One-electron Energy Research Articles

An investigation of the kohn-sham and slater approximations to the hartree-fock exchange potential

The self-consistent-field equations for the ground state of the argon atom have been solved for a range of values of the parameter C in the local approximation: to the Hartree-Fock exchange potential. The orbital solutions with and without the Latter tail modification are used to investigate the relative merits of the Slater (C = 1) and Kohn-Sham (C = 2/3) local exchange potentials. To do this various expectation values are calculated. Total and one-electron binding energies are calculated as the expectation value of the Slater determinant built up from the orbitals and by using the statistical approximation to that expression. It is this statistical approximation for the total energy which Kohn-Sham varied to obtain their local exchange potential. When computing one-electron binding energies for frozen orbitals in the statistical approximation, it is only in the Slater case (C = 1) that the eigen-values of the self-consistent-field equations may be so interpreted. The role of the virial theorem and the improvements yielded by scaling are discussed. The Kohn-Sham model compares better with Hartree-Fock than the Slater model with regard to total energy, one-electron binding energies and the virial theorem. Other choices for C, according to various criteria, are discussed with the view towards a better statistical approximation to the Hartree-Fock exchange potential.

Spin-polarized energy bands in sodium

The augmented-plane-wave (APW) method was used with a spin-dependent potential to calculate the one-electron energies in solid sodium. The charge density from which this potential was obtained was derived from spin-polarized Hartree-Fock-Slater atomic wave functions with fractional occupation of the valence spin orbitals. The atoms were placed on a cesium chloride Bravais lattice with alternating spin density. This “antiferromagnetic” arrangement was maintained with a decreasing moment as the calculation was carried to self-consistency. The possibility of this state having a lower total energy than the “paramagnetic” state is discussed. A “spin-density-wave” potential was introduced into the Hamiltonian and a perturbation calculation was performed. The unperturbed functions used in this calculation were self-consistent solutions of the “paramagnetic” state. By fitting the bands so obtained to the above “antiferromagnetic” bands the amplitude of the “spin-density-wave” in that configuration was determined. The possibility of obtaining a state with a lower total energy by using a “spin-density-wave” with this amplitude and varying wavelength is discussed.

Multicenter distribution of molecular integrals and energy components

A central problem in both semiempirical and ab initio molecular electronic structure theory is treatment of the three- and four-center interactions. A systematic evaluation of multicenter contributions and a search for invariants has been carried out by computing frequency distribution functions for one-, two- versus three-, four-center two-electron integrals and by separating one-, two-center from three-, four-center terms in the one-electron energies of molecular-orbital theory. Data have been obtained for a considerable number of polyatomic molecules and representative results are presented for NH−2, NH3, CH4, H3O+, C2N2, C3H+3, N4, and C6H6.

Application of the multiple scattering Xα method to the dirheniumoctachloride anion Re2Cl 2−8

Crystal field theory predicts the existence of a quadruple bond in Re2Cl2−8. The multiple scattering Xα method has been used to verify this as well as to provide the ordering of both occupied and unoccupied one-electron energy states.

Size quantization effects in thin film Casimir interaction

We investigate the role of size quantization in the vacuum force between metallic films of nanometric thickness. The force is calculated by the Lifshitz formula with the film dielectric tensor derived from the one-electron energies and wavefunctions under the assumption of a constant potential inside the film and a uniform distribution of the positive ion charge. The results show that quantization effects tend to reduce the force with respect to the continuum plasma model. The reduction is more significant at low electron densities and for film size of the order of few nanometers and persists for separation distances up to 10-50 nm. Comparison with previous work indicates that the softening of the boundary potential is important in determining the amount of the reduction. The calculations are extended to treat Drude intraband absorption. It is shown that the inclusion of relaxation time enhances the size quantization effects in the force calculations.

Td B(BO)4−: A Tetrahedral Boron Oxide Cluster Analogous to Boron Hydride Td BH4−

A density functional theory and wave function theory investigation on the geometrical and electronic structures of B5O4(0/-) clusters has been performed in this work. B5O4(-) anion proves to possess a perfect tetrahedral ground state of T(d) B(BO)4(-) ((1)A1) analogous to BH4(-) with four equivalent -BO terminals around the B center, while B5O4 neutral favors a slightly off-planed C(s) B(BO)4 ((2)A') which contains three -BO terminals and one -O- bridge. An intramolecular BO radical transfer occurs from T(d) B(BO)4(-) to C(s) B(BO)4 when one electron is detached from the anion. The one-electron detachment energies of the anion and characteristic stretching vibrational frequencies of -B=O groups at about 2000 cm(-1) have been calculated to facilitate future experimental characterization of these clusters.

