- Email: [email protected]
Potential energies of the 41Π and 51Π states of NaK by polarization labelling spectroscopy and by ab initio calculations
26 November 1999
Chemical Physics Letters 314 Ž1999. 47–51 www.elsevier.nlrlocatercplett
Potential energies of the 41 P and 51 P states of NaK by polarization labelling spectroscopy and by ab initio calculations P. Kowalczyk
a,)
, W. Jastrze¸bski b, A. Pashov b, S. Magnier c , M. Aubert-Frecon ´
d
a
Institute of Experimental Physics, Warsaw UniÕersity, ul. Hoza ˙ 69, 00-681 Warsaw, Poland Institute of Physics, Polish Academy of Sciences, Al. Lotnikow ´ 32 r 46, 02-668 Warsaw, Poland c Laboratoire de Physique Moleculaire et des Collisions, Institut de Physique, Technopole ´ ˆ 2000, 1 Bd. Arago, F-57078 Metz Cedex 3, France Laboratoire de Spectrometrie CNRS et UniÕersite´ Lyon I (UMR 5579), Campus de la Doua, Bat. ´ Ionique et Moleculaire, ´ ˆ 205, 43 Bd. du 11 NoÕembre 1918, F-69622 Villeurbanne Cedex, France b
d
Received 10 March 1999; in final form 24 September 1999
Abstract The 41 P and 51 P states of the NaK molecule have been investigated experimentally by polarization labelling spectroscopy technique and theoretically within the framework of the ab initio pseudopotential method. A very good agreement has been obtained between the calculated potential curves and those determined from experimental observations. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction The NaK molecule has been the subject of intense experimental and theoretical interest, as a prototype of a simple heteronuclear system for which both spectroscopic measurements w1–6x and quantum mechanical calculations w7–9x are feasible. Recently, we have reported an experimental investigation of two highly excited states of NaK, 31 P and 61 Sq w10x, as well as a theoretical study of numerous electronic states of this molecule w11x. In the present Letter we present the first experimental observation of the 41 P and 51 P states of NaK, which correlate with the atom pairs NaŽ32 S1r2 . q KŽ52 P3r2 . and NaŽ32 S1r2 . q KŽ42 D5r2 ., respectively. The analysis of the two ) Corresponding author. Fax: q48-22-6256406; e-mail: [email protected]
band systems, 41 P § X1 Sq and 51 P § X1 Sq, enables us to determine accurate molecular constants and RKR potential curves from the experimental data. The resulting interatomic potentials are compared with the theoretical curves computed using a method developed by Foucrault et al. w12x. Additionally, we calculate the R-dependent transition dipole moments for the 41 P–X1 Sq and 51 P–X1 Sq transitions. 2. Experiment and analysis The experimental apparatus and procedure have been described in our previous papers w10,13x. In brief, we applied the polarization labelling spectroscopy technique with a V-type optical–optical double-resonance excitation scheme. The fixed frequency of the probe laser ŽArq at 514.5, 501.7,
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 1 1 4 3 - 4
P. Kowalczyk et al.r Chemical Physics Letters 314 (1999) 47–51
48
496.5, 488.0 or 476.5 nm. coincided with a set of known transitions in the D1 P § X1 Sq system of NaK, thus labelling the rotational levels involved in the ground state. The pump laser Žpulsed dye laser and XeCl excimer laser system, 2 mJ pulse energy, 0.1 cmy1 spectral linewidth. was tuned across the two investigated band systems, in the spectral range 25 550–28 650 cmy1 . The laser frequency was calibrated against the optogalvanic spectrum of argon and frequency marks were provided by a 0.5 cm long Fabry–Perot ´ interferometer. The accuracy in the determination of absolute wavenumbers was better than 0.1 cmy1 . NaK vapour was produced by heating a mixture of sodium and potassium Žnatural isotopic composition. in a stainless-steel heat pipe. The temperature was maintained at around 4508C and a pressure of 4 Torr of helium buffer gas was established. The rotationally resolved spectra were observed for both 23 Na39 K and 23 Na41 K isotopic species. Analysis of the 41 P § X1 Sq band system was rather straightforward as numerous strong ÕX-progressions have been observed, containing 1387 lines in total. The absolute vibrational numbering in the 41 P state was established using the isotope effect and confirmed by the measured intensity distribution
Table 1 The experimentally determined Dunham coefficients for the 41 P and 51 P states of 23 Na39K Žin cmy1 .. The quoted error s of a constant is one standard deviation Constant
41 P state value
Te Y10 Y20 Y30 Y40 Y01 Y11 Y21 Y31 Y41 Y02 Y12 Y22 Y32 Y03 y 01
26 463.039 87.4288 y0.772039 Ž=10. 0.109963 Ž=10 3 . y0.114430 Ž=10. 0.666289 Ž=10 3 . y0.462093 Ž=10 5 . y0.40950 Ž=10 6 . 0.26779 Ž=10 8 . y0.44046 Ž=10 6 . y0.16315 Ž=10 8 . y0.47976 Ž=10 9 . 0.30259 Ž=10 11 . y0.65251 Ž=10 12 . 0.5956 Ž=10 4 . 0.1015
51 P state
s Ž%.
