Fourier transform spectroscopy and direct potential fit

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Fourier transform spectroscopy and direct potential fit of a shelflike state: ..... material,36 depicted in Fig. 2, and ... 36) and compared with the results of other ap-.
THE JOURNAL OF CHEMICAL PHYSICS 134, 104307 (2011)

Fourier transform spectroscopy and direct potential fit of a shelflike state: Application to E (4)1  + KCs L. Busevica,1 I. Klincare,1 O. Nikolayeva,1 M. Tamanis,1 R. Ferber,1,a) V. V. Meshkov,2 E. A. Pazyuk,2 and A. V. Stolyarov2,b) 1 2

Laser Center, Department of Physics, University of Latvia, 19 Rainis Blvd., Riga LV-1586, Latvia Department of Chemistry, Moscow State University, 119991, GSP-2, Moscow, Leninskie gory 1/3, Russia

(Received 20 December 2010; accepted 9 February 2011; published online 14 March 2011) The paper presents high-resolution experimental study and a direct potential construction of a shelflike state E(4)1  + of the KCs molecule converging to K(42 S) + Cs(52 D) atomic limit; such data are of interest for selecting optical paths for producing and monitoring cold polar diatomics. The collisionally enhanced laser induced fluorescence (LIF) spectra corresponding to both spin-allowed E(4)1  + → X (1)1  + and spin-forbidden E(4)1  + → a(1)3  + transitions of KCs were recorded in visible region by Fourier transform spectrometer with resolution of 0.03 cm−1 . Overall about 1650 rovibronic term values of the E(4)1  + state of 39 K133 Cs and 41 K133 Cs isotopologues nonuniformly covering the energy range [16987, 18445] cm−1 above the minimum of the ground X -state were determined with the uncertainty of 0.01 cm−1 . Experimental data field is limited by vibrational levels v  ∈ [2, 74] with rotational quantum numbers J  ∈ [1, 188]. The closed analytical form for potential energy curve (PEC) based on Chebyshev polynomial expansion (CPE) was implemented to a direct potential fit (DPF) of the experimental term values of the most abundant 39 K133 Cs isotopologue. Besides analyticity, regularity, correct long-range behavior, and nice convergence properties, the CPE form demonstrated optimal balance on flexibility and constraint for the DPF of a shelflike state aggravated by a limited data set. The mass-invariant properties of the CPE PEC were tested by the prediction of rovibronic term values of the 41 K133 Cs isotopomer which coincided with their experimental counterparts with standard deviation of 0.0048 cm−1 . The CPE modeling is compared with the highly flexible pointwise inverted perturbation approach model, as well as with conventional Dunham analysis of restricted data set v  ≤ 50. Reliability of the empirical PEC is additionally confirmed by good agreement between the calculated and experimental relative intensity distributions in the long E(v  ) → X (v  ) LIF progressions. © 2011 American Institute of Physics. [doi:10.1063/1.3561318] I. INTRODUCTION

Research on cold and ultracold molecular gases had proved to be one of the frontier topics in atomic, molecular, and optical physics due to possible application in various areas stretching from ultracold chemistry to quantum computers and tests of standard model, see Refs. 1–3 for a review. As far as ultracold diatomic molecules are concerned, special emphasis has been put in recent years to polar species, in particular, to heteronuclear alkali diatomics. Such diatomics are rather easy producible from trapped ultracold alkali atoms, and they may possess considerable electric dipole moment values that facilitate their manipulation by external electric field. An accurate knowledge of excited electronic states is necessary to select optimal optical paths for obtaining and monitoring cold mixed alkali diatomics of interest. In particular, to exploit photoassociation for efficient production of these species, with further possibilities of their transformation to “absolute” rovibrational ground state with v  = J  = 0, it might be helpful to employ, as intermediate levels, the excited states which exhibit some avoided crossing, leading to double well or shelflike potential energy curves (PECs). Such a) Electronic mail: [email protected]. b) Electronic mail: [email protected].

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a shape of PEC may ensure that the transition probabilities to the ground state does not vanish for comparatively wide range of respective excited and ground state vibrational quantum numbers v  and v  needed for both photoassociation and emission steps. One of the first detailed studies of a shelflike state for an alkali diatomic containing a heavy Rb or Cs atoms was accurate characterization of the C(3)1  + state of the NaRb molecule4 converging to the Na(32 S) + Rb(62 S) atomic asymptote. In this study, besides the empirical PEC up to internuclear distance r = 12.4 Å, promising empirical values of photoassociation rate and of population transfer to the ground state were derived. For heteronuclear alkali diatomics with Cs atom, according to ab initio calculations,5, 6 the lowest shelflike state is the 1  + state that dissociates to the M(n2 S) + Cs(52 D) limit of separated atoms; here, M stands for Li, Na, K, and Rb with n = 2–5, respectively. The state in question is C(3)1  + for LiCs and NaCs, while changing to E(4)1  + for KCs and RbCs. The shelflike character of the 1  + state caused by avoided crossing of ion-pair and valence adiabatic states is expected5 to be most pronounced for a light M-atom such as Li, becoming the smoothest for Rb. Several Cs-containing heteronuclear alkali diatomics were already successfully created in ultracold conditions, see Refs. 7 and 8 for LiCs, Refs. 9 and 10 for NaCs, and