Schottky defects in cubic lattice of SrTiO 3

Structural and electronic properties produced by formation of Schottky defects in cubic structure of SrTiO 3 crystal are investigated by means of a quantum-chemical simulation based on the Hartree–Fock methodology. The occurrence of Sr partial Schottky defect (V Sr+V O) and two types of Ti partial Schottky defects (V Ti+2V O) is modeled using a supercell containing 135 atoms. Vacancy-induced changes in the positions of their neighboring atoms are analyzed in light of the computed electron density redistribution in the defective region of supercell. The observed local one-electron energy levels in the gap between the upper valence band and the conduction band can be attributed to the presence of anion and cation vacancies.

Calculations of electronic structure of the UF6 molecule and the UO2 crystal with a relativistic pseudopotential

The influence of relativistic effects on the properties of uranium hexafluoride was considered. Detailed comparison of the spectrum of one-electron energies obtained in the nonrelativistic (by the Hartree-Fock method), relativistic (by the Dirac-Fock method), and scalar-relativistic (using a relativistic potential of the uranium atom core) calculations was carried out. The methods of optimization of atomic basis in the LCAO calculations of molecules and crystals are discussed which make it possible to consider distortion of atomic orbitals upon the formation chemical bonds. The influence of the atomic basis optimization on the results of scalar-relativistic calculations of the molecule UF6 properties is analyzed. Calculations of the electronic structure and properties of UO2 crystals with relativistic and nonrelativistic pseudopotentials are fulfilled.

Benchmark Dyson Orbital Study of the Ionization Spectrum and Electron Momentum Distributions of Ethanol in Conformational Equilibrium

An extensive study, throughout the valence region, of the electronic structure, ionization spectrum, and electron momentum distributions of ethanol is presented, on the ground of a model that focuses on a mixture of the gauche and anti conformers in their energy minimum form, using weight coefficients obtained from thermostatistical calculations that account for the influence of hindered rotations. The analysis is based on accurate calculations of valence one-electron and shakeup ionization energies and of the related Dyson orbitals, using one-particle Green's Function (1p-GF) theory in conjunction with the so-called third-order Algebraic Diagrammatic Construction scheme [ADC(3)]. The confrontation against available UPS (HeI) measurements indicates the presence in the spectral bands of significant conformational fingerprints at outer-valence ionization energies ranging from approximately 14 to approximately 18 eV. The shakeup onset is located at approximately 24 eV, and a shoulder at approximately 14.5 eV in the He I spectrum can be specifically ascribed to the minor anti (C(s)) conformer fraction. Thermally and spherically averaged Dyson orbital momentum distributions are computed for seven resolvable bands in model (e, 2e) ionization spectra at an electron impact energy of 1.2 keV. A comparison is made with results obtained from standard (B3LYP) Kohn-Sham orbitals and EMS measurements employing a high-resolution spectrometer of the third generation. The analysis is qualitatively in line with experiment and reveals a tremendously strong influence of the molecular conformation on the outermost electron momentum distributions. Quantitatively significant discrepancies with experiment can nonetheless be tentatively ascribed to strong dynamical disorder in the gas phase molecular structure.

First-principles calculations of structural, electronic and optical properties of orthorhombic CaPbO3

Calculations within the density functional theory approach were performed to obtain structural parameters, electronic band structure, carrier effective masses and optical absorption spectra in orthorhombic CaPbO3. Both local density and generalized gradient approximations, LDA and GGA, respectively, were considered. A comparison reveals good agreement of the calculated lattice parameters with experimental results. A direct Γ → Γ one-electron energy band gap of 0.84 eV (0.94 eV) was obtained within the GGA (LDA) level of calculation, in contrast to a previous interpretation of experimental data pointing to a gap of only 0.43 eV. Comparting our results with band gap energies previously obtained for CaXO3 crystals (X = C in calcite, X = Si in wollastonite and X = Ge,Sn,Pb in the orthorhombic phase), we note that the energy gap oscillates, but with an overall trend to decrease, as the atomic number of the X atomic species increases.

