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The 6sς, 6dς, and 7dς Rydberg 1Σ+g States and Two Doubly Excited 1Σ+g States of Na2
Journal of Molecular Spectroscopy 198, 239 –243 (1999) Article ID jmsp.1999.7968, available online at http://www.idealibrary.com on
The 6ss, 6ds, and 7ds Rydberg 1S g1 States and Two Doubly Excited 1S g1 States of Na 2 X. Dai,* H. Chen,* Y. Liu,* J. Li,* J. Xiang,* D. Chen,* Li Li,* ,1 and G.-H. Jeung† *Department of Physics and Center of Atomic and Molecular Sciences, Tsinghua University, Beijing 100084, China; and †Laboratoire Aime´ Cotton (CNRS UPR3321), Baˆt. 505, and ASCI (CNRS UPR9029), Baˆt. 506, Campus d’Orsay, 91405, Orsay, France Received December 17, 1998; in revised form August 25, 1999
Five highly excited 1 S g1 states of Na 2 were observed for the first time by pulsed optical– optical double resonance (OODR) fluorescence excitation spectroscopy. Three of the five states are assigned to the 6s s 1 S g1 , 6d s 1 S g1 , and 7d s 1 S g1 Rydberg states. The other two states are assigned to doubly excited 1 S g1 states. © 1999 Academic Press Key Words: Na 2; 1 S g1 states; Rydberg states; doubly excited states. 1. INTRODUCTION
The Na 2 molecule is interesting in many aspects. Since the A 1 S u1 4 X 1 S g1 and B 1 P u 4 X 1 S g1 transitions are in the dye laser frequency region and its Rydberg states can be reached by two-photon and two-step dye laser excitation, Na 2 has become one of the most intensively studied molecules in molecular spectroscopy. Many new molecular spectroscopic techniques have been developed in the studies of the Na 2 molecule. Because the electronic structure of Na 2 is relatively simple and relatively complete spectroscopic data are available, Na 2 is a good test molecule for theoretical calculations and molecular dynamics. The 1 S g1 Rydberg states of Na 2 have been studied extensively both theoretically and experimentally. Eighty-four electronic states of Na 2 below the 3s 1 5p atomic limit, including 11 1 S g1 states, have been calculated (1, 2). Effantin et al. (3) observed the v 5 0 –10 levels of the (3s 1 4s) 3 1 S g1 state (the number, 3, before 1 S g1 is the energy rank at R e of the states with 1 S g1 symmetry and 3s 1 4s is the dissociation limit) with Fourier transform spectrometry of laser-induced infrared fluorescence. Schawlow and co-authors observed six Rydberg 1 S g1 states (the 4 1 S g1 , 5 1 S g1 , 6 1 S g1 , 7 1 S g1 , 8 1 S g1 , and a higher 1 S g1 state with maximum vibrational quantum numbers of 20, 18, 20, 7, 7, and 1, respectively) by two-step polarization-labeling spectroscopy (4 –5). Wang et al. (6) and Yan et al. (7) probed the high vibrational levels of the 4 1 S g1 and 5 1 S g1 states by optical– optical double resonance (OODR) fluorescence excitation spectroscopy. With an extremely sensitive space-chargelimited diode ionization detector, Tsai et al. studied the longrange part of the 4 1 S g1 , 5 1 S g1 , 6 1 S g1 states (8 –10). Martin et al. (11–12), using OODR multiple photon ionization spectroscopy of a molecular beam, observed and analyzed the 3s 1 ns (n 5 23–40) and 3s 1 nd (n 5 12–39) Rydberg 1 S g1 states. 1
Recently, we carried out perturbation-facilitated OODR (PFOODR) fluorescence excitation spectroscopy via A 1 S u1 ; b 3 P u mixed intermediate levels to probe the high-lying triplet gerade states of Na 2. The 1 3 S g2 (13), 4 3 D g , 7 3 D g , 10 3 D g (14), 4 3 P g , 6 3 P g (15), and several 3 S g1 states (16) have been observed in the energy region 35 000 –39 500 cm 21. Several previously unreported 1 S g1 states were observed via the A 1 S u1 ; b 3 P u mixed intermediate levels and confirmed by OODR excitation through unperturbed A 1 S u1 intermediate levels. This paper reports our experimental observation and analysis of these 1 S g1 states. II. EXPERIMENT
To whom correspondence should be addressed.
