The rotational spectra, potential function, Born-Oppenheimer breakdown, and magnetic shielding of SnSe and SnTe

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1 THE JOURNAL OF CHEMICAL PHYSICS 126, The rotational spectra, potential function, Born-Oppenheimer breakdown, and magnetic shielding of SnSe and SnTe Luca Bizzocchi and Barbara M. Giuliano Dipartimento di Chimica G. Ciamician, Università di Bologna, Via F. Selmi 2, Bologna, Italy Mareike Hess and Jens-Uwe Grabow a Institut für Physikalische Chemie und Elektrochemie, Lehrgebiet A, Universität Hannover, Callinstraße 3-3A, Hannover, Germany Received 27 November 2006; accepted 25 January 2007; published online 20 March 2007 The pure rotational spectra of 27 isotopic species of SnSe and SnTe have been measured in the frequency range of 5 24 GHz using a Fabry-Pérot-type resonator pulsed-jet Fourier-transform microwave spectrometer. Gaseous samples of both chalcogenides were prepared by laser ablation of suitable target rods and were stabilized in supersonic jets of Ar. Global multi-isotopolog analyses of all available high-resolution data produced spectroscopic Dunham parameters Y 01, Y 11, Y 21, Y 31, Y 02, and Y 12 for both species, as well as Born-Oppenheimer breakdown BOB coefficients 01 for Sn, Se, and Te. A direct fit of the same data sets to an appropriate radial Hamiltonian yielded analytic potential energy functions and BOB radial functions for the X 1 + electronic state of both SnSe and SnTe. Additionally, the magnetic hyperfine interaction produced by the dipolar nuclei 119 Sn, 117 Sn, 77 Se, and 125 Te was observed, yielding first determinations of the corresponding spin-rotation coupling constants American Institute of Physics. DOI: / I. INTRODUCTION a Author to whom correspondence should be addressed. Electronic mail: jens-uwe.grabow@pci.uni-hannover.de The heavy groups diatomics chalcogenides constitute a very interesting class of molecules. The bulk materials corresponding to these species are narrow band-gap semiconductors and have recently attracted much attention due to their technological importance in optoelectronic devices for infrared and visible radiation. Studies of these systems have also drawn significant attention in the past. The first observation of the visible band spectra of tin selenide SnSe and tin telluride SnTe dates back to the early 1930s, 1,2 and, in the subsequent years, many other UV and visible band systems of SnSe and SnTe were observed both in emission and in absorption, 3 8 allowing the spectroscopic constants of the low-lying electronic states of these molecules to be determined. Matrix IR spectroscopy, 9 photoelectron spectroscopy, 10 and mass spectrometry thermodynamical studies 11 of tin selenide and tin telluride have also been performed. On the contrary, only few high-resolution spectroscopic investigations of SnSe and SnTe have so far been reported. The pure rotational spectra of both species were first investigated in the late 1960 s by Hoeft 12 and Hoeft and Tiemann 13 using an X-band Stark-modulated microwave spectrometer equipped with a high temperature waveguide absorption cell C. They observed microwave spectra GHz of 29 isotopologs of SnSe and 27 isotopologs of SnTe obtaining rotational parameters and the r e bond length for each isotopic species. The electric dipole moment and the rotational g factors of both molecules were also determined. 14,15 Subsequently, Tiemann et al. 16 measured the millimeter-wave spectra GHz of several groups and groups diatomic molecules, including SnSe 8 isotopologs and SnTe 13 isotopologs. The effect of the breakdown of the Born-Oppenheimer approximation on the rotational constant Y 01 B e was revealed, but no detailed Dunham analysis of the complete multiisotopolog data sets was reported. In the present work we investigated the pure rotational spectra of SnSe and SnTe using a Fourier-transform microwave FT-MW spectrometer. Jet-cooled gaseous samples of tin chalcogenides were obtained by laser ablation of suitable solid precursors. More than 400 rotational lines belonging to 27 isotopologs of both chalcogenides were recorded; vibrationally excited states up to v=13 and v=15 were observed for the most abundant isotopic species of SnSe and SnTe, respectively. The resulting large data sets were first analyzed using a Dunham-type combined-isotopolog parameter-fit model, which yielded highly improved Y lm Dunham constants and Born-Oppenheimer breakdown BOB correction parameters. A direct potential fit DPF analysis was then used to derive a compact analytic potential energy function and associated radial BOB functions for SnSe and SnTe in their ground 1 + electronic state. In addition, the fairly high resolution obtainable with the coaxially oriented beam resonator arrangement COBRA technique makes it possible to observe the small hyperfine splitting of the rotational lines produced by the magnetic coupling of the dipolar nuclei 117 Sn, 119 Sn, 77 Se, and 125 Te. The corresponding spinrotation constants, which were determined for the first time, provide information on the magnetic shielding in the SnSe and SnTe molecules /2007/ /114305/12/$ , American Institute of Physics

2 Bizzocchi et al. J. Chem. Phys. 126, FIG. 1. Color online Details of the laser-ablation pulsed molecular beam source with dc-discharge facility. II. EXPERIMENT The rotational transitions of SnSe and SnTe were recorded using the broadband 2 26 GHz FT-MW spectrometer of the Balle-Flygare type 17 operating in Hannover. The instrument is equipped with a LASER-ablation source in the in the COBRA configuration. 18,19 The solid material was ablated from the rotating rod using a pulsed neodymium-doped neodymium-doped yttrium aluminum garnet Nd:YAG laser =1064 nm,500 mj pulse 1 at a repetition rate of 20 Hz, seeded in Ar stated purity of 3.5, stagnation pressure of 2.5 bars and expanded adiabatically into the Fabry-Pérottype resonator through a pulsed solenoid valve General Valve series 9. The rods, of approximately 30 mm length were made by pressing equal amounts of finely powdered tin and selenium or tin and tellurium, onto a central 1.5 mm diameter stainless steel backbone with the help of a small amount of acrylic acid based glue. The resulting cylinder-shaped sample was then removed from the die, dried for 12 h in air, and slightly sanded to form a 6 mm diameter rod. The rotational spectra of SnSe and SnTe were recorded in the frequency region between 5 and 24 GHz. The beam source is equipped with a modified expansion nozzle with dc-discharge capabilities: the central channel 3 mm in diameter accommodates alternating layers of brass electrodes and Teflon insulators. 