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Investigation of the D 1Π state of NaK by polarisation labelling spectroscopy
Journal of Molecular Spectroscopy 250 (2008) 27–32
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Investigation of the D 1P state of NaK by polarisation labelling spectroscopy A. Adohi-Krou a, W. Jastrzebski b, P. Kowalczyk c, A.V. Stolyarov d, A.J. Ross e,* a
UFR SSMT Université de Cocody, 22 BP 582 Abidjan 22, Republic of Ivory Coast Institute of Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland Institute of Experimental Physics, Warsaw University, ul. Hoz_ a 69, 00-681 Warsaw, Poland d Department of Chemistry, Moscow State University, 119992 GSP-2 Leninskie gory b1/3, Moscow, Russia e Université Lyon 1; CNRS; LASIM UMR 5579, bât. A. Kastler, 43 Bd du 11 novembre 1918, 69622 Villeurbanne, France b c
a r t i c l e
i n f o
Article history: Received 7 January 2008 In revised form 31 March 2008 Available online 14 April 2008 Keywords: Alkali diatomics NaK Analytical potential curve Spin-orbit perturbation Polarisation spectroscopy Dissociation energy
a b s t r a c t Two-colour polarisation labelling experiments measuring the D–X system of NaK have furnished observations of the D 1P state of NaK up to v00 = 42. The last observed level is located 7 cm1 below the Na(3p 2 P3/2) + K(4s) atomic asymptote, 22247.15 cm1 above the minimum of the electronic ground state, clearly indicating the dissociation products of this state. The vibrational progressions all exhibit irregular intervals, predominantly because of strong interactions with the nearby d 3P state, which also dissociates to Na(3p) + K(4s) atoms. The polarisation data have been combined with some resolved fluorescence D–X transitions, and analysed by fitting to spectroscopic parameters and to an analytical potential curve. A full deperturbation treatment has not been attempted, but a ‘robust’ weighting scheme has been used to reduce the influence of levels that cannot be properly represented by a single channel model. Parameters determined in a fit to a potential curve include Te = 20090.18 ± 0.02 cm1, well depth 2157.0 ± 0.3 cm1, Re = 4.1547 ± 0.0002 Å, with an unweighted root mean square error of 0.12 cm1 for 959 data. Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction This paper presents a study of the potential curve of the D 1P state of the NaK molecule extending close to the Na(3p 2P3/2) + K(4s) atomic asymptote. The D 1P state of NaK has been observed in various high-resolution spectroscopic studies. Most of these followed the pioneering work of Breford and Engelke [1], taking advantage of the D–X transitions which can be readily excited with argon-ion laser lines, and which also provide a very important gateway to the triplet manifold of NaK. Although it has allowed observations of lower electronic states (triplets or singlets), no clear picture of the D 1P state itself has ever really emerged, although some unpublished work by Hessel and Giraud-Cotton [2] circulated quite widely, giving a set of parameters describing low vibrational levels of the D 1P state from an analysis of laser induced D–X fluorescence transitions. Breford and Engelke first suggested [3] that the perturbing partner state would be of 3P1 symmetry. This was quickly confirmed by polarisation measurements [4], and the d 3P1 state was explicitly investigated by Kowalczyk by selective (triplet) detection of laser excitation signals [5]. More recent investigations have focused on the interactions of the D 1P state with neighbouring states (the electronic states in this energy region are depicted in Fig. 1). Tamanis and
* Corresponding author. Fax: +33 4 72 44 58 71. E-mail addresses: [email protected], [email protected] (A.J. Ross). 0022-2852/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jms.2008.04.003
co-workers [6] have studied the K-doubling in low-lying excited states of NaK, using argon ion laser lines to excite the molecule in the presence of a scanning radio-frequency electric field (radio-frequency + optical double resonance), showing that the K-doubling in the D 1P state may be attributed to interactions with the lowerlying C1R+ state only. The origin of the K-doubling in the two lowest 1 P states (B 1P and D 1P) was successfully modelled by the ab initio electronic structure calculations of Adamson and coworkers [7]. A careful deperturbation of the rotational structure observed in some levels of the D 1P d 3P complex has been published by Pazyuk and co-workers [8], covering vibrational levels v = 2–18 in the singlet state. We had access to D ? X data from Fourier transform records [9] and photographic measurements [10,2], but the observations were rather sparse; the D 1P state needed to be explored more systematically than was possible with fixed frequency laser lines. The two-colour polarisation labelling technique is well suited to such a purpose, normally probing a range of upper state vibrational levels from a single ro-vibrational level of the electronic ground state. 2. Experiment A brief description of the heatpipe source and optical arrangement used in this work can be found in our work describing the C1R+ state of NaK [11], and a more complete presentation of the experiment is given in an earlier paper [12]. Argon ion laser lines
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A. Adohi-Krou et al. / Journal of Molecular Spectroscopy 250 (2008) 27–32
Fig. 1. Electronic states of NaK in the region of the Na(3s) + K(5s) and Na(3p) + K(4s) asymptotes. Triplet states are indicated with broken lines. The D 1P and d 3P potential curves are from this work, the C1R+ state is taken from Ref. [11]. The curves for the 3 3R+ and 4 1R+ states are from Magnier et al. (Ref. [28]), shifted by 80 cm1 to match the experimental Na(3p) + K(4s) asymptote.
provided some labelling transitions for NaK, but to improve the flexibility of the experiment, a tunable laser source, a single-mode CR 899 ring dye laser operating with Rhodamine 6G dye, was also employed as a probe. It excited known transitions of the B-X (v00 = 0, 2, 3) system in NaK, extending the range of ro-vibrational levels observable in the D 1P state. Transitions from the labelled T 00v;J levels were excited in a ‘V’ type optical-optical double resonance polarisation scheme, using a pulsed ‘pump’ laser (Lumonics HD500 pumped by an excimer laser) operating with Coumarin dyes (C503, C480 and C460) between 18000 and 21900 cm1. Because the B-X system of NaK is quite crowded, more than one series of lines usually appeared in the spectra. Measured widths of unsaturated lines were 0.2 cm1. The calibration of the D–X bands was conservatively estimated to be accurate to ±0.05 cm1, but the combination differences within the D state were so irregular that neither successive vibrational intervals, nor rotational D2F0 (J) intervals gave an immediate, unambiguous assignment. The labelling B-X transitions (measured to within 0.1 cm1 on a HighFinesse wavemeter) allowed us to discriminate between the possible options suggested by the upper state combination differences. Exciting the D–X system from v00 = 0 of the ground state allowed levels 0-30 to be observed in the D state, weaker signals, exciting from v00 = 2, 3 extended the observations to v0 = 42. The fluorescence series clearly converge towards the upper spin-orbit component, Na(3p 2P3/2) + K(4s). The vibrational levels 29–36 were strongly affected by mixing with d 3P, but the last members of the fluorescence progression excited from v00 = 3, J00 = 24 (v0 = 39– 42) appear to be regular: the dominant perturber, d 3P1, dissociates to the lower spin-orbit component. This suggested that it should be possible to define the uppermost part of the potential curve for this state fairly well. Part of this spectrum is shown in Fig. 2. The inset shows the ‘line plus extra-line’ structures seen in some regions, where the expected line position lies close to the midpoint between the two. The set of rovibrational levels observed in this work is indicated in Fig. 3. Although the polarisation labelling signals usually arise from resonances involving the lower state of the labelling transition (the usual ‘V’ configuration), some D ? X transitions were identified as coming from a pump-dump ‘K’ scheme, identifying single rovibrational energy levels of the D 1P state through transitions to known levels of the electronic ground state. These can only occur if the ring dye laser (operating around 17200 cm1) excites NaK in a two-step process, the first from low vibrational levels of the ground state to the B 1P (or C1R+) state, and then from much higher vibrational levels of the ground state (populated by fluorescence from the first step) to the D 1P state,
Fig. 2. R(24), P(24) D 1P X1R+ transitions to high vibrational levels (v0 indicated) in NaK, from v00 = 3, J00 = 24, labelled by Ar+ laser line 5017 Å via P(24) 3-3 D–X. The inset illustrates line/extra line pairs to v0 = 35 (weak extra lines) and v0 = 36, where singlet/triplet mixing is very pronounced.
