Spin orbit interactions, new spectral data, and deperturbation of the coupled b 3 u and A 1 u states of K 2

Transcription

1 JOURNAL OF CHEMICAL PHYSICS VOLUME 117, NUMBER DECEMBER 2002 Spin orbit interactions, new spectral data, and deperturbation of the coupled b 3 u and A 1 u states of K 2 M. R. Manaa University of California, Lawrence Livermore National Laboratory, Energetic Materials Center, L-282, Livermore, California A. J. Ross, F. Martin, and P. Crozet Laboratoire de Spectrométrie Ionique et Moléculaire (UMR 5579 CNRS), Bâtiment Kastler, Université Lyon I, Domaine Scientifique de la Doua, Villeurbanne Cedex, France A. M. Lyyra Department of Physics, Temple University, Philadelphia, Pennsylvania Li Li Department of Physics and Key Lab of Atomic and Molecular Nanoscience, Tsinghua University, Beijing 10084, China C. Amiot Laboratoire Aimé Cotton, CNRS II, Bâtiment 505, Campus d Orsay Cedex, France T. Bergeman Department of Physics and Astronomy, State University of New York, Stony Brook, New York Received 19 August 2002; accepted 30 September 2002 We report calculations of the spin orbit energy as a function of internuclear distance R within the b 3 u state of K 2, and between the b 3 0u and A 1 u states, together with new spectroscopic data on the b state and previously unpublished data on the A state. Both the new data and previous data are fitted to Hamiltonian parameters using the discrete variable representation DVR method. The DVR matrix includes nonrelativistic Born Oppenheimer potentials and spin orbit interactions, which are scaled to match the known asymptotic limits and to best fit the experimental data. We report fitted Dunham coefficients that yield the A and b state potentials by means of the Rydberg Klein Rees method. These parameters thus take into account second-order spin orbit perturbation shifts from the vibrational levels of these two states which are normally not considered in band-by-band fits to spectroscopic data American Institute of Physics. DOI: / I. INTRODUCTION Photoassociation of alkali atom dimers is an important first step in many cold atom and thermal atom experiments. It can lead to decay into bound or continuum ground singlet or triplet states or can be the first step in multiphoton spectroscopy. Often, the first photoassociation step is to states below the lowest S P excitation threshold. Transitions from the X 1 g component of the wave function of two colliding atoms, as well as from the X state in traditional spectroscopy, will excite the u ungerade manifold of states. It is therefore useful to have accurate information on ungerade states below the lowest S P threshold namely, the states normally designated as A 1 u and b 3 u. In this study, we report ab initio calculations of the spin orbit terms both diagonal within the b state and coupling the A and b states, and we present new spectroscopic data on perturbed levels of the b state of K 2 and previously unpublished data on the A state. The data are fit to eigenvalues of a discrete variable representation DVR matrix which incorporates Rydberg Klein Rees RKR potential curves calculated from fitted Dunham coefficients. The spectroscopy of the A 1 u and b 3 u states has been pursued to a varying extent in the various homonuclear alkali dimers. A few comparisons are interesting. For Li 2, consistent with the 2 2 P atomic fine structure splitting of only cm 1, only a few isolated levels in the A state have sufficient triplet character to serve as window levels, through which higher triplet states can be excited. 1 In the earlier work on the Li 2 A and b states 2 and in the most recent analyses, 3,4 it has not been necessary to consider the mutual perturbations. In Na 2 Na 3 2 P fine structure splitting: cm 1, perturbations of the X A magnetic rotation spectrum 5 were identified as perturbations between the A 1 u and b 3 0u states in Refs A b perturbations were later discussed by Refs. 9 and 10. Intercombination bands have been studied by laser-induced fluorescence Perturbation-facilitated triple resonance 17 from higher triplet levels populated through perturbed A-state levels has yielded extensive data, so that coupling elements have been extracted over many vibronic levels. 18 The A-state potential energy curve was extended to R 10.7 Å based on data presented in the thesis of Chawla 19 and to R 40.5 Å by Tiemann et al., 20 who used a Franck Condon pumping scheme between X and A states. A-state Dunham coefficients based on /2002/117(24)/11208/8/$ American Institute of Physics