20 A further Teflon spacer is placed directly after the cathode of the discharge zone see Fig. 1 in order to confine the discharge plasma before expansion. A kv voltage is applied continuously, resulting in an average discharge current of A, depending on the backing pressure. Even though this technique is not strictly required to produce the investigated chalcogenides, it has proven to be very effective in increasing the population of vibrationally excited states. An example is given in Fig. 2, which shows the J =4 3, transition of 120 Sn 80 Se in the vibrational state v=4 recorded using different discharge currents. III. ANALYSIS AND RESULTS A. Observed spectra and assignments The previously reported spectroscopic constants for SnSe Ref. 12 and the Dunham parameters for SnTe Ref. 13 were used to predict the position of the corresponding FIG. 2. Color online Records of the J=4 3 v=4 rotational transition of 120 Sn 80 Se using different discharge currents. Experimental conditions: Backing pressure, 2 bars; excitation frequency, MHz; accumulated experiments, 1000; sampling interval, 100 ns; Fourier transform, 8 K. rotational transitions in the frequency range of our FT-MW spectrometer. Lines produced by the ablation of the appropriate rod were quickly found at the predicted frequencies, including those of some previously unobserved isotopologs or vibrationally excited states. The highest vibrational energy levels observed for SnSe and SnTe were v=13 and v=15, respectively. In total, we recorded 159 rotational transitions of 27 isotopologs of SnSe in the J range of 1 5 and 250 transitions of 27 isotopic species of SnTe in the J range of 1 7. B. Hyperfine structure analysis Each J+1 J transition of SnSe and SnTe containing the uneven mass number A nuclei 117 Sn, 119 Sn, 77 Se, and 125 Te are characterized by a spin-rotation splitting due to the coupling of the nuclear spin angular momentum I with the rotational angular momentum J, following the coupling scheme, F = J + I. 1 The rotational transitions of 119 Sn 77 Se, 117 Sn 77 Se, 119 Sn 125 Te, and 117 Sn 125 Te exhibit quadruplet structures since both atoms possess nuclear spins of I= 1 2. In these cases the coupling scheme adopted was F 1 = J + I 1 ; F = F 1 + I 2. 2 The measured component frequencies were analyzed using the following Hamiltonian: H = B v J 2 D v J 4 + C I Sn I Sn J + C I X I X J, 3 where X=Se and Te. The hyperfine analysis was performed independently for each isotopolog using Pickett s SPFIT

3 Rotational spectra, potential of SnSe and SnTe J. Chem. Phys. 126, TABLE I. Ground-state molecular constants for 11 isotopologs of SnSe containing dipolar nuclei. Numbers in parentheses give one standard deviation in units of the last quoted digit. Isotopolog x a % B 0 MHz D 0 khz C I Sn khz C I Se khz 119 Sn 80 Se Sn 80 Se Sn 77 Se Sn 78 Se Sn 77 Se Sn 78 Se Sn 77 Se Sn 76 Se Sn 82 Se Sn 77 Se Sn 77 Se b a Natural abundance % of the isotopolog. b Constrained. Value derived from the Dunham constants of Table III. program, 21 giving its ground-state rotational constant B 0, quartic centrifugal distortion constant D 0, and the nuclear spin-rotation coupling constants C I of the relevant nuclei. The resulting constants for the 11 isotopologs of SnSe and for the 10 isotopologs of SnTe are collected in Tables I and II, respectively. The results of the hyperfine analyses were used to derive the hypothetically unsplit frequencies for the transitions of these isotopologs, which were then included in the Dunham-type and direct potential analyses described in the next subsections. C. Dunham analysis To the experimental transition frequencies of SnSe and SnTe, a Dunham-type energy level expression 22 was leastsquares fit, incorporating first-order semiclassical mass scaling to allow for a multi-isotopolog analysis. However, since pure mass scaling was insufficient to reproduce the spectrum within experimental accuracy, BOB corrections had to be taken into account. Adopting the procedure proposed by Le Roy, 23 one isotopolog is chosen as the reference species =1 to which the Dunham parameters of all other species are related to. In this formalism, the rovibrational energy of a hyperfine-free isotopolog can be expressed as E v,j = l,m 0,0 Y lm v l J J +1 m, where Dunham coefficients for 1 are generated by Y lm = Y 1 lm + M A M A lm + M B A M lm B B m+l/2 1, 5 in which M A and M B represent the atomic masses of the atoms A and B, respectively, M X =M X M 1 X X=A,B, and is the reduced mass of the isotopolog. Combining Eqs. 4 and 5, the entire multi-isotopolog data set can be expressed in terms of the Dunham coefficients Y 1 lm reference A species =1 and of the mass-dependent correction terms lm and B lm. These latter depend on the choice of the reference isotopolog, and can be related to the Watson-type BOB terms, 24 lm A and lm B, through the equation X X lm = M X 1 lm Y 1 m lm + A lm + B lm 1, 6 e where m e is the electron mass. Only the BOB corrections to the parameter Y 01 were found to be significant in the present investigation. The multi-isotopolog data sets of SnSe and SnTe were analyzed using the program DPARFIT 3.3 by Le Roy, 25 which 4 TABLE II. Ground-state molecular constants for ten isotopologs of SnTe containing dipolar nuclei. Numbers in parentheses give one standard deviation in units of the last quoted digit. Isotopolog x a % B 0 MHz D 0 khz C I Sn khz C I Te khz 119 Sn 130 Te Sn 128 Te Sn 130 Te Sn 128 Te Sn 125 Te Sn 125 Te Sn 126 Te Sn 125 Te Sn 125 Te b Sn 125 Te b a Natural abundance % of the isotopolog. b Constrained. Value derived from the Dunham constants of Table IV.