Fig. 3. Set of ro-vibrational levels observed in the D 1P state of NaK (without distinction of parity). Open circles: ‘V’ polarisation measurements, triangles: ‘K’ polarisation measurements, filled circles: resolved fluorescence data.
with Te = 20090 cm1. Each ‘K’ progression identifies only one level of the upper state. The same is true for the numerous D ? X bands recorded in dispersed laser-induced fluorescence: many lines finally provide relatively little information on the upper state, even allowing for rotational relaxation around the level initially pumped by the laser. Fig. 3 distinguishes between these data types using open circles for ‘V’ polarisation measurements, triangles for ‘K’ measurements and filled circles for (earlier) dispersed fluorescence measurements. In total, 1125 D–X transitions were measured in the polarisation labelling experiment as ‘V’ transitions and 124 as ‘K’ transitions, and roughly 200 fluorescence lines were selected from the available data [2,9,10] to complete these. The input data file, and a list of probe transitions used in the polarisation labelling experiment, are given as supplementary material. 3. Analysis and results The data were treated by weighted least squares fits. The experimental uncertainties were estimated to be 0.025 cm1 for fluorescence lines measured either in Fourier transform records or from photographic plates (this figure reflects the probability of the fixed-frequency argon ion lines having excited a transition in the wings of the Doppler profile), and 0.05 cm1 for the polarisation labelling data. 220 transitions that were observed as (line + extra
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A. Adohi-Krou et al. / Journal of Molecular Spectroscopy 250 (2008) 27–32
line) pairs (see Fig. 2), were deliberately excluded from the fit, as a single state model is obviously inappropriate for such cases. All data were converted to energies with respect to the minimum of the electronic ground state, calculating T00 (v0 ,J00 ) values from the X1R+ potential curve given by Russier-Antoine et al. [13]. This potential curve is believed to give the best available representation for the X1R+ state of NaK, but it was determined from rotational levels J00 < 70. Comparing vibrational energy spacings v = 0–9 measured in D–X fluorescence spectra at J = 94 with intervals calculated from this curve, we found differences up to twice our estimated experimental uncertainty, and therefore worked initially with a data set truncated at J = 70. Frequent avoided crossings between levels of D 1P and d 3P 0,1,2 lead to strong interactions particularly for the higher vibrational levels, where the inner walls of the two states are almost coincident. This has been noted and discussed by Kowalczyk and by Pazyuk et al. in earlier work [5,8]. The lowest vibrational levels of the D 1P state are not always crossed by d 3P in the selected range of J, but the rotational levels are shifted by close-lying triplet levels, so that irregular vibrational patterns were seen in the excitation spectrum even at low v’. No model neglecting these interactions will succeed in reproducing the energy level patterns to experimental accuracy. With insufficient data pertaining to the perturbing state to improve the literature description of the d 3P state and spin-orbit matrix elements coupling it to the D 1P state, we can nevertheless focus on the singlet state, and extract a reasonable description of it from our observations. The D 1P state energies were treated by two leastsquares fits. The first is a direct fit to an analytical potential curve. This gives a useful ‘overall picture’ of the state, although the residual root mean square error does not match the experimental uncertainties. It should, however, supply quite reasonable starting parameters for future deperturbation of this system. The second is a parameter fit, coming much closer to reproducing the measured transitions to the experimental uncertainties. 4. Fit to an analytical potential curve To produce a potential curve for the D 1P state, we performed a least-squares fit to the ‘Morse/Lennard-Jones’ analytical expression given by Hajigeorgiou and Le Roy [14], using Le Roy’s program DPotFit [15]. The Morse/Lennard-Jones potential, VMLJ(R), is expressed as
" V MLJ ðRÞ ¼ De 1
ðRe Þ
6
!
R6
e
#2 /ðRÞyp ðRÞ
ð1Þ
The function /MLJ(R) can be truncated at a low order (NS) of /i for the short-range part of the curve, corresponding to R < Re; higher orders (up to NL) are required for the long-range part, with R > Re./MLR is given by:
/MLJ ðRÞ ¼ ½1 yp ðRÞ
NS X ðor N L Þ
/i yp ðRÞi þ yp ðRÞ/1
i¼0
with yp ðRÞ
p
Rp Re p. Rp þRe
We took p = 2 and optimised /i, NS and NL. We finally selected NS = 2, NL = 8. This expression assures a Morse-type potential close to the equilibrium bond length, but takes a long-range form De CR66 ; as R tends to infinity, the limiting value of the exponent function / is defined by the quantities De,C6 and Re. Reasonable values for De and C6 were available. De (=22247.15 cm1) was calculated from the known ground state dissociation energy of NaK (5273.78 (±0.24) cm1 from Ref. [13]) plus the sodium atomic line (16973.368 cm1) [16], and the C6 coefficient was taken from ab initio calculations by Bussery et al. [17]. Both these parameters
were constrained in the fits. Since the fluorescence series clearly converge towards the upper spin-orbit component, Na(3p 2P3/2) + K(4s), we have taken the C6 coefficient, 5.552 106 cm1 Å6, determined for the attractive X = 1 component of this asymptote in the Hund’s case (c) limit, given in Ref. [17]. Neither of these parameters had a critical influence on the fit, because our data scarcely extend into the asymptotic region of the potential (i.e. beyond the modified Le Roy radius at 12 Å). For this reason, we chose not to use the ‘Morse/Long-Range’ potential form introduced by Le Roy and co-workers [18,19] which extends Eq. (1) to incorporate more than one specified inverse-power term in the long-range expansion, even though this option is coded into the DPotFit program and has been used in recent analyses, notably for Van der Waals bound states of Ca2 [19], and KLi [20], and but also for N2 [18] and MgH [21]. Because we use Eq. (1) with p = 2, the leading correction to the limiting C6/R6 long-range form of our potential happens to be an effective C8/R8 term [19]. We find an effective C8 parameter for the D 1P state of NaK = 3.42 109 cm1 Å8. This is larger than the ab initio value 4.52 108 cm1 Å8 reported in Ref. [17], but as this term simply connects the limiting long range behaviour to the fitted potential in the data region, we claim no improvement on the theoretical value. The (rather small) effects of K doubling in the D 1P state were taken into account by including an additional term to the effective radial Hamiltonian for the levels of e parity (this convention was chosen to 2K 2 ½1 yp ðRÞ qK ½JðJ þ 1ÞK . A remain consistent with Ref. [6]), 2lh R2 single K-doubling parameter, qK, was used. The difference between e and f levels tends to zero at large R. The remaining difficulty was to extract valid information from a system which is so subject to avoided crossings of rotational levels that elimination of ill-fitting data (our usual approach) would lead to an unacceptably reduced data set. (We note the sharp contrast with the analagous D 1P state in NaRb, whose dissociation products in NaRb are Na (3p 2P3/2) + Rb(5s), which has been studied by Docenko and co-workers [22], and where a single potential gave a much more satisfactory representation of most of the data set). We have adopted a ‘robust’ weighting scheme (suggested by Watson [23]) to reduce the influence of outliers, in this case, levels that cannot be properly represented by our single channel model. In this approach, DPotFit performs iterative fits with each datum being weighted as 1=½ðr2i þ ðR=3Þ2 , where R is the residual for that point after the previous iteration. The resulting parameters reproduce 74% of the data set to within 2 standard deviations, and generate 60 severe outliers (jEobs E calcj > 0.25 cm1). The parameters defining the potential curve are given in Table 1 (no uncertainties are quoted for the fitted /i values, since these Table 1 Parameters for the Morse/Lennard–Jones potential curve (Eq. (1)) Parameter
Value
Te (cm1) De (cm1) Re (Å) C6 (cm1 Å6) /0 /1 /2 /3 /4 /5 /6 /7 /8 qK (1/cm1)
20090.18(2) 22247.15 4.15467(24) 5.552 106 2.578981 0.2374 0.884 94.316 669.4 2521.606 4977.1 4725.43 1718.0 1.21(50) 103
Results from a robustly weighted fit of 959 energies, with p = 2, NS = 2, NL = 8. Unweighted rms deviation is 0.12 cm1. De and C6 were constrained to known values (see text).
