One-electron Diatomic Molecules Research Articles

Black hole binaries and light fields: Gravitational molecules

We show that light scalars can form quasibound states around binaries. In the nonrelativistic regime, these states are formally described by the quantum-mechanical Schr\"odinger equation for a one-electron heteronuclear diatomic molecule. We performed extensive numerical simulations of scalar fields around black hole binaries, showing that a scalar structure condenses around the binary---we dub these states ``gravitational molecules.'' We further show that these are well described by the perturbative, nonrelativistic description.

Impact of orbital symmetry on molecular ionization in an intense laser field

We investigate the ionization dynamics of one-electron model diatomic molecules with different orbital symmetries ($1{\ensuremath{\sigma}}_{g}$ and $1{\ensuremath{\sigma}}_{u}$) in a strong laser field via the full three-dimensional time-dependent Schr\odinger equation. It is shown that the ionization probabilities of the molecules with different initial states ($1{\ensuremath{\sigma}}_{g}$ and $1{\ensuremath{\sigma}}_{u}$) oscillate out of phase vs internuclear distance, and the photoelectron spectra of the aforementioned two molecules also show out-of-phase oscillation structures. The above intriguing features are well reproduced by the simulations of the length-gauge strong-field approximation, which can be ascribed to the distinct interferences of electronic wave packets released from different atomic centers.

Solution of the Dirac equation using the Rayleigh-Ritz method: Flexible basis coupling large and small components. Results for one-electron systems.

An algebraic solution of the Dirac equation is reinvestigated. Slater-type spinor orbitals and their corresponding system of differential equations are defined in two- and four-component formalism. They describe the radial function in components of the wave function of the Dirac equation solution to high accuracy. They constitute the matrix elements arising in a generalized eigenvalue equation. These terms are evaluated through prolate spheroidal coordinates. The corresponding integrals are calculated by the numerical global-adaptive method taking into account the Gauss-Kronrod numerical integration extension. Sample calculations are performed using flexible basis sets generated with both signs of the relativistic angular momentum quantum number κ. Applications to one-electron atoms and diatomics are detailed. Variationally optimum values for orbital parameters are obtained at given nuclear separation. Methods discussed in this work are capable of yielding highly accurate relativistic two-center integrals for all ranges of orbital parameters. This work provides an efficient way to overcome the problems that arise in relativistic calculations.

Molecular structure of one-electron diatomic molecules subject to plasma screening and its effect on the dynamics

SynopsisThe effect of plasma screening on the electronic and vibrational structure of one-electron diatomic molecules is analyzed by including screened Coulomb interactions among all their particles in the solution of the Schrödinger equation. The breakdown of the well-known separability of the Schrödinger equation in confocal elliptical coordinates, the lost of long-range Coulomb interaction and the non-degeneracy in both the united atom and separated atom limits bring new elements into the analysis of the modified binding energies (electronic and vibrational), adiabatic correlation rules, molecular Stark mixings due to polarization effects, unexpected presence of molecular shape resonances and changes in non-adiabatic and radiative couplings upon the variation of the screening strength.

Coulomb Sturmians in spheroidal coordinates and their application for diatomic molecular calculations

Coulomb Sturmians are obtained in prolate spheroidal coordinates by separation of variables in the Schrödinger equation and direct solution of the appropriate one-dimensional equations. Molecular orbitals are expressed as linear combinations of the introduced Coulomb Sturmians and some low-lying energy terms and corresponding wave functions are calculated for one-electron diatomic molecules. It is shown that similarity of the one- and two-centre orbitals in spheroidal coordinates, combined with completeness and good convergence properties of Coulomb Sturmians, substantially speeds up convergence and makes the calculated results closer to the exact ones. Application of the elaborated calculating scheme for diatomic many-electron molecules is discussed.

Electron scattering and photoionization of one-electron diatomic molecules

We develop an efficient numerical procedure for solving the scattering problem on one-electron diatomic molecules in the fixed-nuclei approximation. For this purpose, we use the $R$-matrix propagation method together with the discrete variable representation method in prolate spheroidal coordinates. The demonstrated elastic scattering differential cross sections and the photoionization differential cross sections for H${}_{2}^{+}$ and HeH${}^{2+}$ exhibit oscillating structures owing to the two-center nature of the molecules. We analyze them by using the Coulomb corrected independent-atom model to identify the two-center quantum interference, and it is found that multiple-scattering effects become important in the heteronuclear system.

Cooper minima and Young-type interferences in photoionization of one-electron diatomic molecules

We study photon induced electron emission from one-electron diatomic molecular ions. We find an oscillatory behavior of the asymmetry parameter β and of the total cross section for homnuclear molecules. This behavior can be related to the presence of Cooper minima in the partial cross sections and also to coherent electron emission from the two centers of the molecule.