The experimental setup was reported in Refs. (13, 14). Briefly, Lambda Physik EMG202MSC excimer laser pumped two Lambda Physik FL3002E dye lasers simultaneously. One of the dye lasers was used as the PUMP laser, which was operated with intracavity e´talon (0.04 cm 21 linewidth). Another dye laser was used as the PROBE laser without intracavity e´talon (0.2 cm 21 linewidth). Another dye laser was used as the PROBE laser without intracavity e´talon (0.2 cm 21 linewidth). The PUMP laser was operated with DCM dye and the PROBE laser was operated with Coumarin 440, 470, 480, and 500 dyes. The two dye laser beams counterpropagated and superimposed at the center of a cross heatpipe oven. Sodium vapor pressure and temperature were controlled with Ar buffer gas pressure and the temperature was about 500°C with ;1 Torr Ar. The effective length of the sodium vapor region was about 25 cm. The PUMP laser frequency was calibrated by I 2 laser-induced fluorescence lines and the PROBE laser frequency was calibrated using neon lines from optogalvanic spectroscopy in a hollow cathode lamp discharge. The pump laser frequency was tuned to a known transition from the X 1 S g1 ground state to an A 1 S u1 or A 1 S u1 ; b 3 P u mixed intermediate level. The probe laser was then scanned to excite various
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TABLE 1 Summary of Experimental Observations
transitions upward from the selected A 1 S u1 or A 1 S u1 ; b 3 P u mixed state to levels of the 1 S g1 states. Collision-induced fluorescence to X 1 S g1 was filtered with colored glass filters and monitored by a R212 photomultiplier tube (PMT) from a side window. The outputs of the PMT and the neon lamp were sent to a boxcar average, whose outputs were digitized by an A/D converter and acquired with a computer. III. OBSERVATION AND ASSIGNMENT
The intermediate levels in our OODR experiment are A 1S 1 u v 5 1,
J 5 14, 16 (17)
3 A 1S 1 u v 5 3 , b P 0u v 5 10,
A 1S 1 u v 5 6,
J 5 12–14 (18)
J 5 10, 12, 14 (17)
3 A 1S 1 u v 5 8 , b P 0u v 5 14, 3 A 1S 1 u v 5 2 , b P 1u v 5 9,
J 5 10–14 (19) J 5 28 (20).
When an A 1 S u1 ; b 3 P u mixed intermediate level is pumped, transitions into both singlet and triplet states are allowed. However, the OODR signals into singlet levels via a predominant A 1 S u1 level will be much stronger than those via its perturbation partner, the predominant b 3 P u level, and vice versa. So it is easy to distinguish singlet signals from triplet signals. From unperturbed A 1 S u1 intermediate levels, only 1 S g1
and 1 P g states can be reached. Transitions into 1 S g1 states consist of P and R lines, and transitions into 1 P g states consist of P, Q, and R lines. A total of 656 OODR excitation transitions were assigned to transitions into eight 1 S g1 states. Ninety transitions were into the 7 1 S g1 v 5 1–13 levels and 62 transitions were into the v 5 0 –10 vibrational levels of the 8 1 S g1 state (2, 5). Sixty-four transitions were into the v 5 1–9 vibrational levels of the 1 S g1 state, whose v 5 0 –1 levels were observed by Taylor et al. and assigned as the (3s 1 7d) 1 S g1 state (5). The energy positions of the vibrational levels of these three states we observed agree with the predictions of Ref. (5). Five 1S g1 states were observed for the first time. Absolute vibrational numberings of three states were determined by comparing relative intensities of excitation lines observed with calculated Franck–Condon factors of the 1S g1–A 1S u1 systems. These three states were assigned to the 6ss 1S g1, 6ds 1S g1, and the 7ds 1 1 S g Rydberg states according to our n,l labeling (15). The vibrational numberings of the other two states are tentatively determined. These two states cannot be fitted into the Rydberg series and are assigned to doubly excited states. The electronic assignments will be discussed below. Table 1 summarizes the experimental observations of the 1S g1 states. Table 2 gives their Dunham coefficients fitted from term values we observed (21). 1. Electronic Assignment of the Rydberg States Magnier et al. (2) gave potential curves and molecular constants of the 1–11 1 S g1 states of Na 2. The four lower 1 S g1
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THE Na 2 1 S g1 STATES
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TABLE 2 Molecular Constants of the 1S g1 States Observed
Note. All quantities are in cm 21. All Y(0,2) values are calculated with Y(0,2) 5 24Y(0,1) 3 /Y(1,0) 2 . a Calculated and fixed in the fit, Y(1,1) 5 6Y(0,1) 2 (12(2Y(2,0)/Y(0,1)) 1/ 2 )/Y(1,0).
