The Perturbation of the B1Π and C1Σ+ States in KLi

Journal of Molecular Spectroscopy 209, 50–56 (2001) doi:10.1006/jmsp.2001.8389, available online at http://www.idealibrary.com on

The Perturbation of the B 1 Π and C 1 Σ+ States in KLi W. Jastrzebski,∗,1 P. Kowalczyk,† and A. Pashov∗,2 ∗ Institute of Physics, Polish Academy of Sciences, Al.Lotnik´ow 32/46, 02-668 Warsaw, Poland; and †Institute of Experimental Physics, Warsaw University, ul. Ho˙za 69, 00-681 Warsaw, Poland Received February 20, 2001; in revised form June 13, 2001; published online August 30, 2001

Polarization labelling spectroscopy of the B 1 5 ← X1 6 + and C 1 6 + ← X1 6 + band systems in KLi reveals strong mutual perturbation of the B and C states. We show that both excited states can be described in a model comprising numerical IPA potentials for the unperturbed states and only 3 parameters to describe their interaction. Our approach allows to identify many C 2001 Academic Press previously unassigned spectral lines having significant deviation from the “regular” positions. ° Key Words: diatomic molecules; perturbations; inverted perturbation approach.

to the Fourier grid Hamiltonian (FGH) method (2) proposed recently to describe coupled electronic states, although until now the FGH method has not been applied to a case of states interacting via the L-uncoupling operator being of interest in the present case.

I. INTRODUCTION

Perturbations in spectra of diatomic molecules attract the continuing interest of researchers (see, e.g., Refs. (1–6) and references therein). They are often a nuisance, causing apparently irregular shifts of spectral lines, but, if properly analyzed, can also provide a wealth of information about states otherwise inaccessible in experiments. Perturbations are particularly common in the spectra of heteronuclear dimers, where the lower symmetry of the molecule enlarges the possibility of interaction between different molecular states beyond those present in homonuclear systems. As a result almost all the excited states of a given molecule may be perturbed. We experienced such situation in our recent studies of the low-excited B 1 5 and C 1 6 + states of KLi molecule (7, 8), where the perturbed spectrum has frustrated unambiguous rovibrational assignment of numerous observed lines. Some other lines of appreciable intensity, even though assigned, have been arbitrarily dropped from the analysis as not fitting to the expected spectral positions. Thus a considerable part of the spectrum has not been exploited. The present analysis was undertaken to obtain more accurate characteristics of the B and C states and their mutual perturbations. Such interacting systems can be analyzed in the framework of effective Hamiltonian, but they may also be described by numerical potentials or surfaces. The first method leads to a rather tedious level-by-level deperturbation. This paper presents advantages of a global approach to the problem: we show that the existing numerical potentials for the two investigated electronic states require only three parameters to describe many severe local perturbations. This allowed us to assign many more lines in the observed spectra and reproduced most of the experimental observations. Our approach may be considered complementary

II. SUMMARY OF EXPERIMENTAL OBSERVATIONS

The experimental set-ups and principles of the Dopplerfree polarization spectroscopy and the V-type polarization labelling spectroscopy (PLS) techniques used to record KLi spectra have been described in our previous papers (7, 9, 10) and will not be presented here. The measured line positions in the B 1 5 ← X1 6 + and C 1 6 + ← X1 6 + band systems are differences between term values of the upper and lower levels. Since the energies of the lowest vibrational levels in the ground state involved in the observed transitions were determined with very high precision (7), the energies of the levels belonging to the upper states could be calculated. As a result we had at our disposal eigenenergies of 1369 levels (0 ≤ v 0 ≤ 30, 2 ≤ J 0 ≤ 61) in the B 1 5 state and 230 levels (3 ≤ v ≤ 19, 4 ≤ J ≤ 55) in the C 1 6 + state (Table 1). III. MATRIX ELEMENTS OF THE L-UNCOUPLING OPERATOR

The electronic states of 1 6 + and 1 5 symmetry are mixed by the L-uncoupling operator, HL J = −

[1]

If the states are far apart, this interaction results in 3-doubling in the 1 5 state, but if they are close together, local rotational perturbations appear in both states.

