Two-center Problem Research Articles

Hyperspherical Coulomb spheroidal representation in the Coulomb three-body problem

A new representation of the Coulomb three-body wavefunction via the well-known solutions of the separable Coulomb two-centre problem ϕj(ξ, η) = Xj(ξ)Yj(η) is obtained, where Xj(ξ) and Yj(η) are the Coulomb spheroidal functions. Its distinguishing characteristic is the coordination with the asymptotic conditions of the scattering problem below the three-particle breakup. That is, the wavefunction of two colliding clusters in any open channel is the asymptotics of the single, corresponding to that channel, term of the suggested expansion. The effect is achieved due to a new relation between three internal coordinates of a three-body system and the parameters of ϕj(ξ, η). It ensures the orthogonality of ϕj(ξ, η) on a sphere of constant hyperradius, ρ = const, in place of the surface R = |x2 − x1| = const appearing in the traditional Born–Oppenheimer approach. The independent variables ξ and η are the orthogonal coordinates on this sphere with three poles in the coalescence points. They are connected with the elliptic coordinates on the plane by means of a stereographic projection. For the total angular momentum J ⩾ 0 the products of ϕj and the Wigner D-functions form a hyperspherical Coulomb spheroidal (HSCS) basis on a five-dimensional hypersphere, ρ being a parameter. The system of the differential equations and the boundary conditions for the radial functions fJi(ρ), the coefficients of the HSCS decomposition of the three-body wavefunction, are presented.

The magnetic two-centre problem: Stability and critical bonding

Stability properties of the homonuclear diatomic one-electron system are investigated by finite-elements methods. Assuming the Born–Oppenheimer approximation and parallel magnetic fields, the computed critical charges for stability and metastability reveal a surprisingly simple dependence on the magnetic field strength B.

Ionisation of H2 in intense ultrashort laser pulses: parallel versus perpendicular orientation

A theoretical comparison of the electronic excitation and ionisation behaviour of molecular hydrogen oriented either parallel or perpendicular to a linear polarised laser pulse is performed. The investigation is based on a non-perturbative treatment that solves the full time-dependent Schrödinger equation of both correlated electrons within the fixed-nuclei approximation and the dipole approximation. Results are shown for two different laser pulse lengths and intensities as well as for a large variety of photon frequencies starting in the one- and reaching into the six-photon regime. In order to investigate the influence of the intrinsic diatomic two-centre problem even further, two values of the internuclear separation and a newly developed atomic model are considered.

Simple model of scalar-vector interaction for the relativistic two-center problem

The ground-state wave function and the energy term of a relativistic electron moving in the field of two fixed centers, when interaction of this particle with centers is described by two Coulomb and two Coulomb-like scalar potentials are calculated analytically by the linear combination of atomic orbitals method. By means of perturbation theory the asymptotic expansions (at small and large internuclear distances $R$) of the eigenvalues $E(R)$ of the system for arbitrary states are obtained. The dependence of critical charge ${Z}_{\text{cr}}$ and critical distance ${R}_{\text{cr}}$ from the scalar coupling constant was investigated.

The magnetic two-centre problem: Nuclear fusion catalyzed by ultrastrong fields?

Employing the finite-element method (FEM), accurate potential energy curves for the ground state of the homonuclear system consisting of two (static) protons and one electron within a parallel homogeneous magnetic field are computed. The shrinking of the internuclear equilibrium separation with growing magnetic field strength is investigated under quantitative and qualitative aspects. In particular, we study the problem whether this leads to significantly enhanced nuclear fusion rates in the processes d (t, n) α and p (p, e+ ν) d.

Relativistic corrections of order mα6 to the two-centre problem

Effective potentials of the relativistic mα6 order correction for the ground state of the Coulomb two-centre problem are calculated. They can be used to evaluate the relativistic contribution of that order to the energies of hydrogen molecular ions or metastable states of the antiprotonic helium atom, where precision spectroscopic data are available. In our studies we use the variational expansion based on randomly chosen exponents that permits us to achieve high numerical accuracy.

