Electron in the Field of a Molecule with an Electric Dipole Moment

Alhaidari, A. D.; Bahlouli, H.

doi:10.1103/physrevlett.100.110401

Cited by 35 publications

(26 citation statements)

References 25 publications

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“…Despite renewed interest in this problem, no exact analytic solution (aside from our recent contribution [6]) was reported in the literature. This is because, even in the ideal case of a point electric dipole, the interaction includes the intractable noncentral potential 2 cos r θ (in spherical coordinates) which was known not to belong to any of the established classes of exactly solvable potentials.…”

Section: Introductionmentioning

confidence: 99%

Analytic solution of the wave equation for an electron in the field of a molecule with an electric dipole moment

Alhaidari

2008

Annals of Physics

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We relax the usual diagonal constraint on the matrix representation of the eigenvalue wave equation by allowing it to be tridiagonal. This results in a larger solution space that incorporates an exact analytic solution for the non-central electric dipole potential 2 cos r θ , which was known not to belong to the class of exactly solvable potentials. As a result, we were able to obtain an exact analytic solution of the three-dimensional timeindependent Schrödinger equation for a charged particle in the field of a point electric dipole that could carry a nonzero net charge. This problem models the interaction of an electron with a molecule (neutral or ionized) that has a permanent electric dipole moment. The solution is written as a series of square integrable functions that support a tridiagonal matrix representation for the angular and radial components of the wave operator. Moreover, this solution is for all energies, the discrete (for bound states) as well as the continuous (for scattering states). The expansion coefficients of the radial and angular components of the wavefunction are written in terms of orthogonal polynomials satisfying three-term recursion relations. For the Coulomb-free case, where the molecule is neutral, we calculate critical values for its dipole moment below which no electron capture is allowed. These critical values are obtained not only for the ground state, where it agrees with already known results, but also for excited states as well.

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Section: Introductionmentioning

confidence: 99%

Analytic solution of the wave equation for an electron in the field of a molecule with an electric dipole moment

Alhaidari

2008

Annals of Physics

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“…We took (and will continue to take) the Bohr radius, , as the unit of length with m and e  being the mass and charge of the electron. For 0 Q  , the solution obtained in [8] and [21] refers to an electron bound to a neutral molecule with a permanent electric dipole moment d Q (i.e., the dipole-bound anion). In the present study, we extend that work by including higher order contributions coming from the electric quadrupole.…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…subject to the physical boundary conditions were obtained in Refs. [8,21]. Interested readers may consult the cited work and references therein.…”

Section: 4mentioning

confidence: 99%

Electric dipole and quadrupole contributions to valence electron binding in a charge-screening environment

Alhaidari

Bahlouli

2019

Eur. Phys. J. D

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We make a multipole expansion of the atomic/molecular electrostatic charge distribution as seen by the valence electron up to the quadrupole term. The Tridiagonal Representation Approach (TRA) is used to obtain an exact bound state solution associated with an effective quadrupole moment and assuming that the electron-molecule interaction is screened by suborbital electrons. We show that the number of states available for binding the electron is finite forcing an energy jump in its transition to the continuum that could be detected experimentally in some favorable settings. We expect that our solution gives an alternative, viable and simple description of the binding of valence electron(s) in atoms/molecules with electric dipole and quadrupole moments. We also ascertain that our model implies that a pure quadrupole-bound anion cannot exist in such a charge-screening environment.

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“…However, we need to stress that each solution of (3.1.5) will cover part of the energy space complementary to the other one. Luckily, Schrödinger equation has been treated in the past, using the TRA, by different authors including Alhaidari and Bahlouli [14,21,22,28]. We have tabulated few of the solvable potentials of Eq.…”

Section: 1: Scalar Potentialmentioning

confidence: 99%

“…Unfortunately, exact solutions of (3.2.10) cannot be written in a closed form as the recursion relation cannot be compared to any well-known class of orthogonal polynomials contrary to what we had in the previous examples. In fact, the solutions are referred to new polynomials which have been called "dipole polynomials" and have been found in different physical problems like electron in the dipole field and non-central potential problems [21,22]. Moreover, the eigenstates can be evaluated at any order and the energy eigenvalues can be computed numerically with high accuracy.…”

Section: 6)mentioning

confidence: 99%

Analytical Solutions of the 1D Dirac Equation Using the Tridiagonal Representation Approach

Assi¹,

Bahlouli²

2017

JAMP

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This paper aims at extending our previous work on the solution of the one-dimensional Dirac equation using the Tridiagonal Representation Approach (TRA). In the approach, we expand the spinor wavefunction in terms of suitable square integrable basis functions that support a tridiagonal matrix representation of the wave operator. This will transform the problem from solving a system of coupled first order differential equations to solving an algebraic three-term recursion relation for the expansion coefficients of the wavefunction. In some cases, solutions to this recursion relation can be related to well-known classes of orthogonal polynomials whereas in other situations solutions represent new class of polynomials. In this work, we will discuss various solvable potentials that obey the tridiagonal representation requirement with special emphasis on simple cases with spin-symmetric and pseudospin-symmetric potential couplings. We conclude by mentioning some potential applications in graphene.

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Electron in the Field of a Molecule with an Electric Dipole Moment

Cited by 35 publications

References 25 publications

Analytic solution of the wave equation for an electron in the field of a molecule with an electric dipole moment

Analytic solution of the wave equation for an electron in the field of a molecule with an electric dipole moment

Electric dipole and quadrupole contributions to valence electron binding in a charge-screening environment

Analytical Solutions of the 1D Dirac Equation Using the Tridiagonal Representation Approach

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