Atomic-like Orbitals Research Articles

Interaction of Titanium Atoms with the Surface of Perfect and Defective Carbon Nanotubes

The dispersion of metal atoms over the surface of 1D and 2D carbon systems is the most affordable way to control their properties, which are attractive for many applications in electronics, power engineering, and catalysis. In this work, the features of the interaction of titanium atoms with the surface of carbon nanotubes, caused by various structural defects on these surfaces, were studied by first-principles computer simulation based on the density functional theory. Nanotubes (7, 7) and (11, 0) with similar diameters (≈1 nm) but different types of conductivity, metallic and semiconductor, respectively, were chosen for the study. Three types of defects were studied: a single vacancy, a double vacancy, and a topological defect. Two possible orientations of each type of defect relative to the tube axis were considered. We mainly used the basis of atomic-like orbitals (the SIESTA package) and in some test calculations also the basis of plane waves (the VASP package). Computational experiments have shown that the binding energy of Ti atoms with a defect-free nanotube is always lower than with defective ones, regardless of the used approximation for the exchange-correlation functional (LDA or GGA). The binding energies predicted in the LDA approximation are noticeably higher than in the GGA approximation (up to ~15% for the (7, 7) tube and up to ~50% for the (11, 0) tube). The strongest coupling occurs when the titanium atom is adsorbed on a nanotube with a single vacancy. The resulting configuration can be considered as a defect in the substitution of one carbon by a titanium atom.

Automated mixing of maximally localized Wannier functions into target manifolds

Maximally localized Wannier functions (MLWFs) are widely used in electronic-structure calculations. We have recently developed automated approaches to generate MLWFs that represent natural tight-binding sets of atomic-like orbitals; these describe accurately both the occupied states and the complementary unoccupied ones. For many applications, it is required to use MLWFs that describe instead certain target groups of bands: the valence or the conduction bands, or correlated manifolds. Here, we start from these tight-binding sets of MLWFs, and mix them using a combination of parallel transport and maximal localization to construct manifold-remixed Wannier functions (MRWFs): these are orthogonal sets of MLWFs that fully and only span desired target submanifolds. The algorithm is simple and robust, and is showcased here in reference applications (silicon, MoS2, and SrVO3) and in a mid-throughput study of 77 insulators.

ПЕРВОПРИНЦИПНАЯ МОДЕЛЬ ДВУХСЛОЙНОГО ГРАФЕНА

Using the electron density functional theory, numerical modelling of bilayer graphene has been performed. The structure and binding energy of the layers depending on the representation of the wave function of the system have been studied: plane waves (VASP package) and atomic-like orbitals (SIESTA package); and choice of approximation for the exchange-correlation (XC) functional. It has been shown that being in the free form the system creates a stable AB structure of bilayer graphene. The calculation of the layer binding energy has shown that the results of modelling performed with different basis for the wave function are consistent when using XC functionals corresponding to each other and considering the correction to the basis set superposition error in the tight binding method. As expected, the generalized gradient approximation (GGA) has shown underestimated values of the interaction energy of graphene layers. Comparison with experimental data has shown that the energy and geometric characteristics of bilayer graphene are best described by the local electron density approximation (LDA). The 2nd and 3rd generation semi-empirical Grimme corrections for GGA have given estimates of the binding energy higher than LDA, but also close to the experimental results.

Photoionization of the iodine 3d, 4s, and 4p orbitals in methyl iodide.