4 Bizzocchi et al. J. Chem. Phys. 126, TABLE III. Dunham parameters, molecular parameters, and BOB-correction terms for 27 isotopologs of SnSe. Numbers in parentheses give one standard deviation in units of the last quoted digit. Isotopolog x a % Y 01 MHz B e Y 11 MHz e Y 21 khz e Y 31 Hz e Y 02 khz D e Y 12 Hz e 120 Sn 80 Se Sn 80 Se Sn 78 Se Sn 80 Se Sn 78 Se Sn 80 Se Sn 80 Se Sn 78 Se Sn 76 Se Sn 80 Se Sn 82 Se Sn 77 Se Sn 80 Se Sn 76 Se Sn 82 Se Sn 78 Se Sn 77 Se Sn 78 Se Sn 78 Se Sn 76 Se Sn 82 Se Sn 77 Se Sn 78 Se Sn 76 Se Sn 82 Se Sn 77 Se Sn 77 Se a Natural abundance % of the isotopolog. b Adopting 120 Sn 80 Se as the reference isotopolog. Sn 01 = khz b Se 01 = khz b applies Eqs. 4 and 5 in a weighted least-squares routine. The millimeter-wave frequencies previously recorded by Tiemann et al. 16 and Stieda 26 were also included in the analyses: These data comprise 68 lines for 8 isotopologs of SnSe J range of 19 27, v range of 0 8 and 114 lines for 13 isotopologs for SnTe J range of 30 43, v range of 0 6. Different experimental uncertainties i were given to the two sets of data in order to take into account the different measurement precisions: i values of 15 and 0.3 khz were assigned to the previous millimeter-wave measurements 16,26 and to the data of the present work, respectively. 27 Throughout the following, the quality of a fit of N data y i obs with estimated uncertainties i to a given model is represented by the dimensionless root-mean-square deviation d : d = 1 N y calc obs 2 1/2 i y i N i=1 i, 7 in which y calc i is the value of the ith datum calculated from that model. Optimized analyses for tin selenide and tin telluride were achieved by fitting the Dunham coefficients Y 01, Y 11, Y 21, Y 31, Y 02, and Y 12 as well as the BOB-correction terms Sn 01, Se 01, and Te 01. These latter parameters had to be included in the analyses for both component atoms in order to obtain a satisfactory quality of the least-squares fits while adjusting a minimum number of parameters. The overall d of the fits turned out to be for SnSe and for SnTe. Since the ratio B e 2 / e 2 is of the order of 10 8 for both SnSe and SnTe, the Dunham parameters Y 01, Y 11, Y 21, Y 31, Y 02, and Y 12 equal, to a very good approximation, the spectroscopic constants B e, e, e, e, D e, and e, respectively, through which the energy level expression of a vibrating rotor can be expressed as E v,j = G v + B e e v e v e v J J +1 D e + e v J J Tables III and IV summarize the results of the Dunham analyses performed for SnSe and SnTe, respectively. A comparison between the present Dunham constants for the main isotopologs 120 Sn 80 Se and 120 Sn 130 Te and those reported previously by Hoeft 12 and Hoeft and Tiemann 13 is presented in Table V. The Dunham constants obtained in the present work are considerably more precise, having their standard uncertainties reduced by two to three orders of magnitude. The present analyses also yielded precise values of the constants Y 21 for 120 Sn 80 Se and Y 31 for 120 Sn 130 Te, which 8

5 Rotational spectra, potential of SnSe and SnTe J. Chem. Phys. 126, TABLE IV. Dunham parameters, molecular parameters, and BOB-correction terms for 27 isotopolog of SnTe. Numbers in parentheses give one standard deviation in units of the last quoted digit.. Isotopolog x a % Y 01 MHz B e Y 11 MHz e Y 21 khz e Y 31 Hz e Y 02 khz D e Y 12 Hz e 120 Sn 130 Te Sn 128 Te Sn 130 Te Sn 128 Te Sn 126 Te Sn 130 Te Sn 128 Te Sn 126 Te Sn 130 Te Sn 126 Tc Sn 128 Te Sn 130 Te Sn 128 Te Sn 125 Te Sn 130 Te Sn 128 Tc Sn 125 Te Sn 126 Te Sn 130 Te Sn 124 Te Sn 128 Te Sn 124 Tc Sn 126 Te Sn 125 Te Sn 122 Te Sn 125 Te Sn 125 Te a Natural abundance % of the isotopolog. b Adopting 120 Sn 130 Te as the reference isotopolog. Sn 01 = khz b Se 01 = khz b resulted basically undetermined in the earlier investigations. Moreover, the new higher-order Y 12 constant is now determined with uncertainties less than 3% for both SnSe and SnTe. D. Direct potential fit The full Dunham-type treatment described in the previous subsection implies that the spectra of all SnSe and SnTe isotopic species may be fully explained in terms of the properties of a potential energy function and associated massdependent BOB radial strength functions. 23,24,28 In practice, the observed vibrational-rotational level energies of a given isotopolog are accurately described as eigenvalues of the following radial Schrödinger equation: 24,29 2 d 2 2 dr 2 + V ad 1 r + V ad r + 2 J J +1 2 r 2 1+g r v,j r = E v,j v,j r, in which V 1 ad is the total internuclear potential 23 for the reference isotopolog, V ad r is the difference between the ef- 9 TABLE V. Comparison between present and earlier Dunham constants of the most abundant isotopologs 130 Sn 80 Se and 130 Sn 120 Te. Numbers in parentheses give one standard deviation in units of the last quoted digit. 120 Sn 80 Se 120 Sn 130 Te Ref. 12 This work Ref. 13 This work Y 01 MHz Y 11 MHz Y 21 khz Y 31 Hz Y 02 khz Y 12 Hz