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Refs. 11 and 12 for RbCs. Thus it might be of interest to have spectroscopic information on the first shelflike state. Until now experimental data for the state in question have been restricted by the E(4)1  + state of the RbCs molecule.13, 14 The present paper focuses on both experimental and modeling studies of the shelflike E(4)1  + state in the KCs molecule by high-resolution Fourier-transform spectroscopy of the E(4)1  + → X 1  + collision enhanced laser induced fluorescence (LIF). The E(4)1  + state in KCs is promising as a transient state for producing deeply bonded cold diatomic species by the following reasons:

r The high vibrational levels v  close to the dissociation X

limit of the ground electronic X (1)1  + state appeared to be optically reachable from the E(4)1  + state as observed in the present study. r The spin-forbidden E(4)1  + → a(1)3  + LIF progressions to v a levels of the ground triplet a(1)3  + state have been observed, see Refs. 15 and 16. For modeling the respective optical cycles it is necessary to know highly accurate PECs in wide range of interatomic distances for all involved states. The required PECs for the ground singlet X (1)1  + and triplet a(1)3  + states became recently available.15, 16 What is more, a number of LIF progressions originated from the E(4)1  + of KCs were already recorded in those studies. The challenging issue is to find out whether it is possible to describe the observed term values of the regularly perturbed E(4)1  + state by a single PEC within experimental accuracy of about 0.01 cm−1 . A shelflike potential apparently has a too specific set of rovibronic eigenvalues to be adequately processed by the conventional (unconstrained) Dunham series17, 18   i E expt (v  , J  ) = Yi j v  + 12 [J  (J  + 1)] j , (1) i, j

at least in wide range of vibrational v  and rotational J  quantum numbers. Indeed, a large number of molecular constants (the mass-dependent Dunham coefficients Yi j ) are needed for the shelflike potential, and these fitting parameters are expected to loose their initial physical meaning. Furthermore, the ordinary Dunham polynomial expansion (1) has very limited extrapolation properties outside the experimental v  , J  range. In spite of these drawbacks, we have performed the Dunham analysis employing a truncated data set and have constructed the respective Rydberg–Klein–Rees (RKR) PEC. The alternative method for adiabatic PEC construction is a direct potential fit (DPF) of the experimental term values or the respective wave numbers.19 The DPF method is based on the iterative correction of a trial potential U (r ) in the framework of the weighted nonlinear least-squared fitting (NLSF) procedure20  Nexpt  expt calc 2  E − E j j 2 χexpt = , (2) expt σj j=1 expt

expt

where E j are the experimental term values while σ j are their uncertainties. The calculated values E calc are the rovij brational eigenvalues of the radial Schrödinger equation (in

atomic units) 

 J (J + 1) d2 calc φ v J (r ) = 0, (3) + U (r ) + −E − 2μdr 2 2μr 2

where μ is reduced molecular mass and r is internuclear distance. The mass-invariant form of the diabatic PEC U (r ) involved in Eq. (3) tacitly assumes that all possible massdependent intramolecular perturbations induced, for instance, by radial and angular coupling effects,21 can be neglected for the molecular state under study. At the same time, even pronounced mass-invariant homogeneous perturbations, caused by a regular spin–orbit coupling effect, would be effectively absorbed in the empirical PEC. To solve the inverse problem mentioned above, it is necessary to define PEC modeling function in explicit form. In literature there are numerous kinds of functions historically invented for approximation of interatomic PECs (see, for example, Refs. 22–26). All these forms, however, are a compromise between flexibility and constraint. The simple and robust method of the empirical PEC construction is based on a combination of pointwise representation of the potential with natural cubic spline interpolation.27, 28 Due to very high flexibility, this model, which is often called inverted perturbation approach (IPA), has been proved to be ideally suitable for a state having a specific shelflike or double-well shape of its adiabatic PEC (Ref. 29). The IPA model has been exploited in the present study using the programme package elaborated by A. Pashov et al.30 However, extrapolation properties of the pointwise spline-interpolating IPA model outside experimental region are not great. The model has extrapolation drawbacks when the respective experimental data are absent not only for high energies (high v  levels), which is a rather usual case, but also for low or intermediate energies (low v  levels). In contrast to the “spline-pointwise” IPA model, the recently introduced modifications of the Morse/long-range (MLR) potential31, 32 would provide a strong stabilizing constraint on the DPF analysis of the limited vibrational data set.33 However, a practical application of the fully analytical MLR model to the shelflike E(4)1  + state seems to be a challenging issue which is not the purpose of the present study. A way to find a compromise between high elasticity of IPA model and constraint properties of an analytical MLR potential using Chebyshev polynomial expansion (CPE) (Ref. 34) is proposed in the present work. The paper is structured as follows. Section II presents the experiment and preliminary analysis of the recorded spectra. Section III A explains vibrational numbering established by means of IPA PEC fit, while Sec. III B contains description of the analytical CPE method. Analysis of the obtained data starts with conventional Dunham approach to a restricted (v  ≤ 50) data set (Sec. IV A); the resulting CPE PEC of the E(4)1  + state is discussed in Sec. IV B by comparison with the Dunham analysis and the IPA PEC construction. Section IV C contains the comparison of calculated and measured LIF intensity distributions. The paper ends with concluding remarks in Sec. V.

