Type Orbitals Research Articles

Magnetic Coupling in the Ce(III) Dimer Ce2(COT)3.

The monomer [Ce(COT)2]- and the dimer [Ce2(COT)3], with Ce(III) and COT = 1,3,5,7-cyclooctatetraenide, are studied by quantum chemistry calculations. Due to the large spin-orbit coupling, the ground state of the monomer is a strong mixing of σ and π states. The experimental isotropic coupling in the dimer was evaluated by Walter et al. to be J = -7 cm-1 (with a Heisenberg Hamiltonian [Formula: see text]) with a small anisotropic coupling of 0.02 cm-1. The coupling between the two Ce(III) in the dimer is calculated using CI methods. The low energy part of the spectra are modeled by spin Hamiltonians. All spin Hamiltonians parameters are deduced from ab initio calculations. g factors are calculated for both the pseudodoublet of the monomer and the pseudotriplet of the dimer and their sign have been determined. The magnetic coupling in the dimer is rationalized by a model based on crystal field theory. The kinetic and exchange contributions arising from the different configurations to the isotropic and anisotropic couplings are evaluated. It is shown that the main contribution to isotropic coupling is kinetic and originates from the fσ-fσ interaction due to the large transfer integral between those orbitals. However, the fπ-fπ interaction plays a non-negligible role. The anisotropic coupling originates from the difference of exchange energy of states arising from the fσfπ configuration and is, in no matter, related to the anisotropy of the local magnetic moments as already pointed by van Vleck for a fictitious s-p system. The analysis of the natural orbitals evidences a superexchange mechanism through a σCH* orbital of the bridging cycle favored by a local 4fσ/5dσ hybridization and that the δ type orbitals, both the HOMOs of the ligands and the virtual fδ orbitals of the cerium atoms play an important polarization role, and to a less extend the π type orbitals, the HOMOs-1 of the ligands, and the metal fπ orbitals.

Use of lined-up auxiliary functions in the evaluation of overlap integrals over integer and non-integer n Slater type orbitals

Abstract A combined theory for the overlap integrals of integer and non-integer n χ -Slater type orbitals ( χ -ISTO and χ -NISTO) is presented. This approach is based on the use of lined-up auxiliary functions introduced by one of the authors. The suggested theory guarantees a highly accurate calculation of the overlap integrals for the arbitrary values of quantum numbers and locations of orbitals.

The use of quantum damping self-frictional theory in a study of hydrogen-like atomic energies and forces

Abstract By the use of complete orthogonal sets of ψ ( δ ∗ ) -damping self-frictional (DSF) exponential type orbitals ( ψ ( δ ∗ ) -DSFETO) introduced by one of the authors, the quantum DSF study of hydrogen-like atomic energies and forces are suggested. Here, δ ∗ = α ∗ or δ ∗ = p l ∗ , p l ∗ = 2 l + 2 - α ∗ and α ∗ = α - nl ∗ ( z ) are the integer and non-integer DSF quantum numbers which depend on quantities n , l and z , where z = H, He+, Li++, Be+++, B++++, C+++++, N++++++, O+++++++, F++++++++, Ne+++++++++. The average values of potentials, kinetic and total energies and forces of the hydrogen-like atomic systems are calculated. The given values are compared with the results obtained in the non-DSF (NDSF) theory.

LOCALIZED ELECTRONIC SURFACE STATES IN METALLIC STRUCTURES

We present theoretical models to study the localized electronic surface states in metallic structures. The materials under study have been chosen with different types of cubic meshes, fcc, sc and bcc. The calculation method used is closely related to the Linear Combination of Atomic Orbitals (LCAO) in the tight-binding method. We consider three cases: each of the atoms is described by a single atomic orbital of [Formula: see text]-, [Formula: see text]- and [Formula: see text]-type orbitals. In order to solve the rectangular secular equations of the systems under study, the phase field matching method is involved. In particular, we apply our approach to calculate the localized electronic surface states of some metals: (i) Chromium and Silver having, respectively, bcc and fcc structure and described as [Formula: see text]-type orbital. (ii) Nickel with sc crystallization and described by [Formula: see text]-type orbital. (iii) Palladium (Pd) given in fcc crystallization and described by [Formula: see text]-type orbital. The obtained results illustrate spatial edge effects between the bulk modes and the localized electronic states of the metallic surfaces over the three orientations of high symmetry path. We observe many localized states above and below the bulk band range. In addition, the relaxation effect on the surface layer has been investigated to compute the localized electronic surface state in this case and illustrate the lift of the degeneracy compared to the first calculations based on an ordered surface. The spacing geometry caused by the relaxation on the surface has been determined by using the Molecular dynamic algorithm and Morse interatomic potential.