value
0.00007 27 695.839 0.006 84.6274 0.056 y0.38633 0.13 0.14 0.010 0.690070 0.33 y0.44699 2.7 1.0 0.48 0.42 y0.18069 2.2 2.4 2.2 3.9 4.5 1.340
s Ž%. 0.00006 0.007 0.15
0.007 0.088
0.19
0.94
Table 2 Rotationless IPA potential for the 41 P state of NaK Õ
0 1 2 3 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 a
V Žcmy1 . 26 463.039 506.458 592.360 676.820 759.899 841.651 27 001.389 156.430 307.133 453.816 596.754 736.185 872.300 28 005.244 135.117 261.970 385.810 506.597 624.256 738.673 849.695 957.128 29 060.730 160.203 255.193 345.294 430.057 508.999 581.614
R min ˚. ŽA
R max ˚. ŽA
4.0228 3.9169 3.8476 3.7934 3.7479 3.6731 3.6121 3.5601 3.5146 3.4740 3.4374 3.4040 3.3732 3.3447 3.3180 3.2930 3.2693 3.2468 3.2256 3.2057 3.1874 3.1704 3.1546 3.1399 3.1259 3.1126 3.1001 3.0885
4.1829 a 4.3504 4.4887 4.5911 4.6792 4.7591 4.9043 5.0376 5.1634 5.2839 5.4006 5.5143 5.6258 5.7356 5.8443 5.9524 6.0604 6.1689 6.2786 6.3904 6.5051 6.6237 6.7477 6.8786 7.0181 7.1687 7.3331 7.5153 7.7205
The equilibrium distance, R e .
in the vibrational progressions. The vibrational and rotational quantum numbers of the levels observed in this state ranged from 0 to 50 and 7 to 123, respectively. Spectral lines belonging to the 51 P § X1 Sq system appeared weaker Žvide infra. and furthermore they mostly appeared in the region that is strongly contaminated by the Na 2 spectrum. This prevented us from extending our present study beyond vibrational level ÕX s 10 in the 51 P state, with 21 F J X F 105. The data field was limited in this case to 237 lines and the vibrational numbering was determined solely by comparing the observed intensities with calculated Franck–Condon factors. The measured line positions in both band systems were fitted independently to differences in the term values, n s T Ž ÕX , J X . y T Ž ÕY , J Y ., of the upper and
P. Kowalczyk et al.r Chemical Physics Letters 314 (1999) 47–51 Table 3 Rotationless RKR potential for the 51 P state of NaK V Žcmy1 .