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A. Adohi-Krou et al. / Journal of Molecular Spectroscopy 250 (2008) 27–32
Table 2 Dunham parameters for X 1R+ state of NaK, v < 40, (to supply energy origin for dispersed fluorescence and polarisation D–X data) Parameter
Value (cm1)
2r uncertainty
Y1,0 Y2,0 Y3,0 Y4,0 Y5,0 Y6,0
124.02288335 0.494254915 8.616250 104 1.923404 105 8.40000 109 2.41711 109
8.4 104 1.8 104 1.7 105 7.7 107 1.7 108 1.4 1010
Y0,1 Y1,1 Y2,1 Y3,1 Y4,1 Y5,1 Y6,1
9.522927773 102 4.48312540 104 2.483729 106 3.813150 108 4.1950 1010 5.8700 1012 2.15699 1013
4.1 108 3.2 108 9.9 109 1.3 109 7.7 1011 2.1 1012 2.1 1014
Y0,2 Y1,2 Y2,2 Y3,2 Y4,2 Y5,2
2.244992 107 1.60995 109 1.46960 1011 5.49400 1013 2.85675 1014 1.133 1016
2.0 1011 8.1 1012 1.7 1012 1.7 1013 6.4 1015 8.2 1017
Y0,3 Y1,3 Y2,3
4.8750 1013 1.1621 1014 9.2716 1016
3.8 1015 1.0 1015 5.5 1017
Dimensionless root mean square deviation was 1.115, fitting 497 fluorescence lines, plus 88 rotational transitions. The semi-classical Y00 term calculated from Y10, Y20, Y01 and Y11 is 0.026 cm1.
parameters have no independent physical significance). We supply the set of D 1P state energies, referenced to the Na(3p 2P3/2) + K(4s) asymptote, derived from the D–X lines (also given) and the differences between these energies and those calculated from the potential curve, as supplementary material. We also give as supplementary material a pointwise representation of the potential curve (calculated from the parameters listed in Table 2) and the spectroscopic parameters Gv,Bv,Dv . . . etc. generated from it. These can be compared to the numbers obtained in the second fitting approach, which was intended to give better representation of the D 1 P state in terms of root-mean-square deviation, at the expense of a more cumbersome parameter set. 5. Parameter fits Because the potential curve described above does not allow D–X transitions to be readily identified, we also performed parameter fits in which the contribution from distortion terms in the D 1P state were determined from the curve, but the dominant vibrational energies and rotational constants for each vibrational level were determined from the data. To establish an energy origin suitable also at J values up to 110, we first performed a fit to the 40 lowest vibrational energy levels of the ground state. The ‘V’ polarisation data involved only the three lowest levels of the ground state, but some of the photographic dispersed fluorescence, and the ‘K’ polarisation labelling observations, involved higher vibrational levels, often at high J. We processed data available from earlier dispersed fluorescence measurements from the B 1P, D 1P, (Ref. [9]) and A1R+ state [13], collectively assumed to have an experimental uncertainty of 0.004 cm1, since this was the quality of fit cited in Ref. [13], plus the microwave transitions, with uncertainties of 30 KHz, published by Yamada and Hirota [24]. To avoid the parameter correlation problems between ground and excited states suspected to have occurred in Ref. [9], all the A, B and D upper-state term values were treated independently, as though all lines, including rotational relaxation, belonged to distinct fluorescence series. The X state energy levels were expressed by usual Dunham coefficients.
Table 3 Band parameters (in cm1) for the D 1P state of NaK v 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
Gv
Bv
107 Dv
20131.374(46) 20213.084(25) 20293.911(35) 20373.836(30) 20452.751(35) 20530.842a 20608.193(29) 20683.928(09) 20758.710(40) 20833.241(46) 20905.281(33) 20976.525(36) 21046.313(28) 21115.667(46) 21182.983(49) 21248.970(61) 21313.829(52) 21376.689(47) 21438.045(47) 21497.525(46) 21556.067(55) 21612.119(49) 21666.416(43) 21718.610(43) 21769.001(53) 21817.310(82) 21863.286(62) 21906.838(90) 21947.907(96) 21986.622(110) 22022.792(130) 22055.714(140) 22086.110(140) 22113.742(150) 22137.237(130) 22158.179(150) 22171.656(440) 22201.632(1700) 22212.513(1900) 22219.269(110) 22228.062(120) 22234.922(760) 22240.010(760)
0.067305(52) 0.066920(09) 0.066520(39) 0.066077(21) 0.065611(32) 0.06529a 0.064751(23) 0.064311(10) 0.063554(62) 0.063011(47) 0.062715(32) 0.062198(51) 0.062007(36) 0.060875(42) 0.060251(48) 0.059345(87) 0.058748(52) 0.057988(45) 0.057183(51) 0.056611(20) 0.055400(66) 0.054538(47) 0.053533(48) 0.052709(27) 0.051404(59) 0.050149(83) 0.048760(55) 0.047415(94) 0.046005(98) 0.044237(120) 0.042291(140) 0.040768(200) 0.038708(200) 0.036180(220) 0.036064(180) 0.034769(190) 0.043471(1100) 0.003500(100) 0.010850(600) 0.022546(95) 0.020039(120) 0.01761(130) 0.01490(130)
1.703(10) 1.781(7) 1.743(82) 1.767(20) 1.746(28) 1.954a 1.881(36) 1.954(12) 0.455(23) 1.568(130) 1.979(69) 2.066(150) 3.220(110) 2.151(93) 2.229(100) 1.350(270) 2.356(140) 2.310(110) 2.405(130) 3.116(21) 2.727(220) 3.006(130) 3.091(130) 3.929(31} 3.572(160) 3.829(220) 3.676(140) 4.025(280) 4.480(290) 4.052(240) 3.545(400) 4.696(880) 4.521(870) 2.826(960) 13.04(50) 18.69(54) 88.28(820) 248.1(450) 132.7(510) 13.02a 14.70a 17.10a 20.84a
1012 Hv
15.9(23) 2.675(930) 1.184(400) 1.31(110) 6.309(820) 1.141(570) 1.445(640) 10.27(220) 1.27(100) 3.089(700) 3.332(880) 2.800(180) 2.681(940) 4.023(1000) 5.22(120) 5.29(170) 10.44(110) 11.90(220) 13.55(230) 24.66(270) 38.75(300) 39.90(1100) 57.97(1100) 102.5(130)
1236(160) 8174(1200) 4470(1500) 93.55a 140.9a 232.5a 428.8a
Superscript a indicates a value calculated from the potential energy curve, constrained in the fit. Following [6], we have taken the f parity sublevels as a reference, and have used K doubling parameters q = 1.57 105 cm1, qv1 = 5.15 108, qv2 = 4.665 109, and qD = 3.069 1010 in Eq. (1).