2 J. Chem. Phys., Vol. 117, No. 24, 22 December 2002 b 3 u and A 1 u states unperturbed levels are given in Ref. 21. A complete deperturbation analysis of the A b system that includes high resolution polarization spectroscopy data for the A state, v 0 60, is in progress. 22 For potassium K 4 2 P fine structure splitting: cm 1, previous studies 14,23 27 have provided reasonably accurate values for the spin orbit coupling parameters and for the Dunham coefficients, and provide the foundation for the present study. For Rb 2,(5 2 P fine structure splitting: cm 1 Amiot et al. 28 have reported experimental data with considerable irregularity due to the A b perturbations. From theoretical modeling, 28 regularities have been extracted. For the severe perturbations in both Rb 2 and Cs 2, there recently have been penetrating theoretical studies 29,30 of the interactions analogous to those we study here in K 2. Predissociations in Rb 2 Ref. 31 have been quantitatively modeled in Ref. 30. One might also mention studies of analogous states in the heteronuclear alkali dimers. For NaK, the A b perturbation has been studied in Ref. 32, and the b-state fine structure in Ref. 33. Perturbation effects in NaK have been reviewed in Ref. 34. Ab initio calculations of spin orbit elements for NaK have been presented in this reference 34 and also by one of us. 35 One motivation for the present study was the need for better data to model the predissociation effects in the K 2 0 u state near the 4 2 S 4 2 P 3/2 dissociation limit. 37,36 Uncertainties in the potentials limited the accuracy of calculated predissociation widths. To extract potential information comparable to the precision of the spectroscopic data, it is necessary to consider second-order perturbation shifts, which are normally not included in band-by-band data analyses. Perturbative interactions between nearly degenerate levels reflect the off-diagonal spin orbit interaction od (R x ), where R is the internuclear distance and R x is the potential crossing point. However, second-order terms sample od (R) at R values other than R x. Previously, the R dependence of od had been estimated from spectroscopic data. 27 The scaled ab initio calculations of od (R) b 3 0u H so (R) A 1 u, as well as of d (R) b 3 u H so (R) b 3 u, provide additional information on the second-order perturbation shifts and provide more accurate G(v) values. Another limitation on the previous studies of the A b interactions in K 2 was that for b 3 u state levels below v 13, only a few term values had been measured under relatively low resolution, 25 and more extensive precision spectroscopic data were not available. This deficiency is remedied in the present work, which presents new spectroscopic data on b 3 0u levels from v 0 to 24 and also on higher levels, 64 v 118. This information, together with the spin orbit functions and a method for computing the perturbation interactions to all orders, permits more accurate Dunham 38 coefficients to be obtained for both the A and b states of K 2 in this report, the least-squares fit to Dunham parameters extends only to v 90 of the b state. From the Dunham coefficients, accurate potentials may be obtained by widely available RKR Refs. 39 and 40 computer programs. Term values for the model Hamiltonian are calculated by the discrete variable representation. 41,42 Like the Fourier grid Hamiltonian FGH approach used in studies 28 30,43 on Rb 2 and Cs 2 and in a previous study of K 2, 27 DVR provides an accurate representation of the kinetic energy term of the Hamiltonian. Given an appropriate mesh of grid points, the DVR procedure yields all eigenvalues of all coupled states from the diagonalization of a single matrix for each J value. Similar methods have been used also in a study of perturbed states of LiK. 44 Another report is being prepared 45 that will extend the analysis to levels of higher energy. In this report, we present the molecular Hamiltonian in Sec. II. Section III discusses the methods used for calculating the spin orbit interactions and scaling these functions from fits to experimental data. The new spectroscopic data is outlined in Sec. IV. Section V describes the method for calculating term values and performing the fits to Dunham coefficients. Section VI presents results and Sec. VII a discussion of further work in progress. II. MOLECULAR HAMILTONIAN This study of interactions between the A 1 u and b 3 u states of K 2 is limited to observations of levels of parity ( 1) J, the Kronig e parity levels. The Kronig f levels, parity ( 1) J, contain no A-state component and hence do not exhibit the perturbation effects. Furthermore, in view of parity and spin selection rules and intensity distributions, data on the f parity levels were effectively not present in any of the data sets used here. The molecular Hamiltonian includes elements for radial kinetic energy H K, nuclear rotation H rot, and potential energy including spin orbit effects, H V : H R H K H V R H rot R, where R is the internuclear distance. For the Kronig e parity levels studied here, the matrix elements of H V (R) H rot (R) are: 1 1 u 3 0u 3 1u 3 2u V 1 u x 2 B od 0 0 od V 3 u d x 2 B 2xB 0 0 2xB V 3 2 u x 2 B 2 x 2 B x 2 B V 3 u x 2 B d,

3 11210 J. Chem. Phys., Vol. 117, No. 24, 22 December 2002 Manaa et al. FIG. 1. Nonrelativistic Born Oppenheimer K 2 potentials for the A 1 u and b 3 u states, as obtained from the fits to term values in this work. The short dashed lines and numbers indicate vibrational levels. The potential for the B 1 u state is calculated from data given in Ref. 46. The inset shows adiabatic potential functions eigenvalues of H V as a function of R) including the off-diagonal spin orbit interaction, in the vicinity of the avoided crossing between the A 1 u and b 3 0u states. Note that the labels reflect the state character on the left of the avoided crossing. where x J(J 1). In the above, V( 1 u ), V( 3 u ), d, od, and B B(R) 2 /2 R 2 are functions of R ( is the reduced mass. Often in analyzing spectra, one considers expectation values of these quantities over vibronic wave functions, such that the potential functions labeled V in Eq. 2 become G(v) and B(R) becomes B(v). In the DVR procedure, the R-dependent functions enter into the Hamiltonian matrices for given values of J. The kinetic energy Hamiltonian term H K p R 2 /2 (p R is the radial momentum is diagonal in the 2S 1 basis, which is used to write the above matrix and also used for the DVR calculations. To establish the context, we show in Fig. 1 the Born Oppenheimer nonrelativistic potentials V(A 1 u ) and V(b 3 u ) obtained in the least squares fits to term values discussed below. The analogous potential for the B 1 u state, from parameters given in Ref. 46, is also shown. The inset shows the effect of relativistic spin orbit terms that lead to separate potentials for A 1 u, b 3 0u, b 3 1u, and b 3 2u. These coupled potentials are shown near the crossing point, also from parameters derived below. There are spin orbit elements also that couple 3 1u with 1 u and 3 u, which lie higher in energy. According to results from a larger matrix that includes these additional states, these coupling terms introduce shifts of 3 1u levels from 0.07 cm 1 in low v levels of the b state, to 0.2 cm 1 at the upper range of our data. These estimated shifts have not been introduced as corrections to d (R) here. III. AB INITIO CALCULATIONS OF THE SPIN ORBIT INTERACTIONS The zeroth-order Born Oppenheimer wave functions of the b 3 u and A 1 u states were developed from state-averaged multi-configuration self-consistent field SA-MCSF Refs. 47 and 48 configuration interaction CI calculations. The CI description was based on core orbitals FIG. 2. Ab initio spin orbit functions black dots, together with a numerical interpolation short dashed line and the adjusted functions solid line, long dashed line with the experimental asymptotic limit and with a fitted y(r) function see text to agree with the term value data. The points with error bars are the values at R e for d,in a and at the potential crossing point for od,in b, quoted from Ref. 26. consisting of the 1s 3s and 2p orbitals of each potassium atom, while including all single and double excitations from the 14 electrons that were distributed in two active spaces. The first active subspace included the two sets of 3p orbitals, while the second consisted of the 4s and 4p orbitals of each atom. The 12 orbitals with the highest energies were truncated. In D 2h symmetry, the CI description resulted in configuration state function CSF Ref. 49 spaces of dimension and for the 3 u and 1 u states, respectively. The molecular orbitals were determined from a complete active-space SA-MCSCF procedure that consisted of two electrons being distributed in the combined two sets of the 4s and 4p orbitals, and in which three states with equal weight vector of amplitude one were averaged. The molecular orbitals were expanded in terms of the contracted Gaussian basis set 15s12p2d/9s8p2d developed by Partridge et al. 50 Finally, the spin orbit coupling matrix elements 3 u H so 3 u and 3 0u H so 1 u were calculated within the full microscopic Breit Pauli approximation, where both spin orbit and spin other orbit contributions are included. 51 The geometry dependence of the spin orbit induced matrix elements with respect to internuclear distance R is clearly shown in Fig. 2. This variation is attributed to the change in the mixing of atomic orbitals and to the different relative importance of certain CSFs in the CI wave function. We note that an almost identical variation and magnitude of these interactions was noted in recent studies on the NaK