Õ
27 695.839 738.052 821.907 904.989 987.299 28 068.835 149.599 229.591 308.809 387.255 464.929 541.829
0 1 2 3 4 5 6 7 8 9 10 a
R min ˚. ŽA
R max ˚. ŽA
3.9479 3.8408 3.7710 3.7165 3.6710 3.6315 3.5964 3.5647 3.5357 3.5090 3.4841
4.1064 a 4.2805 4.4191 4.5205 4.6068 4.6845 4.7564 4.8242 4.8889 4.9512 5.0115 5.0702
The equilibrium distance, R e .
lower levels, which were expressed by a Dunham expansion of the type T Ž Õ, J . s Te q
Ý Ž Yk l q d yk l . Ž Õ q 12 .
m
m, n
= J Ž J q 1. y L2 1
2
with L s 0 for the X S
q
n
,
state and 1 for the 1 P
49
states. The d ym n constants describe the L doubling in the electronic P states: d s 0 or 1 for f- or e-parity levels, respectively. As the coefficients for the ground state are known with high precision w1x, they were treated as fixed constants in the fits. After removing from the data field a few deviant lines, most probably reflecting perturbations of the excited states, we found that the rms error of both fits was close to 0.06 cmy1 , in agreement with the uncertainty in making a single measurement. The Dunham coefficients relating to the 41 P and 51 P states derived from the final fits are given in Table 1. Once the coefficients in the Dunham expansion had been determined, we applied the RKR method to obtain the rotationless potential curves for the 41 P and 51 P states. The 41 P state potential Žwhere experimental observation extended to high vibrational levels. was further refined by the inverted perturbation approach ŽIPA. technique w14x. Tables 2 and 3 list the energies of the vibrational eigenstates together with the corresponding turning points. Both potential curves reproduce the experimentally derived energies of rovibronic levels in the 41 P and 51 P states within the accuracy of our measurements.
Table 4 Calculated potential energies for the 41 P and 51 P states of NaK
˚. R ŽA 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.0966 4.1673 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 a
˚. R ŽA
V Žcmy1 . 4 1 P state
5 1 P state
37 023.4 34 270.2 32 012.2 30 225.9 28 867.2 27 874.9 187.4 26 753.1 531.6 – 480.4 a 482.3 562.5 714.3 917.6 27 143.8 381.7 624.6 868.3 28 109.5
38 013.3 35 182.8 32 871.1 31 073.6 29 749.8 28 830.5 238.7 27 902.8 761.1 746.6 a – 762.1 870.2 28 040.6 261.6 508.5 770.0 29 038.6 307.9 573.8
Minimum of the potential curve.
6.0 6.2 6.4 6.6 6.8 7.0 7.5 8.0 8.5 9.0 9.5 10.0 11.0 12.0 13.0 15.0 17.0 19.0 21.0 `
V Žcmy1 . 4 1 P state
5 1 P state
28 343.9 567.5 775.1 962.9 29 128.2 269.3 528.1 684.7 779.6 839.5 880.0 908.9 948.0 972.9 989.7 30 008.4 016.6 020.3 022.0 024.9
29 833.8 30 086.2 331.2 568.3 798.1 31 019.2 519.2 909.6 32 170.1 324.9 414.7 469.7 533.7 574.2 605.3 647.0 668.4 678.9 684.5 691.9
50
P. Kowalczyk et al.r Chemical Physics Letters 314 (1999) 47–51
The dissociation energy De of a given 1 P state can be calculated from the relationship Te Ž 1 P . q De Ž 1 P . s De Ž X1 Sq . q D E , where D E is the separation between the atomic asymptotes of the ground X1 Sq state and the excited 1 P state. The dissociation energy of the X1 Sq state, corresponding to the atomic configuration NaŽ32 S 1r2 . q KŽ42 S 1r2 ., has been reported as 5274.9 cmy1 w1x. Making use of the known spacing of the atomic energy levels of potassium D EŽ52 P3r2 – 42 S 1r2 . s 24 720.14 cmy1 and D EŽ42 D5r2 –42 S 1r2 . s 27 397.08 cmy1 w15x, yields DeŽ41 P . s 3532.0 " 0.5 cmy1 and DeŽ51 P . s 4976.2 " 0.5 cmy1 , where the uncertainties are due to the error associated with DeŽX1 Sq .. Thus the Dunham coefficients listed in Table 1 describe ; 88% of the 41 P state potential well depth, whereas for the strongly bound 51 P state they account for only 17%.
