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Vol. 43 No. 2 SCIENCE IN CHINA (Series B) April 2000 The regularities of the Rydberg energy levels of manyvalence electron atom Al ZHENG Nengwu (郑能武) & SUN Yujie (孙育杰) Department of Chemistry, University of Science and Technology of China, Hefei 230026, China Correspondence should be addressed to Zheng Nengwu Received September 9, 1999 Abstract Within the scheme of the weakest bound electron potential model theory, the concept of spectral-level-like series is presented by reasonably classifying the Rydberg energy level of atom Al. Based on this concept, the regularities of the Rydberg energy levels are systematically studied. The deviations of the calculated values from the experimental values are generally about several percent of 1 cm, which is of high accuracy. Keywords: weakest bound electron potential model (WBEPM) theory, spectral-level-like series, multiplet theory of atomic spectra, quantum defects. The energy levels of atoms and ions, especially in high Rydberg states, are an interesting research field concerning physics, astrophysics, chemistry and other high techniques (laser, nuclear fusion, etc.). Since the spectral data of Rydberg states are widely demanded and applied, the corresponding measuring techniques and computing methods are developed quickly. The widely used experimental means are optical double resonance method, level-crossing method[1], two-photon Doppler-free spectroscopy method[2], Stark spectroscopy[3] and millimeter spectroscopy method[4], etc. Due to the eminent contributions of Moore, Corliss, Sugar, Musgrove and Martin, etc., most of the experimental data of energy levels were systematically compiled[5 9] . For the lighter ele- ments that are of interest in the study of celestial bodies, the data are relatively rich. For the elements heavier than Ni (Z>28), only the first three or four energy levels are available, which are involved in the spectra of solar and other stars, especially stars with special chemical characteristics. During the last few years, a great development was made in the study of the abundant energy levels of rare earth elements[10]. However, there are still many blanks left for the high Rydberg energy levels in the periodic table. In respect of the theoretical methods, the relatively refined ways, such as multiconfiguration self consistent field[11], many-body perturbation theory[12], are highly praised. Unfortunately, however, the divergence in the calculation not only makes the computation onerous, but makes it difficult to compute high Rydberg states using these methods. The presentation and development of quantum defect theory (QDT) provided a feasible way to calculate the high Rydberg states theoretically. With QDT, Martin[13] calculated atom Na with high accuracy. However, up to now, its application is still limited to the calculation of one-valence electron systems such as alkali atoms and alkali-like ions. It is still difficult to evaluate the high Rydberg states of many-valence electron atoms (ions) using QDT. In this paper, based on the 114 SCIENCE IN CHINA (Series B) Vol. 43 weakest bound electron potential model[14,15] (abbreviated to WBEPM) theory, the concept of the weakest bound electron was introduced into the many-valence electron systems. Through dividing the electrons in atoms (ions) into the weakest bound electron and non-weakest bound electrons, appropriate Rydberg state sequences were extracted and then the Martin’s expression was extended ito the many-valence electron atom Al. The results are satisfactory. 1 Evaluation of the quantum defect number As early as in 1916, Sommerfeld presented the old quantum defect theory. In 1977 Jaffe and Reinhardt[16] gave a new demonstration to this theory. The energy levels of alkli metal atoms and alkali-metal-like ions can generally be written as follows (in a.u.): En = − 1 1 , 2 (n − δ 1 ) 2 (1) where δl is the quantum defect number, representing the effects of interactions between the valence electron and the atomic core or any short-range spherical perturbation in a many-electron system. For alkali-metal atoms, δl is nearly a constant for a given angular momentum l. There are many ways and formulas to evaluate the quantum defect number. An important formula is Rydberg’s formula: vn = R (n1 − µ1 ) 2 − 1 1 = R 2 − 2 , (n2 − µ 2 ) q p R (2) 2 where vn is frequency of the given spectrum line, R is Rydberg constant, n1 and n2 are principal quantum numbers, and µ1 and µ2 are the corresponding quantum defect numbers. Realizing that p and q in Rydberg’s formula (2) should be the functions n, Ritz