The Doubly Excited States Description of the Negative Hydrogen Ion Using Special Forms of the Hylleraas Type Wave Functions

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1 CHINESE JOURNAL OF PHYSICS VOL. 47, NO. 2 APRIL 2009 The Doubly Excited States Description of the Negative Hydrogen Ion Using Special Forms of the Hylleraas Type Wave Functions M. Biaye, 1, M. Dieng, 2 I. Sakho, 2 A. Konté, 2 A. S. Ndao, 2 and A. Wagué 2 1 Department of Physics and Chemistries, Faculty of Sciences and Technologies of Formation and Education, University Cheikh Anta Diop, Dakar, Senegal 2 Laboratory Atoms Lasers, Department of Physics, Faculty of Sciences and Techniques, University Cheikh Anta Diop, Dakar, Senegal (Received July 18, 2008) The singlet and triplet doubly excited states of the negative hydrogen ion have been described by using special forms of the Hylleraas type wave functions. The energy calculations have been carried out in the framework of the variational method using configuration interaction basis states with a real Hamiltonian. We have obtained a satisfactory agreement between our results and some experimental data and other theoretical results. PACS numbers: v, Jf, Pf I. INTRODUCTION The negative ion of hydrogen continues to be important in atomic physics and astrophysics. Correlations between the two electrons are strong already in the ground state, the only bound state in this three-body system. There are no singly excited states, but there exist doubly excited states, which are embedded in the continuous part of the spectrum of the hydrogen atom and can be detected as resonances in electron-hydrogen scattering or in the photo-detachment cross-section of H. The ground state attracted early interest, especially for the description of stellar atmospheres by Chandrasekhar and others [1 6]. More recently, the rich spectrum of doubly excited states has been central to our understanding of highly-correlated, non-separable problems in quantum physics [7 10]. Furthermore, H has been important in the study of our atmosphere and even more so in the atmosphere of the sun and other stars, as first documented by Chandrasekhar and it has also been central to the development of accelerators. Most experimental and theoretical investigations of double excitation in a negative ion involve the simplest example, the H ion. Experimentally, it is very difficult to observe the resonant states in the negative hydrogen ion H. After the first observation [11] of the two 1 P o resonance states of H between the n = 2 and 3 hydrogen thresholds, to the authors knowledge there is only the observation of the lowest 1 P o resonance state, which was reported by Cohen et al. [12], Halkn et al. [13], and Williams [14]. Recently, the observation of the resonant states of H with high resolution has become possible by the advanced techniques of Doppler-tuned collinear laser spectroscopy. Balling et al. [15] and Andersen et al. [16] have observed the two c 2009 THE PHYSICAL SOCIETY OF THE REPUBLIC OF CHINA

2 VOL. 47 M. BIAYE, M. DIENG, et al. 167 lowest-lying members of the 1 P o dipole series of autodetaching resonances in H below the n = 2 hydrogen threshold by Doppler-tuned collinear laser spectroscopy with a resolution of mev. By two-photon spectroscopy techniques, Stinz et al. [17] and Rislove et al. [18] measured the lowest 1 D e of H with high resolution. More recently Balling et al. [19] and Raarup et al. [20] measured the lowest 1 P o resonance below the H (n = 2) threshold by the momentum spread reduction technique and electron cooling method. The negative hydrogen ion has also been an attractive target for theoretical efforts. Several computational techniques have been applied. Among these computational techniques, we can note the more recent investigations which include the highly accurate calculations of energies and widths for certain states: Ho [21 23], Ivanov and Ho [24], and Ho and Bhatia [25] combine Hylleraas wave functions with the use of the complex rotation method; Lindroth and co-workers [8, 28] combine a discrete numerical basis set with the method of complex rotation. Other recent calculations have been performed by Tang et al. [29] who use a close-coupling method in terms of hyperspherical coordinates, and by Sadeghpour et al. [30] who use the R-matrix method. Ndao et al. [31] also use the Hylleraas type wave functions combined with the diagonalization method. Chen [32] employed the B-spline basis functions to calculate the resonance energy and width with a complex rotation method. We note generally in these theoretical methods the use of high basis sets. The present work is an extension of the earlier calculations [33 35]. We report the results for the low and high-lying singlet and triplet doubly excited nl nl intra-shell states energies of the negative hydrogen ion. A variation method using configuration interaction basis states with a real Hamiltonian is employed in the present investigation. In this work, we also use in the calculations special forms of the Hylleraas-type wave functions constructed in our earlier calculations by using the Hylleraas and the incomplete hydrogenic wave functions. In addition, this work shows the possibility of some singlet and triplet doubly excited intra-shell states description of the negative hydrogen ion by using a small basis set with our theoretical approach: 13 terms for the singlet states and 7 terms for the triplet states. Section II presents the theoretical procedure used in this work. In Section III, a presentation and discussion of our calculations with other theoretical calculations and experimental data are also made. II. THEORY To describe the singlet and triplet doubly excited (nl) 2 and nl nl states of the negative hydrogen ion with n 4, we have used special forms of the Hylleraas-type wave function as follows:

