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 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 1998͒ Zh. E´ ksp. Teor. Fiz. 115, 494–504 ͑February 1999͒ 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- 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. ͓S1063-7761͑99͒00902-6͔

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 and quantum electrodynam- the fine- and hyperfine-structure intervals and other spectro- ics to analyze the results of measurements of the ␦-to-␦(0) scopic constants, such as the exchange energy and the diag- ratio (␦(0)ϭ8772.517(16) MHz is the fine-structure interval 4͒ 3 onal and off-diagonal matrix elements of the –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 Ϫϫ Ϫ 3 ϭ 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 ␦ϭE 3 ϪE 3 amounted to Ϯ5 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 ϭ0,2) 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 cmϪ1 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-independent͒ of order ␣2 Ry ͑Refs. 2 and 3͒, with ␣ The shift and splitting of a level ͉nJLM͘ 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

ϭ ϵ ͑ 1 Note that at M 0 the matrix element VJJϮ1 0 see, e.g., ⌬E ϭϪ ␣ F2, ͑1͒ ͒ 3 nJLM 2 nJLM Ref. 10 , so that the state 3 P1 remains isolated, i.e., does not mix with states with Jϭ0 and Jϭ2. ␣ ͑ ͒ ␦ where the polarizability nJLM contains a scalar component Equation 4 also implies that the minimum value for ␣s ␣t in an electric field ͑anticrossing of the fine-structure sublev- ( nJL) and a tensor component ( nJL), i.e., els͒ is attained at 3M 2ϪJ͑Jϩ1͒ ␣ ϭ␣s ϩ␣t ͑ ͒ ͑0͒ s s t nJLM nJL nJL . 2 2␦ ͑␣ 3 Ϫ␣ 3 ϩ␣ 3 ͒ J͑2JϪ1͒ 3 P0 3 P2 3 P2 Fϭ¯Fϭͱ . ͑6͒ s s t 2 t 2 ͑␣ 3 Ϫ␣ 3 ϩ␣ 3 ͒ ϩ8͑␣ 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 ␣ 3 Ϫ␣ 3 and ␣ 3 in Eqs. ͑4͒ and ͑6͒ is structure of an atomic multiplet͒. 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 ⌬ ϭ Ϫ such summation ͑irrespective of the general approach͒ 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 ʈ⌬ ␦ Ϫ ʈϭ ͑ ͒ det EnJLM JJЈ VJJЈ 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 VJJЈ 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 ␦ on such spectral sums ͑composite matrix elements͒, a method F, we use the solutions of Eq. ͑3͒ and the definition of the for integrating inhomogeneous differential equations and a scalar and tensor components of the polarizability ͑2͒. The method that uses the formalism of Green’s functions. result is In the first approach, the polarizability of the state ͉0͘ is given by the formula 1 2 ␦ϭͱͫ␦͑0͒Ϫ 2͑␣s Ϫ␣s ϩ␣t ͒ͬ ϩ 4͑␣t ͒2 ␣ ϭϪ ⌿͉ ͉␺ ͑ ͒ F 3 3 3 2F 3 ͉ ͘ 2͗ D ͘, 7 2 3 P0 3 P2 3 P2 3 P2 0 0 in which the perturbed wave function ͉⌿͘ satisfies the inho- 1 ͑0͒ 2 s s t Ӎ␦ Ϫ F ͑␣ 3 Ϫ␣ 3 ϩ␣ 3 ͒ mogeneous equation 2 3 P0 3 P2 3 P2 ˆ Ϫ ͒⌿ϭϪ ⌿ ͑ ͒ ͑H E0 D 0 . 8 F4 t 2 ϩ ͑␣ 3 ͒ ϩ•••. ͑4͒ ͑ ͒ ͑ ͒ ˆ ␦͑0͒ 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. ͑8͒, and the calculations can be nal field of the interacting sublevels of the multiplet 3 PJ with a zero projection of total angular momentum, Mϭ0. 