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Fine-Structure Effects in Relativistic Calculations of the Static Polarizability of the Helium Atom A

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Fine-Structure Effects in Relativistic Calculations of the Static Polarizability of the Helium Atom A JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS VOLUME 88, NUMBER 2 FEBRUARY 1999 Fine-structure effects in relativistic calculations of the static polarizability of the helium atom A. Derevianko and W. R. Johnson Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, USA V. D. Ovsyannikov Voronezh State University, 394693 Voronezh, Russia V. G. Pal’chikov*) National Research Institute of Physicotechnical and Radio-Engineering Measurements (State-Owned), 141570 Mendeleevo, Moscow Region, Russia D. R. Plante Department of Mathematics & Computer Science, Stetson University, De Land, Florida 32720, USA G. von Oppen Institute fu¨r Atomare und Analytische Physik, Technische Universita¨t, D-10623 Berlin, Germany ~Submitted 20 July 1998! Zh. E´ ksp. Teor. Fiz. 115, 494–504 ~February 1999! We use the relativistic configuration-interaction method and the model potential method to calculate the scalar and tensor components of the dipole polarizabilities for the excited states 3 3 1s3p P0 and 1s3p P2 of the helium atom. The calculations of the reduced matrix elements for the resonant terms in the spectral expansion of the polarizabilities are derived using two-electron basis functions of the relativistic Hamiltonian of the atom, a Hamiltonian that incorporates the Coulomb and Breit electron–electron interactions. We formulate a new approach to determining the parameters of the Fuss model potential. Finally, we show that the polarizability values are sensitive to the choice of the wave functions used in the calculations. © 1999 American Institute of Physics. @S1063-7761~99!00902-6# 1. INTRODUCTION higher-order perturbation effects, which are needed for a meaningful interpretation of the results of measurements. For The anticrossing of atomic levels in an external field this reason, Schumann et al.1 justified the need to employ constitutes an effective method for precise measurements of methods of quantum mechanics and quantum electrodynam- the fine- and hyperfine-structure intervals and other spectro- ics to analyze the results of measurements of the d-to-d(0) scopic constants, such as the exchange energy and the diag- ratio (d(0)58772.517(16) MHz is the fine-structure interval 4! 3 onal and off-diagonal matrix elements of the spin–orbit cou- in a zero field at the point of anticrossing of the 1s3p P0 1 3 pling operators. In a recent paper, Schumann et al. studied and 1s3p P2 levels of helium. Another interesting result of 23 2 3 5 the 0 0 anticrossing of the 1s3p PJ (J 0,2) levels of that paper was the possibility of studying the effects of spin– helium by methods of high-resolution laser spectroscopy. At spin mixing of helium levels with different values of orbital the anticrossing point, the error in measuring the fine- angular momentum L but the same parity.5,6 structure interval d5E 3 2E 3 amounted to 65 MHz, 3 P0 3 P2 The aim of the present paper is to analyze theoretically and the use of microwave devices makes it possible to reduce the contribution of relativistic effects in calculations of the this value by a factor of at least 100. Nevertheless, the degree scalar and tensor components of dipole polarizabilities, 3 of accuracy already achieved makes it possible to draw which determine the shift and splitting of the 1s3p PJ (J important conclusions concerning the optimum choice 50,2) levels of helium. We employ two alternative ap- of theoretical approaches describing the effect proaches based on the relativistic configuration-interaction of 100– 200 kV cm21 electric fields on the spectrum of the method7 and on the Fuss model potential method.8 helium atom. In particular, the widely used semiempirical approach, which makes it possible to analyze the observed spectrum in 2. ALLOWING FOR RELATIVISTIC EFFECTS IN CALCULATIONS OF DIPOLE POLARIZABILITIES terms of averaged values of the atomic Hamiltonian with OF THE HELIUM ATOM allowance for relativistic corrections ~either spin-dependent or spin-independent! of order a2 Ry ~Refs. 2 and 3!, with a The shift and splitting of a level unJLM& in a uniform the fine-structure constant, does not require allowing for field F is described by the formula 1063-7761/99/88(2)/6/$15.00 272 © 1999 American Institute of Physics JETP 88 (2), February 1999 Derevianko et al. 273 5 [ ~ 1 Note that at M 0 the matrix element VJJ61 0 see, e.g., DE 52 a F2, ~1! ! 