3 168 THE DOUBLY EXCITED STATES... VOL. 47 υ=n l 1 Φ jkmnl l( r 1, r 2 ) = (2r 1 2r 2 ) l (n 2 r0λ 2 2 2r 1 2r 2 ) υ υ=0 υ =n l 1 + (2r 1 2r 2 ) l (n 2 r0λ 2 2 2r 1 2r 2 ) υ υ =0 + (r 1 + r 2 ) j (r 1 r 2 ) k r 1 r 2 m exp( λ(r 1 + r 2 )). (1) Here r 1 and r 2 denote the position of the two electrons; j, k, m are Hylleraas parameters with (j, k, m 0); n is the principal quantum number; l and l are orbital angular momenta; λ is a coefficient defined by λ = Z αnr 0, where Z, α, r 0 are respectively the nucleus charge number, variation parameter, and Bohr radius. The set of parameters (j, k, m ) define the basis states (i.e., the configurations). The even values of k define the symmetric wave functions describing the singlet states, while the odd values define the antisymetric wave functions for the triplet states. The final form of the wave functions including correlation effects due to the mixing of the configurations can be expressed as follows: Ψ nl l( r 1, r 2 ) = jkmα jkm Φ jkmnl l. (3) (2) Here a jkm are the eigenvectors determined by solving the Schrödinger equation: H Ψ nl l( r 1, r 2 ) = E Ψ nl l( r 1, r 2 ), (4) where the Hamilton operator H has the form with H = T + C + W, T = h2 2m ( ); C = ( Ze2 r 1 + Ze2 r 2 ); W = e 2 r 1 r 2 ; (6) where T is the kinetic energy, C is the Coulomb interaction between the atomic nucleus and the two electrons, and W is the Coulomb interaction between electrons. In the Hamilton operator we have neglected all magnetic and relativistic effects together with the motion of the atomic nucleus. In the calculations, we have also neglected the Feschbach shifts due to the interaction between the open and the closed channel, because of the use of the incomplete basis sets of the wave functions. (5)

4 VOL. 47 M. BIAYE, M. DIENG, et al. 169 TABLE I: Comparison of doubly excited 1 Se and 1,3 Po states of the negative hydrogen ion with the other results. Energies E are in ev. TABLE II: Comparison of doubly excited results. Energies E are in ev. 1,3 De states of the negative hydrogen ion with the other III. RESULTS AND DISCUSSIONS Our results are compared in the Tables I, II, and III with some theoretical results available in the literature. Ho [23] for the 3s3p 1 Po and 4s4p 1 Po states, Ho [36] for the 4d2 1 Ge and 4d2 3 Ge states, Ho and Bhatia [25] for the 2p2 1 De and 3p2 1 Do states, Ho and Callaway [37, 41] for some 1 Se, 1,3 Po, 1,3 De, 1,3 Fo, and 1,3 Ge states of the negative hydrogen ion have used the complex rotation method; Lindroth [8] for the 2s2p 1 Po, 3s3p 1 Po, and 4s4p 1 Po states of the negative hydrogen ion has used a discrete numerical basis set combined with the method of complex rotation; Burgers and Lindroth [28] for some 1 Se, 1 Po, 1 De, 1,3 Fo, and 1 Ge states of the negative hydrogen ion have used the complex rotation method in a Sturmian type basis in perimetric coordinates; Kuan et al. [38] for the 2s2p 1 Po and 3s3p 1 Po states of the negative hydrogen ion have used a saddle-point complex rotation method; Sadeghpour et al. [30] for the 2s2p 1 Po, 3s3p 1 Po and 4s4p 1 Po