F and F+ Centers on MgO/Ag(100) or MgO/Mo(100) Ultrathin Films: Are They Stable?

Using density functional theory (DFT) calculations we have investigated the electronic structure and stability of diamagnetic F0 and paramagnetic F+ centers in perfect and stepped ultrathin MgO films on Ag(100) and Mo(100) supports. The results show that when the one-electron energy of an F0 defect state is very close to the metal Fermi level, like in MgO films on Ag(100), the doubly occupied F0 center is metastable and can transform spontaneously into the paramagnetic F+ center by losing one of its electrons to the metal support. The phenomenon is enhanced when the vacancy is closer to the interface or is located at a step edge. On the MgO/Mo(100) films, where the Fermi level is at higher energy, F0 centers are stable and do not show a tendency to transform into F+, but F+ centers are unstable and capture one electron from the Mo metal to generate the corresponding doubly occupied vacancy.

Accurate band gaps and dielectric properties from one-electron theories (abstract only)

For semiconductor modeling, a major shortcoming of density functional theory is that thepredicted band gaps are usually significantly too small. It is generally argued that thisshortcoming is related to the fact that density functional theory is a ground state theory,and as a result, one is not allowed to associate the one-electron energies with the energies ofquasi-particles. Although this fundamental objection is certainly correct, the modeling ofthe positioning of donor and acceptor levels in semiconductors faces serious limitations withpresent density functionals.Several solutions to this problem have been suggested. A particular attractive and fairlysimple one is the inclusion of a small fraction of the non-local exchange in the Hamiltonian(hybrid functionals). This approach leads to sensible band gaps for most semiconductors,but fails for ionic solids. A more reliable approach is via many-electron Green’sfunction techniques, which have made tremendous advances in recent years. Here GW calculations in various flavors are presented for small gap and large gap systems,comprising typical semiconductors (Si, SiC, GaAs, GaN, ZnO, ZnS, CdS andAlP), small gap semiconductors (PbS, PbSe, PbTe), insulators (C, BN, MgO,LiF) and noble gas solids (Ar, Ne). The general finding is that single-shot G0W0 calculations based on wavefunctions obtained from conventionaldensity functional theory yield too small band gaps, whereas G0W0 calculations following hybrid functional calculations tend to overestimate the bandgaps by roughly the same amount. This is at first sight astonishing, since thehybrid functionals yield very good band gaps themselves. The contradiction isresolved by showing that the inclusion of the attractive electron–hole interactions(excitonic effects) is required to obtain good static and dynamic dielectricfunctions using hybrid functionals. The corrections are usually incorporated in GW calculations using ‘vertex corrections’, and in fact inclusion of these vertex correctionsrectifies the predicted band gaps.Finally, in order to remove the dependence on the initial wavefunctions, self-consistent GW calculations are presented, again including an approximate treatment of vertexcorrections. The results are in excellent agreement with experiment, with a few per centdeviation for all materials considered. We conclude that predictive band gapengineering is now possible with the theoretical description approaching experimentalaccuracy.

Structure, optical properties and defects in nitride (III–V) nanoscale cage clusters

Density Functional Theory calculations are reported on cage structured BN, AlN, GaN and InN sub- and low nanosize stoichiometric clusters, including two octahedral families of T(d) and T(h) symmetry. The structures and energetics are determined, and we observe that BN clusters in particular show high stability with respect to the bulk phase. The cluster formation energy is demonstrated to include a constant term that we attribute to the curvature energy and the formation of six tetragonal defects. The (BN)(60) onion double-bubble structure was found to be particularly unstable. In contrast, similar or greater stability was found for double and single shell cages for the other nitrides. The optical absorption spectra have been first characterised by the one-electron Kohn-Sham orbital energies for all compounds, after which we concentrated on BN where we employed a recently developed Time Dependent Density Functional Theory approach. The one-electron band gaps do not show a strong and consistent size dependency, in disagreement with the predictions of quantum confinement theory. The density of excited bound states and absorption spectrum have been calculated for four smallest BN clusters within the first ionisation potential cut-off energy. The relative stability of different BN clusters has been further explored by studying principal point defects and their complexes including topological B-N bond rotational defects, vacancies, antisites and interstititials. The latter have the lowest energy of formation.