3 nJLM 2 nJLM Ref. 10 , so that the state 3 P1 remains isolated, i.e., does not mix with states with J50 and J52. a ~ ! d where the polarizability nJLM contains a scalar component Equation 4 also implies that the minimum value for as at in an electric field ~anticrossing of the fine-structure sublev- ( nJL) and a tensor component ( nJL), i.e., els! is attained at 3M 22J~J11! a 5as 1at ~ ! ~0! s s t nJLM nJL nJL . 2 2d ~a 3 2a 3 1a 3 ! J~2J21! 3 P0 3 P2 3 P2 F5¯F5A . ~6! s s t 2 t 2 ~a 3 2a 3 1a 3 ! 18~a 3 ! As the field strength F increases, the splitting of the level 3 P0 3 P2 3 P2 3 P2 may reach values comparable to the distance between adja- The problem of exact ab initio relativistic calculations of cent levels of the same parity ~the components of the fine s s t the quantities a 3 2a 3 and a 3 in Eqs. ~4! and ~6! is structure of an atomic multiplet!. Hence in the case under 3 P0 3 P2 3 P2 3 3 extremely difficult and involves calculating spectral sums investigation, i.e., the 1s3p P0 and 1s3p P2 levels and a field strength F of several hundred kilovolts per centimeter, over the complete set of unperturbed states. The need for D 5 2 such summation ~irrespective of the general approach! the level shift EnJLM E EnJLM can be found by solving the secular equation emerges in the process of determining the perturbed wave functions or energy shifts of atomic levels in the iD d 2 i5 ~ ! det EnJLM JJ8 VJJ8 0. 3 perturbation-theory setting. In addition to direct summation over the discrete spectrum of intermediate states and integra- Here the finite off-diagonal matrix elements VJJ8 correspond tion over the continuous spectrum of intermediate states, to dipole transitions between the fine-structure components which are extremely involved processes in the relativistic in second-order perturbation theory in the external field F. case, we basically used two methods to effectively calculate To derive a formula describing the dependence of d on such spectral sums ~composite matrix elements!, a method F, we use the solutions of Eq. ~3! and the definition of the for integrating inhomogeneous differential equations and a scalar and tensor components of the polarizability ~2!. The method that uses the formalism of Green’s functions. result is In the first approach, the polarizability of the state u0& is given by the formula 1 2 d5AFd~0!2 2~as 2as 1at !G 1 4~at !2 a 52 Cu uc ~ ! F 3 3 3 2F 3 u & 2^ D &, 7 2 3 P0 3 P2 3 P2 3 P2 0 0 in which the perturbed wave function uC& satisfies the inho- 1 ~0! 2 s s t .d 2 F ~a 3 2a 3 1a 3 ! mogeneous equation 2 3 P0 3 P2 3 P2 ˆ 2 !C52 C ~ ! ~H E0 D 0 . 8 F4 t 2 1 ~a 3 ! 1•••. ~4! ~ ! ~ ! ˆ d~0! 3 P2 In 7 and 8 , D is the dipole moment operator, and H is the relativistic Hamiltonian. This expression allows for the principal, or resonant, part of An important advantage of this method is the possibility the hyperpolarizability ~fourth-order corrections in the exter- of using different expressions for the atomic potential in the ! 3 numerical integration of Eq. ~8!, and the calculations can be nal field of the interacting sublevels of the multiplet 3 PJ with a zero projection of total angular momentum, M50. done not only for a purely Coulomb interaction but in the 7 The contribution of the nonresonant part to the hyperpolar- multiconfiguration interaction approximation, the Hartree– 11 izability is at most a few percent.9 In deriving ~4! we allowed Fock–Dirac approximation, and the relativistic random 12 for the fact that the matrix element V is finite at J85J phase approximation with exchange. The most exact rela- JJ8 13 61, J62 and contains only a tensor part, which depends on tivistic calculations were done by Johnson and Cheng for the projection M of the total angular momentum J.Ifwe the polarizability of the ground state of a heliumlike atom (0) 2 (0) with 2<Z<30, but at present there are no similar results for ignore the multiplet splitting EnJL EnJ8L in comparison to (0) excited states of helium with LÞ0. the energy difference E(0) 2E between different mul- nJL n8J8L8 The effectiveness of the method of Green’s functions is tiplets with n8Þn, the matrix element V can be expressed JJ8 largely determined by the existence of appropriate represen- in terms of the tensor polarizability of the 1s3p 3P state. In 2 tations of these functions. Since in the relativistic case the this approximation the matrix element V02 is given by the ! expressions for the Green’s functions are known only for the formula1 Coulomb field, the use of this approach is restricted to prob- lems in which the difference of the potential and the Cou- F2 52 at ~ ! lomb potential is insignificant or can be taken into account V02 3 3P . 5 A2 2 by perturbation-theory techniques.14,15 To allow for the contribution of relativistic corrections in The difference of the scalar polarizabilities in Eq.
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