A normal weighting scheme ð1=r2i Þ was used, and the fit (1873 data, 130 parameters) returned a dimensionless error of 0.90. The resulting parameters describing levels 0–40 of the X state are listed in Table 2. These were then constrained in a fit of D–X data (dispersed fluorescence plus polarisation labelling measurements) to a set of band constants. The K-doubling in the D 1P state was less well-defined in this work than in the systematic investigations of Tamanis et al. [6], so lambda-doubling parameters were constrained to the values supplied in Ref. [6]. They were assumed to extrapolate reasonably to higher vibrational levels; qv does decrease with increasing vibrational quantum number, as the ab initio predictions of Adamson et al. [7] suggest they should (no quantitative predictions were given in the ab initio work). Following the convention of Ref. [6], D state energies were calculated using the f parity levels as a reference, with e parity levels shifted by Dm, where
1 Dm ¼ q½JðJ þ 1Þ þ qv1 t þ ½JðJ þ 1Þ 2 2 1 2 ½JðJ þ 1Þ þ qd ½JðJ þ 1Þ þ qv2 t þ 2
ð2Þ
A. Adohi-Krou et al. / Journal of Molecular Spectroscopy 250 (2008) 27–32
Once again, obvious line/extra-line pairs were eliminated from the fit. Parameters were not determined for v = 5, because no regular patterns emerged at all for this level. A robustly weighted fit to 1353 data gave a dimensionless error 0.75 and left 158 lines outside twice their estimated uncertainties. The worst outliers form systematic patterns within some vibrational levels (see levels v = 28– 35 for example, in the supplementary material). The band constants thus obtained are listed in Table 3. The distortion constants, Dv and Hv, are clearly ‘fitting parameters’, whose behaviour is quite erratic in some cases. The vibrational intervals, too, are slightly irregular even at the bottom of the potential. Unsurprisingly, a Dunham-type polynomial does not reproduce the data set at all well. 6. Conclusion This work provides a description of most of the vibrational levels of the D 1P state of NaK at rotational resolution. It shows, in sharp contrast to the analogous state of NaRb, studied by Docenko et al. [22], that a standard set of spectroscopic parameters cannot describe the D 1P state of NaK to within experimental uncertainty. The reason for this seems to be the relative positions of the perturbing triplet state. In NaRb, the Hund’s case (a) calculations of Zaitsevskii et al. [25] show that the triplet state is more deeply bound than the singlet, and that the outer limbs of the d 3P and D 1P states remain almost parallel, converging only close to the atomic asymptote. In the case of NaK, the singlet state has the deeper potential well, and the outer limbs of the d 3P and D 1P states cross at about 4.7 Å. The low vibrational levels of the D 1P state of NaK are severely affected by this crossing. Working up the potential well, the perturbations are again strong, but here vibrational overlap becomes very good because the inner walls of the two states are close to one another. If our analytical potential for the singlet state is reasonable, the spin-orbit coupling effects detailed by Pazyuk and co-workers in Ref. [8] should allow us to give good predictions for the severely mixed energy levels of the D,d complex, notably for those observed as line/extra-line pairs, which were excluded from the one-channel data treatment described above. The quality of the partly deperturbed potential curve we propose for the D 1P state, UD(R) was investigated by calculating the expected spin-orbit coupling with nearby d 3P, using the direct channel-coupling (CC) approach. The non-zero elements of the parity-independent 4 4 potential energy matrix Ve/f(R) (adapted from [8] are:
h1 PjHj1 Pi ¼ U D1 P ðRÞ þ BðRÞðx þ 2Þ 3
h3 P0 jHj3 P0 i ¼ U d3 P ðRÞ ASO ðRÞ þ BðRÞðx þ 2Þ h3 P2 jHj3 P2 i ¼ U d3 P ðRÞ þ ASO ðRÞ þ BðRÞðx 2Þ pffiffiffiffiffiffi h3 P0 jHj3 P1 i ¼ BðRÞ 2x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h3 P1 jHj3 P2 i ¼ BðRÞ 2ðx 2Þ 1
3
SO
h PjHj P1 i ¼ n ðRÞ
The results of this coupled channels calculation are at present valid only for bound levels, located below lower dissociation limit of the d 3P state, Na(3P1/2) + K(4S1/2). Although interactions with the other electronic states dissociating to the same Na(3p) + K(4s) atomic asymptote have been ignored, the polarisation labelling measurements for energy levels below 22230 cm1 (including line/extra-line pairs) are recalculated with an unweighted root-mean-square deviation of 0.14 cm1. The majority of poorly-reproduced energies lie above 22200 cm1, where the triplet state is accepted to be least accurate. There is insufficient information in our current set of observations to derive an improved triplet potential curve, plus diagonal and off-diagonal spin-orbit matrix elements (assuming that a two-state model would suffice, which is probably another over-simplification). Nevertheless, we believe that the potential energy curve we have obtained for the D 1P state supplies a reliable starting point for any future work. When more data become available for the triplet state(s), it may be possible to attempt a more refined treatment, either locally, as performed by Eckel et al. [27] for the higher-lying 33P 31P complex in NaK, or to extend the global picture [8] developed at Moscow State University. Acknowledgments This work was partially supported by a grant from the Polish Committee for Scientific Research, No. N202 103 31/0753. We are also grateful for financial support for travel (France/Poland bilateral exchange programmes PAN/CNRS and Polonium). A.V.S. acknowledges support from the Russian Foundation for Basic Research, Grant N06-03-32330. Appendix A. Supplementary data Supplementary data for this article are available on ScienceDirect (www.sciencedirect.com) and as part of the Ohio State University Molecular Spectroscopy Archives (http://msa.lib.ohiostate.edu/jmsa_hp.htm). Supplementary data associated with this article can be found, in the online version, at doi:10.1016/ j.jms.2008.04.003. References [1] [2] [3] [4] [5] [6]
3
h P1 jHj P1 i ¼ U d3 P ðRÞ þ BðRÞðx þ 2Þ
[7] [8]
ð3Þ
where x = J(J + 1), and B(R) = ⁄2/2lR2. The adiabatic potential energy curve for the triplet state Ud3 P (R) is now defined by a hybrid RKR/ab initio potential. The lowest part corresponds to the deperturbed RKR turning points listed in Ref. [8], with a smooth extension to large internuclear distance supplied by recent calculations of Aymar and Dulieu [26], since observations now extend to high vibrational levels. The spin-orbit splitting function of the triplet state, ASO(R), and the off-diagonal spin-orbit coupling function between the singlet and triplet state, nSO(R), were obtained (for R = 2–20 Å) from a large-scale multi-reference configuration interaction calculation, working with effective small-core (9 valence electrons) non-empirical relativistic pseudopotentials.
31
[9] [10] [11] [12] [13] [14] [15]
[16] [17] [18] [19] [20]
E.J. Breford, F. Engelke, Chem. Phys. Lett. 53 (1978) 282–287. M.M. Hessel and S. Giraud-Cotton, unpublished pre-print, (1980). E.J. Breford, F. Engelke, J. Chem. Phys. 71 (1979) 1994–2004. J. McCormack, A.M. McCaffery, Chem. Phys. Lett. 64 (1979) 98–99. P. Kowalczyk, J. Mol. Spectrosc. 136 (1989) 1–11. M. Tamanis, M. Auzinsh, I. Klincare, O. Nikolayeva, R. Ferber, E.A. Pazyuk, A.V. Stolyarov, A. Zaitsevskii, Phys. Rev. A 58 (1998) 1932–1943. S.O. Adamson, A. Zaitsevskii, E.A. Pazyuk, A.V. Stolyarov, M. Tamanis, R. Ferber, R. Cimiraglia, J. Chem. Phys. 113 (2000) 8589–8593. E.A. Pazyuk, A.V. Stolyarov, A. Zaitsevskii, R. Ferber, P. Kowalczyk, C. Teichteil, Mol. Phys. 96 (1999) 955–961. A.J. Ross, C. Effantin, J. D’Incan, R.F. Barrow, Mol. Phys. 56 (1985) 903–912. A. Krou-Adohi, S. Giraud-Cotton, J. Mol. Spectrosc. 190 (1998) 171–188. A.J. Ross, P. Crozet, I. Russier-Antoine, A. Grochola, P. Kowalczyk, W. Jastrzebski, P. Kortyka, J. Mol. Spectrosc. 226 (2004) 95–102. W. Jastrzebski, P. Kowalczyk, Phys. Rev. A 51 (1995) 1046–1051. I. Russier-Antoine, A.J. Ross, M. Aubert-Frécon, F. Martin, P. Crozet, J. Phys. B 33 (2000) 2753–2762. P.G. Hajigeorgiou, R.J. Le Roy, J. Chem. Phys. 112 (2000) 3949–3957. R.J. Le Roy, J.Y. Seto, and Y. Huang, DPotFit: A Computer Program for fitting Diatomic Molecule Spectra to Potential Energy Functions. See http:// leroy.uwaterloo.ca/programs University of Waterloo Chemical Physics Research Report CP-662R, (2006). NIST, Atomic Spectra Database http://physics.nist.gov/cgi-bin/AtData/ display.ksh. B. Bussery, Y. Achkar, M. Aubert-Frécon, Chem. Phys. 116 (1987) 319–338. R.J. Le Roy, Y. Huang, C. Jary, J. Chem. Phys. 125 (2006) 164310. R.J. Le Roy, R.D.E. Henderson, Mol. Phys. 105 (2007) 663–677. H. Salami, A.J. Ross, P. Crozet, W. Jastrzebski, P. Kowalczyk, R.J. Le Roy, J. Chem. Phys. 126 (2007) 194313.