3. Theoretical calculations The potential curves of the 41 P and 51 P states were calculated in the manner described in the previ˚ - R - 21 A. ˚ The ous paper w11x in the range 2.4 A ab initio method employed is based on non-empirical pseudopotentials, parametrized l-dependent polarization potentials and full valence CI calculations. Note that with this method the calculated atomic asymptotes for both the investigated states were placed 20 and 30 cmy1 to above their experimental positions w15x. By matching the atomic asymptote of the calculated ground X1 Sq state w11x with the experimental one, we were able to locate the theoretical potential curves on the absolute energy scale. The resulting molecular potentials are listed in Table 4. To examine the global agreement of the calculated curves with the experimental ones, we determined the energies of the vibrational levels in the theoretical potentials of the 41 P and 51 P states by solving numerically the radial Schrodinger equation using ¨ the Hutson’s code w16x and compared them with the experimental vibrational energies, obtained directly from the RKR potentials. For both states the calculated eigenlevels are consistently too high in the 41 P state by an average of 20.1 cmy1 Žmean deviation
Fig. 1. Theoretical electronic transition dipole moment functions for the 41 P –X1 Sq and 51 P –X1 Sq transitions in NaK.
for Õ s 0–50., in the 51 P state by 49.2 cmy1 Žaverage discrepancy for Õ s 0–10.. We also calculated the electronic transition dipole moment functions DŽ R . for the 41 P–X1 Sq and 51 P–X1 Sq transitions ŽFig. 1.. The low values obtained for the 51 P–X1 Sq system explain the relative weakness of the transition observed in the experiment.
4. Conclusions In this Letter we have presented the experimental and theoretical study of the 41 P and 51 P electronic states in the NaK molecule. The pseudopotential calculations allowed us to reproduce the experimentally determined potential curves with a precision unprecedented for such highly excited molecular states. Since most of the residual discrepancy can be attributed to inaccuracies in the position of the calculated atomic asymptotes of both states, some refinements in the basis set used for calculations should provide even better precision within the same computational method.
Acknowledgements This research was partially sponsored by the KBN grant No. 2 P03B 067 16. We acknowledge the joint French–Polish ‘‘POLONIUM’’ programme, which made our collaboration possible.
P. Kowalczyk et al.r Chemical Physics Letters 314 (1999) 47–51
References w1x A.J. Ross, C. Effantin, J. d’Incan, R.F. Barrow, Mol. Phys. 56 Ž1985. 903. w2x A.J. Ross, R.M. Clements, R.F. Barrow, J. Mol. Spectrosc. 127 Ž1988. 546. w3x S. Kasahara, M. Baba, H. Kato, ˆ J. Chem. Phys. 94 Ž1991. 7713. w4x P. Kowalczyk, J. Chem. Phys. 91 Ž1989. 2779. w5x R.F. Barrow, R.M. Clements, G. Delacretaz, C. Effantin, J. d’Incan, A.J. Ross, J. Verges, ` L. Woste, ¨ J. Phys. B 20 Ž1987. 3047. w6x S. Kasahara, H. Ikoma, H. Kato, ˆ J. Chem. Phys. 100 Ž1994. 63.
51
w7x G.H. Jeung, J.P. Daudey, J.P. Malrieu, Chem. Phys. Lett. 94 Ž1983. 300. w8x W.J. Stevens, D.D. Konowalow, L.B. Ratcliff, J. Chem. Phys. 80 Ž1984. 1215. w9x W. Muller, W. Meyer, J. Chem. Phys. 80 Ž1984. 3311. ¨ w10x A. Pashov, I. Jackowska, W. Jastrze¸bski, P. Kowalczyk, Phys. Rev. A 58 Ž1998. 1048. w11x S. Magnier, Ph. Millie, ´ Phys. Rev. A 54 Ž1996. 204. w12x M. Foucrault, Ph. Millie, ´ J.P. Daudey, J. Chem. Phys. 96 Ž1992. 1257. w13x W. Jastrze¸bski, P. Kowalczyk, Phys. Rev. A 51 Ž1995. 1046. w14x C.R. Vidal, Comments At. Mol. Phys. 17 Ž1986. 173. w15x A.A. Radzig, P.M. Smirnov, Reference Data on Atoms, Molecules and Ions, Springer, Berlin, 1985. w16x J.M. Hutson, J. Phys. B 14 Ž1981. 851.
