CHINESE JOURNAL OF PHYSICS VOL. 43, NO. 2 APRIL Ming-Keh Chen

CHINESE JOURNAL OF PHYSICS VOL. 43, NO. 2 APRIL 2004 Doubly Excited 1,3 P e Resonances in He Between the N=2 and 3 He + Thresholds Ming-Keh Chen Department of Physics, National Chung-Hsing University, Taichung, Taiwan 40227 (Received March 22, 2004) We calculated the energies and widths of seventeen singlet and seventeen triplet P e resonant states of He between the N=2 and 3 thresholds. These states are classified with quantum numbers K, T, and A. For the 1 P e states there are nine members in the 3 (1, 1) n (4 n 12) series and eight members in the 3 ( 1, 1) n (4 n 11) series. There are nine members in the 3 (1, 1) + n ( 3 n 11) series and eight members in the 3( 1, 1) + n ( 3 n 10) series for the 3 P e states. The energies and widths are compared with other theoretical results. Only a few results for low 1,3 P e states can be found in the literature to the author s knowledge, because of the problem of the slow convergence and large uncertainties. Energy values of the high 1,3 P e states are nearly degenerate. We improved the slow convergence and large uncertainties in calculating these nearly degenerate states to obtain stable results. PACS numbers: 31.25.Jf, 32.70.Jz, 32.80.Dz I. INTRODUCTION A doubly-excited two-electron system is a prototype for the theoretical and experimental study of electron-electron correlations. The 3 P e states in H and He below the N=2 threshold has been known [1, 2] for a long time. Both the 1,3 P e states can either decay radiatively to the 1,3 P o states or autoionize. 1 P e states can be obtained by two-photon excitation from the 1 S e ground state. 3 P e states can be formed from the P hydrogen or He + by electron impact and from the 2p 2 3 P e state by two-photon excitation. The experimental results for similar states [3] of N +5 have been obtained from two-electron capture by N +7 using He, H 2, and Ar as targets. Recently, Stinz et al. [4] and Rislove et al. [5] measured the lowest 1 D e of H by two-photon spectroscopy techniques with high resolution. More recently, Iemura et al. [6] measured the ejected electron spectrum of doubly excited states in He produced by slow collisions of He 2+ with Ba. The detailed comparisons between the experimental and ab intio theoretical results are expected to become available. Ho and Bhatia [7] have calculated doubly excited 1,3 P e states in heliumlike systems by the complexrotation method with Hylleraas-type functions. They calculated the lowest five 1 P e and eight 3 P e states of He between the N=2 and 3 He + thresholds. Lindroth [8] calculated the two lowest 3 P e states with a finite discrete spectrum. Elander, Levin, and Yarevsky [9] studied S, P, and D resonant helium by a technique based on the total-angular-momentum representation, the smooth exterior complex-scaling procedure, and the three-dimensional finite-element method. A fourteen-state pseudostate close-coupling calculation has been http://psroc.phys.ntu.edu.tw/cjp 329 c 2005 THE PHYSICAL SOCIETY OF THE REPUBLIC OF CHINA

330 DOUBLY EXCITED 1,3 P e RESONANCES... VOL. 43 carried out for the similar states for N +5 [3] and O 6 [10]. We have developed the saddlepoint complex-rotation method with B-spline functions and successfully obtained accurate results of the doubly-excited H and He [11 15]. In this work, we extended our previous work [16] to calculate the energies and widths of seventeen singlet and eighteen triplet P e states of He between the N=2 and 3 thresholds. II. THEORY We studied doubly excited 1,3 P e resonant He by the saddle-point complex rotation method with B-spline functions [17]. We will briefly summarize the method here. In a configuration interaction scheme, we constructed the wave functions in terms of B splines of order k and the total number N, defined between two end points, r min =0 and r max = R, and built vacancies into the wave functions. With an exponential sequence, we have the trial wave function for a two-electron system Ψ = A(1 P( r 1 )(1 P( r 2 )) C i,j,l1,l 2 Φ i,j ( r 1, r 2 )Yl LM 1,l 2 χ(1,2), (1) i,j,l 1,l 2 with Φ i,j ( r 1, r 2 ) = B i,k(r 1 ) r 1 B j,k (r 2 ) r 2, (2) Y LM l 1,l 2 = m 1,m 2 l 1 l 2 m 1 m 2 LM Y l1,m 1 (ˆr 1 )Y l2,m 2 (ˆr 2 ), (3) and i j jm, (4) where L is the total angular momentum and the numbers i and j are positive integers, which are not larger than N [18], and jm is some selected integer [19, 20]. A is the antisymmetrization operator, χ(1, 2) is the spin wave function, and P( r) is a projection operator. For the present, the 2p orbital is the vacancy orbital. P( r) = φ 2p φ 2p. (5) We assume φ 2p to be hydrogenic with effective nuclear charge q. The saddle-point variation is carried out by first minimizing the energy with respect to C i,j,l1,l 2 and the set of B-spline basis functions, and then maximizing the energy with respect to the effective nuclear charge, q, to obtain the saddle-point energy and wave function. The B-spline basis functions with an exponential knot sequence [18, 21] are employed in the present calculations. After the saddle-point variation is carried out, we calculate the resonance energy and width by a complex-rotation method. The trial wave functions are composed