states observed by Taylor et al. correspond to the 4–7 1 S g1 states, respectively, but the correspondences of the higher two states are not unambiguous (2, 4, 5). Recently we relabeled the 3 P g Rydberg states of Na 2 according to their predominant n, l character at R e ' R e1 (15). This n, l labeling shows more physical significance than their energy ranks. Now we analyze the 1 S g1 states according to their n, l character. Table 3 gives the n, l labels of the 3–16 1 S g1 states. The effective principal quantum number, n*, is defined by a modified Rydberg equation IP 2 E n* 5 5/n* 2 n* 5 @5/~IP 2 E n* !# 1/ 2 n* 5 n 2 d n, l , where IP is the ionization energy (12), 5 is the Rydberg constant, and d n,l is the nearly n-independent quantum defect for the , l (or ,+) Rydberg series. Notice that the ab initio T e values normally lie ;250 cm 21 lower than observed values (2). In our n* calculation, this 250 cm 21 was added if no experimental constants were available. All gerade states have even , (s, d, g, etc.) values. The ns s and nd s Rydberg series, with n* mod 1 values of ;0.7 and ;0.9, respectively, are penetrating Rydberg states and have greater transition strength from the valence A 1 S u1 intermediate state. The ng s series, with n* mod 1 of ;1, is a nonpenetrating Rydberg series and has much smaller transition strength from the A 1 S u1 state (15). According to this labeling, the T e 5 34 977 cm 21 state observed by Taylor et al. is the 5d s 8 1 S g1 state and the state with T e 5 37 050 cm 21 is not the 6d s 13 1 S g1 [the 11 1 S g1 state of Ref. (2)] state, but the 7s s 15 1 S g1 state. Three of the five states we observed are the 6s s 10 1 S g1 , 6d s 13 1 S g1 , and 7d s 16 1 S g1 states, respectively. The 5g s 9 1 S g1 and 6g s 14 1 S g1 states are members of the nonpenetrating ng s series and are normally not observable from valence intermediate states.
2. Electronic Assignment of the Doubly Excited 1S g1 States We observed eight vibrational levels of one 1 S g1 state in the 36 500 –37 400 cm 21 energy region and nine vibrational levels of another 1 S g1 state in the 36 500 –37 500 cm 21 energy region. The energy ranks at R e of these two states are 11 and 12 according to our tentative vibrational numberings. These levels are in the same energy region of the 6s s 10 1 S g1 and 6d s 13 1 S g1 states. These levels are not the high v levels of the 7 1 S g1 and 8 1 S g1 states because the vibrational spacings and B v values are much larger than the predicted values and are not the 7s s 15 1 S g1 and 7d s 16 1 S g1 states either because the lowest levels of these two newly observed states are lower than the potential minima of these two states predicted. None of these two states could be the 5g s 9 1 S g1 nonpenetrating state not only because the nonpenetrating states have very small transition strength from the valence A 1 S u1 state but also because the vibrational spacings and B v values are too large for the 9 1 S g1 levels in this energy region. The 6g s 1 S g1 state is predicted to be in this energy region from the quantum defect of the 5g s 9 1 S g1 state and might become observable through the valence A 1 S u1 state due to the mixing with the 6s s 10 1 S g1 or/and 6d s 13 1 S g1 states. If either of these two states were the 6g s 1 S g1 state, the OODR excitation lines into this nonpenetrating Rydberg state should always be weaker than transitions into the 6s s 10 1 S g1 and/or 6d s 13 1 S g1 states, but this is contrary to our observations. Intensity patterns suggest that mixing with the 10 1 S g1 and/or 13 1 S g1 states is not responsible for the observation of the 11 1 S g1 state, which must therefore be seen in transitions from the A 1 S u1 state because it has an appreciable doubly excited (3p 1 3p) character at short internuclear distance. According to the ab initio calculation, the 7 1 S g1 and 8 1 S g1 states dissociate to the 3p 1 3p atomic limit. The vibrational levels of the 7 1 S g1 and 8 1 S g1 states observed with v max 5 13 for the 7 1 S g1 state and v max 5 10 for the 8 1 S g1 state are low v levels. At small internuclear distance, these two states have a predominant Rydberg character, rather than a doubly excited
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TABLE 3 Labeling of the Rydberg 1S g1 States of Na 2 According to the Predominant øl Character at R e
Note. The upper values are experimental; the lower values are theoretical by Magnier et al. [T e (ab initio) 1 250 cm 21] or predicted by quantum defect of the lower member. a Ref. (2).