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II. EXPERIMENT A. Experimental setup

The experimental setup is the same as was used in our previous studies.15, 16, 35 Briefly, the KCs molecules were produced either in a linear heat-pipe filled with 10 g of potassium (natural isotope mixture) and 7 g of cesium at temperature about 290 o C or in a glass cell filled with potassium and cesium in about 2:1 ratio at temperature about 250 o C. The KCs molecules were excited in a transition E(4)1  + (v  , J  ) ← X (1)1  + (v  , J  = J  ± 1) by dye laser radiation. Backward LIF was sent onto the input aperture of a Fourier transform spectrometer (FTS) Bruker IFS-125HR by a pierced mirror. The LIF spectra were recorded by FTS at a resolution of 0.03 cm−1 . A single mode ring dye laser Coherent 699-21 with Rhodamine 6G dye was used for excitation. Excitation frequencies were controlled by a wavemeter (HighFinesse WS6) with about 0.015 cm−1 accuracy. Laser frequency used in the experiment varied from 16 050 to 17 980 cm−1 . A particular excitation frequency within this range was selected by monitoring the FTS LIF signal in real time at low-resolution “Preview mode”. The LIF spectra in the range from 13 000 to 17 000 cm−1 were detected by a photomultiplier (Hamamatsu R928) operating at room temperature. In order to suppress the He–Ne laser used in the FTS for calibration of the path difference, a notch filter was placed before the detectors. Longpass edge filters were used for elimination of scattered exciting laser light. B. Spectra analysis

We observed LIF progressions to the ground X (1)1  + state which consist of P and R branches only. It indicates that the excited state belongs to the 1  + symmetry. Besides of the dominated spin-allowed E(4)1  + → X (1)1  + transitions, in some spectra we observed progressions at lower frequency range assigned to the spin-forbidden transitions to the ground triplet a(1)3  + state. An example of the observed spectrum is presented in Fig. 1. Assignment of rotational quantum number J  and J  for excited and ground state, respectively, along with the assignment of ground state vibrational quantum number v  , was based on the highly accurate empirical ground state PEC presented in Ref. 16. The term values E(v  , J  ) of the excited state were obtained by adding the observed transition frequencies to a corresponding term value of the ground state level E(v  , J  ), i.e., E(v  , J  ) = ν(v  ,J  )→(v  ,J  ) + E(v  , J  ). The uncertainty of the calculated ground state term values does not exceed 0.003 cm−1 . The uncertainty of determination of the LIF line frequency is about 10% of the spectral resolution. Since the latter was 0.03 cm−1 , our instrumental uncertainty is estimated as 0.003 cm−1 . The full width at half maximum for the Doppler profile at our typical temperatures about 280 ◦ C is 0.017 cm−1 , thus, the uncertainty of upper state level energies is conservatively estimated as 0.010 cm−1 . It should be noted, however, that the shift of the excited level energy determined from weak LIF progressions may be larger, up to 0.02 cm−1 , due to possible excitation at the very edge of Doppler line profile.

FIG. 1. An example of the recorded LIF spectrum from the E(4)1  + state excited by laser frequency 17 620.437 cm−1 . The strongest LIF progression lines originating from the excited level with v  = 25, J  = 82 to the ground X (1)1  + state within the range 13 ≤ v X ≤ 75 are marked in the figure. Weaker transitions to the a(1)3  + state originating from the same excited level are seen at lower frequencies about 13 600 cm−1 . The excitation laser and part of LIF is suppressed by a longpass edge filter for frequencies above 16 500 cm−1 .

III. DIRECT POTENTIAL FIT A. Vibrational assignment and IPA PEC fit

Vibrational assignment of the excited levels was not as straightforward as rotational assignment. Preliminary tentative assignment was done using both term values and relative intensity distributions calculated from the ab initio nonrelativistic PEC given in Ref. 5. However, this preliminary assignment was ambiguous, first, because the inaccuracy in predicted term energies exceeds few vibrational quanta, and, second, because the recorded spectra do not contain a full LIF intensity v  -distribution pattern since the filters were often used. Therefore, vibrational numbering was accomplished in the framework of “spline-pointwise” IPA fitting procedure27, 28 applied to the limited data set v  ≤ 45. The ab initio E(4)1  + state PEC from Ref. 5 was used as initial approximation to start fitting procedure. The fits with varying v  numbering were performed, and only one hypothesis yielded the standard deviation (SD) of the order of experimental uncertainty. As a result of this procedure, the preliminary v  numbering based on the PEC from Ref. 5 had to be diminished by 4. After this was done, increasing of data set up to v  = 74 confirmed the v  -numbering and allowed us to extend the r -range of empirical IPA PEC from 4 to 10.5 Å. A successful IPA PEC fit confirmed us that the energy levels of the E(4)1  + state can be described by a single PEC model. In the data set observed so far the lowest vibrational level was v  = 5. The simulations of Franck–Condon factors (FCF) for the respective E(4)1  + (v  ) − X 1  + (v  ) transitions revealed that the meaningful FCF values for excitation of low vibrational levels v  < 5 take place for absorption transitions from ground state vibrational levels v X ≥ 10. Unfortunately, the excitation frequencies about 16000 cm−1 needed for these transitions induce a very strong K2 LIF in the B 1 u → X 1 g+ transition. Therefore it was not possible

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J. Chem. Phys. 134, 104307 (2011) TABLE I. Experimental rovibronic term values E expt of the 41 K133 Cs isotopologue of the E(4)1  + state. δ values are the differences between the experimental energies and those predicted by means of empirical CPE and IPA PECs, as well as by conventional Dunham model [Eq. (1)], see Table II. All energies are in cm−1 .