Quantum Theory of Orbital Magnetization and Its Generalization to Interacting Systems

Based on standard perturbation theory, we present a full quantum derivation of the formula for the orbital magnetization in periodic systems. The derivation is generally valid for insulators with or without a Chern number, for metals at zero or finite temperatures, and at weak as well as strong magnetic fields. The formula is shown to be valid in the presence of electron-electron interaction, provided the one-electron energies and wave functions are calculated self-consistently within the framework of the exact current and spin-density functional theory.

Theoretical Prediction of Intrinsic Self-Trapping of Electrons and Holes in MonoclinicHfO2

We predict, by means of ab initio calculations, stable electron and hole polaron states in perfect monoclinic HfO2. Hole polarons are localized on oxygen atoms in the two oxygen sublattices. An electron polaron is localized on hafnium atoms. Small barriers for polaron hopping suggest relatively high mobility of trapped charges. The one-electron energy levels in the gap, optical transition energies and ESR g-tensor components are calculated.

Polyoxometalates with Internal Cavities: Redox Activity, Basicity, and Cation Encapsulation in [Xn+P5W30O110](15-n)- Preyssler Complexes, with X = Na+, Ca2+, Y3+, La3+, Ce3+, and Th4+

The Preyssler anion, of general formula [Xn+P5W30O110](15-n)-, is the smallest polyoxometalate (POM) with an internal cavity allowing cation exchange with the solution. The Preyssler anion features a rich chemistry evidenced by its ability to accept electrons at low potentials, to selectively capture various metal cations, and to undergo acid-base reactions. A deep understanding of these topics is herein provided by means of DFT calculations on the title series of compounds. We evaluate the energetics of the release/encapsulation process for several Xn+ cations and identify the effect of the encapsulated ion on the properties of the Preyssler anion. We revisit the relationship between the internal cation charge and the electrochemical behavior of the POM. A linear dependence between the first one-electron reduction energies and the encapsulated Xn+ charge is found, with a slope of 48 mV per unit charge. The protonation also shifts the reduction potential to more positive values, but the effect is much larger. In connection to this, the last proton's pKa = 2 for the Na+ derivative was estimated to be in reasonable agreement with experiment. The electronic structure of lanthanide derivatives is more complex than conventional POM structures. The reduction energy for the CeIV-Preyssler + 1e- --> CeIII-Preyssler process was computed to be more exothermic than that of very oxidant species such as S2Mo18O624-.

Non-iterative approach to the infinite-order two-component (IOTC) relativistic theory and the non-symmetric algebraic Riccati equation

The formal framework of the recently developed infinite-order two-component (IOTC) method of relativistic quantum chemistry is analysed. The former iterative scheme of the IOTC approach is replaced by the non-iterative method which leads to the complete block-diagonalization of the algebraic Dirac Hamiltonian and permits to built the matrix representation of the equivalent two-component one-electron energy operator. The method is equivalent to solving the so-called non-symmetric algebraic Riccati equation. Its solution is illustrated numerically.

Spectroscopic properties of oxygen vacancies in monoclinicHfO2calculated with periodic and embedded cluster density functional theory

We have calculated the electronic structure and spectroscopic properties of the oxygen vacancy in different charge states in the monoclinic phase of $\mathrm{Hf}{\mathrm{O}}_{2}$. Periodic and embedded cluster calculations using density functional theory and a hybrid density functional reproduce the band gap of this material with good accuracy and predict the positions of the one-electron energy levels corresponding to five charge states of the vacancy in the band gap. The optical transition energies as well as optical and thermal ionization energies into the conduction band for all vacancy charge states and the $g$ tensor for electron spin resonance (ESR) active states are calculated. We discuss the relation of the calculated properties to metrology of vacancies using spectroscopic ellipsometry, ESR, and electrical stress measurements.

Redox free energies and one-electron energy levels in density functional theory based ab initio molecular dynamics

Abstract The density functional theory based ab initio molecular dynamics method combines electronic structure calculation and statistical mechanics and should, therefore, ideally be a tool for “first principle” computation of redox free energies in the sense that the redox active solutes and solvent are treated at the same level of theory. In this paper, we give a brief outline how such an approach can be implemented in the framework of the Marcus theory of electron transfer. The method is illustrated and validated using results of previous work. We then continue with a theoretical analysis of the correlation between the energies of one-electron states and redox potentials exploiting the separation in vertical ionization and reorganization contributions inherent in Marcus theory. Testing this relation on the limited set of reactions investigated sofar we find that it is satisfied within the uncertainties of the computation.

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.