valence character because of the mixing with many other Rydberg states [Table 4 of Ref. (22)]. Only the 3p 1 3p 1 3 S g2 state has a predominant doubly excited character at all internuclear distances, probably because it has a very different wavefunction from other valence or Rydberg states which all have the 1 s g molecular orbital as a common bonding orbital. Indeed, this state has about 91% p p 1 p p and 8% d p 1 d p characters near the lower part of the potential well. The ab initio calculation, in particular, based on the linear combination of the atomic orbitals (LCAO) method, is not so suitable for calculating highly excited Rydberg states. First, the basis set used in the present-state ab initio methods is a linear combination of the atomic orbitals (LCAO). Highly excited states require more and more diffuse basis functions, either Gaussian or Slater type. When the internuclear distance is relatively short, the overlap between the two-centered atomic basis sets makes a large overlap and the numerical linear dependency problem causes a computational breakdown (this problem is more severe in the homopolar dimers than in the heteropolar diatomics). This can be avoided by not using too
diffuse atomic basis functions. However, this truncated basis cannot accurately describe the highly excited electronic states, so this cannot be a solution to the linear dependency. The only solution to this problem is to use an elliptic– hyperbolic basis function or a numerical basis sitting on the grids. Unfortunately, no general program using this method exists at present. Second, the configuration interaction requires a good starting vector for each state, i.e., a good initial guess. This is usually provided by a multiconfiguration self-consistent-field (MCSCF) calculation. This means that the active space in the MCSCF should be large enough to include all the important molecular orbitals (MO). In highly excited energy domains, there is usually significant mixing between the valence, singleexcited Rydberg and double-excited Rydberg configuration state functions (CSF). It is not easy to anticipate and include all the important MOs in the MCSCF level. The consequence at the CI is that the program fails to find some roots, i.e., it misses some electronic states, or converges to a different root than initially anticipated (the origin of this problem is the diagonalization of a large matrix). The solution to this problem can only
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THE Na 2 1 S g1 STATES
be expected to come from future progress in numerical analysis. As long as those two problems remain unsolved, the ab initio method cannot be applied to the highly excited electronic states, and one should be very careful in using the ab initio result for those states (which might be proved wrong). In contrast with the ab initio, the multichannel quantum defect theory may be more efficient to calculate highly excited Rydberg states. The mixing between the different types of CSF varies with the internuclear distance (R). So the truly singly excited Rydberg states or doubly excited Rydberg states or valence states cannot be defined rigorously. The spectroscopic constants deduced from low vibrationally excited states (near the potential well minimum) studied in this work provide a practical way to attribute a nonpenetrating Rydberg nature by comparing with the spectroscopic constants of the ground state cation, although this only applies to the nature of the wavefunction near the potential minimum. The 3 S g2 (attractive) and 1 S u2 (repulsive) states made from the 3p 1 3p atomic states are exceptional, where the p 1 p is the predominant CSF and it mixes little with other valence or Rydberg CSFs. The double-excited valence nature of the AOs overlap significantly only at short R and this bonding combination leads to a significantly strong bond as has been explained before (15, 23). The 7 1 S g1 and 8 1 S g1 states, which dissociate into 3p 1 3p, show a large R-dependent mixing. At small R, these two states have a predominant Rydberg character [Table 4 of Ref. (22)]. This can be easily understood in an LCAO-MO argument. The main CSFs for these two states are (3p s 2 3p s ) 2 and (3p p 2 3p p ) 2 , which have a large steric repulsion at short distances. The energy of these states is lowered through mixing with Rydberg CSFs for which the steric repulsion is much smaller due to a nonpenetrating character. IV. CONCLUSION
Five new 1 S g1 states of Na 2 have been observed by OODR spectroscopy. Three of them are assigned to the 6s s 10 1 S g1 , 6d s 13 1 S g1 , and 7d s 16 1 S g1 states, respectively, based on our labeling of the predominant n, l character. Absolute vibrational numbering of the 10 1 S g1 , 13 1 S g1 [the 11 1 S g1 state of Ref. (2)], and 16 1 S g1 states have been determined by comparing the relative intensities of the OODR excitation lines with calculated Franck–Condon factors and Dunham coefficients are obtained. The 11 1 S g1 and 12 1 S g1 states are assigned to the
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doubly excited 1 S g1 states. The state with T e 5 37 046 cm 21 observed by Taylor et al. is reassigned to the 7s s 15 1 S g1 state. ACKNOWLEDGMENTS We thank Professor R. W. Field for helpful discussions. This work was supported by a grant from the National Natural Science Foundation of China. Financial support from NNSF for G.-H. Jeung’s visit to Tsinghua University is also appreciated.