FIG. 2. The data field of experimental rovibronic term values of the KCs E(4)1  + state as a function of J  (J  + 1) values. Black dots denote the experimental term values for 39 K133 Cs isotopologue, red crosses for 41 K133 Cs isotopologue, lines represent the term values from the empirical “splinepointwise” IPA PEC calculated for each fifth vibrational level starting from v  = 0.

to set the required laser frequency by monitoring E(4)1  + (v  < 5) → X 1  + LIF in low-resolution “Preview Mode” of the spectrometer. To facilitate excitation and observation of such weak vibronic transitions, the required term values of the E(4)1  + state were predicted by IPA PEC, as well as in the framework of the conventional Dunham analysis. Both approaches, however, were not able to extrapolate level energies for v  < 5 with the required accuracy (within Doppler width of about 0.02 cm−1 ). Only the CPE model described in Sec. III B provided such accuracy of calculated excitation frequencies that allowed us to observe LIF from low levels v  = 2, 3, and 4. As a result, overall about 1650 rovibronic term values of 39 K133 Cs and 41 K133 Cs isotopologues limited by vibrational levels v  ∈ [2, 74] with rotational quantum numbers J  ∈ [1, 188] were experimentally determined, with the uncertainty of 0.01 cm−1 , for the E(4)1  + state. The data field nonuniformly covers the energy range [16987, 18455] cm−1 above the minimum of the ground X1  + state which corresponds to 87.5% of the depth of the potential well of the E(4)1  + state. The full experimental data set is presented in the supplementary material,36 depicted in Fig. 2, and listed in Table I for the 41 133 K Cs isotopologue. The final IPA PEC UIPA (r ) fit was performed using full experimental data set for the 39 K133 Cs isotopologue. The PEC is determined by 51 points yielding a SD of 0.0058 cm−1 . The IPA PEC is presented in supplementary material (Ref. 36) and compared with the results of other approaches in Sec. IV.

B. Analytical potential based on Chebyshev polynomial expansion

Our current objective is to invent an analytical compact model of interatomic potential providing optimal balance of flexibility and constraint in order to represent a shelflike state

v

J

E expt

δ CPE

δ IPA

δ Dunham

9 10 10 15 11 13 15 14 20 22 22 19 22 23 22 21 28 26 24 17 22 31 35 28 38 45

34 26 31 31 94 81 69 96 63 44 51 88 75 73 89 109 56 84 102 142 128 86 73 133 76 47

17170.104 17187.548 17192.579 17315.189 17351.960 17361.483 17377.879 17427.536 17472.457 17481.323 17491.429 17510.121 17536.812 17551.314 17570.643 17609.771 17610.081 17630.220 17642.322 17656.151 17691.242 17719.527 17757.948 17806.984 17812.962 17881.759

−0.003 −0.003 0.003 0.005 0.004 0.002 −0.004 −0.006 0.002 −0.006 −0.011 −0.007 0.004 −0.002 −0.006 −0.004 −0.001 −0.006 −0.002 −0.007 −0.004 0.005 0.005 0.001 0.006 0.001

−0.003 −0.002 0.003 0.007 0.001 0.002 −0.002 −0.005 0.003 −0.005 −0.012 −0.006 0.003 −0.005 −0.008 −0.006 0.000 −0.005 −0.001 −0.006 −0.002 0.007 0.006 0.003 0.006 0.000

−0.005 −0.004 0.002 0.006 0.007 0.000 −0.008 −0.012 0.001 −0.006 −0.012 −0.008 0.005 −0.002 −0.003 0.001 −0.002 −0.010 −0.003 −0.001 −0.006 0.011 0.009 0.011 0.001 0.003

based on a limited vibrational data set. The target functional should fulfill the following conditions:

r to have correct asymptotic behavior at r → +∞, r to be a linear analytical function with respect to the most fitting parameters,

r to have attractive (predictable) convergence properties. To accomplish this task, the modeling PEC U (r ) for E(4)1  + state was approximated by a continuous analytical function defined on the semiinterval r ∈ [rmin , +∞) as m ck Tk (y p ) , (4) UCPE (r ) = Tdis − k=0 1 + (r/rref )n where Tk (y) are the Chebyshev polynomials of the first kind34 depending on the reduced radial variable y p (r ) ∈ [−1, 1] p

y p (r ; rmin , rref ) =

r p − rref p p , r p + rref − 2rmin

(5)

in which p ∈ [1, 2, . . .] is a small positive integer and rref > rmin ≥ 0 is a reference distance37 chosen as the variable center of expansion. At rmin ≡ 0 the y p exactly coincides with the Surkus variable26 introduced earlier for the ordinary polynomial construction of the generalized potential energy function (GPEF) (Ref. 26) and the MLR potential.31, 32 The invented form (4), being a continuous function of radial coordinate UCPE ∈ C ∞ ([rmin , +∞)), exhibits the long-

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range behavior Cn+ p Cn − n+ p + · · · , (6) n r r where the leading expansion coefficients Cn and Cn+ p have the following forms: UCPE (r ) Tdis −

n Cn = rref

m 

ck ,

(7)

k=0

p

p

n Cn+ p = 2(rmin − rref )rref

m 

ck k 2 .