4 J. Chem. Phys., Vol. 117, No. 24, 22 December 2002 b 3 u and A 1 u states TABLE I. Spin orbit matrix elements in K 2. Columns marked a give the results of the ab initio calculations; columns marked b give values that have been adjusted to be consistent with the known asymptotic values and with the term values, via least squares fitting of the scaling function parameters. d (R) b 3 u H so (R) b 3 u od (R) b 3 0u H so (R) A 1 u R(a 0 ) a b a b system. 34,35 Furthermore, a similar variation with R was obtained for the off-diagonal spin orbit function for Cs At infinite interatomic separation, the b 3 u and A 1 u states correlate with the K(4 2 S) K(4 2 P J ) asymptote, for which the well known fine-structure splitting, 2 P 3/2 2 P 1/2 is cm 1. In comparison, our calculations at R 30a 0 determine a splitting of 2 P 3/2 2 P 1/2 3 3 u H so 3 u cm 1. Thus our results are 24% too small. The major contribution to this error is the neglect of second-order spin orbit effects, which in principle can be determined for the interaction in the matrix of Eq. 2 via a Van Vleck transformation analysis. 52 This, however, requires much more exhaustive calculations which will be reserved for future investigations. Suffice it to note that the second-order correction in the 3 2 state of NaK improved the K fine-structure splitting to 56.7 cm To bring the ab initio functions into agreement with experimental data where available, we multiplied the two ab initio calc,i (R) functions by a scaling function: i (R) y(r) calc,i (R). As discussed above, the asymptotic values must be consistent with the atomic fine-structure interval: d ( ) /3, while od ( ) (2) /3. Also, the values in the bound part of the potentials must be consistent with observed molecular perturbations. The scaling function used in the present work was chosen to be of the form y(r) c 1 c 2 exp( c 3 R). The parameter c 1 in y(r) is such as to achieve the correct value at R. c 3 was chosen so that the term in c 2 was reduced to at R 16 Å, the 4s 4p LeRoy radius. c 2 was adjusted in the least-squares fit to A- and b-state term values. Table I gives the values directly obtained from the ab initio calculations and also for the same values of R, the adjusted values. Figure 2 presents plots of the diagonal and off-diagonal spin orbit functions, a numerical interpolation, and the adjusted functions. For the offdiagonal element, a second adjusted function is shown, in which two exponential terms with varying decay lengths were fitted, simply to indicate the uncertainty in the scaling factor. In fact, parameters for two exponentials in y(r) could not be fitted in a stable way, but fluctuated as the data set was varied. IV. EXPERIMENTAL DATA In this work, we attempted to compile all spectroscopic data previously obtained for the K 2 A and b states together with data newly obtained. Only data from 39 K dimers will be used here, however. In the following, the zero of the energy scale, denoted by E, will be the minimum of the X 1 g state. The available data sets consist of 1 unpublished line measurements used in Ref. 23 approximately 320 term values in the range E cm 1 ; term values from E cm 1, obtained by stimulated emission from the (5d) 1 g state, published in Ref. 26; 3 66 unpublished term values from E cm 1 to cm 1 used for the RKR potentials reported in Ref. 24; 4 70 unpublished term values of perturbed levels of the b 3 0u state between and cm 1, 53 also from stimulated emission from the (5d) 1 g state; 5 term values from 6 unpublished spectral lines given in the thesis of Jong other such term values were not compatible with other data available to us ; 6 previously unpublished data on 330 in general, perturbed levels of the A state, from E to cm 1, obtained in connection with precision measurements of the K 2 X 1 g ground state by Fourier transform spectroscopy; 55, term values of the A and b states in the range E cm 1, 27 obtained by laser excitation