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 ͑4͒ we allowed Fock–Dirac approximation, and the relativistic random 12 for the fact that the matrix element V is finite at JЈϭJ phase approximation with exchange. The most exact rela- JJЈ 13 Ϯ1, JϮ2 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) Ϫ (0) with 2рZр30, but at present there are no similar results for ignore the multiplet splitting EnJL EnJЈL in comparison to (0) excited states of helium with LÞ0. the energy difference E(0) ϪE between different mul- nJL nЈJЈLЈ The effectiveness of the method of Green’s functions is tiplets with nЈÞn, the matrix element V can be expressed JJЈ 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 ϭϪ ␣t ͑ ͒ lomb potential is insignificant or can be taken into account V02 3 3P . 5 ͱ2 2 by perturbation-theory techniques.14,15 To allow for the contribution of relativistic corrections in The difference of the scalar polarizabilities in Eq. ͑4͒, s s calculations of the scalar and tensor components of polariz- ␣ 3 Ϫ␣ 3 , is determined by the contribution of relativ- 3 P0 3 P2 abilities, we used the resonance approximation for the istic effects, which means it is a small quantity of order ␣2. second-order composite matrix elements, i.e., in the spec- 274 JETP 88 (2), February 1999 Derevianko et al. trum of intermediate states we isolated the terms that corre- TABLE I. Reduced matrix elements of the dipole moment operator. spond to transitions in which the Transition Matrix element does not change. 3 → 3 Ϫ For instance, for the scalar polarizability of the triplet 1s3p P0 1s3s S1 6.4797 3 → 3 Ϫ ͉nJLM͘ state, 1s3p P2 1s3s S1 14.489 3 → 3 1s3p P0 1s3d D1 8.2923 3 → 3 2 ͉͗nJLʈrʈnЈJЈLЈ͉͘2 1s3p P2 1s3d D1 1.8542 s 3 → 3 ␣ ϭϪ ͚ , ͑9͒ 1s3p P2 1s3d D2 7.1805 nJL 3͑2Jϩ1͒ Ϫ 3 → 3 nЈJЈLЈ EnJL EnЈJЈLЈ 1s3p P2 1s3d D3 16.994 ␣s(r) ͑ ͒ the resonant term nJL in 9 has the form 2␤2 L͑2JЈϩ1͒ ␣s͑r͒ϭϪ nJL ͫ ␤ nJЈ LϪ1 ͭ ͚ Ϫ R ͑ ͒ nJL 3 Ϫ nJЈ L 1 nJL ⌿ ϭ I F ⌽ ͑ ͒ JЈ EnJL EnJЈ LϪ1 I͑F͒ ͚ Ckl kl . 12 kуl LJ1 2 ͑Lϩ1͒͑2JЈϩ1͒ ⌽ ϫͭ ͮ ͬ ϩ͚ Here the kl are two-particle basis functions with fixed val- Ј Ϫ Ϫ J L 11 JЈ EnJL EnJЈ Lϩ1 ues of total angular momentum J, its projection M, and par- I(F) ity. We found the weighting factors Ckl from the varia- LJ1 2 ϫͫ ␤ nJЈ Lϩ1ͭ ͮ ͬ ͑ ͒ tional principle by using the relativistic no-pair Hamiltonian, ϩ R ͮ . 10 nJЈ L 1 nJL JЈ Lϩ11 which incorporates the Coulomb and Breit electron–electron interaction operators,21,22 averaged over the functions ͑12͒. j j j Here the ͕ 1 2 3 ͖ are Wigner 6 j-symbols,16 and the To exclude the contribution of negative-energy states ͑the m1 m2 m3 nЈJЈLЈ ͑ ͒ positron spectrum͒, the two-particle operators in the relativ- RnJL are radial matrix elements. In Eq. 10 , the effects of singlet–triplet mixing of levels are taken into account by the istic Hamiltonian were multiplied by products of single- parameter particle operators projecting on the subspaces of solutions of the positive-energy .7 ␪ ϭ cos nL , J L, The wave functions ͑12͒ are normalized by the condition ␤ ϭͭ ͑11͒ nJL 1, JÞL, ͉ I͑F͉͒2ϭ ͑ ͒ ␪ 17 ͚ Ckl 1. 