Relativistic density functional calculations using two-spinor minimax finite-element method and linear combination of atomic orbitals for ZnO, CdO, HgO, UubO and Cu2, Ag2, Au2, Rg2

In previous work the authors have presented a highly accurate two-spinor fully relativistic solution of the two-center Coulomb problem utilizing the finite-element method (FEM) and furthermore developed a relativistic minimax two-spinor linear combination of atomic orbitals (LCAO). In the present paper the authors present Dirac-Fock-Slater (DFS-) density functional calculations for two-atomic molecules up to super heavy systems using the fully nonlinear minimax FEM and the minimax LCAO in its linearized approximation (linear approximation to relativistic minimax). The FEM gives highly accurate benchmark results for the DFS functional. Especially considering molecules with up to super heavy atoms such as UubO and Rg2, the authors found that LCAO fails to give the correct systematic trends. The accurate FEM results shed a new light on the quality of the DFS-density functional.

The magnetic two-centre problem: some rigorous properties

We study the quantum mechanical magnetic two-centre problem, i.e., quantum states of an electron within the Coulomb field of two fixed nuclear centres and a homogeneous magnetic field. From the corresponding nonrelativistic Schrodinger equation various characteristic properties are derived. These include the ordering of energy levels and the monotonicity of electronic energies as a function of the nuclear separation if the internuclear axis is parallel to the direction of the B field. For such situations we also obtain lower bounds on the equilibrium separation between the nuclei and establish decay properties of bound state wavefunctions. Moreover, the molecular virial theorem is generalized to encompass the contributions from the magnetic field.

Closed orbits about a massive thin ring

A study of the gravitational interaction of a unit point mass m, and a very thin ring of radius R and mass M (M ≫ m) is presented. The point mass moves in a plane that is perpendicular to that containing the ring. The potential due to the ring is similar in geometry to that of the two-centre problem with two large point masses, if it were possible for the positions of such point masses to be fixed. The scope of the paper is restricted almost exclusively to closed orbits (trajectories), and these display a surprisingly wide range of patterns. A characterization plot, based on the initial conditions of the trajectories, is used to illustrate and aid in the categorization of the trajectories into ‘families’. Figures showing an example member of 11 distinct families and one example trajectory in which the point mass is not restricted to motion in a plane are included. The period of a family of closed trajectories, roughly egg shaped and bound to a particular section of the ring, is found to depend almost linearly on the radius of the orbit, as expected.

Carcass functions in variational calculations for few-body systems

For variational calculations of molecular and nuclear systems involving a few particles, it is proposed to use carcass basis functions that generalize exponential and Gaussian trial functions. It is shown that the matrix elements of the Hamiltonian are expressed in a closed form for a Coulomb potential, as well as for other popular particle-interaction potentials. The use of such carcass functions in two-center Coulomb problems reduces, in relation to other methods, the number of terms in a variational expansion by a few orders of magnitude at a commensurate or even higher accuracy. The efficiency of the method is illustrated by calculations of the three-particle Coulomb systems μμe, ppe, dde, and tte and the fourparticle molecular systems H2 and HeH+ of various isotopic composition. By considering the example of the Λ9Be hypernucleus, it is shown that the proposed method can be used in calculating nuclear systems as well.

Magnetooptical properties of a molecular D 2 − -ion in a quantum wire

In the framework of a model of zero-range potential, the problem of bound states of an electron in the field of two D0 centers (a two-center problem) in a semiconductor quantum wire is considered in the presence of a longitudinal magnetic field. It is shown that the magnetic field produces a significant shift of g and u terms and stabilizes the D 2 − states in quantum wires. It is found that, in the case of transverse polarization of light, the spectral dependence of the photoionization cross section of a D 2 − center exhibits the quantum-confined Zeeman effect with strongly pronounced oscillations of interference nature.

Non-geometrical symmetry and separation of variables in the two-centre problem with a confinement-type potential

An additional spheroidal integral of motion and a group of dynamic symmetry in a model quantum-mechanical problem of two centres eZ_{1}Z_{2}omega with Coulomb and oscillator interactions is obtained, the group properties of its solutions being studied. P(3) \otimes P(2,1), P(5,1) and P(4,2) groups are considered as the dynamic symmetry groups of the problem, among them P(3) \otimes P(2,1) group possessing the smallest number of parameters. The obtained results may appear useful at the calcuations of QQq-baryons and QQg-mesons energy spectra.

Quantization of the potential amplitude in the one-dimensional Schrödinger equation

A consistent scheme is proposed for quantizing the potential amplitude in the one-dimensional Schrodinger equation in the case of negative energies (lying in the discrete-spectrum domain). The properties of the eigenfunctions ϕn(x) and eigenvalues αn corresponding to zero, small, and large absolute values of energy E < 0 are analyzed. Expansion in the set ϕn(x) is used to develop a regular perturbation theory (for E < 0), and a general expression is found for the Green function associated with the time-independent Schrodinger equation. A similar method is used to solve several physical problems: the polarizability of a weakly bound quantum-mechanical system, the two-center problem, and the tunneling of slow particles through a potential barrier (or over a potential well). In particular, it is shown that the transmission coefficient for slow particles is anomalously large (on the order of unity) in the case of an attractive potential is characterized by certain critical values of well depth. The proposed approach is advantageous in that it does not require the use of continuum states.