Ionization of the I 3d, 4s, and 4p orbitals in methyl iodide (CH3I) has been studied by using synchrotron radiation to measure the total ion yield and by recording photoelectron spectra with linearly polarized radiation in two polarization orientations. The complete photoelectron spectrum of CH3I has been recorded at several photon energies, and bands due to the C 1s, I 3d, 4s, 4p, and 4d atomic-like orbitals, as well as the molecular orbitals, have been observed and assigned. In the vicinity of the I 3d5/2 and 3d3/2 ionization thresholds at 626.8 and 638.3 eV, respectively, the ion yield displays weak structure in the pre-edge region due to transitions into valence or Rydberg states, and, at higher energies, a shoulder and a broad maximum attributed to the I 3d5/2 → εf and the I 3d3/2 → εf shape resonances, respectively. The absorption spectrum calculated using time-dependent density functional theory, within the Tamm-Dancoff approximation, has allowed assignments to be proposed for the valence and Rydberg states. The Stieltjes imaging technique has been used to simulate the absorption spectrum above the ionization threshold and indicates that transitions into the f(l = 3) continuum channel dominate. This conclusion has been corroborated by a Continuum Multiple Scattering-Xα (CMS-Xα) calculation. The asymmetric broadening of the photoelectron bands associated with the I 3d orbital, due to post collision interaction, is taken into account in our experimental analysis. Experimentally derived photoelectron anisotropy parameters for the I 3d orbital are in good agreement with the theoretical predictions obtained with the CMS-Xα approach. The I 3d shake-up/shake-off photoelectron spectrum has been recorded, and assignments have been proposed for several of the satellites. The M4N45N45 and M5N45N45 Auger electron yields have been measured, and that for the M5N45N45 decay exhibits a maximum due to interchannel coupling between the 3d5/2 and 3d3/2 continua. The photoelectron band associated with the I 4p orbital has an unusual appearance. Based upon previous theoretical work for the analogous Xe 4p orbital, it appears that the initial I 4p-1 hole state decays rapidly through Coster-Kronig and super-Coster-Kronig transitions. This leads to a redistribution of the spectral intensity associated with the I 4p orbital and results in a photoelectron spectrum containing a single structured band together with an extended continuum. Another continuum is observed on the high binding energy side of the peak due to the 4s orbital, and we assign this to super-Coster-Kronig transitions into the 4p-14d-1 continuum.

Quantum mechanics in an evolving Hilbert space

Many basis sets for electronic structure calculations evolve with varying external parameters, such as moving atoms in dynamic simulations, giving rise to extra derivative terms in the dynamical equations. Here we revisit these derivatives in the context of differential geometry, thereby obtaining a more transparent formalisation, and a geometrical perspective for better understanding the resulting equations. The effect of the evolution of the basis set within the spanned Hilbert space separates explicitly from the effect of the turning of the space itself when moving in parameter space, as the tangent space turns when moving in a curved space. New insights are obtained using familiar concepts in that context such as the Riemann curvature. The differential geometry is not strictly that for curved spaces as in general relativity, a more adequate mathematical framework being provided by fibre bundles. The language used here, however, will be restricted to tensors and basic quantum mechanics. The local gauge implied by a smoothly varying basis set readily connects with Berry's formalism for geometric phases. Generalised expressions for the Berry connection and curvature are obtained for a parameter-dependent occupied Hilbert space spanned by non-orthogonal Wannier functions. The formalism is applicable to basis sets made of atomic-like orbitals and also more adaptative moving basis functions (such as in methods using Wannier functions as intermediate or support bases), but should also apply to other situations in which non-orthogonal functions or related projectors should arise. The formalism is applied to the time-dependent quantum evolution of electrons for moving atoms. The geometric insights provided here allow us to propose new finite-differences time integrators, and also better understand those already proposed.

Large spin splitting in the conduction band of transition metal dichalcogenide monolayers

We study the conduction band spin splitting that arises in transition metal dichalcogenide (TMD) semiconductor monolayers such as MoS$_2$, MoSe$_2$, WS$_2$ and WSe$_2$ due to the combination of spin-orbit coupling and lack of inversion symmetry. Two types of calculation are done. First, density functional theory (DFT) calculations based on plane waves and pseudo potentials that yield large splittings, between 3 and 30 meV. Second, we derive a tight-binding model, that permits to address the atomic origin of the splitting. The basis set of the model is provided by the maximally localized Wannier orbitals, obtained from the DFT calculation, and formed by 11 atomic-like orbitals corresponding to 5$d$ and 3$p$ orbitals of the transition metal (W,Mo) and chalcogenide (S,Se) atoms respectively. In the resulting Hamiltonian we can independently change the atomic spin-orbit coupling constant of the two atomic species at the unit cell, which permits to analyse their contribution to the spin splitting at the high symmetry points. We find that ---in contrast to the valence band--- both atoms give comparable contributions to the conduction band splittings. Given that these materials are most often $n-$doped, our findings are important for developments in TMD spintronics.