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[21] A. Shayesteh, R.D.E. Henderson, R.J. Le Roy, P.F. Bernath, Journal of Physical Chemistry A 111 (2007) 12495–12505. [22] O. Docenko, M. Tamanis, R. Ferber, A. Pashov, H. Knöckel, E. Tiemann, Eur. Phys. J.D 36 (2005) 49–55. [23] J.K.G. Watson, J. Mol. Spectrosc. 219 (2003) 326–328. [24] C. Yamada, E. Hirota, J. Mol. Spectrosc. 153 (1992) 91–95.
[25] A. Zaitsevskii, S.O. Adamson, E.A. Pazyuk, A.V. Stolyarov, O. Nikolayeva, O. Docenko, I. Klincare, M. Auzinsh, M. Tamanis, R. Ferber, and R. Cimiraglia, Phys. Rev. A 63 (2001) art. no.-052504. [26] M. Aymar, O. Dulieu, Mol. Phys. 105 (2007) 1733–1742. [27] S. Eckel, S. Ashman, J. Huennekens, J. Mol. Spectrosc. 242 (2007) 182–194. [28] S. Magnier, M. Aubert-Frécon, P. Millié, J. Mol. Spectrosc. 200 (2000) 96–103.
Contents lists available at ScienceDirect
Journal of Molecular Spectroscopy journal homepage: www.elsevier.com/locate/jms
Investigation of the D 1P state of NaK by polarisation labelling spectroscopy A. Adohi-Krou a, W. Jastrzebski b, P. Kowalczyk c, A.V. Stolyarov d, A.J. Ross e,* a
UFR SSMT Université de Cocody, 22 BP 582 Abidjan 22, Republic of Ivory Coast Institute of Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland Institute of Experimental Physics, Warsaw University, ul. Hoz_ a 69, 00-681 Warsaw, Poland d Department of Chemistry, Moscow State University, 119992 GSP-2 Leninskie gory b1/3, Moscow, Russia e Université Lyon 1; CNRS; LASIM UMR 5579, bât. A. Kastler, 43 Bd du 11 novembre 1918, 69622 Villeurbanne, France b c
a r t i c l e
i n f o
Article history: Received 7 January 2008 In revised form 31 March 2008 Available online 14 April 2008 Keywords: Alkali diatomics NaK Analytical potential curve Spin-orbit perturbation Polarisation spectroscopy Dissociation energy
a b s t r a c t Two-colour polarisation labelling experiments measuring the D–X system of NaK have furnished observations of the D 1P state of NaK up to v00 = 42. The last observed level is located 7 cm1 below the Na(3p 2 P3/2) + K(4s) atomic asymptote, 22247.15 cm1 above the minimum of the electronic ground state, clearly indicating the dissociation products of this state. The vibrational progressions all exhibit irregular intervals, predominantly because of strong interactions with the nearby d 3P state, which also dissociates to Na(3p) + K(4s) atoms. The polarisation data have been combined with some resolved fluorescence D–X transitions, and analysed by fitting to spectroscopic parameters and to an analytical potential curve. A full deperturbation treatment has not been attempted, but a ‘robust’ weighting scheme has been used to reduce the influence of levels that cannot be properly represented by a single channel model. Parameters determined in a fit to a potential curve include Te = 20090.18 ± 0.02 cm1, well depth 2157.0 ± 0.3 cm1, Re = 4.1547 ± 0.0002 Å, with an unweighted root mean square error of 0.12 cm1 for 959 data. Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction This paper presents a study of the potential curve of the D 1P state of the NaK molecule extending close to the Na(3p 2P3/2) + K(4s) atomic asymptote. The D 1P state of NaK has been observed in various high-resolution spectroscopic studies. Most of these followed the pioneering work of Breford and Engelke [1], taking advantage of the D–X transitions which can be readily excited with argon-ion laser lines, and which also provide a very important gateway to the triplet manifold of NaK. Although it has allowed observations of lower electronic states (triplets or singlets), no clear picture of the D 1P state itself has ever really emerged, although some unpublished work by Hessel and Giraud-Cotton [2] circulated quite widely, giving a set of parameters describing low vibrational levels of the D 1P state from an analysis of laser induced D–X fluorescence transitions. Breford and Engelke first suggested [3] that the perturbing partner state would be of 3P1 symmetry. This was quickly confirmed by polarisation measurements [4], and the d 3P1 state was explicitly investigated by Kowalczyk by selective (triplet) detection of laser excitation signals [5]. More recent investigations have focused on the interactions of the D 1P state with neighbouring states (the electronic states in this energy region are depicted in Fig. 1). Tamanis and
* Corresponding author. Fax: +33 4 72 44 58 71. E-mail addresses: [email protected], [email protected] (A.J. Ross). 0022-2852/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jms.2008.04.003
co-workers [6] have studied the K-doubling in low-lying excited states of NaK, using argon ion laser lines to excite the molecule in the presence of a scanning radio-frequency electric field (radio-frequency + optical double resonance), showing that the K-doubling in the D 1P state may be attributed to interactions with the lowerlying C1R+ state only. The origin of the K-doubling in the two lowest 1 P states (B 1P and D 1P) was successfully modelled by the ab initio electronic structure calculations of Adamson and coworkers [7]. A careful deperturbation of the rotational structure observed in some levels of the D 1P d 3P complex has been published by Pazyuk and co-workers [8], covering vibrational levels v = 2–18 in the singlet state. We had access to D ? X data from Fourier transform records [9] and photographic measurements [10,2], but the observations were rather sparse; the D 1P state needed to be explored more systematically than was possible with fixed frequency laser lines. The two-colour polarisation labelling technique is well suited to such a purpose, normally probing a range of upper state vibrational levels from a single ro-vibrational level of the electronic ground state. 2. Experiment A brief description of the heatpipe source and optical arrangement used in this work can be found in our work describing the C1R+ state of NaK [11], and a more complete presentation of the experiment is given in an earlier paper [12]. Argon ion laser lines
28
A. Adohi-Krou et al. / Journal of Molecular Spectroscopy 250 (2008) 27–32
Fig. 1. Electronic states of NaK in the region of the Na(3s) + K(5s) and Na(3p) + K(4s) asymptotes. Triplet states are indicated with broken lines. The D 1P and d 3P potential curves are from this work, the C1R+ state is taken from Ref. [11]. The curves for the 3 3R+ and 4 1R+ states are from Magnier et al. (Ref. [28]), shifted by 80 cm1 to match the experimental Na(3p) + K(4s) asymptote.