Topological Analysis of Integrable Problems of Celestial Mechanics on a Sphere and Pseudosphere

In this paper, we consider the problem of motion of a point on a two-dimensional sphere and pseudosphere (with metric of constant curvature) in the field generated by two attracting centers. This problem is a natural analog of the classical plane two-center problem integrated even by Euler. However, the integrability of similar problems on the sphere and the Lobachevskii plane was established only recently. Therefore, the topological properties of these problems are poorly studied as yet. The present paper fills this gap. We also study the Lagrange problem on the pseudosphere under the action of the Newtonian attraction of the stationary center and one more force constant in magnitude and direction. This problem is a limit case of the two-center problem. In the process of passing to the limit, the second center moves at infinity in the direction of the propulsion force (moreover, its mass grows so that the constancy of the propulsion is ensured, i.e., proportionally to the squared distance). This problem was considered by Lagrange for the first time and reduces to quadratures. The question on gravitational interactions in spaces of constant curvature was posed for the first time by N. N. Lobachevskii, who studied generalizations of the attraction law to the space of constant negative curvature. This theme was further developed in the work of N. E. Zhukovskii [18] on the motion of a pseudospheric (“plane”) plate on the Lobachevskii plane. On a three-dimensional sphere (of constant curvature 1), one also considers an analog of the Newtonian potential generated by a gravitational center, which in this case is cotan θ, where the angle θ is equal to the distance to the center. The integrability of the Kepler problem on the three-dimensional sphere S3 was proved by E. Schrodinger [12]. Schrodinger considered the motion of an electron in the potential field of the nucleus of a hydrogen-like atom from the standpoint of quantum mechanics. Three classical Kepler laws have natural generalizations for spaces of constant curvature. For the Lobachevskii space and for the sphere, these generalizations were formulated by A. N. Chernikov [5] and V. V. Kozlov [8]. In particular, any finite orbit here is an ellipse in one of whose foci the gravitational center (the Sun) is located. Along with this, V. V. Kozlov and O. A. Kharin proved the integrability of the problem on the motion of a material particle on the sphere S2 in the field of two stationary Newtonian centers [9]. In [14–17], the problem of two centers on the three-dimensional sphere S3 and in the three-dimensional Lobachevskii space H3 and also the Lagrange problem (the limit case of the two-center problem when one of the centers is moved off at infinity) in the Lobachevskii space H3 were studied. In these works, the integrabiity of the problems is demonstrated, bifurcation diagrams are constructed, and the classification of regions of feasible motion is carried out. The first step in studying these problems consists of reducing the problem from the three-dimensional to the two-dimensional case, i.e., reducing the number of degrees of freedom. The next step consists of passing to convenient coordinates (in the problems under consideration, for instance, to spherical-conical and pseudo-spherical-conical coordinates which are defined as roots of certain rational functions) in order to reduce the system to the Liouville form.

The problem of two fixed centers: bifurcations, actions, monodromy

A comprehensive analysis of the Euler–Jacobi problem of motion in the field of two fixed attracting centers is given, first classically and then quantum mechanically in semiclassical approximation. The system was originally studied in the context of celestial mechanics but, starting with Pauli’s dissertation, became a model for one-electron molecules such as H 2 + (symmetric case of equal centers) or HHe 2+ (asymmetric case of different centers). The present paper deals with arbitrary relative strength of the two centers and considers separately the planar and the three-dimensional problems. All versions represent non-trivial examples of integrable dynamics and are studied here from the unifying point of view of the energy–momentum mapping from phase space to the space of integration constants. The interesting objects are the critical values of this mapping, i.e., its bifurcation diagram, and their pre-images which organize the foliation of phase space into Liouville–Arnold tori. The classical analysis culminates in the explicit derivation of the action variable representation of iso-energetic surfaces. The attempt to identify a system of global actions, smoothly dependent on the integration constants wherever these are non-critical, leads to the detection of monodromy of a special kind which is here described for the first time. The classical monodromy has its counterpart in the quantum version of the two-center problem where it prevents the assignments of unique quantum numbers even though the system is separable.