REFERENCES 1. G. Jeung, Phys. Rev. A 35, 26 –35 (1987). 2. S. Magnier, Ph. Millie´, O. Dulieu, and F. Masnou-Seeuws, J. Chem. Phys. 98, 7113–7125 (1993). [and references therein] 3. C. Effantin, J. d’Incan, A. J. Ross, R. F. Barrow and J. Verges, J. Phys. B: At. Mol. Phys. 17, 1515–1523 (1984). 4. N. W. Carlson, A. J. Taylor, and A. L. Schawlow, Phys. Rev. Lett. 45, 18 –21 (1980). 5. A. J. Taylor, K. M. Jones, and A. L. Schawlow, J. Opt. Soc. Am. 73, 994 –998 (1983). 6. H. Wang, T. J. Whang, A. M. Lyyra, Li Li, and W. C. Stwalley, J. Chem. Phys. 94, 4756 – 4764 (1991). 7. G. Y. Yan, B. W. Sterling, and A. L. Schawlow, J. Opt. Soc. Am. B 5, 2305–2310 (1988). 8. C. C. Tsai, J. T. Bahns, T. J. Whang, W. C. Stwalley, and A. M. Lyyra, Phys. Rev. Lett. 71, 1152–1153 (1993). 9. C. C. Tsai, J. T. Bahns, and W. C. Stwalley, J. Chem. Phys. 100, 768–774 (1994). 10. C. C. Tsai, J. T. Bahns, and W. C. Stwalley, J. Mol. Spectrosc. 167, 429 – 436 (1994). 11. S. Martin, J. Chevaleyre, M. Chr. Bordas, S. Valignat, and M. Broyer, J. Chem. Phys. 79, 4132– 4141 (1983). 12. C. Bordas, P. Labastie, J. Chevaleyre, and M. Broyer, Chem. Phys. 129, 21–39 (1989). 13. Y. Liu, J. Li, M. Xue, D. Chen, Li Li, and G.-H. Jeung, J. Chem. Phys. 103, 7213–7217 (1995). 14. J. Li, Y. Liu, H. Gao, M. Xue, D. Chen, and Li Li, J. Mol. Spectrosc. 175, 13–20 (1996). 15. Y. Liu, H. Chen, J. Li, D. Chen, Li Li, and R. W. Field, J. Mol. Spectrosc. 192, 32– 40 (1998). 16. X. Dai, J. Li, Y. Liu, H. Chen, D. Chen, and Li Li, unpublished manuscript. 17. Li Li and A. M. Lyyra, unpublished data. 18. K. Shimizu and F. Shimizu, J. Chem. Phys. 78, 1126 –1131 (1983). 19. H. Kato, M. Otani, and M. Baba, J. Chem. Phys. 89, 653– 659 (1988). 20. T.-J. Whang, W. C. Stwalley, Li Li, and A. M. Lyyra, J. Mol Spectrosc. 155, 184 –194 (1992). 21. [The data can be obtained from Li Li by sending a request via electronic mail to [email protected].] 22. G.-H. Jeung and A. J. Ross, J. Phys. B: At. Mol. Phys. 21, 1473–1487 (1988). 23. A. Yiannopoulou, B. Ji, Li Li, M. Li, K. Urbanski, A. M. Lyyra, W. C. Stwalley, and G.-H. Jeung, J. Chem. Phys. 101, 3581–3587 (1994).
Copyright © 1999 by Academic Press
The 6ss, 6ds, and 7ds Rydberg 1S g1 States and Two Doubly Excited 1S g1 States of Na 2 X. Dai,* H. Chen,* Y. Liu,* J. Li,* J. Xiang,* D. Chen,* Li Li,* ,1 and G.-H. Jeung† *Department of Physics and Center of Atomic and Molecular Sciences, Tsinghua University, Beijing 100084, China; and †Laboratoire Aime´ Cotton (CNRS UPR3321), Baˆt. 505, and ASCI (CNRS UPR9029), Baˆt. 506, Campus d’Orsay, 91405, Orsay, France Received December 17, 1998; in revised form August 25, 1999
Five highly excited 1 S g1 states of Na 2 were observed for the first time by pulsed optical– optical double resonance (OODR) fluorescence excitation spectroscopy. Three of the five states are assigned to the 6s s 1 S g1 , 6d s 1 S g1 , and 7d s 1 S g1 Rydberg states. The other two states are assigned to doubly excited 1 S g1 states. © 1999 Academic Press Key Words: Na 2; 1 S g1 states; Rydberg states; doubly excited states. 1. INTRODUCTION
The Na 2 molecule is interesting in many aspects. Since the A 1 S u1 4 X 1 S g1 and B 1 P u 4 X 1 S g1 transitions are in the dye laser frequency region and its Rydberg states can be reached by two-photon and two-step dye laser excitation, Na 2 has become one of the most intensively studied molecules in molecular spectroscopy. Many new molecular spectroscopic techniques have been developed in the studies of the Na 2 molecule. Because the electronic structure of Na 2 is relatively simple and relatively complete spectroscopic data are available, Na 2 is a good test molecule for theoretical calculations and molecular dynamics. The 1 S g1 Rydberg states of Na 2 have been studied extensively both theoretically and experimentally. Eighty-four electronic states of Na 2 below the 3s 1 5p atomic limit, including 11 1 S g1 states, have been calculated (1, 2). Effantin et al. (3) observed the v 5 0 –10 levels of the (3s 1 4s) 3 1 S g1 state (the number, 3, before 1 S g1 is the energy rank at R e of the states with 1 S g1 symmetry and 3s 1 4s is the dissociation limit) with Fourier transform spectrometry of laser-induced infrared fluorescence. Schawlow and co-authors observed six Rydberg 1 S g1 states (the 4 1 S g1 , 5 1 S g1 , 6 1 S g1 , 7 1 S g1 , 8 1 S g1 , and a higher 1 S g1 state with maximum vibrational quantum numbers of 20, 18, 20, 7, 7, and 1, respectively) by two-step polarization-labeling spectroscopy (4 –5). Wang et al. (6) and Yan et al. (7) probed the high vibrational levels of the 4 1 S g1 and 5 1 S g1 states by optical– optical double resonance (OODR) fluorescence excitation spectroscopy. With an extremely sensitive space-chargelimited diode ionization detector, Tsai et al. studied the longrange part of the 4 1 S g1 , 5 1 S g1 , 6 1 S g1 states (8 –10). Martin et al. (11–12), using OODR multiple photon ionization spectroscopy of a molecular beam, observed and analyzed the 3s 1 ns (n 5 23–40) and 3s 1 nd (n 5 12–39) Rydberg 1 S g1 states. 1
Recently, we carried out perturbation-facilitated OODR (PFOODR) fluorescence excitation spectroscopy via A 1 S u1 ; b 3 P u mixed intermediate levels to probe the high-lying triplet gerade states of Na 2. The 1 3 S g2 (13), 4 3 D g , 7 3 D g , 10 3 D g (14), 4 3 P g , 6 3 P g (15), and several 3 S g1 states (16) have been observed in the energy region 35 000 –39 500 cm 21. Several previously unreported 1 S g1 states were observed via the A 1 S u1 ; b 3 P u mixed intermediate levels and confirmed by OODR excitation through unperturbed A 1 S u1 intermediate levels. This paper reports our experimental observation and analysis of these 1 S g1 states. II. EXPERIMENT
To whom correspondence should be addressed.