provided some labelling transitions for NaK, but to improve the flexibility of the experiment, a tunable laser source, a single-mode CR 899 ring dye laser operating with Rhodamine 6G dye, was also employed as a probe. It excited known transitions of the B-X (v00 = 0, 2, 3) system in NaK, extending the range of ro-vibrational levels observable in the D 1P state. Transitions from the labelled T 00v;J levels were excited in a ‘V’ type optical-optical double resonance polarisation scheme, using a pulsed ‘pump’ laser (Lumonics HD500 pumped by an excimer laser) operating with Coumarin dyes (C503, C480 and C460) between 18000 and 21900 cm1. Because the B-X system of NaK is quite crowded, more than one series of lines usually appeared in the spectra. Measured widths of unsaturated lines were 0.2 cm1. The calibration of the D–X bands was conservatively estimated to be accurate to ±0.05 cm1, but the combination differences within the D state were so irregular that neither successive vibrational intervals, nor rotational D2F0 (J) intervals gave an immediate, unambiguous assignment. The labelling B-X transitions (measured to within 0.1 cm1 on a HighFinesse wavemeter) allowed us to discriminate between the possible options suggested by the upper state combination differences. Exciting the D–X system from v00 = 0 of the ground state allowed levels 0-30 to be observed in the D state, weaker signals, exciting from v00 = 2, 3 extended the observations to v0 = 42. The fluorescence series clearly converge towards the upper spin-orbit component, Na(3p 2P3/2) + K(4s). The vibrational levels 29–36 were strongly affected by mixing with d 3P, but the last members of the fluorescence progression excited from v00 = 3, J00 = 24 (v0 = 39– 42) appear to be regular: the dominant perturber, d 3P1, dissociates to the lower spin-orbit component. This suggested that it should be possible to define the uppermost part of the potential curve for this state fairly well. Part of this spectrum is shown in Fig. 2. The inset shows the ‘line plus extra-line’ structures seen in some regions, where the expected line position lies close to the midpoint between the two. The set of rovibrational levels observed in this work is indicated in Fig. 3. Although the polarisation labelling signals usually arise from resonances involving the lower state of the labelling transition (the usual ‘V’ configuration), some D ? X transitions were identified as coming from a pump-dump ‘K’ scheme, identifying single rovibrational energy levels of the D 1P state through transitions to known levels of the electronic ground state. These can only occur if the ring dye laser (operating around 17200 cm1) excites NaK in a two-step process, the first from low vibrational levels of the ground state to the B 1P (or C1R+) state, and then from much higher vibrational levels of the ground state (populated by fluorescence from the first step) to the D 1P state,
Fig. 2. R(24), P(24) D 1P X1R+ transitions to high vibrational levels (v0 indicated) in NaK, from v00 = 3, J00 = 24, labelled by Ar+ laser line 5017 Å via P(24) 3-3 D–X. The inset illustrates line/extra line pairs to v0 = 35 (weak extra lines) and v0 = 36, where singlet/triplet mixing is very pronounced.
Fig. 3. Set of ro-vibrational levels observed in the D 1P state of NaK (without distinction of parity). Open circles: ‘V’ polarisation measurements, triangles: ‘K’ polarisation measurements, filled circles: resolved fluorescence data.
with Te = 20090 cm1. Each ‘K’ progression identifies only one level of the upper state. The same is true for the numerous D ? X bands recorded in dispersed laser-induced fluorescence: many lines finally provide relatively little information on the upper state, even allowing for rotational relaxation around the level initially pumped by the laser. Fig. 3 distinguishes between these data types using open circles for ‘V’ polarisation measurements, triangles for ‘K’ measurements and filled circles for (earlier) dispersed fluorescence measurements. In total, 1125 D–X transitions were measured in the polarisation labelling experiment as ‘V’ transitions and 124 as ‘K’ transitions, and roughly 200 fluorescence lines were selected from the available data [2,9,10] to complete these. The input data file, and a list of probe transitions used in the polarisation labelling experiment, are given as supplementary material. 3. Analysis and results The data were treated by weighted least squares fits. The experimental uncertainties were estimated to be 0.025 cm1 for fluorescence lines measured either in Fourier transform records or from photographic plates (this figure reflects the probability of the fixed-frequency argon ion lines having excited a transition in the wings of the Doppler profile), and 0.05 cm1 for the polarisation labelling data. 220 transitions that were observed as (line + extra
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A. Adohi-Krou et al. / Journal of Molecular Spectroscopy 250 (2008) 27–32
line) pairs (see Fig. 2), were deliberately excluded from the fit, as a single state model is obviously inappropriate for such cases. All data were converted to energies with respect to the minimum of the electronic ground state, calculating T00 (v0 ,J00 ) values from the X1R+ potential curve given by Russier-Antoine et al. [13]. This potential curve is believed to give the best available representation for the X1R+ state of NaK, but it was determined from rotational levels J00 < 70. Comparing vibrational energy spacings v = 0–9 measured in D–X fluorescence spectra at J = 94 with intervals calculated from this curve, we found differences up to twice our estimated experimental uncertainty, and therefore worked initially with a data set truncated at J = 70. Frequent avoided crossings between levels of D 1P and d 3P 0,1,2 lead to strong interactions particularly for the higher vibrational levels, where the inner walls of the two states are almost coincident. This has been noted and discussed by Kowalczyk and by Pazyuk et al. in earlier work [5,8]. The lowest vibrational levels of the D 1P state are not always crossed by d 3P in the selected range of J, but the rotational levels are shifted by close-lying triplet levels, so that irregular vibrational patterns were seen in the excitation spectrum even at low v’. No model neglecting these interactions will succeed in reproducing the energy level patterns to experimental accuracy. With insufficient data pertaining to the perturbing state to improve the literature description of the d 3P state and spin-orbit matrix elements coupling it to the D 1P state, we can nevertheless focus on the singlet state, and extract a reasonable description of it from our observations. The D 1P state energies were treated by two leastsquares fits. The first is a direct fit to an analytical potential curve. This gives a useful ‘overall picture’ of the state, although the residual root mean square error does not match the experimental uncertainties. It should, however, supply quite reasonable starting parameters for future deperturbation of this system. The second is a parameter fit, coming much closer to reproducing the measured transitions to the experimental uncertainties. 4. Fit to an analytical potential curve To produce a potential curve for the D 1P state, we performed a least-squares fit to the ‘Morse/Lennard-Jones’ analytical expression given by Hajigeorgiou and Le Roy [14], using Le Roy’s program DPotFit [15]. The Morse/Lennard-Jones potential, VMLJ(R), is expressed as
" V MLJ ðRÞ ¼ De 1
ðRe Þ
6
!