(8)

k=1

Therefore, the first dispersion coefficient Cn does not depend on the parameter p at all, while the particular p value affects on a period of expansion and higher dispersion coefficients Cn+k× p , where k ∈ [1, 2, . . . , n] is a positive integer. The dissociation energy Tdis , the leading degree of the long-range expansion n, the left boundary of the expansion rmin , the reference distance rref , and the degree of the reduced variable p are assumed to be the fixed parameters of the model. Therefore, the CPE function (4) is linear with respect to most fitting parameters, namely, the Chebyshev expansion coefficients ck . The Chebyshev polynomial expansion is unambiguously related to ordinary polynomials:20 m  k=0

dk y kp =

m 

ck Tk (y p ),

(9)

FIG. 3. Schema of the selected KCs electronic states. The ab initio PECs were calculated in pure (a) Hund’s coupling case in Ref. 5. The shaded area depicts the energy range covered in the present experiment.

k=0

which are widely used in numerous potential energy forms such as Dunham,17 Simons–Thakkar,23, 24 Ogilvie,25 GPEF (Ref. 26), and MLR (Ref. 32). However, in contrast to an ordinary polynomial, the Chebyshev orthogonal polynomial basis set20, 34 provides very attractive convergence properties34 leading to the monotonically decreasing absolute magnitude of the expansion coefficients |ck |. As it was mentioned in Sec. I, the observation of the spinforbidden E(4)1  + − a(1)3  + LIF progressions highlights the pronounced spin–orbit coupling effect in the E(4)1  + state under study. Due to spin–orbit coupling with the triplet (3)3  state converging to the same nonrelativistic dissociation limit K(42 S) + Cs(52 D), see Fig. 3, the empirical potential of the shelflike E(4)1  + state should converge at large internuclear distances to the fine state atomic asymptote K(42 S1/2 ) + Cs(52 D3/2 ). The fixed value of dissociation energy Tdis = De + E 5Cs2 D3/2 corresponding to the fine asymptote (without hyperfine splitting) was determined by the experimental dissociation energy De = 4069.208(40) cm−1 of the KCs ground X -state16 and experimental energy E 5Cs2 D3/2 = 14499.2568 cm−1 of the Cs(52 D3/2 ) term.38 The initial parameters of UCPE (r ) required to start the fitting procedure (2) were estimated from ab initio potential Uab (r ) available for a pure “c” Hund’s coupling case.39 The respective nonempirical long-range coefficient C6 involved into the minimization procedure is available in Refs. 40 and 41. It should be noted that Ref. 41 contains both quasirelativistic and nonrelativistic estimates of the C6 values while Ref. 40 presents only a nonrelativistic estimate. However, the nonrelativistic

estimates of Ref. 41 are about one half of its counterpart from Ref. 40. So, in order to obtain more reliable C6ab coefficient for the quasirelativistic E state, we had averaged the nonrelativistic C6 estimates for the mutually perturbed E(4)1  + and (3)3  states (see Fig. 3) available in Ref. 41. The refined UCPE (r ) parameters were determined during the weighted NLSF procedure (2), where only the experimental expt term values E j of the most abundant 39 K133 Cs isotopologue expt were included, with the uncertainty σ j = 0.007 cm−1 . To extrapolate properly the repulsive wall of the UCPE (r ) PEC into high energy region corresponding to short internuclear distance r j ∈ [2.65, 3.9] Å(outside the experimental range), the ab initio points Uab (r j ) from Ref. 39 were incorporated in the fit  2 Nab  Uab (r j ) − UCPE (r j ) 2 , (10) χab = σ jab j=1 where the uncertainty of the ab initio data was estimated as σ jab = 15 cm−1 . Moreover, to provide a correct smooth behavior of the CPE potential at r → ∞, the constraint (7) on a sum of the expansion coefficients ck was introduced in the fit by a global minimization procedure

2 2 (11) + χab + ((C6ab − C6CPE )/σ6ab )2 , min χexpt where the long-range coefficient C6CPE is calculated according to Eq. (7) while its ab initio counterpart C6ab = 5.986 was estimated with σ6ab = 0.1 (in 107 cm−1 /Å6 ) accuracy, by

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merged C6 coefficients of the E(4)1  + and (3)3  states from 2 2 and χab values were evaluated Refs. 40 and 41. The χexpt by Eq. (2) and Eq. (10), respectively. The minimum of the functional (11) was searched by the Levenberg–Marquardt algorithm.20, 42 It should be noted that according to Eq. (8), a formal consideration of the next dispersion coefficient C8 of the long-range expansion of the KCs E(4)1  + state inevitably requires the parameter p to be equal to 2. However, the longrange tail of the resulting CPE PEC determined with the fixed parameter p = 2 had demonstrated small nonphysical oscillations in the extrapolation region located in the vicinity of the dissociation threshold. Since the accurate long-range behavior of the treated potential is practically immaterial for the present experimental data set (limited by the v  ≤ 74 levels), the optimal values of both parameters p = 4 and rref = 6.48 Å have been determined here by trial-and-error method in order to provide, first of all, the smoothness and compactness of the CPE representation. Therefore, the implicit exclusion of the C8 term was mainly mandated by the limitations of the present data set and, in part, by the restriction of the CPE form.

