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Physical Review A

Coulomb and Coulomb-Stark Green-function approach to adiabatic Rydberg energy levels of alkali-metal–helium systems

E. de Prunelé

Published Jan 1, 1987• 25 citations
Publisher

Abstract

The adiabatic \ensuremath{\Sigma} electronic terms of highly excited states of the He--alkali-metal atom systems are determined as the roots of a transcendental equation whose only required inputs are the alkali-metal quantum defects, the ${e}^{\mathrm{\ensuremath{-}}}$-He s-wave phase shifts and, of course, the internuclear distance R. The results obtained for relatively low excited states are in reasonable agreement with more elaborate calculations. It is found that the notion of adiabatic quantum defects requires R(a.u.)\ensuremath{\lesssim}2n, n being the principal quantum number. The method is generalized to the case where a small external electric field scrE is present. Then the energy levels strongly depend on the angle between scrE and the internuclear axis. Numerical applications to He-H are presented.

DOI
10.1103/physreva.35.496

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The adiabatic \ensuremath{\Sigma} electronic terms of highly excited states of the He--alkali-metal atom systems are determined as the roots of a transcendental equation whose only required inputs are the alkali-metal quantum defects, the e\ensuremath−{e}^{\mathrm{\ensuremath{-}}}e\ensuremath−-He s-wave phase shifts and, of course, the internuclear distance R. The results obtained for relatively low excited states are in reasonable agreement with more elaborate calculations. It is found that the notion of adiabatic quantum defects requires R(a.u.)\ensuremath{\lesssim}2n, n being the principal quantum number. The method is generalized to the case where a small external electric field scrE is present. Then the energy levels strongly depend on the angle between scrE and the internuclear axis. Numerical applications to He-H are presented.
We calculate potential energy curves for heavy alkali-metal dimers, Rb2{\mathrm{Rb}}_{2}Rb2​ and Cs2{\mathrm{Cs}}_{2}Cs2​ in which one of the atoms is in a highly excited Rydberg states. The method combines numerical integration of coupled equations, describing interaction of electron with the ground-state atom in the field of the Coulomb core of the Rydberg atom, with subsequent matching of the obtained wave function with the Coulomb Green's function in the form of the Kirchhoff integral. The spin-orbit interaction for the Rydberg electron is also included. The results show the existence of several groups of states. Most interesting of them are dominated either by the 3S{}^{3}S3S symmetry near the ground-state atom or by the 3PJ{}^{3}{P}_{J}3PJ​ symmetry, J=0,1,2.J=0,1,2.J=0,1,2. All states, except the 3P1{}^{3}{P}_{1}3P1​ state, exhibit oscillatory dependence of energy on the internuclear distance that can support long-range molecular Rydberg states [e.g., ``trilobite states'' for the 3S{}^{3}S3S symmetry, Greene et al., Phys. Rev. Lett. 85, 2458 (2000)]. These states possess large diagonal and transition dipole moments which are expressed analytically in terms of the Coulomb wave functions and calculated in a broad range of internuclear separations.
We introduce a computational method developed for study of long-range molecular Rydberg states of such systems that can be approximated by two electrons in a model potential of the atomic cores. Only diatomic molecules are considered. The method is based on a two-electron RRR-matrix approach inside a sphere centered on one of the atoms. The wave function is then connected to a Coulomb region outside the sphere via a multichannel version of the Coulomb Green's function. This approach is put into a test by its application to a study of Rydberg states of the hydrogen molecule for internuclear distances RRR from 20 to 400 bohrs and energies corresponding to nnn from 3 to 22. The results are compared with previous quantum chemical calculations (lower quantum numbers nnn) and computations based on contact-potential models (higher quantum numbers nnn).