1

E-mail: [email protected]. On leave from the Institute for Scientific Research in Telecommunications, ul.Hajdushka poliana 8, 1612 Sofia, Bulgaria. 2

0022-2852/01 $35.00 C 2001 by Academic Press Copyright ° All rights of reproduction in any form reserved.

1 (J+ L− + J− L+ ). 2µR 2

50

51

MUTUALLY PERTURBED STATES IN KLi

TABLE 1 Identified Energy Levels of the B1 Π and the C 1 Σ+ Electronic States

The L-uncoupling operator is neglected when the radial Schr¨odinger equation for Hund’s coupling case (a) is derived (1). It has only off-diagonal matrix elements which differ from zero for electronic states with 1Ä = 13 = 1 and 1S = 0, with a selection rule for parities of rotational levels involved, e ↔ e and f ↔ f . For a perturbation between 1 6 + and 1 5 states, the e-parity levels of both states are coupled, while the f -parity levels of the 1 5 state remain unaffected. In addition, perturbations obey the general rule 1J = 0, since J2 commutes with total molecular Hamiltonian HTOT . The matrix elements of the L-uncoupling operator can be expressed in case (a) basis functions as (1) LJ 1 + LJ 1 | 5ev 0 J i H56vv 0 J = h 6 v J |H p ≈ −Bvv0 J ηel J (J + 1),

[2]

where Bvv0 J =

¿ ¯ ¯ À ¯ 1 ¯ 1 v J ¯¯ 2 ¯¯ v 0 J 2µ R

[3]

and ηel represents that part of the matrix element which involves only electronic wave functions; to the first approximation it may be treated as R-independent. If the two interacting states are sufficiently separated in energy, the effect of the H L J operator can be treated by the second order perturbation theory. For example, the shift of an e-parity level of the 1 5 state E v50 J is given by δ E v50 J

=

¯ LJ ¯2 ¯ X ¯ H56vv 0J

E v50 J − E v6J Ã ! 2 X Bvv 0J 2 J (J + 1) = ηel 5 6 v E v0 J − E v J v

= qe (v 0 )J (J + 1),

[4]

where summation is performed over all states of 1 6 + symmetry. A similar expression for the energy shifts δ E v6J of the 1 6 + state levels may be written. However, it is difficult to distinguish

experimentally the effect of H L J in this case, because qe (v) is included in the effective rotational constant of the 1 6 + state, Beff = Bv + qe (v) (11). In the 1 5 state the correction [4] lifts degeneracy between the e and f components of each rotational level (3-doubling). In fact the formula [4] may be generalized to include the joint effect of all distant 1 6 ± states on the 1 5 state under consideration. The 1 6 − states contain only levels of f parity, and generate a correction term q f (v 0 )J (J + 1), but only the difference qe (v 0 ) − q f (v 0 ) can be determined experimentally (11). In practice, usually qe (v 0 ) and q f (v 0 ) are very small and can be treated as independent of v 0 . When the potential curves of the 1 6 + and 1 5 states cross, the LJ H operator may lead to local rotational perturbations in both states. If we neglect the influence of neighbouring electronic states and assume that only the H L J operator couples the states, the corrected energy levels can be obtained by diagonalization of H0 + H L J Hamiltonian matrix in the subspace limited to the 1 + 6 and 1 5 states in question. IV. IPA POTENTIAL ENERGY CURVES