IV. ANALYSIS AND DISCUSSION A. Dunham analysis

Due to well known limitations of the conventional (unconstrained) Dunham analysis (1), we did not attempt to fit the overall set of the experimental term values in the framework of Dunham constants Yi j . However, the fully unconstrained Dunham analysis (see Tables II and III) based on the restricted 2 ≤ v  ≤ 50 vibrational set, which partially includes the shelf region, yielded SD 0.0058 cm−1 for the most abundant 39 K133 Cs isotopologue. Furthermore, the isotopically substituted Dunham constants from Table II allowed us to reproduce experimental term values of the 41 K133 Cs isotopologue with SD of 0.0064 cm−1 (see Table I). Both SD values are well consistent with the estimated uncertainty 0.01 cm−1 of the present experiment. The resulting set of Dunham molecular constants was used to construct the pointwise potential up to v  = 50 level by the conventional RKR method. For the 39 K133 Cs isotopologue, the resulting vibrational differences E v  = E v  +1 − E v  and rotational constant Bv  are depicted in Fig. 4, while the respective centrifugal distortion constants (CDCs) are presented in Fig. 5. As it would be expected for a shelflike state, the overall molecular constants demonstrate a peculiar dependence on vibrational quantum number v  . In the intermediate shelflike region, the vibrational constants exhibit “effective” harmonic oscillator behavior since the corresponding vibrational quantum E v  ≈ const [see Fig. 4(a)]. Moreover, as can be seen from Table II, the “mechanical” meaning of some Dunham constants disappears. In particular, the first unharmonic rotational constant Y11 is positive instead of being negative as expected for an attractive diatomic potential of usual shape.17, 18 Furthermore, the derived Dunham molecular constants demonstrate, as it is expected, nonrealistic behavior in the extrapolation region corresponding to v  > 50 and v  < 2.

J. Chem. Phys. 134, 104307 (2011) TABLE II. The mass-dependent 39 K133 Cs Dunham molecular constants Yi j and their uncertainties Yi j (in cm−1 ) of the E(4)1  + state obtained in the framework of the unconstrained Dunham analysis [Eq. (1)] of the restricted 2 ≤ v  ≤ 50 vibrational set of the present experimental rovibronic term values. i

j

Yi j

Yi j

0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2

0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 4 4 4

0.1688093707964885E + 05 0.3080906325327967E + 02 −0.2857642529037892E + 00 0.2525451146308333E − 01 −0.3891088698026048E − 02 0.3451252794646093E − 03 −0.1962525736682913E − 04 0.7222259549274694E − 06 −0.1692525421966681E − 07 0.2434054953588594E − 09 −0.1959592494181685E − 11 0.6770893879572643E − 14 0.1916984501576217E − 01 0.5942540700331354E − 05 −0.2316429843977754E − 04 0.2036674611059575E − 05 −0.1079429572885772E − 06 0.2556681071002156E − 08 −0.9789841116555208E − 12 0.1088940434322790E − 13 0.5980174038960881E − 16 −0.1072240401099699E − 17 0.2514728723346904E − 07 −0.3574893240928863E − 07 0.9138698938518142E − 08 −0.1344375300348306E − 08 0.1243732557647663E − 09 −0.7548630713426814E − 11 0.3023757535328357E − 12 −0.7852489817351228E − 14 0.1264685391677036E − 15 −0.1144147521731941E − 17 0.4435683458340068E − 20 −0.5016003096916856E − 11 0.2931207592885829E − 11 −0.7120379733468684E − 12 0.9692200003508204E − 13 −0.8268761024845887E − 14 0.4609697867022436E − 15 −0.1694579101078259E − 16 0.4053596358000092E − 18 −0.6050875019126259E − 20 0.5106962114131017E − 22 −0.1857703580197183E − 24 −0.4404390049501628E − 17 0.5082586965249937E − 18 −0.9743286833511986E − 20

0.50E − 01 0.34E − 01 0.10E − 01 0.19E − 02 0.21E − 03 0.15E − 04 0.74E − 06 0.24E − 07 0.52E − 09 0.71E − 11 0.56E − 13 0.19E − 15 0.35E − 04 0.19E − 04 0.39E − 05 0.41E − 06 0.23E − 07 0.62E − 09 0.50E − 12 0.15E − 13 0.18E − 15 0.87E − 18 0.97E − 08 0.54E − 08 0.12E − 08 0.15E − 09 0.11E − 10 0.56E − 12 0.19E − 13 0.43E − 15 0.65E − 17 0.57E − 19 0.22E − 21 0.74E − 12 0.39E − 12 0.87E − 13 0.11E − 13 0.83E − 15 0.42E − 16 0.14E − 17 0.33E − 19 0.49E − 21 0.41E − 23 0.15E − 25 0.13E − 17 0.10E − 18 0.18E − 20

B. Potential energy curves

The resulting mass-invariant parameters of the adiabatic CPE potential for the shelflike E(4)1  + state are presented in Table IV. The parameters are included in the tables in the supplementary information as well.36 For convenience, the FORTRAN subroutine providing the CPE PEC construction

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TABLE III. The position of the PEC minimum of the KCs E(4)1  + state predicted in the framework of the empirical CPE, IPA, and Dunham models. The relativistic ab initio data are borrowed from Ref. 39. The electronic energy Te (in cm−1 ) refers to the minimum of the ground X -state, while the equilibrium distance re is in Å.