This paper systematically studies the energy level shift (Stark effect) of the highly excited Rydberg state of cesium (Cs) atoms under the action of an electrostatic field. In view of the limitations of the traditional hydrogen atom Stark perturbation theory in describing alkali metal atoms, we propose a complete set of parametric approximation methods, which fully consider the correction of energy level structure by quantum number deficiency. By introducing the effective principal quantum number n* and the correction coefficient 𝛽l, we extend the fourth-order Stark perturbation formula for hydrogen atoms to the Cs atomic system. The study first reviewed the analytical theory of the Stark effect of hydrogen atoms and derived the fourth-order energy shift expression based on parabolic quantum numbers. Then, the quantum number depletion characteristics of Cs atoms and their influence on Stark matrix elements were analyzed; On this basis, the parametric correction rules of the system were established, including the replacement of the effective quantum number, the determination of the polarizability correction coefficient, and the conditional use of the linear term. We have developed corresponding computing programs that can quickly estimate the energy shift of different quantum states under any electric field. By comparing with the existing experimental data and precise numerical calculation results, the reliability and application scope of this method were verified. Research shows that for the high-Rydberg state with n>20, the relative error of this parametric method under moderate electric field intensity (10-100 V/cm) is very small, making it particularly suitable for rapid estimation in large-scale parameter scanning and experimental design. The work of this paper provides practical tools and theoretical basis for the theoretical research and experimental analysis of the Stark effect of alkali metal atoms.
The high-precision determination of microwave radiation parameters may be based on measurements of the spectral characteristics of radiation transitions between the Rydberg states of atoms. Frequencies and matrix elements are calculated for dipole transitions from even-parity nS1/2 and nD5/2 to odd-parity n′P3/2 and n′F7/2 (where n′ = n, n ± 1, n ± 2) for the Rydberg states of alkali-metal atoms. The matrix elements determine the splitting of Rydberg-state energy levels in the field of a resonance microwave (μw) radiation, which results in the splitting of the resonance in electromagnetic induced transparency (EIT). Numerical computations based on the single-electron quantum defect method (QDM) and the Fues’ model potential (FMP) approach with the use of the most reliable data of the current literature on quantum defect values were performed for the 2S, 2P, 2D and 2F series of the Rydberg states of Li, Na, K, Rb and Cs atoms. The calculated data were approximated by quadratic polynomials of the principal quantum number. The polynomial coefficients were determined with the use of a standard curve-fitting interpolation polynomial procedure for numerically presented functions. The approximation equations may be used for the accurate evaluation of the frequencies and matrix elements of μw transitions in wide ranges of the Rydberg-state quantum numbers n >> 1.
Collisional broadening of nS Rydberg states by K, Rb, and Cs atoms is studied using the adiabatic approach for the $^{3}\mathrm{\ensuremath{\Sigma}}$ quasimolecular state and impulse approximation for all other contributions to the collisional width. The calculations are based on the use of the low-energy phase shifts for electron scattering by alkali metals obtained from ab initio close-coupling calculations and their extrapolation according to the modified effective range theory. We obtain a good agreement for the magnitude of the width with experimental data and satisfactory agreement for the oscillatory part of the width as a function of the principal quantum number. We find that the theory overestimates the amplitude of the oscillations. In the case of Cs as a perturber, the agreement with experimental data is good if we assume the existence of the low-energy 3^{3}3P resonance of ${\mathrm{Cs}}^{\mathrm{\ensuremath{-}}}$. However, recent ab initio bound-state and scattering calculations indicate the existence of the Cs (3{(}^{3}(3P) bound state rather than the resonance.
We use the adiabatic theory in the complex plane of internuclear distance to calculate the Rydberg-state broadening cross sections in the ambient alkali-metal gases. We have found that the imaginary parts of the branch points between the relevant energy curves have an oscillatory structure that is projected into a similar structure in the broadening cross section. The results of the distorted-wave diabatic perturbation theory and the present theory are in excellent agreement for the principal quantum numbers of the Rydberg atoms n30. Unlike the diabatic perturbation theory, the adiabatic approach does not merge with the impulse approximation and fails for higher n.