6 Bizzocchi et al. J. Chem. Phys. 126, fective adiabatic potentials for the isotopolog and for the reference species, and g r is the nonadiabatic centrifugal potential correction function for the isotopolog. Both V ad r and g r are written as the sum of two terms, one for each component atom, 28 V ad r = M A M S ad A r + M B A M S ad B r, B g r = M A R ad A r + M B R ad B r, M A M B 10a 10b where the empirical functions used to define R ad A/B r and S ad A/B r are presented below. A DPF of eigenvalue differences derived from Eq. 9 to experimental frequencies gives very compact sets of empirical parameters and radial correction functions which may be used to predict other system properties. 29 The DPF analyses reported herein were performed using Le Roy s DPOTFIT program. 30 The model used for the reference-isotopolog potential energy function of both SnSe and SnTe, V 1 ad r, isthe so-called extended Morse oscillator, or the EMO p potential form V EMOp r = D e 1 e y p r r e 2, 11 where D e is the well depth, r e is the equilibrium distance, and the exponent coefficient in Eq. 11 is expressed by the power series expansion, N EMOp = i y i p, i=0 12 in which the variable y p is a version of the generalized expansion variable of Šurkus et al., 31 y p r = rp p r e r p p + r. 13 e The expanded Morse oscillator model implemented in the DPOTFIT code allows us to use different polynomial orders for r in Eq. 12 at short range N S and long range N L, that is, when r r e and r r e, respectively. The resulting model function can thus be defined as EMO p N S,N L. Following Refs. 32 and 33, the radial strength functions for the potential energy and centrifugal BOB corrections are written in the forms X r = 1 y p r u X i y p r i + u X y p r, S ad R ad N i=0 N X r = 1 y p r t X i y p r i + t X y p r, i=0 14a 14b for atom X=A,B. The use of these forms makes it possible to account explicitly for the values of the radial BOB functions at r e and at the potential asymptote, which are for S ad X r, u N 0 and u N, respectively. A number of fits to the rotational data sets of SnSe and SnTe were carried out employing a range of EMO p N S,N L TABLE VI. Results of fits to the high-resolution rotational data of SnSe and SnTe using different models. Note that the adiabatic BOB radial functions associated with each EMO p N L,N L has the form S ad X = 1 y p u X 1 y p r. Model No. of fitted parameters d SnSe a d SnTe a Dunham fit EMO 1 0, EMO 1 3, EMO 2 3, EMO 3 3, EMO 1 4, EMO 2 4, EMO 3 4, EMO 1 5, EMO 2 5, EMO 3 5, a Dimensionless root-mean-square deviation of the fit. models for the potential energy function and the BOB radial strength functions; a summary of the results is presented in Table VI. Very recent multireference double-excitation configuration interaction MRDCI calculations 34,35 yielded ground-state D e values of cm 1 for SnSe and cm 1 for SnTe; thus, the DPF analyses carried out for both chalcogenides were performed with the well depth kept constrained at these ab initio values. Inspection of Table VI shows that a satisfactory fit d 1 to the present set of high-resolution data may be achieved for both SnSe and SnTe, adopting a number of different potential models. The quality of the fits generally degrades with increasing p, confirming that values of p 1, while helping make potential function more stable at large r, typically produce poorer fits. 28 It is worth noting that this behavior is not observed for the EMO p 5,5 models, whose d tends to its limiting value irrespective of the p adopted for representing the radial variable. For the purpose of comparison, Table VI includes also the results of DPF fits performed using the EMO 1 0,0 model, which correspond to a simple Morse-type function see Eqs The very poor fits obtained well illustrate the deficiency of the pure Morse model to reproduce the shape of the potential energy curve, even in the proximity of its minimum, while adopting the MRDCI D e values. Since it requires the same number of parameters of the corresponding Dunham analysis, we chose an EMO 1 4,4 potential and the associated p=1 BOB functions to represent the X 1 + electronic states of SnSe and SnTe. Higher p values, required for the correct long-range behavior of models that adopt a more flexible potential function, 36 are not necessary in the present case since no oscillatory behavior was observed. The nonadiabatic centrifugal BOB corrections had no effect in the fit, so that the g r function defined in Eq. 10b was omitted from the model. The resulting parameters are collected in Table VII. It is to be noted that the values of d for the direct potential fits are slightly larger than those of the corresponding Dunham fits; however, this is to be expected because these latter do not require the mechanical consistency between B v and the centrifugal distortion constants, which is naturally imposed in the DPF analysis.

