses strong catalytic properties for CONO(x) removal and H(2) production. Clusters often show more intriguing functionalities than their bulk counterparts. Here, the geometric and electronic structures of Ce(n)O(m) (n=1-4,m=2n-1,2n) clusters are studied for the first time using the projected augmented wave method in density functional theory with detailed assessment of the exchange-correlation functional and the Hubbard parameter U. We note that the U value strongly affects the electronic structures of the oxygen-deficient Ce(n)O(2n-1) clusters, though less so on the stoichiometric Ce(n)O(2n). Furthermore, the local density approximation (LDA)+U method is more accurate than the generalized gradient approximation+U in describing the localization of the 4f electrons of the Ce(n)O(m) clusters. The calculated vibration frequency of the CeO molecule with the LDA+U (U=4 eV) is 818.4 cm(-1), in close agreement with experimental values of 820-825 cm(-1) for the low lying states. Different optimal U values were noted for the ceria cluster (4 eV) and its bulk (6 eV), due to quantum-size and geometric effects. The largely reduced formation energy of an oxygen vacancy indicates that the catalytic effect of the Ce(n)O(m) clusters are far greater than bulk CeO(2).     

Geometries and binding energies of small beryllium clusters using various levels of approximations for exchange and correlation: a comparative study

Electronic structure and geometries of small (n≦4) neutral, singly, and doubly ionized clusters of beryllium atoms have been studied using ab initio quantum chemical and local density techniques. In the quantum chemical method the exchange potential is treated exactly in the Hartree-Fock scheme while the correlation correction is incorporated through perturbative procedure. In the density functional approach the exchange and correlation potentials are treated in the local spin density approximation. To facilitate an unambiguous comparison between these methods we have used the same basis functions and numerical procedures. All the investigations yield nearly identical geometries. However, the binding energies using these methods can vary by as much as 2 eV and this variation is comparable to what one obtains using different basis functions.     

Why the generalized gradient approximation works and how to go beyond it

The local spin density (LSD) approximation, while of only moderate accuracy, has proven extremely reliable over three decades of use. We argue that any gradient-corrected functional should preserve the correct features of LSD even if the system under study contains no regions of small density gradient. The Perdew-Wang 1991 (PW91) functional respects this condition, while, e.g., the Lee-Yang-Parr (LYP) correlation functional violates it. We extend this idea to the next generation of density functionals, those which incorporate exact exchange via the optimized effective potential (OEP), with a model in which the correlation hole is constructed from the exact exchange hole. The resulting exchange-correlation hole is deeper and less diffuse than the exact exchange hole. We denote such a functional as “locally correlated Hartree-Fock” and list a variety of conditions such a functional should satisfy. We demonstrate the promise of this approach with a crude simple model. © 1997 John Wiley & Sons, Inc.     

First-principles local density approximation + U and generalized gradient approximation + U study of plutonium oxides

The electronic structure and properties of PuO2 and Pu2O3 have been studied from first principles by the all-electron projector-augmented-wave method. The local density approximation+U and the generalized gradient approximation+U formalisms have been used to account for the strong on-site Coulomb repulsion among the localized Pu 5f electrons. We discuss how the properties of PuO2 and Pu2O3 are affected by the choice of U as well as the choice of exchange-correlation potential. Also, oxidation reaction of Pu2O3, leading to formation of PuO2, and its dependence on U and exchange-correlation potential have been studied. Our results show that by choosing an appropriate U, it is promising to correctly and consistently describe structural, electronic, and thermodynamic properties of PuO2 and Pu2O3, which enable the modeling of redox process involving Pu-based materials possible.     

Nonlocal functionals for exchange and correlation in density functional theory: Application to atoms and to small atomic clusters

The local density approximation (LDA ) to exchange and correlation effects has well-known limitations. The nonlocal weighted density approximation (WDA ) corrects some of those defects. This is illustrated here by applications to free atoms and small atomic clusters. The WDA also induces a nonlocal kinetic energy functional that is tested for atoms. © 1995 John Wiley & Sons, Inc.     