A theoretical model to compute the localized electronic states at the surface of hexagonal structures with different coupling orbitals

We present a theoretical model to compute the localized electronic states at the surface of an hexagonal compact packing (hcp) structures. The system under study is assimilated to a surface layer that ends a 3D semi-infinite hcp system, in [100] direction.To compute the model, we have used the phase field matching method (PFMM) along with the linear combination of atomic orbitals (LCAO) in the tight-binding approach. The PFMM helps to solve the rectangular secular equations generated by the lack of symmetry on the surface. Moreover, the localized surface states of the electrons on the surface were found by making use of the scattering reflection probabilities, as given in the Landauer and Büttiker formalism.In this work, we have studied three cases: each atom, in the surface, is described by the interaction energies of type f-, p-f- and s-p-f- coupling orbitals. Practically, our model calculations were applied to compute the localized electronic states of the following elements: (i) Gadolinium (Gd) described as f-like orbital. (ii) Lutetium (Lu) given as p-f- type orbitals. (iii) Thallium (Tl) described by s-p-f- type orbitals.The results show many localized surface branches above and down the bulk bands. Some of these states are resonant since they are located in the bulk range, and others are proper surface states since they are observed outside the bulk band range. In addition, we have investigated the relaxation effect on the surface by calculating the localized electronic surface states in the case of f-coupling orbitals. The spacing geometry caused by the relaxation on the surface has been determined by using the Molecular Dynamic algorithm. In this case, the results detail the apparition of more localized electronic states inside and outside the bulk range while integrating the relaxation effects.

Dependence of electron impact differential cross sections on the ionic charge to radius ratio for the Al3+(2p) and Be2+(1s) ions.

The triple differential cross sections (TDCSs) have been obtained for the electron impact ionization of ionic targets, Al3+(2p) and Be2+(1s), having nearly the same ratio of ionic charge to radius. In the first of this kind of study, the trends of cross sections have been found to match to a greater extent despite ionization taking place from the ionic targets having considerable difference in nuclear charges as well as the ionization taking place from different types of orbitals, p-orbital and s-orbital. The trends of TDCSs have not been found to agree considerably for the neutral Al (3p) and Be (2s) targets.

Impact of Coulomb potential on peak structures arising in momentum and low-energy photoelectron spectra produced in strong-field ionization of laser-irradiated atoms

The phenomenon of pronounced peak structure(s) of longitudinal momentum distributions as well as a spike-like structure of low-energy spectra of photoelectrons emitted from laser-irradiated Ar and Ne atoms in a single ionization process is theoretically studied in the tunneling and multiphoton regimes of ionization. The problem is addressed assuming only the direct above-threshold ionization (ATI) as a physical mechanism underlying the phenomenon under consideration (viz. solely contributing to observed photoelectron momentum distributions (PMD)) and using the Coulomb–Volkov (CV) ansatz within the frame of conventional strong-field approximation (SFA) applied in the length-gauge formulation. The developed CV-SFA approach also incorporates the density functional theory essentially exploited for numerical composition of initial (laser-free) atomic state(s) constructed from atomic orbitals of Gaussian type. Our presented CV-SFA based (and laser focal-volume averaged) calculation results proved to be well reproducing both the pronounced double-peak and/or ATI-like multi-peak structure(s) experimentally observed in longitudinal PMD under conditions of tunneling and/or multiphoton regime, respectively. In addition, our CV-SFA results presented for tunneling regime also suggest and remarkably reproduce a pronounced structure observed in relevant experiments as a ‘spike-like’ enhanced maximum arising in low-energy region (around the value of about 1 eV) of photoelectron spectra. The latter consistency allows to identify and interpret these results as the so-called low-energy structure (LES) since the phenomenon proved to appear as the most prominent if the influence of Coulomb potential on photoelectron continuum states is maximally taken into account under calculations (viz. if the parameter Z in CV’s functions is put equal to 1). Moreover, the calculated LES proved to correspond (viz., established as closely related) to the mentioned double-peak structure arising in the low-momentum region ( a.u.) of longitudinal PMDs calculated under condition of the tunneling regime. Thus, the phenomena under consideration can be well understood and adequately interpreted beyond the terms and/or concepts of various different alternative strong-field approaches and models (such as e.g., extensively invoked and exploited nowadays though, more sophisticated SFA-based ‘rescattering’ mechanism) compared to which, the currently applied CV-SFA model (through the same underlying physical mechanism of solely direct ATI suggested) is additionally able to provide and reveal an intimate and transparent interrelation between the phenomena of LES and double-peak structure arising in PMDs observed in the tunneling regime.