The potential energy curves of the B and C states were calculated using our modification of the Inverted Perturbation Approach (IPA) method which is described in detail in Refs. (12, 13), so only the essential features are presented here. The method requires an approximate potential U0 (R) for the investigated state and calculates a correction δU (R) which allows the set of eigenvalues {E v J } obtained by solving the Schr¨odinger equation with U0 (R) + δU (R) to agree with the set of experimental eigenexp values {E v J } in the least squares approximation (LSA) sense. In our realization of the method, the correction to the potential and the potential itself are defined as sets of points connected with cubic spline function. This turns out to be a flexible way of description of various functions and the IPA method has been successfully applied even for electronic states with “exotic” shape of the potential energy curve (e.g., with two minima (14) and “shelf” region (15)). IV.1. The B1 5 State Initially the IPA program was used for construction of an accurate potential curve of the B 1 5 state over the whole region of the observed energies. Owing to the strong coupling between the B 1 5 and C 1 6 + states by the L-uncoupling operator it is not possible to describe the difference between the e- and f parity levels of the B state simply by a 3-doubling constant q, as has been done, for example, for the 31 5 state in NaK (12). The IPA potential curve was therefore calculated from only f -parity energy levels, which are not perturbed by the C 1 6 + state. The fitting procedure resulted in an IPA potential curve, listed in Table 2, which reproduces 358 f -parity levels corresponding to v 0 ≤ 18 with a standard deviation of 0.07 cm−1 for the PLS

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52

JASTRZEBSKI, KOWALCZYK, AND PASHOV

TABLE 2 The IPA Potential Curve Describing the f -Parity Energy Levels of the B1 Π State for v0 ≤ 18a

FIG. 2. A fragment of the polarization labeling spectrum of the B ← X system in KLi (obtained with linearly polarized λ = 565.91 nm probe laser). Perturbation of v 0 = 5, e-parity levels of the B 1 5 state by v = 6 levels of the C 1 6 + state causes doubling of the corresponding P and R rotational lines. The predicted unperturbed positions of lines are marked with asterisks.

a

For interpolation of the potential to an arbitrary middle point the natural cubic spline (21) should be used. The experimental level energies can be retrieved when solving the Schr¨odingera equation by the Numerov–Cooley method (22) with 0.003 A grid space.

data and 0.0057 cm−1 for the Doppler-free data (dimensionless standard error (16) σ¯ f = 0.88). For v 0 ≥ 18 some levels deviated significantly from their expected positions. This was the case, for instance, for levels with v 0 = 19, 20, which had to be almost totally excluded from the fit. The discrepancies probably originate from interactions between the B 1 5 state and

the neighboring b3 5 and c3 6 + states in KLi, pointed out in (17). Therefore we restricted the validity of our fit only up to v 0 = 18. It is worth noting that B 1 5 ∼ b3 5 and B 1 5 ∼ c3 6 + perturbations involving both e- and f -parity levels of the B state have been observed in the analogous NaK molecule (18–20). In the next step the IPA potential was employed to compute energies of e-parity levels in the B 1 5 state. Figure 1 illustrates typical shifts of the calculated level positions from the experimental ones, here for v 0 = 0. It can be clearly seen that rotational levels around J = 29 and J = 58 are affected by strong perturbation. The same perturbation manifests itself in the spectra by doubling of the rotational lines in regions where perturbation culminates (Fig. 2). IV.2. The C1 6 + State

FIG. 1. Shifts δ E of the experimental e-parity levels of the B 1 5 state in KLi for v 0 = 0 from their positions calculated with the IPA potential (open triangles). Dots present the corresponding shifts calculated in Section 5.

As in the case of e-parity levels in the B 1 5 state, the C 1 6 + state levels cannot be described simply by a potential curve, because of the strong B 1 5 ∼ C 1 6 + interaction. Far from the centers of perturbation the correction to the unperturbed energy levels (v, J ) may be approximated by q(v)J (J + 1) as in case of 3-doubling (see Section 3). The q(v) dependence, however, has to be explicitly taken into account, because the interacting states overlap in the energy scale. Nevertheless, within the accuracy of the PLS measurement, it has been possible to construct an effective potential curve, incorporating the q(v)J (J + 1) correction (Table 3). It reproduces the positions of 220 experimental energy levels of the C state with a standard deviation of 0.07 cm−1 . We excluded 10 experimental levels from the IPA fit, because of their large shifts; they were later taken into account in the deperturbation analysis.

