Eigenvalues and retardation effects in the n=10 states of helium

High-precision variational eigenvalues are obtained for the 10D, 10F, 10G and 10H states of helium by the application of multiple basis set methods. The accuracy is sufficient to provide the first definitive test of asymptotic expansion methods extensively developed by Drachman for Rydberg states of high angular momentum. The results are also compared with recent high-precision measurements for the 10G-1OH and IOF-1OG transition frequencies, and interpreted in terms of predicted long-range retardation corrections. Bethe logarithms are calculated and Lamb shift corrections included. Small residual discrepancies persist which could be explained by uncalculated radiative shifts. The suggestion by Kelsey and Spruch (1978) that Casimir-Polder retardation effects might be observable as energy shifts in the Rydberg states of helium has led to a concerted effort to observe the shifts (Palfrey et a1 1984, Hessels et a1 1987, 1990), particularly in the n = 10 manifold of states. A parallel development of theory (Au et a1 1984, Babb and Spruch 1988, Au 1989) provides accurate predictions for the shifts. However, retardation effects are revealed only to the extent that all the ordinary (non-retarded) effects of comparable size are known and can be subtracted from the observations. The most accurate available calculations for the fine-structure splittings in the n = 10 states of helium are the asymptotic calculations of Drachman (1982, 1985, 1988) derived from a core polarisation model. His expansions become increasingly accurate with increasing angular momentum L, but contain uncertainties larger than the retardation shift for L < 5 where the shift is large enough to be readily observable. High-precision calculations are available for low-lying states, but a rapid loss of accuracy with increasing n has until now prevented the direct application of variational methods to Rydberg states. The purpose of this letter is to report on the application of recently developed variational techniques (Drake 1987, 1988, Drake and Makowski 1988) to the 10D, 10F, 10G and 10H states of helium. These are the first variational calculations for any Rydberg states as high as n = 10, and the first for G and H states ( L = 4 and 5). Convergence of the total non-relativistic energies for the 10H states to a few parts in 10” makes these the most accurately determined two-electron states. The results allow retardation effects to be extracted from the experimental data to the full extent of the experimental precision. They also allow a precise assessment of the accuracy of asymptotic expansion methods. The principal features of the calculation are as follows. The solutions to the non-relativistic two-electron Schrodinger equation are expanded in a basis set of 0953-4075/89/230651+ 07$02.50 @ 1989 IOP Publishing Ltd L65 1 L652 Letter to the Editor Hylleraas-type functions of the form a$)kr ;4r t2 exp(-a,rl prrz)( l \ ' ) , l y ) ; L ) rf exchange (1) where rI2 = Ir, r21, the are linear variational coefficients sild (l\'), l y ) , L ) denotes a vector-coupled product of solid spherical harmonics with angular momenta 1'1'' and l y ) for the two electrons to form a state with total angular momentum L. The values of Z\" and l y ) required for completeness of the basis set are (2) (l'" / ( f ) 1 9 2 1 = (0, L ) , (1, L-11,. * * ([L/21, L-[L/21) for t = 1 ,2 , . . . [L/2]+ 1, where [ ] denotes 'greatest integer in'. The novel features of the calculation which lead to a dramatic improvement in accuracy, especially for Rydberg states, are (i) the screened hydrogenic function ls(Z, r , )nL(Z 1, r2 ) , where 2 is the nuclear charge, is included explicitly in the basis set; (ii) the terms with (0, L ) angular symmetry are 'doubled' in that each combination of powers i, j , k in ( 1 ) is included twice with different non-linear parameters a,, p,; and (iii) a complete optimisation of all the a,, p, is performed by calculating analytically the derivatives dE/da , and aE/ap , , and locating the zeros. One set of (0 , L ) terms represents the asymptotic behaviour of the wavefunction ( a 1 = 2, p1 = 1/ n ) , and the other the inner correlation effects. To these are added a further set with the same a , , p, and powers as the asymptotic (0, L ) set, but with (1, L1) angular symmetry. Without these terms, convergence becomes poor for large basis sets when the mass polarisation operator is included in the Hamiltonian. To summarise, the basis set contains the terms with the first two angular sets having identical non-linear parameters. Except for the truncations described below, all combinations of powers are included in (1) such that i + j + k S N, and the convergence studied as N is progressively increased. The truncations are i s 3 and k ~ 2 in set A, and i + j + k + I i j l s N for k 2 2 in sets B to X . These were carefully checked to ensure that they did not affect the convergence of the eigenvalues to within the final accuracy quoted. The truncation for sets B to X , first suggested by Kono and Hattori (1986), only alters the order in which terms are added as N increases, and so does not disrupt the ultimate completeness of the basis set. The largest basis sets contain 790, 732, 733 and 785 terms, corresponding to an N,,,, of 12, 11, 10 and 10 for the D, F, G and H states respectively. The final non-relativistic eigenvalues, obtained by extrapolating successive differences as N increases, are listed in table 1. Since the estimated uncertainties are the ehtire amount of the extrap.olation, the variational bound corresponding to the largest N,,, basis set can be recovered by adding the uncertainty to the tabulated eigenvalue. To obtain mass polarisation corrections, all calculations were repeated with the ( p / M)p, p z mass polarisation operator included explicitly in the Hamiltonian and the coefficient E g ) in the expansion Eh4 = [ E m + ( P / M ) h * P 2 ) + ( P / w z w 1 2 R M (3) determined by differencing. Here, RM = (1 p / M ) R m is the reduced mass Rydberg and p = Mme/( M + me) is the reduced mass of the electron. A very useful and sensitive test of the accuracy is provided by comparing the spin-averaged l?g) = f [E!$( 'L) + EE)( 'L)] as obtained above from the variational Letter to the Editor L653 Table 1. Non-relativistic eigenvalues for the 10D, 10F, 10G and 10H states of4He, expressed as a correction P E , to the screened hydrogenic energy E,, = -2.005 au. The AE, results include the mass polarisation operator in the Hamiltonian H = H,+ (+/ M ) p , * p 2 with +/ M = 1.370 7456 20 x The last column gives matrix elements of m ? ( r , ) for finite nuclear mass. State A E , ( 1 OW8 au) A E , (lo-' au) .rr( 8 ( r l ) ) 4 (IO-' au)a