13 where nL is the singlet–triplet mixing angle. Chang noted kуl that ␪ is almost entirely independent of n, and the typical nL The single-particle basis orbitals employed in the values of ␪ at Lϭ1,2,3 are 0.02°, 0.5°, and 30°, respec- nL configuration-interaction method incorporates the s, p, d, f , tively. The tensor part of the polarizability has a similar and g partial waves, with a spline approximation used for structure and differs from ͑10͒ only in angular coefficients; each wave. Estimates of the convergence rate of the method for the sake of brevity we will not write it here. ͑for calculations of given accuracy͒ can be found in Refs. 13 To calculate ␣s(r) and ␣t(r) we used exact relativistic nJL nJL and 22. results for the radial integrals, while for the energy denomi- The results of relativistic calculations for the reduced nators we used precise experimental data.18 Calculations of matrix elements ͑without allowance for retardation effects the other terms in ͑9͒ with nЈÞn were done with nonrelativ- for the dipole moment operator͒ are listed in Table I. istic values for the radial matrix elements of the dipole mo- In Tables II and III we list the results for the contribu- ment operator. To this end we used exact numerical data for tions of the intermediate S and D states in calculations of the the oscillator strengths of the s–p and p–d transitions in scalar polarizabilities of the 1s3p 3P (Jϭ0,2) states of he- helium calculated with multiparameter variational wave J lium. The difference between the tensor part of the polariz- functions.19,20 ability of the 1s3p 3P level and the scalar part is that the This approach is applicable primarily because the contri- 2 contributions of the intermediate states 1snЈs 3S and bution of states with nЈϭn in ͑9͒ is numerically predominant 1 1snЈd 3D have opposite signs and that the total contribution and amounts to roughly 95% in the case of excited 1s3p 3P 1 J of the 1snЈd 3D states contains an additional numerical fac- (Jϭ0,2) levels of helium ͑see Tables II and III below͒. 3 tor, Ϫ2/7. Table IV summarizes the results of numerical calcula- 3. SELECTION OF THE BASIS WAVE FUNCTIONS AND tions of the scalar and tensor components of helium polariz- DISCUSSION OF THE RESULTS OF CALCULATIONS abilities. The dependence of ␦ on F near the anticrossing In recent years the configuration-interaction method has point is plotted in Fig. 1. ⌬␣ been successfully used to obtain precise wave functions and Note that the difference remains almost the same if we use nonrelativistic variational values for the resonant ma- matrix elements for with a small number of , s s trix elements in ͑10͒: ␣ 3 ϭ17 207, ␣ 3 ϭ17 198, and e.g., heliumlike systems. In the present paper we use the 3 P0 3 P2 technique developed by Johnson et al.7,13 to calculate the re- ⌬␣ϭ9. The explanation is that the principal contribution to duced matrix elements corresponding to the resonant terms ⌬␣ is provided by the relativistic corrections for the fine in the expansion ͑10͒. structure of the levels in the energy denominators of ͑12͒ The wave function of the initial ͑I͒ and final (F) states rather than the relativistic corrections for the matrix elements can be written of the dipole moment operator ͑see Table I͒. JETP 88 (2), February 1999 Derevianko et al. 275