5 170 THE DOUBLY EXCITED STATES... VOL. 47 states of the negative hydrogen ion have used the method of hyperspherical coordinates; Tang et al. [29] for the 2s2p 1 P o, 3s3p 1 P o, and 4s4p 1 P o states of the negative hydrogen ion have used the close coupling method in terms of hyperspherical coordinates. TABLE III: Comparison of doubly excited 1,3 F o and 1,3 G e states of the negative hydrogen ion with the other results. Energies E are in ev. TABLE IV: Comparison of doubly excited 1 P o and 1 D e states of the negative hydrogen ion with the experiments. Energies E are in ev. The energy is measured from the ground state of H. We have also compared our results in Table IV with the experimental values of MacArtur et al. [39] for the 2s2p 1 P o state, Hamm et al. [11] for the 3s3p 1 P o state, Cohen et al. [12] for the 3s3p 1 P o state, Halka et al. [13] for the 3s3p 1 P o and 4s4p 1 P o states, and Rislove et al. [18] for the 2p 2 1 D e state. For the comparison with theoretical results, we used for energy conversion 1 a.u. = 2 Ry = ev. For comparison with experiments, we used for the H ground state energy, E = a.u. [40].

6 VOL. 47 M. BIAYE, M. DIENG, et al. 171 Table I lists our calculated energies and compares them with the previous calculations by Bürgers and Lindroth, Sadeghpour et al., Ho, Lindroth, Tang et al., Kuan et al., and Ho and Callaway. We note a generally satisfactory agreement between the present calculations and those of the other authors, although we can find a slight difference between our calculations and those of Bürgers and Lindroth for the 3s 2 1 S e state and Ho and Callaway for the 3s3p 3 P o state. In the Table II we compare our results with those of Bürgers and Lindroth, Ho and Bhatia, and Ho and Callaway. We note generally a satisfactory agreement between our results and those of the other authors. For the 3p 2 1 D e, 3p 2 3 D e and 4p 2 3 D e states, we note a slight disagreement between our present results and those of the other authors. Table III compares our results with the other theoretical calculations of Ho, Ivanov and Ho, Bürgers and Lindroth, and Ho and Callaway. We note generally a good agreement between our results and the other calculations. We can also note for the 3p3d 3 F o, 4p4d 3 F o, and 4s4f 3 F o states a slight disagreement between our results and those of Bürgers and Lindroth and Ho and Callaway. The disagreements noted between our results and those of the other calculations can be explained by the fact that we have neglected in the present calculations the Feschbach shifts. These disagreements can be explained also by the choice of the angular part of the wave functions used for the description of the doubly excited states of the negative hydrogen ion. In Table IV a comparison is made between our calculations and the experiments by MacArtur et al., Hamm et al., Cohen et al., Halka et al., and Rislove et al. for the 2s2p 1 P o, 3s3p 1 P o, 4s4p 1 P o, and 2p 2 1 D e states of the negative hydrogen ion. We find a good agreement between our calculations and the experimental results available in the literature. For the 2s2p 1 P o and 4s4p 1 P o states, the degree of agreement between the present calculations and the two experimental results of MacArtur et al. and Halka et al. is the same. We note also that the degree of agreement for the 3s3p 1 P o state between the present calculations and the three experimental results of Hamm et al., Cohen et al., and Halka et al. is the same. In summary, we have calculated the singlet and triplet doubly excited states energies of the negative hydrogen ion by using special forms of the Hylleraas type-wave functions. The calculations have been done in the framework of the variation method using configuration interaction basis states with a real Hamiltonian. We note here generally a satisfactory agreement between the present results and the other available calculations. For the experimental results, we find a good agreement. Acknowledgements The authors would like to thank the Swedish International Development Agency (SIDA) and the International Centre for Theoretical Physics (ICTP) for support.