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53

MUTUALLY PERTURBED STATES IN KLi

TABLE 3 The IPA Potential Curve of the C 1 Σ+ State for v ≤ 19 a

between the states under consideration (B and C) and the other states are eliminated. Therefore instead of diagonalizing the matrix including elements evaluated for basis functions of all states, we diagonalize rather a matrix limited to the subspace of the B and C states, with elements equal to (23) Hi j = hi|H0 | ji + hi|H L J | ji

£ ¤ X hi|H L J |kihk|H L J | ji E i0 + E 0j − 2E k0 ¡ ¢¡ ¢ + , 2 E i0 − E k0 E 0j − E k0 k6=i, j

a For interpolation of the potential to an arbitrary middle point the natural cubic spline (21) should be used. The experimental level energies can be retrieved when solving the Schr¨odingera equation by the Numerov–Cooley method (22) with 0.003 A grid space.

In Sections 4.1 and 4.2 IPA potential curves for the B 1 5 and C 6 + states in KLi were determined. Up to this point all the energy levels apparently affected by perturbations were excluded from consideration. In the next step we took into account the interaction between the B 1 5 and C 1 6 + states in order to reproduce positions of all the experimental energy levels, including those labeled as “perturbed.” The nuclear wavefunctions of the B 1 5 and C 1 6 + states can be calculated numerically from the IPA potentials. Our calculation has been conservatively limited to v 0 ≤ 15 e-parity levels levels of the B 1 5 state, which are sufficiently far from the energy region where perturbations by states other than C 1 6 + can occur; all the observed levels of the C 1 6 + state (v ≤ 19) have been included. If the L-uncoupling operator alone is responsible for the B 1 5 ∼ C 1 6 + interaction, true energy levels can be found by diagonalizing an energy matrix (written in a Hund’s case (a) basis) whose off-diagonal elements are those of the LLJ uncoupling operator, H56v 0 v J (see Eq. [2]). The only unknown LJ factor in H56v is η , because Bv0 v J may be calculated with 0v J el the wave functions obtained by solving the Schr¨odinger equation with the IPA potentials. Strictly speaking, the interactions with other states via the Luncoupling operator cannot be completely neglected. This operator mixes energy levels of the B 1 5 state also with those of other neighboring 1 6 + states (for example A1 6 + ), including energy levels of the C 1 6 + state above v = 19. If the above interactions are small, they can be incorporated by correcting the B ∼ C matrix elements by means of the Van Vleck transformation (1, 23). This transforms the original Hamiltonian H = H0 + H L J to a new Hamiltonian in which first-order interactions 1

where i and j stand for wave functions of the B or C states, while k denotes all other states. The energies E i0 are those calculated from H0 for each state, i.e., E i0 = hi|H0 |ii. In analogy to Section 3, the sum in Eq. [5] may be replaced by q(v1 , v2 )J (J + 1), where v1 and v2 are the vibrational numbers of states for which the corresponding matrix element is calculated. Because of the symmetry properties of the H L J operator (the selection rule 13 = ±1) this sum is different from zero only within the submatrices of the B or C states. In this way the matrix to be diagonalized for each J value can be written symbolically as E

V. DEPERTURBATION ANALYSIS

[5]

      

δ5

LJ

LJ

LJ

δ5

δ5 .. .

δ5

LJ

LJ

LJ

δ5

δ5

E 515 + δ5

LJ

LJ

LJ

LJ

LJ

LJ

E 60 + δ6

δ6

LJ

LJ

LJ

δ6

δ6 .. .

δ6

LJ

LJ

LJ

δ6

δ6

E 619 + δ6

50

+ δ5

    , [6]   

where X

δ5 =

k6=5v10 ,5v20

¡ ¢ h5v10 |H L J |kihk|H L J |5v20 i E 5v10 + E 5v20 − 2E k ¡ ¢¡ ¢ × 2 E 5v10 − E k E 5v20 − E k = q 5 (v10 , v20 )J (J + 1), X δ6 =

[7]

k6=6v1 ,6v2

×

h6v1 |H L J |kihk|H L J |6v2 i(E 6v1 + E 6v2 − 2E k ) ¡ ¢¡ ¢ 2 E 6v1 − E k E 6v2 − E k