7 Rotational spectra, potential of SnSe and SnTe J. Chem. Phys. 126, TABLE VII. Parameters defining the EMO 1 4,4 potential energy function and BOB radial function for the X 1 + state of SnSe and SnTe. Numbers in parentheses are one standard deviations in units of the last quoted digit. Note that the adiabatic BOB radial functions used have the form S ad X = 1 y 1 u X 1 y 1 r. IV. DISCUSSION A. General topics The vibrational constants Y 10 e and Y 20 e x e cannot be determined from a set of pure rotational data; however, assuming a Morse potential function, their values may be estimated from the experimentally derived Y 01, Y 11, and Y 02 parameters using the empirical relationships of Kratzer 37 and Pekeris, 38 whose forms are 39 Y 10 Y 20 4Y 3 01 Y 02 1/2 Y 11 3 Y02 SnSe, 15a + Y01 2 SnTe D e cm a b r e pm Sn u 1 cm Se/Te u 1 cm d a Ab initio value from Ref. 34. b Ab initio value from Ref b The results obtained for the most abundant isotopolog of SnSe and SnTe are presented in Table VIII; the quoted uncertainties were evaluated, applying the law of error propagation to Eqs. 15. The literature values of e and e x e derived from electronic spectroscopy 40 for the most abundant isotopolog 120 Sn 80 Se and 120 Sn 130 Te are also included in Table VIII for comparison. TABLE VIII. Dunham parameters Y 10 and Y 20, molecular parameters e and e x e for the most abundant isotopologs of SnSe and SnTe. 1 errors in parentheses are reported in units of the last quoted digit. Isotopolog Y 10 cm 1 e e cm 1 Expt. a Y 20 cm 1 e x e e x e cm 1 Expt. a 120 Sn 80 Se Sn 80 Se Sn 78 Se Sn 80 Se Sn 78 Se Sn 130 Te Sn 128 Te Sn 130 Te Sn 128 Te Sn 126 Te a Experimental values from Ref. 40. TABLE IX. Isotopically independent parameters U lm, Watson-type Born- Oppenheimer breakdown parameter X 01, and Born-Oppenheimer bond length r BO e for SnSe and SnTe. 1 errors in parentheses are reported in units of last quoted digit. Instead of using Eqs. 4 and 5, the multi-isotopolog data sets of SnSe and SnTe can also be represented, adopting the following level expression: U E v,j = lm l,m m+l/2 1+ m e A M A lm + m e B M lm B v l J J +1 m, 16 which was originally introduced by Ross et al. 41 The parameters X lm appearing in Eq. 16 are the Watson-type BOB terms, 24 which do not depend on the choice of the reference isotopic species; they are related to the X lm parameters through Eq. 6. The isotopically independent coefficients U lm, in turn, can be derived from the corresponding Dunham constants of the =1 reference isotopolog through the following relation: 23 U lm = m+l/2 1 Y 1 lm + A lm + B lm. 17 Although the form of Eq. 4 has proven to be preferable for a number of reasons, 23,24 this alternate factorization is still widely used. The isotopically independent coefficients U lm for SnSe and SnTe were calculated from the Y lm parameters of Tables III and IV, and are reported in Table IX. B. Potential function SnSe SnTe U 01 MHz u U 11 MHz u 3/ U 21 MHz u U 31 MHz u 5/ U 02 MHz u U 12 khz u 5/ U 10 GHz u 1/ U 20 GHz u A B BO r e pm Figure 3 shows the potential energy curve for the X 1 + electronic state calculated using the parameters listed for SnSe in Table VII. The inset with the expanded scale illustrates the part of the potential curve in the proximity of the equilibrium distance r e. Unlike Dunham s expansion, the DPF approach treats the dissociation energy D e as a fitting parameter. In the present case however, the spectroscopic data span only the lower part of the well depth as Fig. 3 illustrates ; thus, a direct determination of D e from the leastsquares fit may produce unreliable results. Furthermore, it has been recently pointed out 36 that the availability of a reliable value of the dissociation energy as a constraint in the DPF analysis is very important to ensure a realistic behavior of the potential function in the extrapolation region.