7 172 THE DOUBLY EXCITED STATES... VOL. 47 References Electronic address: [1] S. Chandrasekhar, Astrophys. J. 100, 176 ( 1944). [2] S. Chandrasekhar, Astrophys. J. 102, 223 ( 1945). [3] S. Chandrasekhar and F. H. Breen, Astrophys. J. 104, 430 ( 1946). [4] S. Chandrasekhar and G. Herzberg, Phys. Rev. 98, 1050 (1955). [5] S. Chandrasekhar and M. K. Krogdahl, Astrophys. J. 98, 205 ( 1943). [6] S. Chandrasekhar and G. Münch, Astrophys. J. 104, 446 (1946). [7] T.-Z.Tang, Y. Wakabayashi, M. Matsuzawa, S. Watanabe, and I. Shimamura, Phys. Rev. A 49, 1021 (1994). [8] E. Lindroth, Phys. Rev. A 52, 2737 (1995). [9] M.-K. Chen, J. Phys. B 30, 1669 (1997). [10] T.-T. Glen, J. Phys. B 31, L1001 (1998). [11] M. E. Hamm et al., Phys. Rev. Lett. 43, 1715 (1979). [12] S. Cohen et al., Phys. Rev. A 36, 4728 (1987). [13] M. Halka et al., Phys. Rev. A 44, 6127 (1991). [14] J. F. Williams, J. Phys. B 21, 2107 (1988). [15] P. Balling et al., Phys. Rev. Lett. 77, 2905 (1996). [16] H. H. Andersen et al., Phys. Rev. Lett. 79, 4770 (1997). [17] A. Stinz et al., Phys. Rev. Lett. 75, 2924 (1995). [18] D. C. Rislove et al., Phys. Rev. A 58, 1889 (1998). [19] P. Balling et al., Phys. Rev. A 61, (2000). [20] M. K. Raarup et al., Phys. Lett. 85, 4028 (2000). [21] Y. K. Ho, Phys. Rev. A 19, 2347 (1979). [22] Y. K. Ho, J. Phys. B 23, L71 (1990). [23] Y. K. Ho, Phys. Rev. A 45, 148 (1992). [24] I. A. Ivanov and Y. K. Ho, Chin. J. Phys. 39, 415 (2001). [25] Y. K. Ho and A. K. Bhatia, Phys. Rev. A 42, 1119 (1990). [26] Y. K. Ho, Phys. Rev. A 23, 2137 (1981). [27] Y. K. Ho, J. Phys. B 12, 387 (1979). [28] A. Bürgers and E. Lindroth, Eur. Phys. J. D 10, 327 (2000). [29] J.-Z. Tang et al., Phys. Rev. A 49, 1021 (1994) [30] H. R. Sadeghpour, C. H. Green, and M. Cavagnero, Phys Rev. A 45, 1587 (1992). [31] A. S. Ndao, A. Wagué, N. A. B. Faye, and A. Konte, Eur. Phys. J. D 5, 327 (1999). [32] M.-K. Chen, Eur. Phys. J. D 21, 13 (2002). [33] M. Biaye, A. Konte, N. A. B. Faye, and A. Wague, Eur. Phys. J. D 13, 21 (2001). [34] M. Biaye, A. Konte, A. S. Ndao, N. A. B. Faye, and A. Wague, Physica Scripta 71, 39 (2005). [35] M. Biaye, A. Konte, A. S. Ndao, and A. Wague, Physica Scripta 72, 373 (2005). [36] Y. K. Ho, Z. Phys. D 11, 277 (1989). [37] Y. K. Ho and J. Callaway, Phys. Rev. A 27, 1887 (1983). [38] W. H. Kuan, T. F. Jian, and K. T. Chung, Phys. Rev. A. 60, 364 (1999). [39] D. W. MacArtur et al., Phys. Rev. A 32, 1921 (1985). [40] G. W. F. Drake, Nucl. Instrum. Method Phys. Res. Sect. B 31, 7 (1988). [41] Y. K. Ho and J. Callaway, Phys. Rev. A 34, 130 (1986).