Te re

CPE

IPA

Dunham

ab initio

16 881.038 5.381

16 880.746 5.382

16 880.937 5.373

16 926 5.435

for arbitrary internuclear distance is also given in the supplementary material. Furthermore, the tables in the supplementary material contain both experimental and reproduced (CPE) rovibronic term values of the 39 K133 Cs isotopologue combined with their residuals. The empirical PEC UCPE (r ) (see Fig. 6) defined analytically by expression (4) on the interval r ∈ [2.54, ∞) Å reproduces 1620 experimental term values of the 39 K133 Cs isotopologue (see Fig. 2) with SD of 0.0053 cm−1 . To confirm the mass-invariant properties of the derived PECs, rovibronic term values of the 41 K133 Cs isotopologue have been calculated and compared with their experimental counterparts. The term values calculated by means of the CPE and IPA models coincide with the experimental data given in Table I with SD of 0.0048 and 0.0050 cm−1 , respectively. In these calculations, only the reduced mass μ of the 41 K133 Cs isotopologue,

FIG. 5. The centrifugal distortion constants Dv  , Hv  , and L v  estimated for the 39 K133 Cs E(4)1  + state in the framework of CPE, IPA, and Dunham models.

entering explicitly in the operators of vibrational and rotational kinetic energy of the radial equation (3), was substituted. In absence of any additional adjustment of the parameters, the observed agreement can be considered as excellent. The empirical PEC UCPE (r ) differs by 1 to 3 vibrational quanta of the E(4)1  + state from its ab initio counterpart Uab (r ) corresponding to a pure “c” Hund’s coupling case39 [see Fig. 7(a) and Table III]. The UCPE (r ) agrees with the

FIG. 4. The vibrational quantum E v  = E v  +1 − E v  (a) and rotational constants Bv  (b) obtained for the 39 K133 Cs E(4)1  + state in the framework of CPE, IPA, and Dunham models. The insets demonstrate declining of Dunham based estimates for v  > 50.

FIG. 6. The empirical mass-invariant potential energy curve UCPE (r ) derived for the KCs E(4)1  + state in the framework of CPE model (4). Horizontal solid lines denote the present experimental energy region used for the UCPE (r ) construction.

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J. Chem. Phys. 134, 104307 (2011)

TABLE IV. The resulting mass-invariant parameters of the CPE potential for the KCs E(4)1  + state defined by expression (4) on the semiinterval [rmin , +∞). The dissociation energy Tdis referring to the minimum of the ground X 1  + state and the expansion coefficients ck are given in cm−1 while the rmin and rref parameters in Å. Symbol † denotes the fixed parameters. †T dis

18 568.466

†n

6 4

†p †r †r

2.65 6.48

min ref

−454.602 −4550.618 4491.176 −1740.666 2738.181 −1628.837 757.107 −1254.274 466.520 −372.279 523.291 −48.995 293.384 −98.701 80.620 −104.074 34.624 −37.820 42.711 2.874 30.759 6.232 11.492 1.437 1.928

c0 c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13 c14 c15 c16 c17 c18 c19 c20 c21 c22 c23 c24

conventional RKR PEC constructed by Dunham constants from Table II within few wave numbers [see Fig. 7(b)]. Except the extrapolation region located near PEC’s minimum, the UCPE (r ) and UIPA (r ) functions almost coincide inside the experimental data region basically limited by the interval of internuclear distances r ∈ [4.0, 10.5] Å [Fig. 7(c)]. It should be noted that the regularization procedure based on minimization of the functional29 rmax  min |UIPA (r )|2 dr, (12) rmin

has been required to construct a realistic IPA curve in the interval r ∈ [5.00, 5.85] Å located near the minimum of the po | of the IPA tential. The minimization of third derivatives |UIPA PEC with respect to internuclear distance constrains here the resulting curve to a parabolic form. To estimate rovibronic energy levels for arbitrary v  and  J values, it is often convenient to avoid at all the explicit numerical solution of the radial equation (3). For this purpose, the extensive table of vibrational, rotational, and CDCs Dv  , Hv  , and L v  have been generated for individual vibrational levels v  ≤ 74 of the E(4)1  + state by means of

FIG. 7. Differences between the empirical E(4)1  + state PEC UCPE (r ) and ab initio relativistic Uab (r ) from Ref. 39 (a), as well as its present empirical URKR (r ) (b), and UIPA (r ) (c) counterparts. Here Uab (r ) was uniformly vertically shifted to match the experimental fine atomic asymptote K(42 S1/2 ) + Cs(52 D3/2 ).