Ab initio modeling of Raman and infrared spectra of calcite

Calcite is the most thermodynamically stable polymorphic phase of the CaCO3 crystal. It is widely used in modern optical instruments operating in the infrared and visible wavelengths of electromagnetic radiation. In particular, due to its anisotropic properties, calcite is used in polarization optics devices. Ab initio quantum mechanical modeling of the Raman and infrared spectra of calcite makes it possible to better understand the structure and nature of the chemical bonds of the compound, and find the optimal conditions for the effective use of unique properties of calcite in photonics tools. The calculations are performed within the framework of the density functional theory in the CRYSTAL program, using the “hybrid” B3LYP functional and the all-electronic bases of the STO-6G, PO-TZVP and BSD atomic orbitals of the Gaussian type. The obtained results for the elastic constants, Raman and infrared spectra of calcite agree satisfactorily with the available experimental data for the basic sets of POB-TZVP and BSD.

Molecules mimicking atoms: monomers and dimers of alkali metal solvated electron precursors

Tetra-amino lithium and sodium complexes M(NH3) (M = Li, Na) have one or two electrons that occupy diffuse orbitals distributed chiefly outside the M(NH3) core. The lowest-energy 1s, 1p, and 1d orbitals follow Aufbau principles found earlier for beryllium tetra-ammonia complexes. Two ground state M(NH3)4 complexes can bind covalently by coupling their 1s1 electrons into a σ-type molecular orbital. The lowest excited states of the [M(NH3)4]2 species are obtained by promoting one or two electrons from this σ to other bonding or anti-bonding σ and π-type molecular orbitals. The electronic structure of solvated electron precursors provides insights into chemical bonding between super-atomic species that are present in concentrated alkali-metal-ammonia solutions.

Low energy electron-impact study of AlO using the R-matrix method

This comprehensive study reports the electron-impact on the open shell AlO molecule at low energy (less than 10 eV) using the R-matrix method. We present the elastic (integrated and differential), momentum-transfer, electronic excitation and ionisation cross sections; along with effective collision frequency over a wide electron temperature range (1000–100 000 K). Correlations via a configuration interaction technique are used to represent the target states. Calculations are performed in the static-exchange and 24-target states close-coupling approximation at the experimental bond length of 1.6178 A. We have used different basis sets 6-311G*, double zeta, polarization (DZP), cc-pCVTZ to represent our target states. We have chosen the Gaussian Type Orbitals (GTOs) basis set DZP to represent the atomic orbitals which gave the best one-electron properties of the molecule. The calculated dipole moment (1.713 au), rotational constant (0.641399 cm-1) and the vertical excitation energies are in concurrence with the best available data. The continuum electron is also represented by GTOs and is placed at the center of mass of the molecule. Resonance analysis is carried out to assign the resonance parameters and the parentage of detected resonances by fitting the eigenphase sums to the Breit–Wigner profile. Our study has detected three core-excited shape resonances in the 24-state model. We detect a stable bound state of AlO- of 1 A 1 symmetry having configuration 1σ 2 … 7σ 21π 42π 4 with a vertical electronic affinity value of 2.59 eV which is in good accord with the experimental value of 2.6 ± (0.01) eV. The ionisation cross sections are calculated using the Binary-Encounter-Bethe Model in which Hartree–Fock molecular orbitals at self-consistent level are used to calculate kinetic and binding energies of the occupied molecular orbitals. We include partial waves up to g-wave beyond which Born closure method is employed to obtain converged cross sections.

Exact solution for the hydrogen atom confined by a dielectric continuum and the correct basis set to study many-electron atoms under similar confinements

The Schrödinger equation associated to the hydrogen atom confined by a dielectric continuum is solved exactly and suggests the appropriate basis set to be used when an atom is immersed in a dielectric continuum. Exact results show that this kind of confinement spread the electron density, which is confirmed through the Shannon entropy. The basis set suggested by the exact results is similar to Slater type orbitals and it was applied on two-electron atoms, where the H− ion ejects one electron for moderate confinements for distances much larger than those commonly used to generate cavities in solvent models.

Correction to: On the effectiveness of exponential type orbitals with hyperbolic cosine functions in atomic calculations

The original version of this article unfortunately contained an error. The first author’s name was missing in Reference 22.