Application of Complete Orthonormal Sets of Ψα-Exponential-Type Orbitals to Accurate Ground and Excited States Calculations of One-Electron Diatomic Molecules Using Single-Zeta Approximation

The applicability of the complete orthonormal sets of ψα-exponential-type orbitals introduced by one of the authors to the study of electronic structure of one electron diatomic molecules is demonstrated using single-zeta approximation. As an example of application, the calculations have been performed for α, π and δ states of one electron homo- and hetero-nuclear diatomic molecules H+2 and HeH2+, respectively. The calculation results are presented. The values for these molecules obtained in eight-digit accuracy are close to the results of solution presented in literature.

Use of basis sets of Ψα-exponential type orbitals in calculation of electronic energies for one-electron diatomic molecules by single-zeta approximation

The use of complete orthonormal sets of Ψα-exponential type orbitals (Ψα-ETOs) is investigated for electronic structure calculations of one-electron diatomic molecular systems. The linear combinations of atomic orbitals (LCAOs) method is used to determine the electronic energy levels of one-electron homonuclear (He3+2, Li5+2, Be7+2 and B9+2) and heteronuclear (HeH2+, LiH3+, BeH4+ and BH5+) molecules by employing the same screening constant for a given set of atomic orbitals of each molecular function. For each molecule, the calculations are carried out for the lowest two energy terms of σ, π and δ states using the values of nuclear separations 0.01⩽R⩽20, and Ψα-ETOs basis sets with α=1, 0 and −1. The results are in good agreement with those found in the literature. The strategies developed in this work can be useful in the study of multielectronic molecules with the help of Ψα-ETOs basis sets.

The ionization of one-electron diatomic molecules in strong and short laserfields

The dynamics of multiphoton processes with strong fields and short pulsesin one-electron diatomic molecules is investigated, solving the timedependent Schrödinger equation. We propose a spectral type method, whichuses L2 functions and prolate spheroidal coordinates. The approach isapplied to the cases of two- and three-photon ionization of H2+,with a photon energy of 0.6 au and parallel polarization. In the three-photonionization case, above threshold ionization occurs. We investigate, in particular,the limit of perturbation theory, and the photoelectron spectrum is calculated forthe different open channels. The case of perpendicular polarization, whereone-photon absorption is important, is investigated. Finally, we determine thefinal vibrational states of the nuclei for the case of two-photon ionization.

A connection between quantum critical points and classical separatracies of electronic states

Wave functions for one-electron diatomic molecules such as H2+ and HeH2+ are analyzed by Bader’s atoms in molecules method. The locations of the degenerate axial critical points in the electron density generated from sigma states arising from the n=1–6 united atom manifolds are shown to correspond well with the boundaries of domains obtained solely from a classical description of the electron motion. The relationship clarifies the connection between the atomic and molecular regimes of classical trajectories and the quantum description of the states. In particular, the classical transition from atomic to molecular character roughly corresponds to the appearance of a critical point in the electron density located on the internuclear axis between the nuclei. The global aspects of the relationship between the classical and quantum descriptions helps to demonstrate the classical framework of the quantum picture.

Bound primitive semiclassical diatomic electronic states

We report the first examples of bound molecular electronic states computed using the primitive semiclassical approximation. This method is based directly on classical actions derived from classical trajectories, and as such is closely related to the Bohr method for atoms. The examples chosen are neutral one-electron diatomic molecules, with fractional nuclear charges; both polar and non-polar cases are considered. It is the reduction of the nuclear repulsion from that experienced by one-electron cations which allow this semiclassical method to find bound states. The errors in the bond length and energies decrease to less than a percent as the quantum numbers are increased; Rydberg states are computed with much higher accuracy. The implications for interpretation of chemical bonding are briefly discussed.

Multiphoton ionization ofone-electron diatomic molecules: the discretization approach forthe two-centre problem

We propose a finite-element approach with prolate spheroidalcoordinates to calculate, for one-electron diatomic molecules,ac Stark shifts and multiphoton ionization cross-sections inlowest-order perturbation theory. We use B-spline expansions torepresent both angular and radial parts of the wavefunctions.The method takes advantage of the fact that the Schrödinger equation isseparable with prolate spheroidal coordinates. The effectivecompleteness of the basis set leads to an accuraterepresentation of the molecular spectrum. Partialcross-sections, associated with each open continuum, arecalculated. Particular attention is placed on the problem ofthe normalization of the discretized continua. We show that theproper normalization is simply obtained from a density ofstates, thus the explicit evaluation of the continua at largedistances is not necessary. Accurate values of energies,shifts, one-, four- and six-photon ionization cross-sections are obtained in lowest-order perturbation theory and they arecompared with other calculations when available.

Semiclassical energies of low-lying states of one-electron diatomics

Primitive semiclassical (PSC) calculations of the electronic, potential and kinetic energies are carried out for several low-lying states of homonuclear and heteronuclear one-electron diatomic molecules in the fixed nucleus approximation. The results are in fair agreement with the quantum values, in particular reproducing all the trends. We point out the limitations of the method in describing the important features. The PSC approach provides a specific interpretation of bonding in one-electron molecules, and the results emphasize the importance of tunneling.