X-ray scattering and density-functional theory calculations to study the presence of hydrogen-bonded clusters in liquid N-methylacetamide

A structural investigation of liquid N-methylacetamide (NMA) is performed by x-ray scattering and density functional theory (DFT). Experimental data are analyzed to yield the total structure function SM(Q) and the pair correlation function g(r). The DFT calculations, using the standard triple zeta valence basis set augmented by a diffuse function for carbon, nitrogen and oxygen atoms, are performed on the one hand to study the structure and stability of the two possible conformers cis and trans. On the other hand, they are meant to examine some possible clusters which may describe the intermolecular arrangement in liquid NMA. Among two series of dimers and trimers associations, the spectra are particularly interpreted in terms of: Trans NMA dimers and trimers which resemble the short-range crystal structure, mixed cis and trans trimers and cis cyclic trimers. The H-bonding parameters and the intermolecular energy for each model are described.     

Communication: Effect of the orbital-overlap dependence in the meta generalized gradient approximation

We study for the first time the effect of the dependence of meta generalized gradient approximation (MGGA) for the exchange-correlation energy on its input, the kinetic energy density, through the dimensionless inhomogeneity parameter, α, that characterizes the extent of orbital overlap. This leads to a simple MGGA exchange functional, which interpolates between the single-orbital regime, where α = 0, and the slowly varying density regime, where α ≈ 1, and then extrapolates to α → ∞. When combined with a variant of the Perdew-Burke-Ernzerhof GGA correlation, the resulting MGGA performs equally well for atoms, molecules, surfaces, and solids.     

A computational investigation of copper-doped germanium and germanium clusters by the density-functional theory

The geometries, stabilities, and electronic properties of Ge(n) and CuGe(n) (n = 2-13) clusters have been systematically investigated by using density-functional approach. According to optimized CuGe(n) geometries, growth patterns of Cu-capped Ge(n) or Cu-substituted Ge(n+1) clusters for the small- or middle-sized CuGe(n) clusters as well as growth patterns of Cu-concaved Ge(n) or Ge-capped CuGe(n-1) clusters for the large-sized CuGe(n) clusters are apparently dominant. The average atomic binding energies and fragmentation energies are calculated and discussed; particularly, the relative stabilities of CuGe10 and Ge10 are the strongest among all different sized CuGe(n) and Ge(n) clusters, respectively. These findings are in good agreement with the available experimental results on CoGe10- and Ge10 clusters. Consequently, unlike some transition metal (TM)Si12, the hexagonal prism CuGe12 is only low-lying structure; however, the basket-like structure is located as the lowest-energy structure. Different from some TM-doped silicon clusters, charge always transfers from copper to germanium atoms in all different sized clusters. Furthermore, the calculated highest occupied molecular orbital and lowest unoccupied molecular orbital (HOMO-LUMO) gaps are obviously decreased when Cu is doped into the Ge(n) clusters, together with the decrease of HOMO-LUMO gaps, as the size of clusters increases. Additionally, the contribution of the doped Cu atom to bond properties and polarizabilities of the Ge(n) clusters is also discussed.     

Interplay between bonding and magnetism in the binding of NO to Rh clusters

We have studied the binding of NO to small Rh clusters, containing one to five atoms, using density functional theory in both spin-polarized and non-spin-polarized forms. We find that NO bonds more strongly to Rh clusters than it does to Rh(100) or Rh(111), suggesting that Rh clusters may be good catalysts for NO reduction. However, binding to NO also quenches the magnetism of the clusters. This (local) effect results in reducing the magnitude of the NO binding energy, and also washes out the clear size-dependent trend observed in the nonmagnetic case. Our results illustrate the competition present between the tendencies to bond and to magnetize, in small clusters.     

Semiempirical estimates of the correlation energy in small clusters of hydrogen atoms

The Pamuk EPCE-F2σ method is applied to neutral and charged clusters composed from 2–9 hydrogen atoms. The range of applicability of the method is demonstrated with H₂, H ₃ ⁺ , and H₃ by comparing the results with the reported rigorous SCF and CI calculations. Predictions of the correlation energy were made for larger hydrogen atom systems, the emphasis being laid in the discussion on H₄, H ₅ ⁺ , and H₆.     