R-matrix calculations of electron impact electronic excitation of BeH

The R-matrix method is used to perform high-level calculations of electron collisions with beryllium mono-hydride at its equilibrium geometry with a particular emphasis on electron impact electronic excitation. Several target and scattering models are considered. The calculations were performed using (1) the UKRMol suite which relies on the use of Gaussian type orbitals (GTOs) to represent the continuum and (2) using the new UKRMol+ suite which allows the inclusion of B-spline type orbitals in the basis for the continuum. The final close-coupling scattering models used the UKRMol+ code and a frozen core, valence full configuration interaction, method based on a diffuse GTO atomic basis set. The calculated electronic properties of the molecule are in very good agreement with state-of-the-art electronic structure calculations. The use of the UKRMol+ suite proved critical since it allowed the use of a large R-matrix sphere (35 Bohr), necessary to contain the diffuse electronic states of the molecule. The corresponding calculations using UKRMol are not possible due to numerical problems associated with the combination of GTO-only continuum and a large R-matrix sphere. This work provides the first demonstration of the utility and numerical stability of the new UKRMol+ code. The inelastic cross sections obtained here present a significant improvement over the results of earlier studies on BeH.

On the effectiveness of exponential type orbitals with hyperbolic cosine functions in atomic calculations

Unconventional basis functions, constructed from exponential type orbitals (ETOs) with hyperbolic cosine functions, are applied to Roothaan-Hartree-Fock calculations of atoms within the minimal basis sets framework. The most popular ETOs, Slater type orbitals, B functions and \(\psi ^{(\alpha ^*)}\) functions with \(\alpha ^*=2\), and two types of hyperbolic cosine functions, \(\cosh (\beta r)\) and \(\cosh (\beta r+\gamma )\), are used in this work. The performance of the present basis functions is investigated and compared to the conventional double-zeta Slater-type basis set and numerical Hartree-Fock results. The improvement in the atomic energies clearly demonstrates how the accuracy increases when we move from ETO to ETO with hyperbolic cosine basis functions. The resulting improved minimal basis sets can also be useful in molecular calculations.

Self-Friction Power Series and Their Use for Construction of One-Range Addition Theorems of Noninteger Slater Functions and Coulomb–Yukawa Like Potentials

Using complete orthogonal \( \mathcal{L}^{{\left( {p_{l}^{*} } \right)}} \)-self-friction polynomials (\( \mathcal{L}^{{\left( {p_{l}^{*} } \right)}} \)-SFPs) in standard convention introduced by the author, the formulas for transformation of noninteger type functions to the power series are presented, where \( p_{l}^{*} = 2l + 2 - \alpha^{*} \) and α * is the integer (\( \alpha^{*} = \alpha ,\; - \infty < \alpha \le 2 \)) or noninteger (\( \alpha^{*} \ne \alpha ,\; - \infty < \alpha^{*} < 3 \)) SF quantum number. With the help of these power series, the one-center one-range addition theorems for χ-Slater type orbitals and V-Coulomb–Yukawa like potentials with noninteger indices (χ-NISTOs and V-NICYPs) are established. As an application, the atomic nuclear attraction integrals of χ-NISTOs and V-NICYPs are investigated.

Self-friction power series and their use for evaluation of atomic nuclear attraction integrals with noninteger indices of Slater orbitals and Coulomb–Yukawa-like potentials

Using complete orthogonal [Formula: see text]-Self-Friction Polynomials ([Formula: see text]-SFPs) introduced by one of the authors, the analytical and power series formulas for SF atomic nuclear attraction integrals over [Formula: see text]-noninteger Slater type orbitals ([Formula: see text]-NISTOs) and [Formula: see text]-noninteger Coulomb–Yukawa-like potentials ([Formula: see text]-NICYPs) are presented, where [Formula: see text] are the integer ([Formula: see text] or noninteger ([Formula: see text] SF quantum numbers and [Formula: see text]. As an application, the computer calculations for dependence of the atomic nuclear attraction integrals over [Formula: see text]-NISTOs and [Formula: see text]-NICYPs functions are presented.

Investigation of the encapsulation of metal cations (Cu2+, Zn2+, Ca2+ and Ba2+) by the dipeptide Phe-Phe using natural bond orbital theory and molecular dynamics simulation.