the present UCPE (r ) and UIPA (r ) PECs. We have used for this procedure the Hutson’s algorithm43 realized in LeRoy’s LEVEL 8 programme suit.44 The resulting vibrational quantum E v  and rovibrational constants Bv  , Dv  , Hv  , and L v  for the 39 K133 Cs isotopologue are depicted in Figs. 4 and 5 while their overall set (including the 41 K133 Cs isotopologue) is given in the tables in the supplementary material.36 In contrast to Dunham analysis, the UCPE (r ) and UIPA (r ) PECs yield very close E v  and Bv  values even for v  > 74 (see Fig. 4). However, the respective CDCs clearly diverge as the vibrational quantum number increases (Fig. 5). This is caused by a different implicit effect of the higher lying bound and embedded continuum vibrational states. Furthermore, rather small but still pronounced discontinuous behavior of the high order CDC L v  (Fig. 5) seems to be attributed to the discontinuous higher derivatives of the “spline-pointwise” UIPA (r ) function with respect to r . As it would be expected from general convergence properties of the Chebyshev orthogonal polynomial expansion,34 the resulting coefficients |ck | smoothly decrease as their ordering number k increases (see Table IV), in contrast to the coefficients of ordinary polynomial expansion |dk | evaluated by Eq. (9) (see Fig. 8). It is interesting to notice that the ratio rref /re ≈ 1.20 of the present reference and equilibrium distances (see Tables III and IV) agree with the empirical rule rref /re ≈ 1.1 ÷ 1.5 recently established for optimal rref values of the enhanced MLR model.37 The present optimal parameter p = 4 is also well correlating with the

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J. Chem. Phys. 134, 104307 (2011)

FIG. 8. Comparison (in log scale) of the coefficients derived for Chebyshev ck [Eq. (4)] and ordinary dk [Eq. (9)] polynomial expansion of the resulting CPE PEC as dependent on their ordering number k.

recommended q = 3, 4, and 5 values of the MLR potential.37, 45 It should be noted that the compactness of the CPE form strongly depends on the optimal choice of p and rref parameters. C. Intensity distributions in the E (4)1  + → X (1)1  + LIF progressions

A nodal structure of strongly oscillating rovibrational wave functions φ v J (r ) in Eq. (3) is very sensitive to a shape of the exploited PEC (Ref. 46). Therefore, a comparison of the experimental relative intensity distributions in a band structure of E(v  , J  ) → X (v  , J  ) LIF progressions, originating from a sufficiently high v  -level of the upper state to a wide range of ground state vibrational levels v  , with their theoretical counterparts would be an additional test of the present CPE analysis. The required transition probabilities I E→X (v  ; v  ) were estimated as



2

    φ Ev J d E X φ vX J dr

, (13) I E→X ∼ ν E4 X

0

assuming that the detected signal is proportional to the intensity of the incoming fluorescence light. Here, d E X (r ) is the ab initio transition dipole moment47, 48 and ν E X   = E Ecalc (v  , J  ) − E calc X (v , J ) is the transition wave numand eigenfuncber. The rovibronic energies E Ecalc , E calc X     tions φ Ev J (r ), φ vX J (r ) were obtained by solving the radial equation (3) with the present UCPE (r ) for the upper E(4)1  + state and highly accurate UIPA (r ) PEC from Ref. 15 for the ground X (1)1  + state. Both experimental and calculated intensities obtained for P- and R-branches were averaged since they were typically very close to each other.

FIG. 9. Normalized experimental and calculated by Eq. (13) intensity distributions in LIF progressions originating from v  = 42, 48, and 55 vibrational levels of the upper E(4)1  + -state.

Figure 9 demonstrates excellent agreement between experimentally observed intensities and the calculated ones.

V. CONCLUDING REMARKS

r Collisionally

enhanced laser induced fluorescence spectra corresponding to both spinallowed E(4)1  + → X (1)1  + and spin-forbidden E(4)1  + → a(1)3  + transitions of the KCs dimer were recorded in visible region by Fourier transform spectrometer. Overall about 1650 rovibronic E(4)1  + state term values of 39 K133 Cs and 41 K133 Cs isotopologues were obtained with typical uncertainty of 0.01 cm−1 . The present experimental data field within vibrational levels 2 ≤ v  ≤ 74 nonuniformly covers about 87.5% of the depth of the potential well. r The closed analytical form for a potential energy curve based on Chebyshev polynomial expansion was implemented to a direct potential fit of the experimental term values of the most abundant 39 K133 Cs isotopologue. The invented CPE form had demonstrated optimal balance between flexibility and constraint for the direct potential fit of a shelflike state aggravated by a limited vibrational data set. There is a real advantage to use the Chebyshev polynomial expansion which is providing a compact representation and attractive convergence properties.

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r The mass-invariant properties and good agreement between simulated and experimental intensity distributions in the LIF progressions originating from high rovibronic levels of the E(4)1  + state justifies reliability of the derived empirical PEC. r Representation of the E(4)1  + state by a single massinvariant PEC is sufficient to reproduce the term values with experimental accuracy of about 0.01 cm−1 of all observed rovibronic levels, including the ones giving spin-forbidden transitions to a 3  + state. This fact indicates that partial triplet character of the E(4)1  + state is caused most probably by spin–orbit interaction with adjacent (2, 3)3  states (see Fig. 3).

ACKNOWLEDGEMENTS

We are indebted to Dr. Asen Pashov for useful remarks and fruitful discussion, as well as for providing his software for IPA PEC fitting and spectra analysis, and to Professor Eberhard Tiemann for his program for Dunham constants fit. We are grateful to Dr. Andrei Jarmola for providing the laser diodes built in home made external cavity resonators and to Andris Berzins for his help in spectra analysis. The support from ESF Grant No. 2009/0223/1DP/1.1.1.2.0/09/APIA/VIAA/008 and the Latvian Science Council, Grant No. 09.1036 is gratefully acknowledged by Riga team. Moscow team thanks for support the Russian Foundation for Basic Researches (Grant No. 1003-00195-a) and MSU Priority Direction 2.3. 1 Cold

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