Communication: A global hybrid generalized gradient approximation to the exchange-correlation functional that satisfies the second-order density-gradient constraint and has broad applicability in chemistry

We extend our recent SOGGA11 approximation to the exchange-correlation functional to include a percentage of Hartree-Fock exchange. The new functional, called SOGGA11-X, has better overall performance for a broad chemical database than any previously available global hybrid generalized gradient approximation, and in addition it satisfies an extra physical constraint in that it is correct to second order in the density-gradient.     

Link between the kinetic‐ and exchange‐energy functionals in the generalized gradient approximation

An approximate kinetic‐energy functional of the generalized gradient approximation form was derived following the “conjointness conjecture” of Lee, Lee, and Parr. The functional shares the analytical form of its gradient dependency with the exchange‐energy functionals of Becke and Perdew, Burke, and Ernzerhof. The two free parameters of this functional were determined using the exact values of the kinetic energy of He and Xe atoms. A set of 12 closed‐shell atoms was used to test the accuracy of the proposed functional and more than 30 others taken from the literature. It is shown that the conjointness conjecture leads to a very good class of kinetic‐energy functionals. Moreover, the functional developed in this work is shown to be one of the most accurate despite its simple analytical form. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

A small angle X-ray scattering study of polyethylene fibres,using the two-dimensional correlation function

Summary By Fourier transformation of the intensity distribution in the small angle X-ray scattering diagram of a polymer fibre one obtains the corresponding two-dimensional correlation function. This is compared with correlation functions calculated on the basis of different models of the semi-crystalline morphology. Here, two steps may be distinguished: in the first, the correctness of a model may be checked with the aid of qualitative features, such as the relative positions and sizes of positive and negative areas; in the second a quatitative agreement is pursued by fixing the various parameters involved. This method was applied to cold drawn low density polyethylene before and after annealing at different temperatures. In the case of the non-annealed sample, the microparacrystalline model, discussed byHosemann andLoboda-Cackovic (J. Appl. Cryst.11, 540 (1978)) was found to give the best fit. Annealing at 80 °C or higher temperatures seems to lead to what has been indicated as a three-dimensional chess-board structure; it consists of fibrils in which crystalline and amorphous regions alternate in such a way that the crystalline regions in a fibril are adjacent to amorphous regions in neighbouring fibrils.

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Zusammenfassung Durch Fourier-Transformation der Intensitätsverteilung im Kleinwinkelstreudiagramm von Polymerfasern bekommt man die zugehörige zweidimensionale Korrelationsfunktion. Diese wird verglichen mit Korrelationsfunktionen, die für verschiedene Modelle der teilkristallinen Morphologie berechnet wurden.Hier werden zwei Schritte unterschieden. Im ersten Schritt wird die Richtigkeit eines Modells überprüft mit Hilfe von qualitativen Merkmalen, wie z. B. die relativen Lagen und Größen der positiven und negativen Bereiche. Im zweiten Schritt wird eine quantitative Übereinstimmung dadurch angestrebt, daß die unterschiedlich bezüglichen Parameter festgestellt werden. Diese Methode wurde angewandt bei kaltverstrecktem Polyäthylen niedriger Dichte, sowohl vor wie nach Temperung bei verschiedenen Temperaturen. Im Falle einer nicht getemperten Probe zeigte sich, daß das mikrokristalline Modell, vorgeschlagen von Hosemann und Loboda-Cackovic (J. Appl. Cryst.11, 540 (1978)), am besten zutrifft. Temperung bei 80°C oder bei höheren Temperaturen scheint zu einer Struktur zu führen, die als dreidimensionale Schachbrett-Struktur bezeichnet wurde. Sie besteht aus Fibrillen, in denen kristalline und amorphe Bereiche abwechseln, und zwar so, daß die kristallinen Bereiche in einer Fibrille nur von amorphen Bereichen in nächstliegenden Fibrillen umgeben sind.