Electronic structures of AlMoOy− (y = 1–4) determined by photoelectron spectroscopy and density functional theory calculations

Vibrationally-resolved photoelectron spectra of AlMoO(y)(-) (y = 1-4) are presented and analyzed in conjunction with density functional theory computational results. The structures determined for the AlMoO(y) anion and neutral clusters suggest ionic bonding between Al(+) and a MoO(y)(-) or MoO(y)(-2) moiety, and point to the relative stability of Mo=O versus Al=O bonds. The highest occupied and partially occupied orbitals in the anions and neutrals can be described as Mo atomic-like orbitals, so while the Mo is in a higher oxidation state than Al, the most energetically accessible electrons are localized on the molybdenum center.

Parameterization of tight-binding models from density functional theory calculations

We present a rigorous bottom-up approach for the derivation of the electronic structure part of tight-binding (TB) models from density functional theory (DFT) calculations. The approach is based on a simultaneous optimization and projection of atomic-like orbitals on self-consistent DFT wave functions and is universally applicable to elements and compounds in arbitrary structural arrangements. The quality and transferability of the derived TB bond and overlap integrals are demonstrated for the examples of a covalent semiconductor (carbon), a transition metal (titanium), and a binary compound with mixed metallic-covalent bonding (TiC). The method can serve as a transparent and physically justified coarse-graining scheme for the construction of nonorthogonal and orthogonal TB total-energy models as well as the closely related bond-order potentials.

Effect of HubbardUon the construction of low-energy Hamiltonians for LaMnO3via maximally localized Wannier functions

We use maximally localized Wannier functions to construct tight-binding (TB) parameterizations for the e_g bands of LaMnO_3 based on first principles electronic structure calculations. We compare two different ways to represent the relevant bands around the Fermi level: i) a d-p model that includes atomic-like orbitals corresponding to both Mn(d) and O(p) states in the TB basis, and ii) an effective e_g model that includes only two e_g-like Wannier functions per Mn site. We first establish the effect of the Jahn-Teller distortion within the d-p model, and then compare the TB representations for both models obtained from GGA+U calculations with different values of the Hubbard parameter U. We find that in the case of the d-p model the TB parameters are rather independent on the specific value of U, if compared with the mean-field approximation of an appropriate multi-band Hubbard Hamiltonian. In contrast, the U dependence of the TB parameters for the effective e_g model cannot easily be related to a corresponding mean-field Hubbard model, and therefore these parameters depend critically on the specific value of U, and more generally on the specific exchange-correlation functional, used in the electronic structure calculation.

Relativistic calculations of the charge-transfer probabilities and cross sections for low-energy collisions of H-like ions with bare nuclei

A new method for solving the time-dependent two-center Dirac equation is developed. The time-dependent Dirac wave function is represented as a sum of atomic-like Dirac-Sturm orbitals, localized at the ions. The atomic orbitals are obtained by solving numerically the finite-difference one-center Dirac and Dirac-Sturm equations with the potential which is the sum of the exact reference-nucleus potential and a monopole-approximation potential from the other nucleus. An original procedure to calculate the two-center integrals with these orbitals is proposed. The approach is tested by calculations of the charge transfer and ionization cross sections for the H(1s)--proton collisions at proton energies from 1 keV to 100 keV. The obtained results are compared with related experimental and other theoretical data. To investigate the role of the relativistic effects, the charge transfer cross sections for the Ne^{9+}(1s)--Ne^{10+} (at energies from 0.1 to 10 MeV/u) and U^{91+}(1s)--U^{92+} (at energies from 6 to 10 MeV/u) collisions are calculated in both relativistic and nonrelativistic cases.

Anomalous Hall conductivity: Local orbitals approach

A theory of the anomalous Hall conductivity based on the properties of single site orbitals is presented. Effect of the finite electron life time is modeled by energy fluctuations of atomic-like orbitals. Transition from the ideal Bloch system for which the conductivity is determined by the Berry phase curvatures to the case of strong disorder for which the conductivity becomes dependent on the relaxation time is analyzed. Presented tight-binding model gives by the unified way experimentally observed qualitative features of the anomalous conductivity in the so called good metal regime and that called as bad metal or hopping regime.