VOL. 43 MING-KEH CHEN 331 TABLE I: The expectation values of cos(θ 12 ), r <, r >, r, and r 2 (in a.u.). state cos(θ 12 ) r < r > r r 2 cos(θ 12 ) r < r > r r 2 1 P e n 3(1, 1) n 3( 1, 1) n 4 0.313 5.66 14.6 10.1 133 0.227 5.33 17.0 11.1 173 5 0.319 5.82 24.8 15.3 357 0.279 5.50 30.1 17.8 522 6 0.322 5.84 38.2 22.0 827 0.300 5.58 46.1 25.8 1205 7 0.324 5.83 54.6 30.2 1677 0.311 5.63 65.0 35.3 2384 8 0.325 5.82 74.1 40.0 3074 0.317 5.65 87.0 46.3 4246 9 0.327 5.81 96.6 51.2 5212 0.321 5.67 112 58.8 7012 10 0.328 5.81 122 64.0 8317 0.323 5.68 140 72.7 10932 11 0.329 5.80 151 78.3 12649 0.325 5.68 171 88.3 16296 12 0.329 5.80 182 94.8 18503 - - - - - 3 P e n 3(1, 1) + n 3( 1, 1) + n 3 0.270 5.44 8.73 7.08 58.1 0.091 5.14 8.87 7.00 58.7 4 0.306 6.09 15.8 11.0 161 0.237 5.50 20.2 12.9 246 5 0.316 5.98 26.4 16.2 411 0.279 5.61 34.0 19.8 669 6 0.320 5.90 40.1 23.0 920 0.299 5.66 50.7 28.2 1464 7 0.323 5.84 56.9 31.4 1829 0.309 5.69 70.4 38.0 2799 8 0.325 5.81 76.7 41.3 3305 0.316 5.72 93.0 49.4 4871 9 0.326 5.79 99.6 52.7 5547 0.319 5.73 119 62.2 7907 10 0.327 5.77 125 65.6 8789 0.322 5.75 147 76.6 12166 11 0.328 5.76 154 80.0 13286 - - - - - of the saddle-point wave functions (Eq. (1)) and the open-channel components. We choose the open-channel components [22] to be: with Ψ open = Aψ 2p ( r 1 ) KO kc=l C kc u kc ( r 2 )Y L,M 1,L χ(1,2), (6) u kc ( r i ) = r kc i e βr i, (7) and Y LM 1,L = m 1,m 2 1Lm 1 m 2 LM Y 1,m1 (ˆr 1 )Y L,m2 (ˆr 2 ), (8) where ψ 2p is the 2p radial wave function of hydrogen. The non-negative integer, KO, is chosen to be large enough to ensure the accuracy of the resonance energy and width in

332 DOUBLY EXCITED 1,3 P e RESONANCES... VOL. 43 TABLE II: Energies and widths for the doubly excited 1 P e resonances between the N=2 and 3 thresholds (in a.u.). state 3(1, 1) n 3( 1, 1) n n E width E width present 4 0.2789924 4.4366(-5) 0.2593508 1.2663(-6) [7] 0.278993 4.43(-5) 0.259352 1.259(-6) [25] 0.278887 0.259211 present 5 0.2536386 2.6438(-5) 0.2445129 8.752(-7) [7] 0.2536387 2.6383(-5) 0.2445136 8.75(-7) [25] 0.253541 0.244425 present 6 0.2422374 1.5406(-5) 0.2372820 5.828(-7) [7] 0.242237 1.6(-5) [25] 0.242159 0.237185 present 7 0.2360988 9.508(-6) 0.2331141 3.956(-7) [25] 0.236052 0.233032 present 8 0.2324109 6.138(-6) 0.2304758 2.784(-7) present 9 0.2300209 4.280(-6) 0.2286958 2.18(-7) present 10 0.2283835 3.080(-6) 0.2274268 1.60(-7) present 11 0.2272127 2.214(-6) 0.2265128 1.20(-7) present 12 0.2263462 1.698(-6) the calculation by the complex-rotation method. In the complex-rotation calculation, each radial coordinate r i in u kc ( r i ) of the open-channel components, Ψ open, takes the form r i e iθ. Ho and Bhatia [7] found that some energy values of 1,3 P e appear as nearly degenerate. The problem of slow convergence and higher uncertainties was encountered in the presence of the nearly degenerate states. In order to improve the accuracy, as in our previous work, we constructed the close-channel components with the saddle-point wave functions of all the states which are nearly degenerate in the complex-rotation calculations [15]. III. RESULTS AND DISCUSSION In the present work, the values of R for the end points are chosen to be 400 a.u. so that the saddle-point energies converge to within the uncertainty of 10 8 a.u. We included 5 7 and 5 8 partial waves in calculating the 1 P e and 3 P e states, respectively, to ensure that the saddle-point energies converged within the uncertainty of 10 8 a.u. For each partial wave, our results converge to within 10 8 a.u. with increasing order k and total number N of the B-spline. We calculated the expectation value of the angle between the two electrons, θ 12, the average value of r for the inner electron and outer electron, r <, r >, and the average values of r and r 2, r, r 2, which are shown in Table I. From the expectation values of r < and r >, we can assign the principle quantum numbers of the inner and outer