of a beam of K 2 molecules; and 8 new data described below. There is some duplication in the preceding term value counts. A schematic summary of the available data is shown in Fig. 3. The experimental data sets listed above all show evidence of the abundant spin orbit perturbations in low-lying levels of the A 1 u state. Our recent experiments aimed to exploit some of these gateways in order to access triplet gerade states of K 2, which fluoresce to the b 3 u state. In principle, this closely resembles work performed on the lithium dimer, 4 but with one major restriction. With only one tunable laser, we required a chance coincidence allowing us to excite X 1 g A 1 u /b 3 u 2 3 g with a single color double resonance. Having scanned about 1000 cm 1 with a singlemode Ti:sapphire laser, we found one transition strong enough to allow fluorescence to be recorded at high resolution 0.06 cm 1 on a Fourier spectrometer, namely, cm 1, which excites the P(67) line of the 16-1 band of the A X system, followed by the R(66) line from b 3 0u,v 26 to an undetermined v of 2 3 g. Bright yellow fluorescence occurred to the continuum of a 3 u, while Fourier transform spectra revealed two pieces of a progression of the 2 3 g b 3 0u system, beginning around cm 1,atv 0, and terminating at 7200 cm 1. The two pieces were unfortunately separated by very strong A X emission, which completely dominated the region cm 1, so that no measurements could be made for many

5 11212 J. Chem. Phys., Vol. 117, No. 24, 22 December 2002 Manaa et al. TABLE II. Dunham coefficients for the K 2 A 1 u and b 3 u states. All data are in cm 1. The figure in square brackets indicates the power of ten by which the previous figure is to be multiplied, and the figures in parentheses are the uncertainties of the last digit. These parameters apply to b-state energies up to cm 1 and A-state energies up to cm 1, the limit of the A-state data. They are to be used to generate RKR potentials, as stated in the text. The digits given are needed to reproduce the fit results. Due to correlation between parameters, the uncertainty limit quoted by the fitting program in many cases is comparable to the parameter value. Therefore uncertainties are given only for the low order parameters for comparison with previous results. Y ij A 1 u b 3 u Previous work Ref. 26 T e Y 10 ( e ) Y 20 ( e x e ) Y 30 ( e y e ) Y 01 (B e ) Y 11 ( e ) FIG. 3. Summary of the experimental data available for this study. Solid dots denote term values for which the major component was the indicated state (b 3 1u, b 3 0u, or A 1 u ). Crosses in a and b denote term values such that the fraction of the indicated state was at least intermediate vibrational levels. The triplet emission in the cm 1 region was severely affected by spinorbit interactions with A 1 u, while the infrared part of the spectrum was much less perturbed. V. TERM VALUE CALCULATION AND FITTING PROCEDURE Data arising from transitions to the ground state were converted to excited-state term values using ground-state E(v,J) term values from previously determined parameters for the X state. 55,56 Data obtained from transitions from the 2 3 g state were introduced in the form of combination differences. The experimental term values or combination differences were introduced into a least-squares fit to Dunham constants for the A and b states and to parameters in the spin orbit scaling functions. The Dunham coefficients Table II are not fit directly to the data, but indirectly via RKR potentials and DVR eigenvalues. The four-channel, coupled-potential DVR matrix consists of the elements given in Eq. 2 plus the kinetic energy matrix. The elements given in Eq. 2 are both diagonal and off-diagonal in the case a basis states, labeled by 2S 1, but diagonal in i and j, where i, j label the mesh points. The kinetic energy matrix is diagonal in 2S 1, but full in i and j, 41 thereby representing the Laplacian to high accuracy. For greater efficiency, we used the scaled variable y, such that R r 0 /y 2 r s, as in Ref. 42, so that the density of mesh points is greater for R near R e the potential minima than at large R, where the wavelength is greater. r 0 and r s may be chosen to optimize the scaling and were typically 0.8 and 3.0 Å, respectively. Depending on the maximum energy, from 160 to 350 mesh points were required for each of four channels. More mesh points are required as the maximum energy increases because the required range in R is greater and also because the distance between nodes for the most energetic wave