TABLE II. Contributions of the S and D states of the intermediate spectrum to the polarizability of the 3 1s3p P0 state of helium.

3 ␭ ϭ 3 ␭ ϭϪ 3 3 nЈ 1snЈs S1 ( 0 0.698) 1snЈs S1 ( 0 0.302) 1snЈs S1 1snЈd D1 2 Ϫ0.34 Ϫ1.67 Ϫ1.57 3 Ϫ2377.93 Ϫ2682.56 Ϫ2638.45 18 742.21 4 451.87 319.37 312.17 623.53 5 25.38 18.27 17.92 86.01 6 6.36 4.60 4.72 27.16 7 2.59 1.88 1.85 12.29 8 1.34 0.97 0.96 6.72 9 0.79 0.57 0.57 4.13 10 0.51 0.37 0.34 2.59

Total Ϫ1888.6 Ϫ2337.7 Ϫ2301.50 19 540.63

␭ To test the above results, we did alternative polarizabil- radial quantum number, l the effective orbital angular mo- ␯ ity calculations using the method of the Green’s functions of mentum, and nl the effective principal quantum number in an optical electron to sum over the complete intermediate- the formula for the energy of the atomic state ͉nl͘, states spectrum in ͑9͒. Z2 Note that the correct selection of the initial analytical E ϭϪ . ͑17͒ nl ␯2 representation of the Green’s functions GE(r1 ,r2) plays an 2 nl important role in specific polarizability calculations, since it Here and in ͑15͒, Z is the charge of the residual ion. The makes it possible to obtain the result in the form most ratio- explicit expression for R (r) coincides in form with hydro- nal and convenient for further applications. In this paper we nl genlike wave functions:23 have taken the Green’s function for the Fuss model potential 3/2 from Ref. 14. The angular part of GE(r1 ,r2) is simply the 2Z 2Zr product of spherical harmonics, while for the radial part R ͑r͒ϭ U ͩ ͪ . ͑18͒ nl ␯2 nrl ␯ nl nl gl(E;r1 ,r2) we have taken an expansion in Sturm functions, 15 ␭ which have only a discrete spectrum: The parameter l (l may represent a specific set of spin– orbit quantum number, in addition to representing a specific ͑ ͒ϭ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͒ ͑ ͒ GE r1 ,r2 ͚ gl E;r1 ,r2 Y lm n1 Y lm* n2 , 14 angular momentum can be found by comparing 17 with lm experimental values of the lowest state of a valence electron ϱ ͑ ͒ 4Z U ͑2Zr /␯͒U ͑2Zr /␯͒ with a given l see Ref. 23 . The radial quantum number nr ͑ ͒ϭ kl 1 kl 2 ͑ ͒ of this state is assumed to be zero. As shown by Simons,8 ␭ gl E;r1 ,r2 ␯ ͚ ϩ␭ ϩ Ϫ␯ , 15 l kϭ0 k l 1 represents the entire experimental spectrum of the atom ͑ ␭ where ␯ϭZ/ͱϪ2E, and fairly well in most cases the weak dependence of l on the position of the can be ignored͒. k! ␭ x 2␭ ϩ1 With such a definition of ␭ for atomic series whose U ͑x͒ϭͱ x l expͩ Ϫ ͪ L l ͑x͒. l kl ⌫ ϩ ϩ ␭ ͒ k ͑k 2 2 l 2 lowest states are the ground and the metastable, the error in ͑16͒ calculating the radial matrix elements ͗nl͉rL͉nЈl͘Lу1 with ͑ ͒ The radial wave functions R (r) are obtained from the resi- wave functions 18 may reach 50%. In view of this we nl ␭ dues of the Green’s function at the poles of formulated a modified approach to the definition of nr and l ␯ϭ␯ ϭ ϩ␭ ϩ ϭ in these series,24 which allowed us to significantly refine the gl(E;r1 ,r2): nl nr l 1, with nr 0,1,2,... the calculations of the spectroscopic characteristics of atoms in ground and excited states. Here the radial quantum number TABLE III. Contributions of the S and D states of the intermediate spectrum of the lowest state ͑ground or metastable͒ of the series is 3 to the polarizability of the 1s3p P2 state of helium. assumed to be unity, so that the effective orbital angular

3 3 3 3 nЈ 1snЈs S1 1snЈd D1 1snЈd D2 1snЈd D3 2 Ϫ1.57 TABLE IV. Scalar and tensor polarizabilities of the helium atom. 3 Ϫ2638.76 187.32 2809.37 15 735.95 4 312.15 6.23 93.52 523.74 Quantity Numerical value 5 17.92 0.86 12.90 72.25 6 4.72 0.27 4.07 22.81 s ␣ 3 17 203 3 P0 7 1.85 0.12 1.84 10.32 ␣s 17 191 3 3P 8 0.96 0.07 1.01 5.65 2 s s ⌬␣ϭ␣ 3 Ϫ␣ 3 10 9 0.57 0.04 0.62 3.47 3 P0 3 P2 ␣t 351.65 3 3P 10 0.34 0.03 0.39 302.17 2 Ϫ4 ¯F 0.29 343ϫ10 Total Ϫ2301.83 194.94 2923.72 16 376.36 ϭ150.99 kV cmϪ1 276 JETP 88 (2), February 1999 Derevianko et al.

TABLE V. Scalar and tensor polarizabilities of the helium atom calculated by the model potential method.

Quantity Numerical value

s ␣ 3 17 266 3 P0 ␣s 17 255 3 3P 2 s s ⌬␣ϭ␣ 3 Ϫ␣ 3 11 3 P0 3 P2 ␣t 374.16 3 3P 2 Ϫ4 ¯F 0.28 458ϫ10 ϭ146.45 kV cmϪ1