VOL. 43 MING-KEH CHEN 333 TABLE III: Energies and widths for the doubly excited 3 P e resonances between the N=2 and 3 thresholds (in a.u.). state 3(1, 1) + n 3( 1, 1) + n n E width E width present 3 0.3360894 4.49570(-3) 0.2911514 7.4392(-5) [7] 0.3360879 4.4887(-3) 0.29115828 7.40(-5) [8] 0.33609 4.49(-3) 0.29115 7.41(-5) [9] 0.336089 4.49(-3) 0.291157 7.4(-5) [25] 0.336622 0.290666 present 4 0.2715578 1.79170(-3) 0.2535726 2.3728(-5) [7] 0.2715572 1.7887(-3) 0.25357465 2.3523(-5) [25] 0.271566 0.253369 present 5 0.2509318 8.5647(-4) 0.2419564 1.4620(-5) [7] 0.2509315 8.55(-4) 0.2419583 1.44(-5) [25] 0.250877 0.241818 present 6 0.2409237 4.5584(-4) 0.2358935 9.518(-6) [7] 0.24091 4.5(-4) 0.235894 8.0(-6) [25] 0.240904 0.235788 present 7 0.2353965 2.678(-4) 0.2322705 6.448(-6) [25] 0.235358 present 8 0.2319708 1.6860(-4) 0.2299237 4.60(-6) present 9 0.2297413 1.1374(-4) 0.2283137 3.28(-6) present 10 0.2281904 7.988(-5) 0.2271616 2.82(-6) present 11 0.2270719 5.860(-5) electrons approximately. From the expectation values of θ 12 and with the help of Lin s results [23, 24], these states are classified with quantum numbers K, T, and A. For the 1 P e states, there are nine members in the 3 (1,1) n ( 4 n 12) series and eight members in the 3 ( 1,1) n ( 4 n 11) series. There are nine members in the 3 (1,1) + n ( 3 n 11) series and eight members in the 3 ( 1,1) + n ( 3 n 10) series for the 3 P e states. The series 3 (1,1) +, n have the approximate configuration of 3pnp. The series 3 ( 1,1) n +, have the approximate configuration of 3dnd. The energies and widths are shown in Tables II and III for 1 P e and 3 P e He, respectively. The widths of the series with the approximate configuration of 3pnp are much larger. Only a few results for low 1,3 P e states can be found in the literature to the author s knowledge, because of the problem of the slow convergence and large uncertainties. The energy values of high 1,3 P e states are nearly degenerate. They are states 3 (1,1) +, n+1 and 3( 1,1) n +,, the series with K=1 and K= 1, which appear almost in pairs. We improved the slow convergence and large uncertainties in calculating these nearly degenerate states by the complex-rotation method to obtain stable results. The accuracy is one or two digits more, after we constructed the close-channel components with the saddle-point wave functions of all the states, which are nearly degenerate, in the