The experimental setup was reported in Refs. (13, 14). Briefly, Lambda Physik EMG202MSC excimer laser pumped two Lambda Physik FL3002E dye lasers simultaneously. One of the dye lasers was used as the PUMP laser, which was operated with intracavity e´talon (0.04 cm 21 linewidth). Another dye laser was used as the PROBE laser without intracavity e´talon (0.2 cm 21 linewidth). Another dye laser was used as the PROBE laser without intracavity e´talon (0.2 cm 21 linewidth). The PUMP laser was operated with DCM dye and the PROBE laser was operated with Coumarin 440, 470, 480, and 500 dyes. The two dye laser beams counterpropagated and superimposed at the center of a cross heatpipe oven. Sodium vapor pressure and temperature were controlled with Ar buffer gas pressure and the temperature was about 500°C with ;1 Torr Ar. The effective length of the sodium vapor region was about 25 cm. The PUMP laser frequency was calibrated by I 2 laser-induced fluorescence lines and the PROBE laser frequency was calibrated using neon lines from optogalvanic spectroscopy in a hollow cathode lamp discharge. The pump laser frequency was tuned to a known transition from the X 1 S g1 ground state to an A 1 S u1 or A 1 S u1 ; b 3 P u mixed intermediate level. The probe laser was then scanned to excite various
239 0022-2852/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.
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DAI ET AL.
TABLE 1 Summary of Experimental Observations
transitions upward from the selected A 1 S u1 or A 1 S u1 ; b 3 P u mixed state to levels of the 1 S g1 states. Collision-induced fluorescence to X 1 S g1 was filtered with colored glass filters and monitored by a R212 photomultiplier tube (PMT) from a side window. The outputs of the PMT and the neon lamp were sent to a boxcar average, whose outputs were digitized by an A/D converter and acquired with a computer. III. OBSERVATION AND ASSIGNMENT
The intermediate levels in our OODR experiment are A 1S 1 u v 5 1,
J 5 14, 16 (17)
3 A 1S 1 u v 5 3 , b P 0u v 5 10,
A 1S 1 u v 5 6,
J 5 12–14 (18)
J 5 10, 12, 14 (17)
3 A 1S 1 u v 5 8 , b P 0u v 5 14, 3 A 1S 1 u v 5 2 , b P 1u v 5 9,
J 5 10–14 (19) J 5 28 (20).
When an A 1 S u1 ; b 3 P u mixed intermediate level is pumped, transitions into both singlet and triplet states are allowed. However, the OODR signals into singlet levels via a predominant A 1 S u1 level will be much stronger than those via its perturbation partner, the predominant b 3 P u level, and vice versa. So it is easy to distinguish singlet signals from triplet signals. From unperturbed A 1 S u1 intermediate levels, only 1 S g1
and 1 P g states can be reached. Transitions into 1 S g1 states consist of P and R lines, and transitions into 1 P g states consist of P, Q, and R lines. A total of 656 OODR excitation transitions were assigned to transitions into eight 1 S g1 states. Ninety transitions were into the 7 1 S g1 v 5 1–13 levels and 62 transitions were into the v 5 0 –10 vibrational levels of the 8 1 S g1 state (2, 5). Sixty-four transitions were into the v 5 1–9 vibrational levels of the 1 S g1 state, whose v 5 0 –1 levels were observed by Taylor et al. and assigned as the (3s 1 7d) 1 S g1 state (5). The energy positions of the vibrational levels of these three states we observed agree with the predictions of Ref. (5). Five 1S g1 states were observed for the first time. Absolute vibrational numberings of three states were determined by comparing relative intensities of excitation lines observed with calculated Franck–Condon factors of the 1S g1–A 1S u1 systems. These three states were assigned to the 6ss 1S g1, 6ds 1S g1, and the 7ds 1 1 S g Rydberg states according to our n,l labeling (15). The vibrational numberings of the other two states are tentatively determined. These two states cannot be fitted into the Rydberg series and are assigned to doubly excited states. The electronic assignments will be discussed below. Table 1 summarizes the experimental observations of the 1S g1 states. Table 2 gives their Dunham coefficients fitted from term values we observed (21). 1. Electronic Assignment of the Rydberg States Magnier et al. (2) gave potential curves and molecular constants of the 1–11 1 S g1 states of Na 2. The four lower 1 S g1
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THE Na 2 1 S g1 STATES
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TABLE 2 Molecular Constants of the 1S g1 States Observed
Note. All quantities are in cm 21. All Y(0,2) values are calculated with Y(0,2) 5 24Y(0,1) 3 /Y(1,0) 2 . a Calculated and fixed in the fit, Y(1,1) 5 6Y(0,1) 2 (12(2Y(2,0)/Y(0,1)) 1/ 2 )/Y(1,0).