Chemical Physics Letters 314 Ž1999. 47–51 www.elsevier.nlrlocatercplett
Potential energies of the 41 P and 51 P states of NaK by polarization labelling spectroscopy and by ab initio calculations P. Kowalczyk
a,)
, W. Jastrze¸bski b, A. Pashov b, S. Magnier c , M. Aubert-Frecon ´
d
a
Institute of Experimental Physics, Warsaw UniÕersity, ul. Hoza ˙ 69, 00-681 Warsaw, Poland Institute of Physics, Polish Academy of Sciences, Al. Lotnikow ´ 32 r 46, 02-668 Warsaw, Poland c Laboratoire de Physique Moleculaire et des Collisions, Institut de Physique, Technopole ´ ˆ 2000, 1 Bd. Arago, F-57078 Metz Cedex 3, France Laboratoire de Spectrometrie CNRS et UniÕersite´ Lyon I (UMR 5579), Campus de la Doua, Bat. ´ Ionique et Moleculaire, ´ ˆ 205, 43 Bd. du 11 NoÕembre 1918, F-69622 Villeurbanne Cedex, France b
d
Received 10 March 1999; in final form 24 September 1999
Abstract The 41 P and 51 P states of the NaK molecule have been investigated experimentally by polarization labelling spectroscopy technique and theoretically within the framework of the ab initio pseudopotential method. A very good agreement has been obtained between the calculated potential curves and those determined from experimental observations. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction The NaK molecule has been the subject of intense experimental and theoretical interest, as a prototype of a simple heteronuclear system for which both spectroscopic measurements w1–6x and quantum mechanical calculations w7–9x are feasible. Recently, we have reported an experimental investigation of two highly excited states of NaK, 31 P and 61 Sq w10x, as well as a theoretical study of numerous electronic states of this molecule w11x. In the present Letter we present the first experimental observation of the 41 P and 51 P states of NaK, which correlate with the atom pairs NaŽ32 S1r2 . q KŽ52 P3r2 . and NaŽ32 S1r2 . q KŽ42 D5r2 ., respectively. The analysis of the two ) Corresponding author. Fax: q48-22-6256406; e-mail: [email protected]
band systems, 41 P § X1 Sq and 51 P § X1 Sq, enables us to determine accurate molecular constants and RKR potential curves from the experimental data. The resulting interatomic potentials are compared with the theoretical curves computed using a method developed by Foucrault et al. w12x. Additionally, we calculate the R-dependent transition dipole moments for the 41 P–X1 Sq and 51 P–X1 Sq transitions. 2. Experiment and analysis The experimental apparatus and procedure have been described in our previous papers w10,13x. In brief, we applied the polarization labelling spectroscopy technique with a V-type optical–optical double-resonance excitation scheme. The fixed frequency of the probe laser ŽArq at 514.5, 501.7,
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 1 1 4 3 - 4
P. Kowalczyk et al.r Chemical Physics Letters 314 (1999) 47–51
48
496.5, 488.0 or 476.5 nm. coincided with a set of known transitions in the D1 P § X1 Sq system of NaK, thus labelling the rotational levels involved in the ground state. The pump laser Žpulsed dye laser and XeCl excimer laser system, 2 mJ pulse energy, 0.1 cmy1 spectral linewidth. was tuned across the two investigated band systems, in the spectral range 25 550–28 650 cmy1 . The laser frequency was calibrated against the optogalvanic spectrum of argon and frequency marks were provided by a 0.5 cm long Fabry–Perot ´ interferometer. The accuracy in the determination of absolute wavenumbers was better than 0.1 cmy1 . NaK vapour was produced by heating a mixture of sodium and potassium Žnatural isotopic composition. in a stainless-steel heat pipe. The temperature was maintained at around 4508C and a pressure of 4 Torr of helium buffer gas was established. The rotationally resolved spectra were observed for both 23 Na39 K and 23 Na41 K isotopic species. Analysis of the 41 P § X1 Sq band system was rather straightforward as numerous strong ÕX-progressions have been observed, containing 1387 lines in total. The absolute vibrational numbering in the 41 P state was established using the isotope effect and confirmed by the measured intensity distribution
Table 1 The experimentally determined Dunham coefficients for the 41 P and 51 P states of 23 Na39K Žin cmy1 .. The quoted error s of a constant is one standard deviation Constant
41 P state value
Te Y10 Y20 Y30 Y40 Y01 Y11 Y21 Y31 Y41 Y02 Y12 Y22 Y32 Y03 y 01
26 463.039 87.4288 y0.772039 Ž=10. 0.109963 Ž=10 3 . y0.114430 Ž=10. 0.666289 Ž=10 3 . y0.462093 Ž=10 5 . y0.40950 Ž=10 6 . 0.26779 Ž=10 8 . y0.44046 Ž=10 6 . y0.16315 Ž=10 8 . y0.47976 Ž=10 9 . 0.30259 Ž=10 11 . y0.65251 Ž=10 12 . 0.5956 Ž=10 4 . 0.1015
51 P state
s Ž%.