= q 6 (v1 , v2 )J (J + 1), p L J = −ηel Bv0 v J J (J + 1),

[8] [9]

and E 5v0 , E 6v are the unperturbed energies of e-parity levels in the B and C states, respectively. The 5 and 6 indices stand for all quantum numbers describing the B and the C states except v 0 and v. Because the neighbouring states, considered in Eqs. [7] and [8], are quite remote from the energy levels of the B and C states

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54

JASTRZEBSKI, KOWALCZYK, AND PASHOV

taken into account in [6], the q 5 (v10 , v20 ) and q 6 (v1 , v2 ) coefficients are almost constant with only a very weak dependence on v 0 and v. It is therefore possible to approximate them with a finite power series in v 0 and v: X 0 p 0q q5 q 5 (v10 , v20 ) = pq v1 v2

izes H, namely E = T† HT,

E denoting the diagonal matrix of eigenvalues E icalc . ∂H/∂α j symbolizes the matrix whose elements are given by ¯ À ¿ ¯ ¯ ∂H ¯ 0 ¯ ¯ vJ , vJ ¯ ∂α j ¯

pq

= q 6 (v1 , v2 ) =

5 q00

X

+

5 0 q01 (v1

+

v20 )

+ ···

[10]

q6 pq v1 v2 p

q

pq 6 6 = q00 + q01 (v1 + v2 ) + · · ·

[11]

Here we take into account that the quantity q 5 (v10 , v20 ) is sym5 5 = q10 . The metric with respect to v10 and v20 (see Eq. [7]), so q01 6 same is valid also for q (v1 , v2 ). The rovibrational energy levels of the B 1 5 and C 1 6 + states can thus be described from: • the IPA potential curves which correspond to the unperturbed states; • an ηel constant, which represents the electronic part of the H L J operator; and 6 • a set of coefficients {q 5 pq } and {q pq } which describe the influence of all states other than B and C. The values of the unknown parameters {α j } (where {α j }, j = 6 1, 2, . . . stands for ηel , {q 5 pq }, and {q pq }) should minimize the difference between the experimental energy levels and those obtained by diagonalization of the Hamiltonian [6]. Since the eigenvalues E icalc of matrix [6] vary non-linearly with {α j }, an appropriate LSA technique is required (see, e.g., (24)). In such cases one usually starts from some trial values of the parameters {α 0j } and then improves them iteratively. By expanding the dependence of the eigenvalues on {α j } in a power series it is possible to write ¯ k X ∂ E icalc ¯¯ E icalc (α) ≈ E icalc (α 0 ) + · 1α j . [12] ∂α j ¯α0j j=1 Now the system of linear equations exp

Ei

= E icalc (α),

[13]

exp

where E i are the experimental energies, can be solved in the LSA sense; the most efficient is here the Singular Value Decomposition (SVD) technique (25). The procedure is repeated using the corrected parameters α 0j + 1α j as trial values until the merit function χ 2 stops decreasing significantly. For evaluation of the first derivatives in [12] the Hellman–Feynman theorem is applied, according to which · µ ¶ ¸ ∂ E icalc ∂H = T† T . ∂α j ∂α j ii

[16]

|v J i and |v 0 J i being the C and B basis functions. It follows that the corresponding matrices can be written symbolically as ! Ã √ 0 −Bv0 v J J (J + 1) ∂H , [17] = √ ∂ηel 0 −Bv0 v J J (J + 1) Ã ! J (J + 1) 0 ∂H = , [18] 5 ∂q00 0 0 ! Ã J (J + 1)(v10 + v20 ) 0 ∂H , [19] = 5 ∂q01 0 0 µ 0 ∂H = 6 0 ∂q00

¶ 0 , J (J + 1)