states observed by Taylor et al. correspond to the 4–7 1 S g1 states, respectively, but the correspondences of the higher two states are not unambiguous (2, 4, 5). Recently we relabeled the 3 P g Rydberg states of Na 2 according to their predominant n, l character at R e ' R e1 (15). This n, l labeling shows more physical significance than their energy ranks. Now we analyze the 1 S g1 states according to their n, l character. Table 3 gives the n, l labels of the 3–16 1 S g1 states. The effective principal quantum number, n*, is defined by a modified Rydberg equation IP 2 E n* 5 5/n* 2 n* 5 @5/~IP 2 E n* !# 1/ 2 n* 5 n 2 d n, l , where IP is the ionization energy (12), 5 is the Rydberg constant, and d n,l is the nearly n-independent quantum defect for the , l (or ,+) Rydberg series. Notice that the ab initio T e values normally lie ;250 cm 21 lower than observed values (2). In our n* calculation, this 250 cm 21 was added if no experimental constants were available. All gerade states have even , (s, d, g, etc.) values. The ns s and nd s Rydberg series, with n* mod 1 values of ;0.7 and ;0.9, respectively, are penetrating Rydberg states and have greater transition strength from the valence A 1 S u1 intermediate state. The ng s series, with n* mod 1 of ;1, is a nonpenetrating Rydberg series and has much smaller transition strength from the A 1 S u1 state (15). According to this labeling, the T e 5 34 977 cm 21 state observed by Taylor et al. is the 5d s 8 1 S g1 state and the state with T e 5 37 050 cm 21 is not the 6d s 13 1 S g1 [the 11 1 S g1 state of Ref. (2)] state, but the 7s s 15 1 S g1 state. Three of the five states we observed are the 6s s 10 1 S g1 , 6d s 13 1 S g1 , and 7d s 16 1 S g1 states, respectively. The 5g s 9 1 S g1 and 6g s 14 1 S g1 states are members of the nonpenetrating ng s series and are normally not observable from valence intermediate states.
2. Electronic Assignment of the Doubly Excited 1S g1 States We observed eight vibrational levels of one 1 S g1 state in the 36 500 –37 400 cm 21 energy region and nine vibrational levels of another 1 S g1 state in the 36 500 –37 500 cm 21 energy region. The energy ranks at R e of these two states are 11 and 12 according to our tentative vibrational numberings. These levels are in the same energy region of the 6s s 10 1 S g1 and 6d s 13 1 S g1 states. These levels are not the high v levels of the 7 1 S g1 and 8 1 S g1 states because the vibrational spacings and B v values are much larger than the predicted values and are not the 7s s 15 1 S g1 and 7d s 16 1 S g1 states either because the lowest levels of these two newly observed states are lower than the potential minima of these two states predicted. None of these two states could be the 5g s 9 1 S g1 nonpenetrating state not only because the nonpenetrating states have very small transition strength from the valence A 1 S u1 state but also because the vibrational spacings and B v values are too large for the 9 1 S g1 levels in this energy region. The 6g s 1 S g1 state is predicted to be in this energy region from the quantum defect of the 5g s 9 1 S g1 state and might become observable through the valence A 1 S u1 state due to the mixing with the 6s s 10 1 S g1 or/and 6d s 13 1 S g1 states. If either of these two states were the 6g s 1 S g1 state, the OODR excitation lines into this nonpenetrating Rydberg state should always be weaker than transitions into the 6s s 10 1 S g1 and/or 6d s 13 1 S g1 states, but this is contrary to our observations. Intensity patterns suggest that mixing with the 10 1 S g1 and/or 13 1 S g1 states is not responsible for the observation of the 11 1 S g1 state, which must therefore be seen in transitions from the A 1 S u1 state because it has an appreciable doubly excited (3p 1 3p) character at short internuclear distance. According to the ab initio calculation, the 7 1 S g1 and 8 1 S g1 states dissociate to the 3p 1 3p atomic limit. The vibrational levels of the 7 1 S g1 and 8 1 S g1 states observed with v max 5 13 for the 7 1 S g1 state and v max 5 10 for the 8 1 S g1 state are low v levels. At small internuclear distance, these two states have a predominant Rydberg character, rather than a doubly excited
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DAI ET AL.