Empirical LCAO parameters for π molecular orbitals in planar organic molecules

A novel parametrization within a simplified LCAO model (a type of Hückel model) is presented for the description of π molecular orbitals in organic molecules containing π-bonds between carbon, nitrogen, or oxygen atoms with sp2 hybridization. It is shown that the model is quite accurate in predicting the energy of the highest occupied π orbital and the first π–π* transition energy for a large set of organic compounds. Four empirical parameter values are provided for the diagonal matrix elements of the LCAO description, corresponding to atoms of carbon, nitrogen with one pz electron, nitrogen with two pz electrons, and oxygen. The bond-length dependent formula of Harrison (proportional to 1/d 2) is used for the non-diagonal matrix elements between neighbouring atoms. The predictions of our calculations have been tested against available experimental results in more than sixty organic molecules, including benzene and its derivatives, polyacenes, aromatic hydrocarbons of various geometries, polyenes, ketones, aldehydes, azabenzenes, nucleic acid bases and others. The comparison is rather successful, taking into account the small number of parameters and the simplicity of the LCAO method, involving only pz atomic-like orbitals, which leads even to analytical calculations in some cases.

Recent developments in tight-binding approaches for nanowires

Full-band nanowire simulations pose significant computational challenges. Nanowires and nanostructures in general have many interfaces, may be composed of alloys, and feature confinement on a scale of a few tens of nanometers. The empirical tight-binding approach is well-suited for modeling these devices: Its basis consists of atomic-like orbitals with limited-range interactions and reasonably-sized basis sets can accurately reproduce the bands of a wide range of semiconductors. The method easily accommodates strain and electromagnetic fields. Over the years the application of the tight-binding approach to nanodevices such as superlattices and resonant-tunneling diodes has led to the development of many useful computational techniques. Recently, its application to random-alloy nanowire calculations has led to the development of approximate bandstructure methods superior to the Virtual-Crystal Approximation for these nanostructures, and its use in nanowire transmission calculations has led to a highly efficient method for transmission calculations. I discuss tight-binding models generally and then give a more in-depth discussion of the recent developments in tight-binding models as applied to nanowires.

The Use of the “+<i>U</i>” Correction in Describing Defect States at Metal Oxide Surfaces: Oxygen Vacancies on CeO<sub>2</sub> and TiO<sub>2</sub>, and Li-doping of MgO

There are many examples of defects in strongly-correlated metal oxides for which density functional theory predicts electronic structures that qualitatively disagree with experimental data. This behaviour arises from the self-interaction error inherent to standard density functionals, and is demonstrated by both p- and n-type systems where the defect state is a small polaron associated with host lattice atoms. An approximate correction is to describe the electron—electron interactions in the orbitals of interest within the DFT+U formalism. This gives improved descriptions for systems where the states of interest are well represented by atomic-like orbitals. The qualitative failure of standard DFT and corresponding improvement achieved with DFT+U is illustrated for cases where the defect state is primarily associated with localised cation f and d states (O vacancies in CeO2 and TiO2) and anion p states (Li-doped MgO). [DOI: 10.1380/ejssnt.2009.389]

The nature of the chemical bond

In this paper we briefly review the basic requirements that must be satisfied by any wave function representing many-electron systems. Following that, we examine the conditions under which the classical concepts of molecular structure, chemical structure and chemical bond can be translated into a quantum-mechanical language. Essential to this aim is the utilization of an independent particle model (IPM) for a many-electron system. In spite of the great popularity of the Hartree-Fock (HF) model only Valence-Bond (VB) type wave functions with optimized, singly occupied and non necessarily orthogonal atomic-like orbitals, can provide a quantum-mechanical translation of the classical concepts of chemical structure and chemical bond, although the HF model can still be useful for translating the concept of molecular structure. Finally, a quantum-dynamical-type of analysis allows us to conclude that, from the quantum mechanical point of view, the chemical bond is a consequence of interference effects. From the energetic point of view, the interference effect responsible for the bond formation manifests itself as a reduction of the kinetic energy of the electrons as the bond is formed.

[formula omitted]: New material. A DFT study

In the present work we have determined the band structure and the densities of states (DOS) of Tb0.5Bi0.5MnO3 in cubic phase using the density functional theory (DFT). The determination of the lattice constant was made by minimizing the energy as a function of volume. The internal coordinate of the O-atom was optimized. From these optimizations we have found that the lattice constant is 8.482Å and the bond length between Tb and O-atoms is 2.193Å and the one for Bi and O-atoms is 2.051Å. From a study of the DOS of f-Tb, eg and t2g–Mn atomic-like orbitals for each spin polarization we conclude that there exists a permanent magnetic moment of about 16Bohr magneton per unit cell. Also, a metallic behavior is displayed, and thus it is expected that ferroelectricity will disappear. Inside the conduction band, next to Fermi level, two peaks in the DOS-spin-down are observed, which are due to f-Tb. Therefore, conduction electrons will be spin-down polarized and this shows that Tb0.5Bi0.5MnO3 has possible applications in spintronics.