value
0.00007 27 695.839 0.006 84.6274 0.056 y0.38633 0.13 0.14 0.010 0.690070 0.33 y0.44699 2.7 1.0 0.48 0.42 y0.18069 2.2 2.4 2.2 3.9 4.5 1.340
s Ž%. 0.00006 0.007 0.15
0.007 0.088
0.19
0.94
Table 2 Rotationless IPA potential for the 41 P state of NaK Õ
0 1 2 3 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 a
V Žcmy1 . 26 463.039 506.458 592.360 676.820 759.899 841.651 27 001.389 156.430 307.133 453.816 596.754 736.185 872.300 28 005.244 135.117 261.970 385.810 506.597 624.256 738.673 849.695 957.128 29 060.730 160.203 255.193 345.294 430.057 508.999 581.614
R min ˚. ŽA
R max ˚. ŽA
4.0228 3.9169 3.8476 3.7934 3.7479 3.6731 3.6121 3.5601 3.5146 3.4740 3.4374 3.4040 3.3732 3.3447 3.3180 3.2930 3.2693 3.2468 3.2256 3.2057 3.1874 3.1704 3.1546 3.1399 3.1259 3.1126 3.1001 3.0885
4.1829 a 4.3504 4.4887 4.5911 4.6792 4.7591 4.9043 5.0376 5.1634 5.2839 5.4006 5.5143 5.6258 5.7356 5.8443 5.9524 6.0604 6.1689 6.2786 6.3904 6.5051 6.6237 6.7477 6.8786 7.0181 7.1687 7.3331 7.5153 7.7205
The equilibrium distance, R e .
in the vibrational progressions. The vibrational and rotational quantum numbers of the levels observed in this state ranged from 0 to 50 and 7 to 123, respectively. Spectral lines belonging to the 51 P § X1 Sq system appeared weaker Žvide infra. and furthermore they mostly appeared in the region that is strongly contaminated by the Na 2 spectrum. This prevented us from extending our present study beyond vibrational level ÕX s 10 in the 51 P state, with 21 F J X F 105. The data field was limited in this case to 237 lines and the vibrational numbering was determined solely by comparing the observed intensities with calculated Franck–Condon factors. The measured line positions in both band systems were fitted independently to differences in the term values, n s T Ž ÕX , J X . y T Ž ÕY , J Y ., of the upper and
P. Kowalczyk et al.r Chemical Physics Letters 314 (1999) 47–51 Table 3 Rotationless RKR potential for the 51 P state of NaK V Žcmy1 .