TABLE 3 Labeling of the Rydberg 1S g1 States of Na 2 According to the Predominant øl Character at R e
Note. The upper values are experimental; the lower values are theoretical by Magnier et al. [T e (ab initio) 1 250 cm 21] or predicted by quantum defect of the lower member. a Ref. (2).
valence character because of the mixing with many other Rydberg states [Table 4 of Ref. (22)]. Only the 3p 1 3p 1 3 S g2 state has a predominant doubly excited character at all internuclear distances, probably because it has a very different wavefunction from other valence or Rydberg states which all have the 1 s g molecular orbital as a common bonding orbital. Indeed, this state has about 91% p p 1 p p and 8% d p 1 d p characters near the lower part of the potential well. The ab initio calculation, in particular, based on the linear combination of the atomic orbitals (LCAO) method, is not so suitable for calculating highly excited Rydberg states. First, the basis set used in the present-state ab initio methods is a linear combination of the atomic orbitals (LCAO). Highly excited states require more and more diffuse basis functions, either Gaussian or Slater type. When the internuclear distance is relatively short, the overlap between the two-centered atomic basis sets makes a large overlap and the numerical linear dependency problem causes a computational breakdown (this problem is more severe in the homopolar dimers than in the heteropolar diatomics). This can be avoided by not using too
diffuse atomic basis functions. However, this truncated basis cannot accurately describe the highly excited electronic states, so this cannot be a solution to the linear dependency. The only solution to this problem is to use an elliptic– hyperbolic basis function or a numerical basis sitting on the grids. Unfortunately, no general program using this method exists at present. Second, the configuration interaction requires a good starting vector for each state, i.e., a good initial guess. This is usually provided by a multiconfiguration self-consistent-field (MCSCF) calculation. This means that the active space in the MCSCF should be large enough to include all the important molecular orbitals (MO). In highly excited energy domains, there is usually significant mixing between the valence, singleexcited Rydberg and double-excited Rydberg configuration state functions (CSF). It is not easy to anticipate and include all the important MOs in the MCSCF level. The consequence at the CI is that the program fails to find some roots, i.e., it misses some electronic states, or converges to a different root than initially anticipated (the origin of this problem is the diagonalization of a large matrix). The solution to this problem can only
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THE Na 2 1 S g1 STATES
be expected to come from future progress in numerical analysis. As long as those two problems remain unsolved, the ab initio method cannot be applied to the highly excited electronic states, and one should be very careful in using the ab initio result for those states (which might be proved wrong). In contrast with the ab initio, the multichannel quantum defect theory may be more efficient to calculate highly excited Rydberg states. The mixing between the different types of CSF varies with the internuclear distance (R). So the truly singly excited Rydberg states or doubly excited Rydberg states or valence states cannot be defined rigorously. The spectroscopic constants deduced from low vibrationally excited states (near the potential well minimum) studied in this work provide a practical way to attribute a nonpenetrating Rydberg nature by comparing with the spectroscopic constants of the ground state cation, although this only applies to the nature of the wavefunction near the potential minimum. The 3 S g2 (attractive) and 1 S u2 (repulsive) states made from the 3p 1 3p atomic states are exceptional, where the p 1 p is the predominant CSF and it mixes little with other valence or Rydberg CSFs. The double-excited valence nature of the AOs overlap significantly only at short R and this bonding combination leads to a significantly strong bond as has been explained before (15, 23). The 7 1 S g1 and 8 1 S g1 states, which dissociate into 3p 1 3p, show a large R-dependent mixing. At small R, these two states have a predominant Rydberg character [Table 4 of Ref. (22)]. This can be easily understood in an LCAO-MO argument. The main CSFs for these two states are (3p s 2 3p s ) 2 and (3p p 2 3p p ) 2 , which have a large steric repulsion at short distances. The energy of these states is lowered through mixing with Rydberg CSFs for which the steric repulsion is much smaller due to a nonpenetrating character. IV. CONCLUSION
Five new 1 S g1 states of Na 2 have been observed by OODR spectroscopy. Three of them are assigned to the 6s s 10 1 S g1 , 6d s 13 1 S g1 , and 7d s 16 1 S g1 states, respectively, based on our labeling of the predominant n, l character. Absolute vibrational numbering of the 10 1 S g1 , 13 1 S g1 [the 11 1 S g1 state of Ref. (2)], and 16 1 S g1 states have been determined by comparing the relative intensities of the OODR excitation lines with calculated Franck–Condon factors and Dunham coefficients are obtained. The 11 1 S g1 and 12 1 S g1 states are assigned to the
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doubly excited 1 S g1 states. The state with T e 5 37 046 cm 21 observed by Taylor et al. is reassigned to the 7s s 15 1 S g1 state. ACKNOWLEDGMENTS We thank Professor R. W. Field for helpful discussions. This work was supported by a grant from the National Natural Science Foundation of China. Financial support from NNSF for G.-H. Jeung’s visit to Tsinghua University is also appreciated.
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