Complexes of the dipeptide phenylalanine-phenylalanine (Phe-Phe) with divalent metal cations (Cu2+, Zn2+, Ca2+ and Ba2+) were studied at the B3LYP and MP2 levels of theory with the basis sets 6-311++G(d,p) and 6-31 + G(d) in the gas phase. The relative energies of these complexes indicated that cation-π bidentate/tridentate conformations are more favourable than other conformations with uncoordinated rings. These findings were confirmed by the calculated values of thermodynamic parameters such as the Gibbs free energy. Natural bond orbital (NBO) analysis was carried out to explore the metal-ligand coordination in Phe-Phe-Cu2+/Zn2+ complexes. Possible orbital transitions, types of orbitals and their occupancies were determined for a range of Phe-Phe-Cu2+/Zn2+ complexes. The charge transfer involved in various orbital transitions was explored by considering the second-order perturbation energy. NBO analysis revealed that the change transfer is stronger when the metal cation uses both the 4s + 4p subshells rather than just its 4p subshell. We also performed molecular dynamics (MD) simulations to check the stability and consistency of the most favourable binding motifs of Cu2+, Zn2+, Ca2+ and Ba2+ with Phe-Phe over time. The structures of the Phe-Phe-Cu2+/Zn2+/Ca2+/Ba2+ complexes obtained using MD simulation were found to be in good agreement with those obtained in the DFT-based calculations. Graphical Abstract Conformational search on encapsulation of divalent metal cations (Ca2+, Zn2+, Ca2+, Ba2+) by the Phe-Phe dipeptide.

Direct Pathway to Molecular Photodissociation on Metal Surfaces Using Visible Light.

We demonstrate molecular photodissociation on single-crystalline metal substrates, driven by visible-light irradiation. The visible-light-induced photodissociation on metal substrates has long been thought to never occur, either because visible-light energy is much smaller than the optical energy gap between the frontier electronic states of the molecule or because the molecular excited states have short lifetimes due to the strong hybridization between the adsorbate molecular orbitals (MOs) and metal substrate. The S-S bond in dimethyl disulfide adsorbed on both Cu(111) and Ag(111) surfaces was dissociated through direct electronic excitation from the HOMO-derived MO (the nonbonding lone-pair type orbitals on the S atoms (nS)) to the LUMO-derived MO (the antibonding orbital localized on the S-S bond (σ*SS)) by irradiation with visible light. A combination of scanning tunneling microscopy and density functional theory calculations revealed that visible-light-induced photodissociation becomes possible due to the interfacial electronic structures constructed by the hybridization between molecular orbitals and the metal substrate states. The molecule-metal hybridization decreases the gap between the HOMO- and LUMO-derived MOs into the visible-light energy region and forms LUMO-derived MOs that have less overlap with the metal substrate, which results in longer excited-state lifetimes.

Theory and Applications of Generalized Pipek-Mezey Wannier Functions.

The theory for the generation of Wannier functions within the generalized Pipek-Mezey approach (Lehtola, S.; Jónsson, H. J. Chem. Theory Comput. 2014, 10, 642) is presented and an implementation thereof is described. Results are shown for systems with periodicity in one, two, and three dimensions as well as isolated molecules. The generalized Pipek-Mezey Wannier functions (PMWF) are highly localized orbitals consistent with chemical intuition where a distinction is maintained between σ- and π-orbitals. The PMWF method is compared with the so-called maximally localized Wannier functions (MLWFs) that are frequently used for the analysis of condensed matter calculations. Whereas PMWFs maximize the localization criterion of Pipek and Mezey, MLWFs maximize that of Foster and Boys and have the disadvantage of mixing σ- and π-orbitals in many cases. The PMWF orbitals turn out to be as localized as the MLWF orbitals as evidenced by cross-comparison of the values of the PMWF and MLWF objective functions for the two types of orbitals. Our implementation in the atomic simulation environment (ASE) is compatible with various representations of the wave function, including real-space grids, plane waves, and linear combinations of atomic orbitals. The projector-augmented wave formalism for the representation of atomic core electrons is also supported. Results of calculations with the GPAW software are described here, but our implementation can also use output from other electronic structure software such as ABINIT, NWChem, and VASP.

Evaluation of Self-Friction Three-Center Nuclear Attraction Integrals with Integer and Noninteger Principal Quantum Numbers n over Slater Type Orbitals

We have proposed a new approach to evaluate self-friction (SF) three-center nuclear attraction integrals over integer and noninteger Slater type orbitals (STOs) by using Guseinov one-range addition theorem in standard convention. A complete orthonormal set of Guseinov ψα exponential type orbitals (ψα-ETOs, α=2,1,0,-1,-2,…) has been used to obtain the analytical expressions. The overlap integrals with noninteger quantum numbers occurring in SF three-center nuclear attraction integrals have been evaluated using Qnsq auxiliary functions. The accuracy of obtained formulas is satisfactory for arbitrary integer and noninteger principal quantum numbers.