8 Bizzocchi et al. J. Chem. Phys. 126, FIG. 3. Color online Potential energy function determined for the X 1 + electronic state of SnSe, with energy levels indicating the experimental domain. By performing the DPF analysis to the SnSe and SnTe sets of data, treating the dissociation energy in the EMO 1 4,4 model as a free parameter, we obtained D e values of cm 1 for SnSe and cm 1 for SnTe. The corresponding theoretically computed MRCI values 34,35 are about 10% smaller, while the dissociation energies which can be extrapolated from the isotopically independent coefficients U lm of Table IX through the relation for a Morse oscillator, D e h U 2 10, 18 4U 20 are quite similar to the DPF values, yielding and cm 1 for SnSe and SnTe, respectively. These values can be thought as maximum limits of the corresponding dissociation energies if these data are unavailable from other sources. C. Internuclear distance The equilibrium rotational constants determined in the present work have been used to perform a reevaluation of the equilibrium internuclear distance r e for SnSe and SnTe. Due to the high accuracy of the experimental Y 01 parameters, considerable care must be taken in evaluating uncertainties in the derived structure and in interpreting them. This is particularly relevant in the present case because it is possible to present equilibrium internuclear distances up to ten figures. From the Y 01 values of Tables III and IV, equilibrium bond length for each isotopic species can be derived by the use of the equation r e = C 2 Y 01 1/2, 19 where is the reduced mass of the isotopolog and C 2 is the conversion factor C 2 = 1017 N A h 8 2 = pm MHz 1/2 u 1/2. 20 The coefficient C 2 has been evaluated using the fundamental constants recommended by CODATA in the 2005 paper of Mohr and Taylor. 42 The resulting r e for both tin chalcogenides are given in Table X; the quoted uncertainties are of the order of 0.01 ppm and simply reflect the uncertainties in the Dunham constants Y 01, C 2, and the atomic masses. 43 Inspection of Table X shows a variation in bond lengths with isotopolog over pm, which is nearly two orders of magnitude larger than their estimated errors and can be interpreted as a clear indication that the limit of the Born- Oppenheimer approximation has been reached. The isotopically independent, or Born-Oppenheimer bond lengths, r BO e, for SnSe and SnTe can be determined from the U 01 parameters given in Table IX using the equation of Bunker, 44 r e BO = N Ah 8 2 U The bond lengths obtained for SnSe and SnTe are reported in the last row of Table IX, with the quoted uncertainties reflecting the standard errors of the corresponding U 01 parameters. It is worth noting that the values of r BO e for SnSe and SnTe differ substantially from the r e distances derived by the direct potential fit and listed in Table VII. These latter, in turn, are placed approximately in the middle of the interval over which the r e, calculated using Eq. 19, are spread. This discrepancy was somewhat expected since the energy level expression Eq. 16 is based on the clamped nuclei potential energy function, 24 V CN r, instead of the effective adiabatic potential for the reference isotopolog V 1 ad r used in the present DPF analyses. D. Born-Oppenheimer breakdown Table IX reports the Watson-type BOB-correction terms X X 01 for Sn, Se, and Te derived from the corresponding 01 through Eq. 6. The coefficients X lm are adimensional, do not depend on the choice of the reference isotopolog, and thus allow for a direct comparison with results obtained for other molecules. Each BOB term can be explicitly separated into three contributions, 45 A 01 = A 01 ad + A 01 Dunh + A 01 nad = A 01 ad + D 1 Y g J B. 22 m e B e m p There is an equivalent definition for B 01.InEq. 22 A 01 ad is the total adiabatic correction to the Born-Oppenheimer approximation; the second term, called the Dunham term, depends on the Dunham correction to Y 01 which has the form 22 Y D 01 = B 3 e e 2 Y 10 2 Y 21 4Y a 1 Y 20 3Y 01 8a 1 6a a 1 3, 23 where a 1, the first expansion parameters of the Dunham potential, can be expressed in terms of the Dunham coefficients by a 1 =Y 11 Y 10 / Y The use of the Y lm constants of Tables III, IV, and VIII gives Dunham corrections of khz for 120 Sn 80 Se and khz for 120 Sn 130 Te.

9 Rotational spectra, potential of SnSe and SnTe J. Chem. Phys. 126, TABLE X. Equilibrium bond lengths derived for SnSe and SnTe isotopologs. 1 errors in perentheses are reported in units of the last quoted digit. AB r e pm AB r e pm 120 Sn 80 Se Sn 80 Se Sn 78 Se Sn 80 Se Sn 78 Se Sn 80 Se Sn 80 Se Sn 78 Se Sn 76 Se Sn 80 Se Sn 82 Se Sn 77 Se Sn 80 Se Sn 76 Se Sn 82 Se Sn 78 Se Sn 77 Se Sn 78 Se Sn 78 Se Sn 76 Se Sn 82 Se Sn 77 Se Sn 78 Se Sn 76 Se Sn 82 Se Sn 77 Se Sn 77 Se Sn 130 Te Sn 128 Te Sn 130 Te Sn 128 Te Sn 126 Te Sn 130 Te Sn 128 Te Sn 126 Te Sn 130 Te Sn 126 Te Sn 128 Te Sn 130 Te Sn 128 Te Sn 125 Te Sn 130 Te Sn 128 Te Sn 125 Te Sn 126 Te Sn 130 Te Sn 124 Te Sn 128 Te Sn 124 Te Sn 126 Te Sn 125 Te Sn 122 Te Sn 125 Te Sn 125 Te The third term of Eq. 22 represents the nonadiabatic contribution to the Born-Oppenheimer breakdown: 1 g J B, g J being the rotational g factor, is the isotopically independent value of 1 g J referred to nucleus B as the origin, 45 1 g J B = 1 g J + 2c Am p M A. 24 M A + M B Here, m p is the proton mass and c A is the formal charge on the atom A: c