[20]

and so on. The steps of the fitting procedure are: 1. solution of the Schr¨odinger equation with the IPA potentials to determine unperturbed energy levels and wave functions; 2. calculation of the Bv0 v J integrals with the unperturbed wave functions; 3. diagonalization of the Hamiltonian matrix [6] with a trial set of parameters {α 0 } and determination of the transformation matrix T; 4. evaluation of the derivatives ∂ E icalc /∂α j using Eqs. [14] and [17]–[20]; 5. solution of the system of linear equations [13] in the LSA sense to obtain corrections 1α j to the trial values of {α 0j }; 6. calculation of new trial values of E icalc using parameters {α 0j + 1α j }. Steps (3)–(6) are iterated until the fit has converged. Applying this procedure to the B and C state energy levels in KLi we tried different combinations of parameters to describe the expansion of q 5 (v10 , v20 ) and q 6 (v1 , v2 ) and found that the only meaningful one was 5 5 0 + q01 (v1 + v20 ) q 5 (v10 , v20 ) = q00

q 6 (v1 , v2 ) = 0.

[14]

Here E icalc is the ith eigenvalue of the Hamiltonian H, α j is some parameter, and T is the transformation matrix which diagonal-

[15]

[21]

Adding more qi j parameters failed to improve the r.m.s. deviation of the fit; moreover the parameters were not statistically determined. A total of 678 e-parity experimental energy levels

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MUTUALLY PERTURBED STATES IN KLi

v0

55

FIG. 3. The calculated (dots) and experimental (open triangles) shifts δ E of the e-parity levels in the B 1 5 state of KLi from their unperturbed positions for = 1, 3, 4, 5, 14, 15. Note different vertical scales on the diagrams.

of the B 1 5 state and 223 of the C 1 6 + state were fitted. 27 levels from the B 1 5 state and 7 from the C 1 6 + state were excluded from the fit, because their deviation from the calculated positions exceeded twice the experimental error. The standard deviation of the fit with parameters listed in Table 4 amounts to 0.07 cm−1 . It should be mentioned that the present value for ηel = 0.741

is in good agreement with the value ηel = 0.77 obtained in (7 ) from deperturbation of v 0 = 0–2 levels observed at high resolution. Figures 1 and 3 compare calculated and experimental shifts of the B state energy levels from their unperturbed positions (obtained from the IPA potential).

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56

JASTRZEBSKI, KOWALCZYK, AND PASHOV

TABLE 4 Parameters of the Hamiltonian Used to Describe the B1 Π ∼ C 1 Σ+ Interaction in KLi

ACKNOWLEDGMENTS This work has been funded in part by Grant 2 P03B 067 16 from the Polish Committee for Scientific Research. The authors thank Dr. Amanda Ross for critically reading the manuscript and for valuable suggestions.

REFERENCES

VI. DISCUSSION

We have attempted to describe the observed energy levels of the B 1 5 and C 1 6 + states in KLi within a model that includes the IPA potential curves of the unperturbed states and some parameters taking into account the electronic part of the offdiagonal matrix elements of the H L J operator (ηel ) and possible 5 5 interaction with the neighbouring electronic states (q00 and q01 ). With the values of the parameters presented in Table 4 it was possible to reproduce the positions of 1259 energy levels of the B 1 5 (678 e parity levels corresponding to v 0 ≤ 15 and 358 f parity levels, v 0 ≤ 18) and C 1 6 + (223 levels, v ≤ 19) states with a standard deviation of 0.07 cm−1 . The model allowed us to identify previously unassigned lines in the experimental spectra having significant deviations from the expected “regular” positions. Some 10 experimental levels are still about 0.4 cm−1 from the calculated positions. Although their number is small compared with the total number of fitted levels, an extension of the applied model might still be necessary. For the first five vibrational levels of the B state, the model predicted the deviations of level energies from their unperturbed values correctly. It is worth noting that for v 0 = 0, 1, and 2 highly accurate energies of levels (from Ref. (7 )) were included in the fit. However, for some levels with v 0 ≥ 5, the model gave smaller deviations around the perturbation maxima than those observed experimentally (see for example v 0 = 5, 14, and 15 in Fig. 3), although perturbations were correctly localized. One possibility is that the assumption about weak dependence of the electronic factor ηel on R may not be valid. To overcome this problem, the ηel (R) dependence might be parameterized and the new parameters added to those which are to be determined in the fitting procedure. In addition, a more detailed description of perturbations between the B and the C states in KLi would require more abundant and accurate experimental data in order to construct better IPA potentials for the unperturbed states and to clarify the influence of the neighboring electronic states on the B 1 5 and the C 1 6 + states.