On the Low-Frequency Interaction between a Central Dyadic Wave and a Spherical Cavity

The dyadic fundamental solution of the linearized theory of elasticity in a homogeneous and isotropic medium is considered as a point-generated incident dyadic field. This central field is disturbed by a small spherical cavity within the medium of propagation. This two-center interaction problem provides a general dyadic scattering problem which includes the classical elasticity problems as a special case, obtained via contractions with specific directions. An analytic solution of the low-frequency approximations can be obtained via a generalization from the vector displacement field to a dyadic displacement field. Using this generalization the zeroth- and the first-order approximation of the near field are obtained explicitly. In the far field the leading nonvanishing approximation for the radial and the tangential scattering amplitudes as well as that for the scattering cross-section are also evaluated. The analytic results obtained are then compared against relative results in classical elasticity obtained via the boundary element method and an amazing coincidence is observed when the radius of the sphere is small with respect to the wavelengh of the central incident field.

Two-spinor fully relativistic finite-element (FEM) solution of the two-center Coulomb problem

A highly accurate 2-spinor fully relativistic solution of the two-center Coulomb problem utilizes the finite-element method (FEM). Like in previous work on the 4-spinor Dirac equation we use elliptic–hyperbolic coordinates, global factors, further one-dimensional singular coordinate transformations and extrapolation techniques. In the same grid 2-spinor energies for H 2 + are up to four orders more accurate than 4-spinor values, for the super-heavy Th 2 179+ up to one order and one generally finds, high convergence orders similar to a non-relativistic approach. The 2-spinor values converge from above, exhibit remarkable properties and the calculations are considerably faster and more stable.

Facility location and the geometric minimum-diameter spanning tree

Let P be a set of n points in the plane. The geometric minimum-diameter spanning tree (MDST) of P is a tree that spans P and minimizes the Euclidean length of the longest path. It is known that there is always a mono- or a dipolar MDST, i.e., a MDST with one or two nodes of degree greater 1, respectively. The more difficult dipolar case can so far only be solved in slightly subcubic time. This paper has two aims. First, we present a solution to a new data structure for facility location, the minimum-sum dipolar spanning tree (MSST), that mediates between the minimum-diameter dipolar spanning tree and the discrete two-center problem (2CP) in the following sense: find two centers p and q in P that minimize the sum of their distance plus the distance of any other point (client) to the closer center. This is of interest if the two centers do not only serve their customers (as in the case of the 2CP), but frequently have to exchange goods or personnel between themselves. We show that this problem can be solved in O( n 2log n) time and that it yields a factor-4/3 approximation of the MDST. Second, we give two fast approximation schemes for the MDST, i.e., factor-(1+ ε) approximation algorithms. One uses a grid and takes O ∗(E 6−1/3+n) time, where E=1/ ε and the O ∗ -notation hides terms of type O(log O(1) E). The other uses the well-separated pair decomposition and takes O( nE 3+ Enlog n) time. A combination of the two approaches runs in O ∗(E 5+n) time. Both schemes can also be applied to MSST and 2CP.

Quantum monodromy in the two-centre problem

Using modern tools from the geometric theory of Hamiltonian systems it is shown that electronic excitations in diatoms which can be modelled by the two-centre problem exhibit a complicated case of classical and quantum monodromy. This means that there is an obstruction to the existence of global quantum numbers in these classically integrable systems. The symmetric case of H+2 and the asymmetric case of H He++ are explicitly worked out. The asymmetric case has a non-local singularity causing monodromy. It coexists with a second singularity which is also present in the symmetric case. An interpretation of monodromy is given in terms of the caustics of invariant tori.

Resonant phenomena in classical dynamics of three-body Coulomb systems.

We consider dynamics of a planar three-body Coulomb system similar to a hydrogen molecular ion (heavy-light-heavy particles). The system has three degrees of freedom. In the limit of infinitely heavy nuclei the system is reduced to the famous two-center problem which is integrable. When masses of heavy particles are finite, one degree of freedom in the Hamiltonian system corresponds to slow nuclei motion, while other two degrees of freedom correspond to fast electron motion. The averaging method predicts that actions of "fast" motions of the system with frozen nuclei are approximate integrals of the full system (adiabatic invariants). However, during slow evolution of the "heavy" subsystem certain resonance conditions can be satisfied. We study the phenomena of capture into resonances and scattering on resonances which can lead to destruction of adiabatic invariance in the system.