function This work T e Y Y Y Y Y Y Y Y Y Y 10, Y 11, Y 12, Y 13, Y 14, Y Y Y Y Y decreases. See Ref. 43 for a similar but possibly more efficient procedure. To the extent that the mesh points are adequate, diagonalization of the DVR matrices generates all eigenvalues for a given J. For example, up to E cm 1, there are approximately 370 eigenvalues for the A b system for each J. Thus this approach requires that a large matrix be diagonalized for each value of J for which there are data. The advantage is that second-order perturbation shifts, including centrifugal distortion effects, are included without requiring fits to additional parameters beyond the Dunham Y i0 and Y i1 parameters. From the Dunham coefficients, the nonrelativistic Born Oppenheimer potentials V( 1 u ) and V( 3 u ) in Eq. 2 are calculated in the form of RKR turning points for each vibrational level and are then interpolated using six-point Lagrange interpolation. Typically five turning points beyond the limit of the fitted data were calculated by the RKR routine. Beyond this, the potential was extrapolated to smaller R by a function of the form A B/R n, where n was typically equal to 6. At larger R, the ab initio potentials of Magnier 57 were shifted to match the outermost RKR turning point and the long-range dispersion potential with current values 59 of

6 J. Chem. Phys., Vol. 117, No. 24, 22 December 2002 b 3 u and A 1 u states the C n coefficients, which were used beyond the so-called Le Roy radius. With this procedure, the DVR eigenvalues were not sensitive to details of the extrapolated portions of the potential. The fitting process started at low energies to fit loworder Dunham coefficients, then gradually worked up in energy steps of cm 1. It is essential to determine low order parameters accurately by fits to term values relatively low in each well before going to higher energies. As the energy of the uppermost level in the fit increases, the number of required Dunham coefficients increases. When the uncertainty in the highest-order Y i0 or Y i1 parameter becomes less than some arbitrary fraction typically chosen to be 0.1 of this parameter, then the next-higher-order parameter is added into the fit. For each pass of the fit, term values for approximately 130 J values in the range J in the available data were calculated by matrix diagonalization. At the highest maximum energies, this procedure of fitting Dunham parameters via RKR potentials and the DVR matrix eigenvalues encountered convergence difficulties. As is well known, 38,58 the Dunham expansion about R R e diverges at R 0 and hence has a radius of convergence of R e. The outer turning point of v 90 of the b state does occur at 1.92R e, which is rather close to 2R e, where one might expect the Dunham expansion to diverge. However, in practice for single-channel problems the Dunham expansion has been used quite far beyond this point for example, in Ref. 59. The convergence difficulties in this work are more likely associated with the sparse data available at higher energies and with the task of computing numerical derivatives in the fitting procedure. These difficulties were much less severe below E cm 1, where A-state data are available. Despite attempts to optimize the parameter increments when computing derivatives, ultimately these problems and the resultant convergence difficulties limited our ability to fit highorder terms in the Dunham expansion. The energy maximum was thus limited to cm 1 in this work. Even up to this point, a fit with all required parameters simultaneously varied did not converge, but instead it was necessary to fit parameters in selected groups. Two sample energy regions are shown in Figs. 4 and 5. In Fig. 4, the data would be sufficient to extract G(v) and B(v) values as well as perturbation interaction terms for these levels of the A and b states, as was done in Refs. 23 and 26. By contrast, the more sparse data shown in Fig. 5 would make such an approach difficult. However, the data suffice to extract Dunham coefficients that effectively average over adjacent vibrational levels in this region of energy, taking into account the R-dependent spin orbit coupling term od (R). VI. RESULTS In this work, we have fit available data on the K 2 A state up to E cm 1 (v 88, and b states up to E cm 1 (v 93. There was one higher A-state data point, at E cm 1, but this point could not be accurately fit; the minimum residual was 0.9 cm 1 due to the absence of nearby A-state data. Beyond E cm 1, the Dunham- RKR representation of the