Relativistic electron in the field of finite electric dipole

Abstract

The problem on electron motion under the action of finite electric dipole ranked among classic problems of nonrelativistic quantum mechanics. It has applications in the theory of mesomolecules and in the physics of polar molecules. The energy levels and other characteristics of the nonrelativistic electron‐dipole system have been investigated by a number of authors analytically [1], as well as numerically [2, 4‐6]. It is well known [1‐ 3, 5] that a certain critical value of the dipole moment, d c = Zr , exists, below which an electron lacks bound states in the field of a dipole, i.e., the electron cannot be held by the dipole field, when the separations between the poles of the dipole are less than r c = d c / Z = 0.64 au. Solving the corresponding relativistic problem (i.e., the problem on a relativistic electron motion in the field of a finite dipole) presents a complicated mathematical problem, because variables in the Dirac equation with the two-center potential cannot be separated (in distinction to the Schrodinger equation) in any orthogonal coordinates. In this paper, the energy term close to the continuum is calculated analytically by sewing together the logarithmic derivatives of the asymptotic solution. This method has earlier been successfully employed by V.S. Popov for solving the Dirac equation for the symmetrical problem of two Coulomb centers with equal values and signs of the center’s charges. In this paper, we generalize the results of [1], where the problem on nonrelativistic electron motion in the field of a finite electric dipole has been solved, to the relativistic case. In doing so, we employ the abovementioned method of sewing together the asymptotics. This method has been employed in [7, 10] for solving the Coulomb two-center problem with the center’s charges being equal both in the value and the sign. We consider the motion of relativistic electron in the field of finite electric dipole consisting of the charges + Z α and — Z α , which are spaced at the distance R . The potential of such a system has the form