Semiclassical electronic eigenvalues for charge asymmetric one-electron diatomic molecules: general method and sigma states

The primitive semiclassical method is applied to the motion of the electron in an asymmetrically charged one-electron diatomic molecule in the fixed nucleus approximation. Using the Einstein-Keller-Brillouin rules, semiclassical approximations to the eigenvalues are found for planar motion (σ states). The general method is applied to the one-electron diatom HeH 2+. The agreement between the classical approximations and the quantum eigenvalues is qualitatively good and often quantitatively good, to within a fraction of a percent. For the two stable states considered, the semiclassical results fail at the onset of chemical bonding, emphasizing the importance of tunnelling. In contrast to the charge symmetric molecules, for some non-bonding states the primitive semiclassical results are good for all internuclear separations. An interesting scaling law is derived between states of a molecule with a common ratio of quantum numbers: (2n + 1) 2 E n(R n) = (2m + 1) 2 E m(R m), where (2m + 1) 2 R n = (2n + 1) 2 R m.

Erratum: Charge exchange between bare beryllium and boron with metastable hydrogen atoms at low energies

Theoretical results for [sup 9]Be[sup 4+]-H(2[ital s]) and [sup 11]B[sup 5+]-H(2[ital s]) partial and total charge-exchange cross sections at low relative velocities (0.027--0.32 a.u.) are given, using the Landau-Zener method. The necessary molecular parameters for this method are obtained from the exact one-electron diatomic molecule (OEDM) molecular energies. The partial cross sections which are mainly populated are those corresponding to the principal quantum number of separated atoms [ital n]=5 for Be[sup 3+] and [ital n]=6 for B[sup 4+]. The total cross sections show a rather strong increase with increasing collision energies and a quite large maximum. They lie between the cross sections corresponding to the neighbor reactions for the incident nuclei with [ital Z]=3 (Li), [ital Z]=6 (C), and [ital Z]=7 (N). The atomic collisions with beryllium are very important in plasma tokamaks.

Fully numerical soluti ons of molecular Dirac equations for highly charged one-electron homonuclear diatomic molecules

The two-centre four-component Dirac equations for the homonuclear one-electron systems H 2 +, Ne 2 19+, Ca 2 39+, Zn 2 59+, Zr 2 79+, Sn 2 99+, Nd 2 119+, Yb 2 139+, Hg 2 159+, Th 2 179+, and Fm 2 199+ are solved numerically by a finite-difference approach. Accurate values for the non-relativistic and relativistic orbital energies are given as benchmarks for these molecules. The assumed bond lengths are 2/ Z au where Z is the nuclear charge. For the lowest σ g orbital of the Th 2 179+ molecule, the orbital energy is 14.6 au above the global basis-set result while for the lowest orbital of σ u symmetry the present orbital energy is 19.1 au below the global basis-set results. The relativistic corrections to the quadrupole moment of the σ g orbital are given.

A highly accurate calculation for the electronic states of H +2 in B-spline basis

B-spline basis is successfully applied in this study towards the Schrödinger equation of one-electron diatomic molecules in spheroidal coordinates. 14-figure accuracy was obtained for the eigen-energies of the lowest states of H + 2 and HeH 2+. Those results were compared with the best published results. The quadrupole moments of H + 2 were also calculated by the eigenfunctions along with 13-figure accuracy having been achieved.

Charge exchange between bare beryllium and boron with metastable hydrogen atoms at low energies.

Theoretical results for $^{9}\mathrm{Be}^{4+}$-H(2s) and $^{11}\mathrm{B}^{5+}$-H(2s) partial and total charge-exchange cross sections at low relative velocities (0.027--0.32 a.u.) are given, using the Landau-Zener method. The necessary molecular parameters for this method are obtained from the exact one-electron diatomic molecule (OEDM) molecular energies. The partial cross sections which are mainly populated are those corresponding to the principal quantum number of separated atoms n=5 for ${\mathrm{Be}}^{3+}$ and n=6 for ${\mathrm{B}}^{4+}$. The total cross sections show a rather strong increase with increasing collision energies and a quite large maximum. They lie between the cross sections corresponding to the neighbor reactions for the incident nuclei with Z=3 (Li), Z=6 (C), and Z=7 (N). The atomic collisions with beryllium are very important in plasma tokamaks.

Chemical binding from the infinite dimensional limit

The electronic structure of molecules calculated in the limit of infinite spatial dimensions is related to properties in three-dimensional space. In the spirit of the electrostatic theorem, we introduce the concept of antibinding and binding regions to visualize chemical binding. In addition, criteria are developed to determine the ionic or covalent character of the bond. The computational effort requires only the minimization of a multidimensional potential surface at the infinite dimensional limit, yet detailed information can be extracted. Numerical examples are given for the one-electron diatomic molecule