Role of charge transfer configurations in LaMnO3, CaMnO3, and CaFeO3

A simple scheme is proposed to analyze the N-electron wave function obtained in embedded cluster calculations in valence bond terms such as ligand-to-metal charge transfer and non-charge-transfer determinants. The analysis is based on a unitary transformation of pairs of natural orbitals to optimal atomiclike orbitals. The procedure is applied to compare the degree of ionicity in NiO and MnO and to explain the existence or absence of Jahn-Teller distortions in LaMnO3, CaMnO3, and CaFeO3. The authors find that the ground state of LaMnO3 is dominated by non-charge-transfer configurations, whereas the charge transfer configurations dominate the ground state wave function in the other two perovskites.

Construction of Wannier functions from localized atomiclike orbitals

The problem of construction of the Wannier functions (WFs) in a restricted Hilbert space of eigenstates of the one-electron Hamiltonian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}$ (forming the so-called low-energy part of the spectrum) can be formulated in several different ways. One possibility is to use the projector-operator techniques, which pick up a set of trial atomic orbitals and project them onto the given Hilbert space. Another possibility is to employ the downfolding method, which eliminates the high-energy part of the spectrum and incorporates all related to it properties into the energy-dependence of an effective Hamiltonian. We show that by modifying the high-energy part of the spectrum of the original Hamiltonian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}$, which is rather irrelevant to the construction of the WFs in the low-energy part of the spectrum, these two methods can be formulated in an absolutely exact and identical form, so that the main difference between them is reduced to the choice of the trial orbitals. Concerning the latter part of the problem, we argue that an optimal choice for the trial orbitals can be based on the maximization of the site-diagonal part of the density matrix. This idea is illustrated on a simple toy model consisting of only two bands, as well as on a more realistic example of ${t}_{2g}$ bands in ${\mathrm{V}}_{2}{\mathrm{O}}_{3}$. Using the model analysis, we explicitly show that a bad choice of the trial orbitals can be linked to the discontinuity of phase of the Bloch waves in the reciprocal space, which leads to the delocalization of the WFs in the real space. Nevertheless, such a discontinuity does not necessary contribute to the matrix elements of $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}$ in the Wannier basis. Similar tendencies are seen in more realistic calculations for the ${t}_{2g}$ bands in ${\mathrm{V}}_{2}{\mathrm{O}}_{3}$, though with some variations caused by the multi-orbital effects. An analogy with the search of the ground state of a many-electron system is also discussed.

Optimized atomic-like orbitals for first-principles tight-binding molecular dynamics

We analyze the optimization of atomic-like minimal basis sets for the hydrocarbons and for materials made up only of C atoms, e.g. C-nanotubes. In our approach the optimized numerical atomic orbitals (NAOs) are obtained as a linear combination of only two primitive NAOs. We find that the optimized basis sets yield an important lowering of the total energy, and bondlengths in very good agreement with the experimental evidence. Also, we find that a good “universal” minimal basis set for the hydrocarbons and C-only materials can be obtained using this simple method. The approach discussed in this paper is a promising tool for the simulation of complex organic materials, beyond the hydrocarbons, using optimized minimal basis sets.

Origin of the negative volume magnetostriction of the intermetallic compound GdAl 2

Spontaneous volume magnetostriction of the intermetallic compound GdAl 2 with a cubic C15 Laves structure is studied from first principles using the local spin-density approximation combined optionally with an open-core treatment of the atomic-like Gd-4f orbitals. Local magnetic moments and equilibrium lattice parameters are evaluated for two spin structures, namely for the ferromagnetic ground state and for a state with randomly oriented Gd moments. The resulting spontaneous volume magnetostriction compares well with experimentally observed values; its negative sign however cannot be ascribed to a variation of the size of the local Gd moment but has its origin in the volume dependence of exchange interactions between the Gd moments. The Curie temperature and its pressure dependence are derived from the calculated pair exchange interactions as well; their values are found in reasonable agreement with experiment supporting thus the above explanation of the anomalous magnetoelastic behavior of the GdAl 2 compound.