cally suggested that p and q are in the form of infinite series 1 1 v n = R 2 − 2 , q p p = n1 + a1 + q = n2 + a2 + b1 n12 b2 n 22 + c1 n14 + c2 n 24 + + [17] theoreti- (3.1) d1 n16 d2 n 26 + L, (3.2) + L. (3.3) This is the well-known form of Ritz formula, which is now still applied in many fields. Only using the first two terms of (3.2) and (3.3), will Ritz formula reduce to Rydberg’s formula. With a slight transformation, one can get the quantum defect number µ1 µ1 = n1 − p = − a1 − b1 n12 − c1 n14 − d1 n16 − L. (4) Ritz formula can be used to precisely evaluate the spectral data of lower energy levels of alkali atoms; for high Rydberg states so doing is very inconvenient. In 1980, a qualification for Ritz No. 2 REGULARITIES OF RYDBERG ENERGY LEVELS 115 formula was made by Martin[13], who set δ=n-n*, m=n-δ0, where δ0 is the quantum defect number of the lowest energy level in a Rydberg state series. Thus δ = a + bm −2 + cm −4 + dm −6 , (5) δ is the quantum defect corresponding to an energy level with principal quantum number n. With (5), Martin evaluated the energy levels of several Rydberg series of atom Na, with deviations from experimental results within only several percent of 1 cm-1. In addition, there are other methods to calculate the quantum defect number, including the formula presented by Hicks[17], the dynamical improvement of quantum defects by Ganesh Vaidyanathan, et al.[18], the core polarization qualification for quantum defects in high-angularmomentum states of alkali atoms by Freeman et al.[19], perturbation WKB calculation of quantum defects of alkali atoms by Lu[20]. 2 The weakest bound electron and spectral-level-like series The weakest bound electron is such an electron that most weakly connects with the system among all the electrons in the system. So it is also the most active electron that is excited or ionized most easily in the system. In the systems of alkali atoms or alkali-like ions with one valence electron, the weakest bound electron assuredly is the only valence electron, while in the systems with many-valence electrons such as atoms C and Al, one has to determine which of the electrons is the weakest bound electron by separating the valence electrons. Let us take atom C as an example. The ground electronic configuration of atom C is [He]2s22p2. Its valence electrons are two electrons in 2s orbits and two in 2p orbits, in which the two 2p electrons are easier to be excited with the same probability. In the viewpoint of stepwise excitation or ionization, however, the two electrons in 2p orbits can only be excited or ionized one by one. Then the first excited or ionized electron can be viewed as the electron that most weakly connects with the system. So one can appoint the excited 2p electron as the weakest bound electron in the present system. Exciting the weakest bound electron to various energy levels will produce various electronic configuration series, such as series 2p3s, 2p4s, 2pns, and series 2p3d, 2p4d, 2pnd, etc. Note in an elec- tronic configuration series, the only difference lies in principal quantum number n of the weakest bound electron. As is well known, in an electronic configuration series each electronic configuration may usually possess some spectral terms and further several spectral levels. Therefore, the concept of “spectral-level-like series” is suggested here: a spectral-level-like series is a series composed of energy levels with the same spectral level symbol in a given electronic configuration series of a system, such [He]2s23pns (n 3) ns 3P0 as one spectral-level-like series of atom C. Note that a spectral-level-like series always depends on a given electronic configuration series, because the same spectral term can be generated from different electronic configurations. Starting with this definition, it is easy to define a spectral-level-like series for the electronic configuration series 3s3p, 3s4p 3snp of atom C in double excited state. 116 SCIENCE IN CHINA (Series B) Vol. 43 According to multiplet theory for the atomic spectra[21], the energy level T, illustrated by L-S coupling scheme, can be represented as a function of parameters of spectral level, T = T (nl , config. - nwbe, L, S , J , Z ), (6) where n, l are principal quantum number and angular quantum number of the weakest bound electron, respectively; the config.-nwbe means the electronic configuration of the non-weakest bound electrons; the whole of nl and config.