Õ
27 695.839 738.052 821.907 904.989 987.299 28 068.835 149.599 229.591 308.809 387.255 464.929 541.829
0 1 2 3 4 5 6 7 8 9 10 a
R min ˚. ŽA
R max ˚. ŽA
3.9479 3.8408 3.7710 3.7165 3.6710 3.6315 3.5964 3.5647 3.5357 3.5090 3.4841
4.1064 a 4.2805 4.4191 4.5205 4.6068 4.6845 4.7564 4.8242 4.8889 4.9512 5.0115 5.0702
The equilibrium distance, R e .
lower levels, which were expressed by a Dunham expansion of the type T Ž Õ, J . s Te q
Ý Ž Yk l q d yk l . Ž Õ q 12 .
m
m, n
= J Ž J q 1. y L2 1
2
with L s 0 for the X S
q
n
,
state and 1 for the 1 P
49
states. The d ym n constants describe the L doubling in the electronic P states: d s 0 or 1 for f- or e-parity levels, respectively. As the coefficients for the ground state are known with high precision w1x, they were treated as fixed constants in the fits. After removing from the data field a few deviant lines, most probably reflecting perturbations of the excited states, we found that the rms error of both fits was close to 0.06 cmy1 , in agreement with the uncertainty in making a single measurement. The Dunham coefficients relating to the 41 P and 51 P states derived from the final fits are given in Table 1. Once the coefficients in the Dunham expansion had been determined, we applied the RKR method to obtain the rotationless potential curves for the 41 P and 51 P states. The 41 P state potential Žwhere experimental observation extended to high vibrational levels. was further refined by the inverted perturbation approach ŽIPA. technique w14x. Tables 2 and 3 list the energies of the vibrational eigenstates together with the corresponding turning points. Both potential curves reproduce the experimentally derived energies of rovibronic levels in the 41 P and 51 P states within the accuracy of our measurements.
Table 4 Calculated potential energies for the 41 P and 51 P states of NaK
˚. R ŽA 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.0966 4.1673 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 a
˚. R ŽA
V Žcmy1 . 4 1 P state
5 1 P state
37 023.4 34 270.2 32 012.2 30 225.9 28 867.2 27 874.9 187.4 26 753.1 531.6 – 480.4 a 482.3 562.5 714.3 917.6 27 143.8 381.7 624.6 868.3 28 109.5
38 013.3 35 182.8 32 871.1 31 073.6 29 749.8 28 830.5 238.7 27 902.8 761.1 746.6 a – 762.1 870.2 28 040.6 261.6 508.5 770.0 29 038.6 307.9 573.8
Minimum of the potential curve.
6.0 6.2 6.4 6.6 6.8 7.0 7.5 8.0 8.5 9.0 9.5 10.0 11.0 12.0 13.0 15.0 17.0 19.0 21.0 `
V Žcmy1 . 4 1 P state
5 1 P state
28 343.9 567.5 775.1 962.9 29 128.2 269.3 528.1 684.7 779.6 839.5 880.0 908.9 948.0 972.9 989.7 30 008.4 016.6 020.3 022.0 024.9
29 833.8 30 086.2 331.2 568.3 798.1 31 019.2 519.2 909.6 32 170.1 324.9 414.7 469.7 533.7 574.2 605.3 647.0 668.4 678.9 684.5 691.9
50
P. Kowalczyk et al.r Chemical Physics Letters 314 (1999) 47–51
The dissociation energy De of a given 1 P state can be calculated from the relationship Te Ž 1 P . q De Ž 1 P . s De Ž X1 Sq . q D E , where D E is the separation between the atomic asymptotes of the ground X1 Sq state and the excited 1 P state. The dissociation energy of the X1 Sq state, corresponding to the atomic configuration NaŽ32 S 1r2 . q KŽ42 S 1r2 ., has been reported as 5274.9 cmy1 w1x. Making use of the known spacing of the atomic energy levels of potassium D EŽ52 P3r2 – 42 S 1r2 . s 24 720.14 cmy1 and D EŽ42 D5r2 –42 S 1r2 . s 27 397.08 cmy1 w15x, yields DeŽ41 P . s 3532.0 " 0.5 cmy1 and DeŽ51 P . s 4976.2 " 0.5 cmy1 , where the uncertainties are due to the error associated with DeŽX1 Sq .. Thus the Dunham coefficients listed in Table 1 describe ; 88% of the 41 P state potential well depth, whereas for the strongly bound 51 P state they account for only 17%.