R6
e
#2 /ðRÞyp ðRÞ
ð1Þ
The function /MLJ(R) can be truncated at a low order (NS) of /i for the short-range part of the curve, corresponding to R < Re; higher orders (up to NL) are required for the long-range part, with R > Re./MLR is given by:
/MLJ ðRÞ ¼ ½1 yp ðRÞ
NS X ðor N L Þ
/i yp ðRÞi þ yp ðRÞ/1
i¼0
with yp ðRÞ
p
Rp Re p. Rp þRe
We took p = 2 and optimised /i, NS and NL. We finally selected NS = 2, NL = 8. This expression assures a Morse-type potential close to the equilibrium bond length, but takes a long-range form De CR66 ; as R tends to infinity, the limiting value of the exponent function / is defined by the quantities De,C6 and Re. Reasonable values for De and C6 were available. De (=22247.15 cm1) was calculated from the known ground state dissociation energy of NaK (5273.78 (±0.24) cm1 from Ref. [13]) plus the sodium atomic line (16973.368 cm1) [16], and the C6 coefficient was taken from ab initio calculations by Bussery et al. [17]. Both these parameters
were constrained in the fits. Since the fluorescence series clearly converge towards the upper spin-orbit component, Na(3p 2P3/2) + K(4s), we have taken the C6 coefficient, 5.552 106 cm1 Å6, determined for the attractive X = 1 component of this asymptote in the Hund’s case (c) limit, given in Ref. [17]. Neither of these parameters had a critical influence on the fit, because our data scarcely extend into the asymptotic region of the potential (i.e. beyond the modified Le Roy radius at 12 Å). For this reason, we chose not to use the ‘Morse/Long-Range’ potential form introduced by Le Roy and co-workers [18,19] which extends Eq. (1) to incorporate more than one specified inverse-power term in the long-range expansion, even though this option is coded into the DPotFit program and has been used in recent analyses, notably for Van der Waals bound states of Ca2 [19], and KLi [20], and but also for N2 [18] and MgH [21]. Because we use Eq. (1) with p = 2, the leading correction to the limiting C6/R6 long-range form of our potential happens to be an effective C8/R8 term [19]. We find an effective C8 parameter for the D 1P state of NaK = 3.42 109 cm1 Å8. This is larger than the ab initio value 4.52 108 cm1 Å8 reported in Ref. [17], but as this term simply connects the limiting long range behaviour to the fitted potential in the data region, we claim no improvement on the theoretical value. The (rather small) effects of K doubling in the D 1P state were taken into account by including an additional term to the effective radial Hamiltonian for the levels of e parity (this convention was chosen to 2K 2 ½1 yp ðRÞ qK ½JðJ þ 1ÞK . A remain consistent with Ref. [6]), 2lh R2 single K-doubling parameter, qK, was used. The difference between e and f levels tends to zero at large R. The remaining difficulty was to extract valid information from a system which is so subject to avoided crossings of rotational levels that elimination of ill-fitting data (our usual approach) would lead to an unacceptably reduced data set. (We note the sharp contrast with the analagous D 1P state in NaRb, whose dissociation products in NaRb are Na (3p 2P3/2) + Rb(5s), which has been studied by Docenko and co-workers [22], and where a single potential gave a much more satisfactory representation of most of the data set). We have adopted a ‘robust’ weighting scheme (suggested by Watson [23]) to reduce the influence of outliers, in this case, levels that cannot be properly represented by our single channel model. In this approach, DPotFit performs iterative fits with each datum being weighted as 1=½ðr2i þ ðR=3Þ2 , where R is the residual for that point after the previous iteration. The resulting parameters reproduce 74% of the data set to within 2 standard deviations, and generate 60 severe outliers (jEobs E calcj > 0.25 cm1). The parameters defining the potential curve are given in Table 1 (no uncertainties are quoted for the fitted /i values, since these Table 1 Parameters for the Morse/Lennard–Jones potential curve (Eq. (1)) Parameter
Value
Te (cm1) De (cm1) Re (Å) C6 (cm1 Å6) /0 /1 /2 /3 /4 /5 /6 /7 /8 qK (1/cm1)
20090.18(2) 22247.15 4.15467(24) 5.552 106 2.578981 0.2374 0.884 94.316 669.4 2521.606 4977.1 4725.43 1718.0 1.21(50) 103
Results from a robustly weighted fit of 959 energies, with p = 2, NS = 2, NL = 8. Unweighted rms deviation is 0.12 cm1. De and C6 were constrained to known values (see text).
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A. Adohi-Krou et al. / Journal of Molecular Spectroscopy 250 (2008) 27–32
Table 2 Dunham parameters for X 1R+ state of NaK, v < 40, (to supply energy origin for dispersed fluorescence and polarisation D–X data) Parameter
Value (cm1)
2r uncertainty
Y1,0 Y2,0 Y3,0 Y4,0 Y5,0 Y6,0
124.02288335 0.494254915 8.616250 104 1.923404 105 8.40000 109 2.41711 109
8.4 104 1.8 104 1.7 105 7.7 107 1.7 108 1.4 1010
Y0,1 Y1,1 Y2,1 Y3,1 Y4,1 Y5,1 Y6,1
9.522927773 102 4.48312540 104 2.483729 106 3.813150 108 4.1950 1010 5.8700 1012 2.15699 1013
4.1 108 3.2 108 9.9 109 1.3 109 7.7 1011 2.1 1012 2.1 1014
Y0,2 Y1,2 Y2,2 Y3,2 Y4,2 Y5,2
2.244992 107 1.60995 109 1.46960 1011 5.49400 1013 2.85675 1014 1.133 1016
2.0 1011 8.1 1012 1.7 1012 1.7 1013 6.4 1015 8.2 1017
Y0,3 Y1,3 Y2,3
4.8750 1013 1.1621 1014 9.2716 1016
3.8 1015 1.0 1015 5.5 1017
Dimensionless root mean square deviation was 1.115, fitting 497 fluorescence lines, plus 88 rotational transitions. The semi-classical Y00 term calculated from Y10, Y20, Y01 and Y11 is 0.026 cm1.
parameters have no independent physical significance). We supply the set of D 1P state energies, referenced to the Na(3p 2P3/2) + K(4s) asymptote, derived from the D–X lines (also given) and the differences between these energies and those calculated from the potential curve, as supplementary material. We also give as supplementary material a pointwise representation of the potential curve (calculated from the parameters listed in Table 2) and the spectroscopic parameters Gv,Bv,Dv . . . etc. generated from it. These can be compared to the numbers obtained in the second fitting approach, which was intended to give better representation of the D 1 P state in terms of root-mean-square deviation, at the expense of a more cumbersome parameter set. 5. Parameter fits Because the potential curve described above does not allow D–X transitions to be readily identified, we also performed parameter fits in which the contribution from distortion terms in the D 1P state were determined from the curve, but the dominant vibrational energies and rotational constants for each vibrational level were determined from the data. To establish an energy origin suitable also at J values up to 110, we first performed a fit to the 40 lowest vibrational energy levels of the ground state. The ‘V’ polarisation data involved only the three lowest levels of the ground state, but some of the photographic dispersed fluorescence, and the ‘K’ polarisation labelling observations, involved higher vibrational levels, often at high J. We processed data available from earlier dispersed fluorescence measurements from the B 1P, D 1P, (Ref. [9]) and A1R+ state [13], collectively assumed to have an experimental uncertainty of 0.004 cm1, since this was the quality of fit cited in Ref. [13], plus the microwave transitions, with uncertainties of 30 KHz, published by Yamada and Hirota [24]. To avoid the parameter correlation problems between ground and excited states suspected to have occurred in Ref. [9], all the A, B and D upper-state term values were treated independently, as though all lines, including rotational relaxation, belonged to distinct fluorescence series. The X state energy levels were expressed by usual Dunham coefficients.
Table 3 Band parameters (in cm1) for the D 1P state of NaK v 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
Gv
Bv
107 Dv
20131.374(46) 20213.084(25) 20293.911(35) 20373.836(30) 20452.751(35) 20530.842a 20608.193(29) 20683.928(09) 20758.710(40) 20833.241(46) 20905.281(33) 20976.525(36) 21046.313(28) 21115.667(46) 21182.983(49) 21248.970(61) 21313.829(52) 21376.689(47) 21438.045(47) 21497.525(46) 21556.067(55) 21612.119(49) 21666.416(43) 21718.610(43) 21769.001(53) 21817.310(82) 21863.286(62) 21906.838(90) 21947.907(96) 21986.622(110) 22022.792(130) 22055.714(140) 22086.110(140) 22113.742(150) 22137.237(130) 22158.179(150) 22171.656(440) 22201.632(1700) 22212.513(1900) 22219.269(110) 22228.062(120) 22234.922(760) 22240.010(760)
0.067305(52) 0.066920(09) 0.066520(39) 0.066077(21) 0.065611(32) 0.06529a 0.064751(23) 0.064311(10) 0.063554(62) 0.063011(47) 0.062715(32) 0.062198(51) 0.062007(36) 0.060875(42) 0.060251(48) 0.059345(87) 0.058748(52) 0.057988(45) 0.057183(51) 0.056611(20) 0.055400(66) 0.054538(47) 0.053533(48) 0.052709(27) 0.051404(59) 0.050149(83) 0.048760(55) 0.047415(94) 0.046005(98) 0.044237(120) 0.042291(140) 0.040768(200) 0.038708(200) 0.036180(220) 0.036064(180) 0.034769(190) 0.043471(1100) 0.003500(100) 0.010850(600) 0.022546(95) 0.020039(120) 0.01761(130) 0.01490(130)
1.703(10) 1.781(7) 1.743(82) 1.767(20) 1.746(28) 1.954a 1.881(36) 1.954(12) 0.455(23) 1.568(130) 1.979(69) 2.066(150) 3.220(110) 2.151(93) 2.229(100) 1.350(270) 2.356(140) 2.310(110) 2.405(130) 3.116(21) 2.727(220) 3.006(130) 3.091(130) 3.929(31} 3.572(160) 3.829(220) 3.676(140) 4.025(280) 4.480(290) 4.052(240) 3.545(400) 4.696(880) 4.521(870) 2.826(960) 13.04(50) 18.69(54) 88.28(820) 248.1(450) 132.7(510) 13.02a 14.70a 17.10a 20.84a
1012 Hv
15.9(23) 2.675(930) 1.184(400) 1.31(110) 6.309(820) 1.141(570) 1.445(640) 10.27(220) 1.27(100) 3.089(700) 3.332(880) 2.800(180) 2.681(940) 4.023(1000) 5.22(120) 5.29(170) 10.44(110) 11.90(220) 13.55(230) 24.66(270) 38.75(300) 39.90(1100) 57.97(1100) 102.5(130)
1236(160) 8174(1200) 4470(1500) 93.55a 140.9a 232.5a 428.8a
Superscript a indicates a value calculated from the potential energy curve, constrained in the fit. Following [6], we have taken the f parity sublevels as a reference, and have used K doubling parameters q = 1.57 105 cm1, qv1 = 5.15 108, qv2 = 4.665 109, and qD = 3.069 1010 in Eq. (1).