-nwbe jointly represent the electronic configuration of an atom (ion) system; L, S and J are the total orbital angular quantum number, the total spin angular quantum number and total angular quantum number respectively; Z is the nuclear change number. For a spectral-level-like series as we defined, config-nwbe L, S, J and Z of each member of the series are all the same. Then the corresponding energy level T becomes a unique functional of config. And for the unique function of principal quantum number n of the weakest bound electron, we express the relationship as follows: T = Tl ,config - nwbe, L , S , J , Z [n] (7.1) T = T [n], (7.2) or abbreviate it to where T represents energy level; n is the principal quantum number of the weakest bound electron. In WBEPM theory, the relationship between spectral level and the effective parameters of the weakest bound electron has been provided[15] RZ ′ 2 . n′ 2 T= where T denotes spectral level, Z (8) and n are the effective nuclear charge and effective principal quantum number of the weakest bound electron, respectively, and R is Rydberg constant. As a matter of fact, for a spectral-level-like series of a many-electron atom (ion), the effective nuclear charge Z in eq. (8) is apparently not 1, even so is the case of the alkali atoms. The situation is more complex in many-valence electron systems. However, in terms of the weakest bound electron, we continue following the concept of quantum defects theory and attribute various effects to the change in the effective principal quantum number n, i.e. the change in the quantum defect number. The following equation expresses the transformation between WBEPM theory and QDT: Z ′ / n ′ = Z net / n * , (9) where Znet=1, n*= n ′ / z ′ . Then (8) becomes T= 2 RZ net , (10) n* = n − δ , (11) n *2 where δ is the quantum defect number. Thus in the framework of WBEPM theory, through sepa- No. 2 REGULARITIES OF RYDBERG ENERGY LEVELS 117 ration of the weakest bound electron and non-weakest bound electrons, using the concept of spectral-level-like series, one can extend the Martin’s expression from alkali atomic systems to manyvalence electron system Al and predict the energy levels of high Rydberg states of many-valence electron Al. 3 Examples and analysis According to the above discussion, we calculated three spectral-level-like series 3s2ns(n ns2S1/2, 3s2np(n 4) np2P1/2 and 3s2nd(n 4) 4) nd2D3/2 of atom Al. The experimental values were selected from ref. [8], the energy unit is wave number (in cm-1). Extending the Martin’s expression to this many-valence electron atomic system, we have δ = a + bm −2 + cm −4 + dm −6 , (12) where δ=n-n*, m=n-δ0, δ0 is the quantum defect number of the lowest energy level in a given series, and a, b, c, d are fitted parameters (table 1). Table 1 The fitted parameters in Martin’s expression for three spectral-level-like series of atom Al Al: 3s2ns(ns2S1/2) Al:3s2np(np2P1/2) Al:3d2ns(nd2D3/2) a 1.757 5 1.275 3 0.977 8 b 0.205 9 0.252 1 -2.685 7 c 0.337 7 0.243 2 -72.002 5 d -0.322 0 0.693 2 425.128 1 δ0 1.812 4 1.492 3 0.368 1 Using the extended Martin’s expression to calculate the quantum defect numbers of each energy level in a given spectral-level-like series, one can predict the energy of each level with (10). In tables 2 4, the calculated level energy values are given. For comparison, the corresponding experimental values are also listed. Table 2 Comparison between the calculated and experimental energy values of spectral-level-like series Al: 3s2ns (n 4) ns2S1/2/cm-1 n Eexp Ecal δexp δcal 4 22 930.614 1.812 4 22 930.574 1.812 4 5 10 588.957 1.780 8 10 588.960 1.780 8 6 6 133.968 1.770 3 6 134.970 1.770 3 7 4 005.248 1.765 7 4 006.245 1.765 7 8 2 821.137 1.763 2 2 821.137 1.763 2 9 2 094.474 1.761 6 2 094.481 1.761 7 10 1 616.470 1.760 6 1 616.491 1.760 7 11 1 285.370 1.760 6 1 285.328 1.760 0 12 1 046.470 1.759 7 1 046.448 1.759 6 13 868.470 1.759 1 868.483 1.769 2 14 732.370 1.759 1 732.347 1.759 0 15 625.870 1.758 6 625.887 1.758 7 16 541.070 1.758 6 541.063 1.758 6 17 472.385 1.758 4 18 416.000 1.758 3 19 369.141 1.758 2 20 329.775 1.758 2 25 203.144 1.757 9 30 137.581 1.757 8 35 99.305 1.757 7 40 75.035 1.757 7 50 47.152 1.757 6 118 SCIENCE IN CHINA (Series B) Vol. 43 Table 3 The comparison between the calculated and experimental energy values of spectral-level-like series Al: 3s2np(n 3)np2P1/2/cm-1 δexp Ecal δcal 3 48 278.370 1.492 3 48 278.650 1.492 4 4 15 328.566 1.324 4 15 328.628 1.324 4 5 8 006.405 1.297 8 8 006.408 1.297 8 6 4 943.357 1.288 4 4 943.356 1.288 4 7 3 358.716 1.284 0 3 358.625 1.283 9 8 2 431.097 1.281 4 9 1 841.231 1.279 9 10 1 442.811 1.278 9 15 582.692 1.276 7 20 313.012 1.276 1 35 96.486 1.275 6 n Eexp Table 4 The comparison between the calculated and experimental energy values of