FIG. 1. Dependence of ␦ ͑in atomic units͒ on the electric field strength F in cant figures cancel out. This loss of accuracy can be avoided the vicinity of anticrossing ¯F. if we expand the polarizabilities as functions of the energy of an atomic level in a Taylor series. The calculation then re- duces to finding the energy derivatives of the polarizabilities, momentum ␭ is equal to ␯ Ϫ2, with ␯ the effective prin- l g g and these can be expressed in terms of third-order dipole cipal quantum number of the lowest state. Note that the wave matrix elements with two Green’s functions, which, in par- function of the lowest level at n ϭ1 coincides precisely with r ticular, enter into the expression for the hyperpolarizability the wave function obtained by the quantum-defect method in of the atomic states.26 atoms.25 It is no accident that the numerical results in Tables IV Thus, the sets of states of an atom with spin–orbit quan- and V arer close, since the model potential method yields a tum numbers of the ground and metastable levels are, strictly correct dependence in the higher-order matrix elements for speaking, incomplete since they do not contain states with the energy of the atomic levels,23 provided that we use the n ϭ0. Hence the radial Green’s functions in the subspaces of r exact ͑experimental͒ values for the energies of the fine- the series in question contain additional ‘‘imaginary’’ terms structure sublevels. This condition makes it possible to take with n ϭ0 and an effective principal quantum number ␯ r im into account the contributions of relativistic and correlation ϭ␯ Ϫ1. The binding energy of the ‘‘imaginary’’ state, de- g effects in the model potential method. Furthermore, using fined in ͑17͒, is almost ten times higher than the excitation this method, one can easily show that the states belonging to energy of any level in a series, so that its contribution to the a complete set and not taken into account in relativistic cal- optical-transition amplitude can be ignored. culations ͑the states belong to the continuum and to the dis- Note also that the wave function of a valence electron in crete spectrum with nЈϾ10) contribute no more than 0.2% state ͉nl͘ has the correct sign in the asymptotic region to the numerical values of the quantities considered here. ͑which provides the principal contribution to multiple matrix The present work was made possible by the financial elements͒ only if we multiply it by an addition phase factor support the Russian Fund for Fundamental Research ͓Grant (Ϫ1)k, where kϭnϪn ϪlϪ1. r 97-02-16407and an international grant with the German Re- The second and third columns of Table II contain the ͑ ͔͒ Ј 3 search Society 96-02-00257, 436 RUS 113/164/O R,S and contributions of the intermediate 1sn s S1 states to the sca- ͑ 3 the U.S. National Science Foundation Grant PHY 95- lar polarizability of the 1s3p P0 state calculated by the tra- ͒ ␭ 13179 . ditional and modified approaches, which yield values of 0 equal to 0.698 and Ϫ0.302, respectively. A comparison with ͒ ͑ * E-mail: [email protected] the data of precise variational calculations the fourth col- [email protected] umn in Table II͒ suggests that the discrepancy of the final 1͒Here and in the text that follows we use the atomic system of units. ␭ ϭϪ results is less than 2% if the value 0 0.302 is used, ␭ ϭ 1 while calculations with 0 0.698 yield an error exceeding R. Schumann, M. Dammasch, U. Eichmann, Y. Kriescher, G. Ritter, and 20%. G. von Oppen, J. Phys. B 30, 2581 ͑1997͒. 2 D. R. Cok and S. R. Lundeen, Phys. Rev. A 19, 1830 ͑1979͒. The results of polarizability calculations based on the 3 D. R. Cok and S. R. Lundeen, Phys. Rev. A 24, 3283 ͑1981͒. use of Green’s functions in the method of the Fuss model 4 H. D. Yang, P. McNicholl, and H. Metcalf, Phys. Rev. A 53, 1725 ͑1996͒. potential are listed in Table V. The numerical discrepancies 5 V. G. Pal’chikov and G. von Oppen, Phys. Scr. 52, 366 ͑1995͒. 6 ´ of the data of Tables IV and V are due, on the one hand, to V. G. Pal’chikov and G. von Oppen, Zh. Eksp. Teor. Fiz. 110, 876 ͑1996͒ ͓JETP 83, 481 ͑1996͔͒. the semiempirical approximation of the model potential 7 W. R. Johnson, D. R. Plante, and J. Sapirstein, Adv. At., Mol., Opt. Phys. method and, on the other, to the allowance for the contribu- 35, 255 ͑1995͒. tion of the continuous spectrum in the Green’s functions 8 G. Simons, J. Chem. Phys. 55, 756 ͑1971͒. 9 method. Note that the results for the difference of scalar A. Derevianko, W. R. Johnson, V. D. Ovsiannikov, V. G. Pal’chikov, D. R. Plante, and G. von Oppen, in Proc. Int. Conf. At. Phys. (ICAP-98, polarizabilities and for the tensor polarizability of the Windsor) ͑1998͒, p. 168. 3 10 1s3p P2 state differ from those in Table IV only by 10%. N. L. Manakov and V. D. Ovsiannikov, J. Phys. B 10, 569 ͑1977͒. To obtain numerical values of the difference of scalar polar- 11 W. R. Johnson and C. D. Lin, Phys. Rev. A 9, 1486 ͑1974͒. 12 izabilities with an accuracy of about 1%, the calculations of W. R. Johnson, D. Kolb, and K. N. Huang, At. Data Nucl. Data Tables 28, 333 ͑1983͒. the radial integral should be done with an accuracy of five or 13 W. R. Johnson and K. T. Cheng, Phys. Rev. A 53, 1375 ͑1996͒. six figures, since, as Table V implies, the first three signifi- 14 S. A. Zapryagaev, N. L. Manakov, and V. G. Pal’chikov, Theory of One- JETP 88 (2), February 1999 Derevianko et al. 277

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