b-state potential was unstable in the fitting procedure. Analytic functions for the potential, as FIG. 4. Solid dots show the experimental term values. The lines connect calculated term values for integral J). The new data on the b 3 0u state complements the older data on perturbed levels of the A 1 u state, which in this region is taken from Ref. 26. in Ref. 27, appear to provide more stable fits in the high energy regime. The upper limit of E cm 1 for the present fit to the b-state data is 935 cm 1 below the 4 2 S 4 2 P 1/2 threshold. Here 1802 term values or combination differences were used in the fit to 42 parameters, including one parameter each labeled c 2 above for scaling the spin orbit functions d (R) and od (R) in addition to the c 1 parameters, which were adjusted to give the correct asymptotic values. We have made unweighted and weighted nonlinear leastsquares fits to the term values and combination differences. For each, we were obliged to delete approximately 30 out of 1830 data points with residuals greater than 0.15 cm 1,in some cases from states in which the data were sparse and perturbing levels could not be accurately characterized. After removing these data points, the unweighted fit to the data set described above gave an rms residual of cm 1. For the weighted least-squares fit, we assumed nominal uncertainties 0 of cm 1 for data sets 1 5, cm 1 for sets 6 and 7, and cm 1 for the new data set 8, each according to the stated calibration and measurement FIG. 5. As in the previous figure, but for a higher energy, showing less complete data for perturbed levels of the A 1 u state from Refs. 55 and 56, plus new data on b 3 0u.

7 11214 J. Chem. Phys., Vol. 117, No. 24, 22 December 2002 Manaa et al. uncertainties. The weights are equal to 1/ 2 0.) The rms ratio of the residual to 0 was 1.73, rather than the ideal value of unity. This may reflect again that the data were in many cases too sparse to accurately characterize perturbation regions, that our model is slightly deficient, or that there are slight calibration errors in the various data sets. A more careful test of our Hamiltonian model would require data on b 3 u 1 and 2 levels, and on regions of sparse data indicated in Fig. 3. Including the B 1 u and (2) 3 u states in the Hamiltonian matrix might improve the model, but would require significantly more computational time. Residuals of up to 0.03 cm 1 in the low vibrational levels of the A state did not significantly diminish when a few additional Y ij parameters were added. The quality of the weighted least-squares fit is exhibited in an EPAPS file. 60 For each input term value, this file gives the J value, the calculated fitted term value, the residual, 0 for that particular data set, the eigenvalue fraction for the A 1 u, b 3 0u, and b 3 1u components, and the vibrational quantum number for the largest component. A second EPAPS file gives the deperturbed G(v) and B(v) parameters and the RKR turning points for the A- and b-state levels, and also the spin orbit functions on a finer, more extensive mesh than in Table I. The spin orbit scaling parameters c 1 to bring the ab initio calculated values into agreement with the known asymptotic limiting values were and for the diagonal and off-diagonal spin orbit functions, respectively. With c a 1 0 in both, the fitted values of c 2 were for d ( od ), with uncertainties of in each case. Fitted Dunham parameters from the weighted leastsquares fit are presented in Table II. We emphasize that since these parameters are used to produce RKR potential functions that are employed with the spin orbit functions in the DVR matrix, they include second-order perturbation effects within the A b manifold of vibrational levels. Thus to reproduce the observations or to use these parameters to calculate a term value of interest, one needs the spin orbit functions and RKR turning points, 60 rotational terms in the Hamiltonian as given in Eq. 2, and the DVR or FGH method for calculating eigenvalues for the coupled potentials problem. In principle, a good approximation would be to compute all the vibronic wave functions and eigenvalues for the four coupled components for the chosen J value, calculate the perturbation interactions, and then to calculate second-order perturbation sums. The number of digits presented for each parameter is such as to permit calculations to cm 1. Uncertainties stated by the fitting program are given for low-order parameters, but for high-order parameters, these uncertainties are large compared with the