spectral-level-like series Al: 3s2nd(n 3)nd2D3/2/cm-1 n Eexp Ecal δcal 3 15 842.935 0.368 2 δexp 15 842.880 0.368 1 4 9 348.965 0.573 9 9 195.529 0.545 5 5 6 044.648 0.739 2 6 044.625 0.739 2 6 4 111.953 0.834 0 4 113.287 0.834 8 7 2 934.206 0.884 5 2 934.202 0.884 5 8 2 184.946 0.913 1 2 184.649 0.912 6 9 1 684.950 0.929 8 1 684.948 0.929 8 10 1 337.400 0.941 7 1 337.196 0.941 0 11 1 086.070 0.948 1 1 086.199 0.948 7 12 899.670 0.955 8 899.413 0.954 2 13 757.270 0.962 1 756.789 0.958 2 14 645.770 0.964 2 645.487 0.961 3 15 557.070 0.964 7 556.994 0.963 7 16 485.370 0.963 7 485.494 0.965 6 17 427.470 0.977 7 426.908 0.967 2 18 378.270 0.967 6 378.308 0.968 4 19 337.670 0.972 7 337.550 0.969 5 20 303.035 0.970 4 21 273.552 0.971 1 22 248.169 0.971 7 23 226.160 0.972 3 24 206.952 0.972 7 25 190.091 0.973 2 30 130.256 0.974 6 50 45.661 0.976 7 4 Discussion Tables 2 4 show that the calculated values are very close to the corresponding experimental No. 2 REGULARITIES OF RYDBERG ENERGY LEVELS 119 values, with an absolute deviation generally no more than several percent of 1 cm-1 except for level 4d2D3/2. The much higher energy levels, without the corresponding experimental values being compared, are predicted and listed in the tables. Considering the high accuracy of the foregoing calculated values, the forecasted values are reliable. The results prove that the extended Martin’s expression is suitable for the spectral-level-like series of the many-valence electron atom Al. The study on how to extend the Martin’s expression to other many-valence electron systems is in progress. The related work will be reported in succession. In SCF methods, the concept of indistinguishability of all electrons in a system prevails. In pseudopotential model based on the behavior of the valence electron there is the idea of separation of valence electrons and core electrons; in QDT, the valence shell and the core are also treated separately in one-valence electron atoms (ions); while in WBEPM theory, through separation of the weakest bound electron and non-weakest bound electrons, one can solve the one-electron Schrödinger equation of the weakest bound electron. This is very convenient for the discussion about the excited states of atomic systems. It is these conceptual breakthroughs and the advancement of the concept of spectral-level-like series within an electronic configuration series that make it possible to transform (9) and extend Martin’s expression. This paper provides a new way of studying the regularities in the Rydberg energy levels of many-valence electron atoms Al. Acknowledgements This work was supported by the Research Foundation of Ministry of Science and Technology and the National Natural Science Foundation of China (Grant No. 59872039). References 1. Svanberg, S., Tsekeris P., Happer, W., Hyperfine-structure studies of highly excited D and F levels in alkali atoms using a cw[continuous wave]tunable dye laser, Phys. Rev. Lett., 1973, 30: 817. 2. Harvey, K. C., Stoicheff, B. P., Fine structure of the n2D series in rubidium near the ionization limit, Phys. Rev. Lett., 1977, 38: 537. 3. Ducas, T. W., Zimmerman, M. L., Infrared stark spectroscopy of sodium Rydberg states, Phys. Rev. A, 1977, 15: 1523. 4. Fabre, C., Haroche, S. Goy, P., Millimeter spectroscopy in sodium Rydberg states: Quantum defect, fine-structure, and polarizability measurements, Phys. Rev., A, 1978, 18: 229. 5. Moore, C., Atomic Energy Levels, Washington D. C.: NSRDS-NBS 35, 1971, 1 6. 7. 8. 9. 10. 309. Corliss, C., Sugar, J., Energy levels of potassium, J. Phys. Chem. Ref. Data, 1979, 8: 1111. Sugar, J., Musgrove, A., Energy levels of krypton Kr I through Kr XXXVI, J. Phys. Chem. Ref. Data, 1991, 20: 859. Martin, W. C., Zalubas, R., Energy levels of aluminum, J. Phys. Chem. Ref. Data, 1979, 8: 820. Martin, W. C., Zalubas, R., Energy levels of silicon, J. Phys. Chem. Ref. Data, 1983, 12: 325. Pell. F. E, Gardant, N., Auzel, F., Effect of excited-state population density on nonradiative multiphonon relaxation rates of rare-earth ions, J. Opt. Soc. Am., B, 1998, 15: 667. 11. Roos, B. O., Advances in Chemical Physics, Vol. 69, New York: Wiley, 1987, 339 348. 12. Lauderdale, W. J., Stanton J. F., Gauss, J. et al., Many-body perturbation theory with a restricted open-shell Hartree-Fock reference, J. Chem. Phys., 1992, 97: 6606. 13. Martin, W. C., Series formulas for the spectrum of atomic sodium (Na I), J. Opt. Soc. Am., 1980, 70: 784. 14. Zheng, N. W., Xin, H. W., Successive ionization potentials of 4f n electrons within WBEPM theory, J. Phys. B: At. Mol. 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