A normal weighting scheme ð1=r2i Þ was used, and the fit (1873 data, 130 parameters) returned a dimensionless error of 0.90. The resulting parameters describing levels 0–40 of the X state are listed in Table 2. These were then constrained in a fit of D–X data (dispersed fluorescence plus polarisation labelling measurements) to a set of band constants. The K-doubling in the D 1P state was less well-defined in this work than in the systematic investigations of Tamanis et al. [6], so lambda-doubling parameters were constrained to the values supplied in Ref. [6]. They were assumed to extrapolate reasonably to higher vibrational levels; qv does decrease with increasing vibrational quantum number, as the ab initio predictions of Adamson et al. [7] suggest they should (no quantitative predictions were given in the ab initio work). Following the convention of Ref. [6], D state energies were calculated using the f parity levels as a reference, with e parity levels shifted by Dm, where
1 Dm ¼ q½JðJ þ 1Þ þ qv1 t þ ½JðJ þ 1Þ 2 2 1 2 ½JðJ þ 1Þ þ qd ½JðJ þ 1Þ þ qv2 t þ 2
ð2Þ
A. Adohi-Krou et al. / Journal of Molecular Spectroscopy 250 (2008) 27–32
Once again, obvious line/extra-line pairs were eliminated from the fit. Parameters were not determined for v = 5, because no regular patterns emerged at all for this level. A robustly weighted fit to 1353 data gave a dimensionless error 0.75 and left 158 lines outside twice their estimated uncertainties. The worst outliers form systematic patterns within some vibrational levels (see levels v = 28– 35 for example, in the supplementary material). The band constants thus obtained are listed in Table 3. The distortion constants, Dv and Hv, are clearly ‘fitting parameters’, whose behaviour is quite erratic in some cases. The vibrational intervals, too, are slightly irregular even at the bottom of the potential. Unsurprisingly, a Dunham-type polynomial does not reproduce the data set at all well. 6. Conclusion This work provides a description of most of the vibrational levels of the D 1P state of NaK at rotational resolution. It shows, in sharp contrast to the analogous state of NaRb, studied by Docenko et al. [22], that a standard set of spectroscopic parameters cannot describe the D 1P state of NaK to within experimental uncertainty. The reason for this seems to be the relative positions of the perturbing triplet state. In NaRb, the Hund’s case (a) calculations of Zaitsevskii et al. [25] show that the triplet state is more deeply bound than the singlet, and that the outer limbs of the d 3P and D 1P states remain almost parallel, converging only close to the atomic asymptote. In the case of NaK, the singlet state has the deeper potential well, and the outer limbs of the d 3P and D 1P states cross at about 4.7 Å. The low vibrational levels of the D 1P state of NaK are severely affected by this crossing. Working up the potential well, the perturbations are again strong, but here vibrational overlap becomes very good because the inner walls of the two states are close to one another. If our analytical potential for the singlet state is reasonable, the spin-orbit coupling effects detailed by Pazyuk and co-workers in Ref. [8] should allow us to give good predictions for the severely mixed energy levels of the D,d complex, notably for those observed as line/extra-line pairs, which were excluded from the one-channel data treatment described above. The quality of the partly deperturbed potential curve we propose for the D 1P state, UD(R) was investigated by calculating the expected spin-orbit coupling with nearby d 3P, using the direct channel-coupling (CC) approach. The non-zero elements of the parity-independent 4 4 potential energy matrix Ve/f(R) (adapted from [8] are:
h1 PjHj1 Pi ¼ U D1 P ðRÞ þ BðRÞðx þ 2Þ 3
h3 P0 jHj3 P0 i ¼ U d3 P ðRÞ ASO ðRÞ þ BðRÞðx þ 2Þ h3 P2 jHj3 P2 i ¼ U d3 P ðRÞ þ ASO ðRÞ þ BðRÞðx 2Þ pffiffiffiffiffiffi h3 P0 jHj3 P1 i ¼ BðRÞ 2x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h3 P1 jHj3 P2 i ¼ BðRÞ 2ðx 2Þ 1
3
SO
h PjHj P1 i ¼ n ðRÞ
The results of this coupled channels calculation are at present valid only for bound levels, located below lower dissociation limit of the d 3P state, Na(3P1/2) + K(4S1/2). Although interactions with the other electronic states dissociating to the same Na(3p) + K(4s) atomic asymptote have been ignored, the polarisation labelling measurements for energy levels below 22230 cm1 (including line/extra-line pairs) are recalculated with an unweighted root-mean-square deviation of 0.14 cm1. The majority of poorly-reproduced energies lie above 22200 cm1, where the triplet state is accepted to be least accurate. There is insufficient information in our current set of observations to derive an improved triplet potential curve, plus diagonal and off-diagonal spin-orbit matrix elements (assuming that a two-state model would suffice, which is probably another over-simplification). Nevertheless, we believe that the potential energy curve we have obtained for the D 1P state supplies a reliable starting point for any future work. When more data become available for the triplet state(s), it may be possible to attempt a more refined treatment, either locally, as performed by Eckel et al. [27] for the higher-lying 33P 31P complex in NaK, or to extend the global picture [8] developed at Moscow State University. Acknowledgments This work was partially supported by a grant from the Polish Committee for Scientific Research, No. N202 103 31/0753. We are also grateful for financial support for travel (France/Poland bilateral exchange programmes PAN/CNRS and Polonium). A.V.S. acknowledges support from the Russian Foundation for Basic Research, Grant N06-03-32330. Appendix A. Supplementary data Supplementary data for this article are available on ScienceDirect (www.sciencedirect.com) and as part of the Ohio State University Molecular Spectroscopy Archives (http://msa.lib.ohiostate.edu/jmsa_hp.htm). Supplementary data associated with this article can be found, in the online version, at doi:10.1016/ j.jms.2008.04.003. References [1] [2] [3] [4] [5] [6]
3
h P1 jHj P1 i ¼ U d3 P ðRÞ þ BðRÞðx þ 2Þ
[7] [8]
ð3Þ
where x = J(J + 1), and B(R) = ⁄2/2lR2. The adiabatic potential energy curve for the triplet state Ud3 P (R) is now defined by a hybrid RKR/ab initio potential. The lowest part corresponds to the deperturbed RKR turning points listed in Ref. [8], with a smooth extension to large internuclear distance supplied by recent calculations of Aymar and Dulieu [26], since observations now extend to high vibrational levels. The spin-orbit splitting function of the triplet state, ASO(R), and the off-diagonal spin-orbit coupling function between the singlet and triplet state, nSO(R), were obtained (for R = 2–20 Å) from a large-scale multi-reference configuration interaction calculation, working with effective small-core (9 valence electrons) non-empirical relativistic pseudopotentials.
31
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