A = D /er e, with D being the dipole moment and c A +c B =0. Experimental values for the dipole moment 14 and for the rotational g factor 15 are available for both SnSe and SnTe; thus, the nonadiabatic terms X 01 nad can be readily evaluated. Using Eq. 22 and the experimentally derived values of the X 01 for SnSe and SnTe, all the contributions to the BOB correction were evaluated; the results are collected in Table XI. It is very difficult to obtain an independent check of the values of the adiabatic BOB terms X 01 ad derived in this TABLE XI. Contribution to the Born-Oppenheimer breakdown parameters for SnSe and SnTe. errors in parentheses are reported in units of the last quoted digit. X X/AB 01 X 01 Dunh X 01 nad X 01 ad Sn/SnSe Sn/SnTe Se/SnSe Te/SnTe way; nevertheless, their order of magnitude can be estimated from the g J factors. Using the wobble-stretch theory 46 and 2 the estimation of the matrix element of the operator Lˆ x +Lˆ 2 y, 45 one obtains an upper limit for X 01 ad, X 01 ad 1 2 Y 1 02 E 2 2hY 01 Z A 2 M + Z B 2 A M g J 25 B 1 m p, where E is the energy difference between the electronic ground state and the first 1 excited state, and Z A and Z B are the nuclear charges of atoms A and B, respectively. Using experimental values of E, 40,47 Eq. 25 gives values of for SnSe and for SnTe, confirming that the experimentally derived values of X 01 ad presented in Table XI are of the right order of magnitude and the finite volume effect of the nuclei is entirely negligible for both tin chalcogenides. E. Spin-rotation constants The newly determined spin-rotation coupling constants of the dipolar nuclei 117 Sn, 119 Sn, 77 Se, and 125 Te for the molecules SnSe and SnTe are given in Tables I and II, respectively. For a diatomic molecule the spin-rotation constant C I i due to the nucleus i is proportional to the product g i B, where g i is the nuclear g factor of the nucleus and B is the rotational constant. Table XII collects the values of the ratio C I i /g i B 0 for the different isotopologs of the SnSe and

10 Bizzocchi et al. J. Chem. Phys. 126, TABLE XII. Value of the ratio C I i /g i B 0 for various SnSe and SnTe isotopologs. 1 errors in parentheses are reported in units of the last quoted digit. AB C I Sn /g Sn B C I Se /g Se B AB C I Sn /g Sn B C I Te /g Te B Sn 80 Se Sn 130 Te Sn 80 Se Sn 128 Te Sn 77 Se Sn 130 Te Sn 78 Se Sn 128 Te Sn 77 Se Sn 125 Te Sn 78 Se Sn 125 Te Sn 77 Se Sn 126 Te Sn 76 Se Sn 125 Te Sn 82 Se Sn 125 Te Sn 77 Se Sn 125 Te Sn 77 Se Average Average SnTe molecules: as expected, there is a good general agreement within each set of data, reflecting the similarity in the electronic structure between isotopologs. In a diatomic molecule, C I i for a given nucleus i can be expressed as the sum of nuclear and electronic contributions, 48,49 C I i = C i nucl + C i elec. 26 The nuclear term depends only on the nuclear position and can be evaluated from the molecular geometry using the relation: 49 C nucl i = 0 e 2 g i BZ 4 m p R, 27 where 0 is the vacuum permeability, e is the elementary charge, m p is the proton mass, g i is the nuclear g factor, B is the rotational constant, R is the internuclear separation, and Z is the atomic number of the second nucleus. In Eq. 27 all the quantities are expressed in Systeme International SI units. From the experimental values of the spin-rotation constants, both nuclear and electronic parts of C I i are calculated through Eqs. 26 and 27 and are reported in Table XIII. Following the theory depicted by Flygare, 49 and neglecting any relativistic effect, these results may be used to extract information on the magnetic shielding of SnSe and SnTe. i The average magnetic shielding av determines the chemical shift which is obtainable from NMR measurements; this parameter is divisible into two parts, 49 i av = i d + i p. 28 For a particular nucleus in a diatomic molecule, the paramagnetic contribution i p is directly proportional to C elec i, 49 m p elec i C i p = 3m e g i B, 29 where m e is the electron mass and the other quantities have been defined previously. The diamagnetic part i d in turn, is related to the nuclear contribution to the spin-rotation constant by 49 m p nucl i d = atom C i d 3m e g i B, 30 where atom d is the free atom diamagnetic shielding, which can be estimated by theoretical calculations and is usually obtainable from tables. However, the reliability of the derived i d depends mostly on that of i d atom and care must be taken in using tabulated values of the latter since typically the calculations do not take into account relativistic effects, which become appreciable for atom having Z 30. Nonrelativistic calculations 50 give values of 5086, 2998, and 5362 ppm for Sn, Se, and Te, respectively, while a successive relativistic Hartree-Fock-Slater computation 51 quoted by Fuller 52 gives corrected i d atom of 6203, 3327, and 6639 ppm respectively. However, owing to the inherent uncertainty of these atomic values, the resulting values of the derived shielding constants should be viewed cautiously. TABLE XIII. Nuclear and electronic contributions to the experimental spin-rotation constants and corresponding paramagnetic shielding for various isotopologs of SnSe and SnTe. Sn X a Isotopolog C Sn khz C nucl Sn khz C elec Sn khz p ppm C X khz C nucl X khz C elec X khz p ppm 119 Sn 80 Se Sn 80 Se Sn 77 Se Sn 130 Te Sn 130 Te Sn 125 Te a In this table X=Se and Te.