accuracy needed to reproduce the term value calculation and sometimes comparable to the magnitude of the parameters themselves. The potentials for the A and b states obtained from these Dunham coefficients have already been shown in Fig. 1. Also, the fitted spin orbit interactions obtained by fitting scaling functions for each of the ab initio H so (R) functions have been shown in Fig. 2 above. The Dunham-RKR approach as employed here makes possible a direct comparison of the present results with results from some of the previous 14,23 26 studies of the coupled A b system in K 2. The magnitude of the second-order shifts for the A state can be judged from the differences between the present and previous results, which were based on fits to results of Y ij parameters to G(v) and B(v) parameters obtained in band-by-band fits to rotational energies. The differences between present and previous values for T e and e for the b state may be attributed to the new data available on 0 v 24 b 3 0u levels. VII. CONCLUSIONS This study has three basic objectives. The first is to report calculations of the spin orbit functions d (R) b 3 u H so (R) b 3 u diagonal within the K 2 b state and od (R) b 3 0u H so (R) A 1 u, obtained by ab initio calculations in conjunction with fits to a scaling function using spectroscopic data. Both functions d (R) and od (R) exhibit notable dips as a function of R that could not easily be extracted from the experimental data alone. The fitted scaling functions preserve and, to some extent, corroborate these features. Second, we report new spectroscopic data on low and high vibrational levels of the K 2 b 3 0u state, 0 v 24, and 65 v 118, of which levels v 93 were used in the least-squares fits reported here. Third, we report an alternative method for analyzing and fitting perturbed spectroscopic data, using DVR matrices to calculate term values. In light of similar efforts elsewhere, the third point requires clarification. Our goals here were to corroborate and scale the ab initio H so (R) functions and to extract Dunham parameters for comparison with results of previous studies that fit vibrational and rotational constants for each vibronic state. In place of Dunham coefficients and RKR potentials, an analytic potential function, as in Refs. 61 or 27, can be used. Analytic potential functions require fewer fitted parameters and converge more easily when data at higher energies are included. Such an approach will be used in a separate report that carries the analysis of K 2 A- and b-state data to levels closer to the dissociation limit. 45 The gaps in our knowledge of the K 2 A and b states suggest possible directions of future work. In the compilation presented here, there are few data for b 3 1u and essentially none for b 3 2u levels. Over significant energy regions, data for A 1 u and b 3 0u levels are rather sparse, as indicated in Fig. 3. An accurate analysis of higher energy levels of this system would require that information on B 1 u and 3 u states be included in the Hamiltonian to obtain the correct fine structure at the dissociation limit as in Ref. 36. Some further progress can be achieved with the presently available data, possibly including data on energy levels near the dissociation limit. 37 Additional precision studies near the K S 4 2 P dissociation limit under way in the laboratory of Tiemann 62 will provide a useful complement to the information presented here. In summary, using new and previously unpublished spectroscopic data, we have shown that ab initio diagonal

8 J. Chem. Phys., Vol. 117, No. 24, 22 December 2002 b 3 u and A 1 u states and off-diagonal spin orbit functions can be used with DVR matrix computations of term values to extract accurate Dunham coefficients, hence accurate potentials, for perturbatively interacting diatomic states. The particular states studied here are of interest in the excitation of cold colliding K atoms. Accurate potentials and spin orbit coupling functions make possible improved estimates of transition strengths, lifetimes, and predissociation rates. ACKNOWLEDGMENTS The authors thank Ch. Lisdat, E. Tiemann, and W. C. Stwalley for valuable communications. The new Fourier transform data were obtained through collaboration between Temple and Lyon I Universities, with financial support from an NSF-CNRS grant. The participation of L. Li was made possible by funds from the Lagerqvist Research Fund of Temple University. The work at Stony Brook was supported by ONR. 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