3. Theoretical calculations The potential curves of the 41 P and 51 P states were calculated in the manner described in the previ˚ - R - 21 A. ˚ The ous paper w11x in the range 2.4 A ab initio method employed is based on non-empirical pseudopotentials, parametrized l-dependent polarization potentials and full valence CI calculations. Note that with this method the calculated atomic asymptotes for both the investigated states were placed 20 and 30 cmy1 to above their experimental positions w15x. By matching the atomic asymptote of the calculated ground X1 Sq state w11x with the experimental one, we were able to locate the theoretical potential curves on the absolute energy scale. The resulting molecular potentials are listed in Table 4. To examine the global agreement of the calculated curves with the experimental ones, we determined the energies of the vibrational levels in the theoretical potentials of the 41 P and 51 P states by solving numerically the radial Schrodinger equation using ¨ the Hutson’s code w16x and compared them with the experimental vibrational energies, obtained directly from the RKR potentials. For both states the calculated eigenlevels are consistently too high in the 41 P state by an average of 20.1 cmy1 Žmean deviation
Fig. 1. Theoretical electronic transition dipole moment functions for the 41 P –X1 Sq and 51 P –X1 Sq transitions in NaK.
for Õ s 0–50., in the 51 P state by 49.2 cmy1 Žaverage discrepancy for Õ s 0–10.. We also calculated the electronic transition dipole moment functions DŽ R . for the 41 P–X1 Sq and 51 P–X1 Sq transitions ŽFig. 1.. The low values obtained for the 51 P–X1 Sq system explain the relative weakness of the transition observed in the experiment.
4. Conclusions In this Letter we have presented the experimental and theoretical study of the 41 P and 51 P electronic states in the NaK molecule. The pseudopotential calculations allowed us to reproduce the experimentally determined potential curves with a precision unprecedented for such highly excited molecular states. Since most of the residual discrepancy can be attributed to inaccuracies in the position of the calculated atomic asymptotes of both states, some refinements in the basis set used for calculations should provide even better precision within the same computational method.
Acknowledgements This research was partially sponsored by the KBN grant No. 2 P03B 067 16. We acknowledge the joint French–Polish ‘‘POLONIUM’’ programme, which made our collaboration possible.
P. Kowalczyk et al.r Chemical Physics Letters 314 (1999) 47–51
References w1x A.J. Ross, C. Effantin, J. d’Incan, R.F. Barrow, Mol. Phys. 56 Ž1985. 903. w2x A.J. Ross, R.M. Clements, R.F. Barrow, J. Mol. Spectrosc. 127 Ž1988. 546. w3x S. Kasahara, M. Baba, H. Kato, ˆ J. Chem. Phys. 94 Ž1991. 7713. w4x P. Kowalczyk, J. Chem. Phys. 91 Ž1989. 2779. w5x R.F. Barrow, R.M. Clements, G. Delacretaz, C. Effantin, J. d’Incan, A.J. Ross, J. Verges, ` L. Woste, ¨ J. Phys. B 20 Ž1987. 3047. w6x S. Kasahara, H. Ikoma, H. Kato, ˆ J. Chem. Phys. 100 Ž1994. 63.
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w7x G.H. Jeung, J.P. Daudey, J.P. Malrieu, Chem. Phys. Lett. 94 Ž1983. 300. w8x W.J. Stevens, D.D. Konowalow, L.B. Ratcliff, J. Chem. Phys. 80 Ž1984. 1215. w9x W. Muller, W. Meyer, J. Chem. Phys. 80 Ž1984. 3311. ¨ w10x A. Pashov, I. Jackowska, W. Jastrze¸bski, P. Kowalczyk, Phys. Rev. A 58 Ž1998. 1048. w11x S. Magnier, Ph. Millie, ´ Phys. Rev. A 54 Ž1996. 204. w12x M. Foucrault, Ph. Millie, ´ J.P. Daudey, J. Chem. Phys. 96 Ž1992. 1257. w13x W. Jastrze¸bski, P. Kowalczyk, Phys. Rev. A 51 Ž1995. 1046. w14x C.R. Vidal, Comments At. Mol. Phys. 17 Ž1986. 173. w15x A.A. Radzig, P.M. Smirnov, Reference Data on Atoms, Molecules and Ions, Springer, Berlin, 1985. w16x J.M. Hutson, J. Phys. B 14 Ž1981. 851.