11 Rotational spectra, potential of SnSe and SnTe J. Chem. Phys. 126, TABLE XIV. Contributions to magnetic shieldings and shielding spans of the nuclei in SnSe and SnTe. Sn X a Isotopolog p ppm d ppm av ppm ppm p ppm d ppm av ppm ppm 119 Sn 80 Se Sn 80 Se Sn 77 Se Sn 130 Te Sn 130 Te Sn 125 Te a In this table X=Se and Te. On the other hand, the span of the shielding tensor, which is a measure of its asymmetry, is independent of d i atom and should be reliable to within a few ppm. For a diatomic molecule the span is given by 49,53 i = i p + m p C i 2m e g i B, 31 where the span for the nucleus i, i i, is defined by i, with i and i being the components of the shielding tensor parallel and perpendicular to the molecular axis, respectively. 54 i p is a relatively small correction term known as the quadrupole term, 55,56 which can be neglected at the current level of approximation. Combining the relations and using Eq. 31, one can evaluate the magnetic shielding, including the diamagnetic and paramagnetic parts, and the corresponding span. All these results are reported in Table XIV. V. CONCLUSIONS The rotational spectra of SnSe and SnTe in their 1 + electronic ground state have been measured in the 5 24 GHz frequency range using a cavity pulsed-jet Fourier-transform microwave spectrometer. The collected data were analyzed in a global multi-isotopolog fit, resulting in accurate rovibrational Dunham constants for 27 isotopologs of both chalcogenides. Correction terms due to the breakdown of the Born-Oppenheimer approximation have been found to be significant for the constant Y 01 B e. Direct potential fits to the present sets of high-resolution data provide analytic potential functions for the X 1 + electronic states of SnSe and SnTe. The Dunham analysis yielded Watson-type BOB parameters Sn 01 and Se 01,or Sn 01 and Te 01, which are very similar in magnitude for both molecules, indicating that the field shift effect arising from the finite size of the moderately heavy Sn and Te nuclei is entirely negligible. In addition a very precise evaluation of the equilibrium bond distances r e and isotopically independent Born- Oppenheimer bond length r BO e have been derived. The hyperfine splitting produced by the magnetic coupling due to the I= 1 2 nuclei 117 Sn, 119 Sn, 77 Se, and 125 Te has been observed, yielding novel determinations of the corresponding nuclear spin-rotation constants. These magnetic hyperfine data have been used to estimate NMR shielding parameters of both tin chalcogenides. ACKNOWLEDGMENTS The authors are grateful to R. J. Le Roy for providing us a modified version of his DPOTFIT 1.0 code. One of the authors L.B. thanks the University of Bologna for financial support Marco Polo Project. Another author J.-U.G. gratefully acknowledges support from the Deutsche Forschungsgemeinschaft DFG and the Land Niedersachsen. 1 G. D. Rochester and H. G. Howell, Univ. of Durham Phil. Soc. Proc. 9, J. W. Walker, J. W. Straley, and A. W. Smith, Phys. Rev. 15, R. F. Barrow, Proc. Phys. Soc. London 52, R. F. Barrow and E. E. Vago, Proc. Phys. Soc. London 55, R. F. Barrow and E. E. Vago, Proc. Phys. Soc. London 56, D. Sharma, Proc. Natl. Acad. Sci. India A 14, D. Sharma, Nature London 157, E. E. Vago and R. F. Barrow, Proc. Phys. Soc. London 58, C. P. Marino, J. D. Guerin, and E. R. Nixon, J. Mol. Spectrosc. 51, L.-S. Wang, B. Niu, Y. T. Lee, and D. A. Shirley, J. Chem. Phys. 92, R. Colin and J. Drowart, Trans. Faraday Soc. 60, J. Hoeft, Z. Naturforsch. A 21A, J. Hoeft and E. Tiemann, Z. Naturforsch. A 23A, J. Hoeft, F. J. Lovas, E. Tiemann, and T. Törring, Z. Naturforsch. A 24A, R. Honerjäger and R. Tischer, Z. Naturforsch. A 32A, E. Tiemann, H. Arnst, W. U. Stieda, T. Törring, and J. Hoeft, Chem. Phys. 67, T. J. Balle and W. H. Flygare, Rev. Sci. Instrum. 52, J.-U. Grabow and W. Stahl, Z. Naturforsch., A: Phys. Sci. 45, J.-U. Grabow, W. Stahl, and H. Dreizler, Rev. Sci. Instrum. 67, J.-U. Grabow, E. S. Palmer, M. C. McCarthy, and P. Thaddeus, Rev. Sci. Instrum. 76, H. M. Pickett, J. Mol. Spectrosc. 148, J. L. Dunham, Phys. Rev. 41, R. J. Le Roy, J. Mol. Spectrosc. 194, J. K. G. Watson, J. Mol. Spectrosc. 80, R. J. Le Roy, DPARFIT 3.3, a computer program for fitting multi-isotopolog diatomic spectra; University of Waterloo Chemical Physics Research Report No. CP-660, W. U. Stieda, Ph.D. thesis, Freie Universität Berlin, Germany, See EPAPS Document No. E-JCPSA for a table of all observed rotational transitions of 27 SnSe and SnTe isotopologs. This document can be reached via a direct link in the online article s HTML reference section or via the EPAPS homepage epaps.html. 28 R. J. Le Roy, D. R. T. Appadoo, K. Anderson, and A. Shayesteh, J. Chem. Phys. 123, J. K. G. Watson, J. Mol. Spectrosc. 223, R. J. Le Roy, J. Y. Seto, and Y. Huang, DPOTFIT 1.0, a computer program for fitting diatomic molecular spectra to potential energy functions; Uni-