1. H. Lefebvre-Brion and R. W. Field, “Perturbations in the Spectra of Diatomic Molecules.” Academic Press, New York, 1986. 2. O. Dulieu and P. S. Julienne, J. Chem. Phys. 103, 60–66 (1995). 3. C. Amiot, O. Dulieu, and J. Verg`es, Phys. Rev. Lett. 83, 2316–2319 (1999). 4. P. Cacciani and V. Kokoouline, Phys. Rev. Lett. 84, 5296–5299 (2000). 5. Y. Kimura, H. Lefebvre-Brion, S. Kasahara, H. Katˆo, M. Baba, and R. Lefebvre, J. Chem. Phys. 113, 8637–8642 (2000). 6. Jianyuan Shang, Jianbing Qi, Li Li, and A. Marrjatta Lyyra, J. Mol. Spectrosc. 203, 255–263 (2000). 7. V. Bednarska, A. Ekers, P. Kowalczyk, and W. Jastrz¸ebski, J. Chem. Phys. 106, 6332–6337 (1997). 8. A. Pashov, W. Jastrz¸ebski, and P. Kowalczyk, Chem. Phys. Lett. 292, 615– 620 (1998). 9. W. Jastrz¸ebski and P. Kowalczyk, Phys. Rev. A 51, 1046–1051 (1995). 10. A. Pashov, I. Jackowska, W. Jastrz¸ebski, and P. Kowalczyk, Phys. Rev. A 58, 1048–1054 (1998). 11. G. Herzberg, “Molecular Spectra and Molecular Structure I. Spectra of Diatomic Molecules.” Van Nostrand, New York, 1955. 12. A. Pashov, W. Jastrz¸ebski, and P. Kowalczyk, Comput. Phys. Commun. 128, 622–634 (2000). 13. A. Pashov, W. Jastrz¸ebski, and P. Kowalczyk, IPA package. Program manual, 1999. Available on request from the CPC Program Library or from the authors. 14. I. Jackowska, W. Jastrz¸ebski, R. Ferber, O. Nikolayeva, and P. Kowalczyk, Mol. Phys. 89, 1719–1724 (1996). 15. A. Pashov, W. Jastrz¸ebski, and P. Kowalczyk, J. Chem. Phys. 113, 6624– 6628 (2000). 16. J. Y. Seto, R. J. Le Roy, J. Verg`es, and C. Amiot, J. Chem. Phys. 113, 3067–3076 (2000). 17. W. Jastrz¸ebski and P. Kowalczyk, Spectrochim. Acta Part A 54, 459–463 (1998). 18. P. Kowalczyk, J. Chem. Phys. 91, 2779–2789 (1989). 19. K. Ishikawa, T. Kumauchi, M. Baba, and H. Katˆo, J. Chem. Phys. 96, 6423– 6432 (1992). 20. R. Ferber, E. A. Pazyuk, A. V. Stolyarov, A. Zaitsevskii, P. Kowalczyk, H. Chen, H. Wang, and W. C. Stwalley, J. Chem. Phys. 112, 5740–5750 (2000). 21. C. De Boor, “A Practical Guide to Splines,” Applied Mathematical Sciences, Vol. 27. Springer-Verlag, New York, 1978. 22. J. W. Cooley, Math. Comput. 15, 363–374 (1961). 23. J. T. Hougen, “The Calculation of Rotational Energy Levels and Rotational Line Intensities in Diatomic Molecules.” U.S. Govt. Printing Office, Washington, D.C., 1970. 24. W. F. Rowe and E. B. Wilson, J Mol. Spectrosc. 56, 163–165 (1975). 25. W. H. Press, S. A. Teukolski, W. T. Vetterlingand, and B. P. Flannery, “Numerical Recipes in Fortran 77.” Cambridge Univ. Press, Cambridge, UK, 1992.

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