334 DOUBLY EXCITED 1,3 P e RESONANCES... VOL. 43 TABLE IV: Quantum defect (µ) and reduced widths for the doubly excited 1,3 P e resonances between the N=2 and 3 thresholds (in a.u.). state(n) µ reduced width µ reduced width 1 P e 3(1, 1) n 3( 1, 1) n present 4 1.032 1.141(-3) 0.330 6.259(-5) [25] 1.030 0.323 present 5 1.011 1.675(-3) 0.264 9.298(-5) [25] 1.004 0.254 present 6 1.002 1.924(-3) 0.238 1.115(-4) [25] 0.992 0.219 present 7 0.997 2.056(-3) 0.225 1.230(-4) [25] 0.987 0.199 present 8 0.995 2.110(-3) 0.217 1.312(-4) present 9 0.993 2.197(-3) 0.212 1.48(-4) present 10 0.992 2.251(-3) 0.199 1.51(-4) present 11 0.990 2.220(-3) 0.205 1.51(-4) present 12 0.989 2.267(-3) 3 P e 3(1, 1) + n 3( 1, 1) + n present 3 0.905 4.137(-2) 0.307 1.453(-3) [25] 0.909 0.297 present 4 0.817 5.781(-2) 0.0064 1.511(-3) [25] 0.817 0.0067 present 5 0.827 6.225(-2) 0.0336 1.865(-3) [25] 0.822 0.0513 present 6 0.829 6.302(-2) 0.047 2.105(-3) [25] 0.827 0.0710 present 7 0.839 6.261(-2) 0.0541 2.41(-3) [25] 0.830 present 8 0.838 6.193(-2) 0.0574 2.41(-3) present 9 0.845 6.168(-2) 0.0599 2.44(-3) present 10 0.847 6.125(-2) 0.0612 2.87(-3) present 11 0.846 6.135(-2) complex-rotation calculations. In Tables II and III, we compared our results with those of Ho and Bhatia [7], Lindroth [8], Elander, Levin, and Yarevsky [9], and Lipsky et al. [25]. Ho and Bhatia [7] calculated the lowest five 1 P e and eight 3 P e states of He below the N=3 threshold. They obtained pretty good results. The accuracy of their results deteriorated more rapidly than ours for the highly excited states. Ho and Bhatia [7] have pointed out the problem of the nearly

VOL. 43 MING-KEH CHEN 335 degenerate states. Lipsky et al. [25] only calculated the energies of eight 1 P e and nine 3 P e states. The agreement between our results and those of Ho and Bhatia [7], Lindroth [8], Elander, Levin, and Yarevsky [9], and Lipsky et al. [25] is good. Iemura et al. [6] did not classify the (3PnP) 1,3 P e states from their observed values. No experimental results of (3PnP) 1,3 P e and (3PnP) 1,3 P e can be found to the author s knowledge. In Table IV, the quantum defects (µ) and the reduced widths (n µ) 3 Γ are calculated. They converge well with increasing n, the principal quantum number of the outer electron. The convergence is better for the series with K=1 than that for the series with K= 1. The small quantum defects (µ) for the series of triplet states with K= 1 converge not as well as for the other series. Our results on quantum defects compare well with those of Lipsky et al [25]. The agreement is better for the series with K=1, whose quantum defects are larger. This shows that our energies and widths for the highly excited states are reliable because of the good convergence for the quantum defects and reduced widths. We improved the accuracy of the resonant energies and widths of the nearly degenerate 1,3 P e states of helium and extended the calculations to study the highly excited states. Acknowledgments 005. This work is supported by National Science Council Grant No. NSC90-2112-M-005- References [1] A. K. Bhatia, Phys. Rev. A 2, 1667 (1970). [2] D. W. F. Drake, Phys. Rev. Lett. 24, 126 (1970). [3] D. H. Oza et al., J. Phys. B 21, L131 (1988). [4] A. Stinz et al., Phys. Rev. Lett. 75, 2924 (1995). [5] D. C. Rislove et al., Phys. Rev. A 58, 1889 (1998). [6] K. Iemura et al., Phys. Rev. A 64, 062709 (2001). [7] Y. K. Ho and A. K. Bhatia, Phys. Rev. A 47, 2628 (1993). [8] E. Lindroth, Phys. Rev. A 49, 4473 (1994). [9] N. Elander, S. Levin, and E. Yarevsky, Phys. Rev. A 67, 062508 (2003). [10] P. Moretto-Capella et al., J. Phys. B 22, 271 (1989). [11] M.-K. Chen, J. Phys. B 30, 1669 (1997). [12] M.-K. Chen, Phys. Rev. A 56, 4537 (1997). [13] M.-K. Chen, Phys. Rev. A 60, 2565 (1999). [14] M.-K. Chen, J. Phys. B 32, L487 (1999). [15] M.-K. Chen, Nucl. Phys. A 684c, 684 (2001). [16] M.-K. Chen, Nucl. Instr. And Meth. B 205, 78 (2003). [17] C. deboor, A Practical Guide to Splines (New York: Springer, 1978). [18] M. -K. Chen and C. -S. Hsue, J. Phys. B 25, 4059 (1992). [19] M. -K. Chen, J. Phys. B 27, 865 (1994).

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