Sub-Doppler Resolution Infrared Molecular-beam Spectroscopy Stark Effect Measurement of the Dipole Moment of Hydrogen Fluoride and Hydrogen Cyanide in Excited Vibrational States BY T. E. GOUGH, R. E. MILLER * AND G. SCOLES~ Guelph-Waterloo Centre for Graduate Work in Chemistry, Department of Chemistry, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 Received 19th January, 1981 Sub-Doppler beam-calorimetric infrared spectroscopy is used to determine the dipole moments of hydrogen fluoride and hydrogen cyanide in vibrationally excited states. For the former pi = 1.872 f 0.003 D while for the latter pool = 3.012 * 0.002 D. The results for hydrogen fluoride are used to revise the literature value of p e for this molecule to 1.803 * 0.002 D. In recent years there has been a considerable amount of interest, both experimental and theoretical, in obtaining the dipole moments of diatomic molecules expressed as Taylor series in their internuclear separations.In particular, hydrogen fluoride has received much attention because of its use in chemical lasers: these lasers operate on several vibrationally excited states so that the computation of expected gain co- efficients requires knowledge of the dipole-moment function at internuclear distances appreciably removed from equilibrium. The experimental determination of the dipole-moment function involves the measurement of dipole-moment matrix elements. Off-diagonal dipole-moment matrix elements may be obtained from the intensities of vibration-rotation transitions, while the diagonal elements obviously involve measurements of the dipole moments of vibrationally excited molecules.The latter results have traditionally been obtained from Stark-split radiofrequency and microwave spectroscopy. To date the most precise measurements have been made using the molecular-beam electric resonance technique. Using such techniques Muenter and Klemperer,' and later Muenter,' have measured the dipole moment of hydrogen fluoride in its ground vibrational state as po = 1.826 18 D. In these experiments excited vibrational states were not sufficiently populated to allow measurement of the vibrational dependence of the dipole moment. A direct extrapolation to determine p e , the dipole moment at the potential-energy minimum was, therefore, not possible. Accordingly, Muenter and Klemperer' measur.ed puo for deuterium fluoride as 1.8881 D, and after assuming that the isotopic substitution does not modify the electronic structure evaluated pe, for both species, as 1.7965 D.In a related paper, Kaiser3 has cautioned against such an application of the Born-Oppenheimer approximation, showing that p e for hydrogen chloride is The uncertainty in p e is important because Sileo and Cool4 have made extensive * Physics Department, University of Waterloo, Waterloo, Ontario, Canada N2L 3G I . Present address : Research School of Physics, Australian National University, Canberra, ATC, Australia. t Also Physics Department, University of Waterloo, Waterloo, Ontario, Canada N2L 3G 1. D greater than p, for deuterium chloride.78 SUB-DOPPLER RESOLUTION SPECTROSCOPY F- CENTRE LASER DETECTORS measurements of the intensities of infrared emission from several vibrationally excited states of hydrogen fluoride. These results allow evaluation of the first derivatives of the dipole moment with respect to internuclear distance; however, in order to complete characterisation of the dipole moment function a value of ,ue must be supplied.In the present study we use sub-Doppler beam-calorimetric spectroscopy (the laser bolometric method) to study the infrared spectrum of hydrogen fluoride. In par- ticular, the measurement of Stark splittings of the R l component of the funda- mental vibrational transition of hydrogen fluoride is used to obtain a value for the dipole moment of this molecule in its u = 1 state, with an accuracy comparable to conventional microwave techniques.From this value of ,ul and the literature value of ,uo an improved value of pe is obtained. Similar experiments performed on the v3 transition of hydrogen cyanide will be reported but not analysed in detail. ----+ EXPERIMENTAL APPARATUS A N D RESULTS The apparatus used in the present study has been previously discussed in some detaiL5-’ Therefore, only significant modifications implemented since those reports are presented here. The infrared spectrum of the molecular beam is obtained using a 2 K doped silicon bolometer to measure Fig. 1 shows a schematic diagram of the present set-up. I Kr’ LASER FIG. 1 .-Schematic diagram of the apparatus used for sub-Doppler beam-calorimetric spectroscopy.T .E. GOUGH, R. E. MILLER AND G . SCOLES 79 energy imparted to the beam by an appropriately tuned infrared laser. The N.E.P. of the bolometer is W Hz-+ so that the apparatus is sufficiently sensitive to detect a flux of lo7 excited molecules per second impinging upon the bolometer. Further- more, the amount of scattered laser radiation reaching the bolometer is negligible, so that the laser and bolometric detector are only coupled through the excited molecules of the beam. As a result, the background problems which plague most traditional absorption experiments are eliminated. The sensitivity of the experiment allows the use of a highly collimated molecular beam which, when crossed orthogonally by laser radiation, provides a spectral resolution far superior to that obtainable in bulk gas experiments. Thus, the observed linewidth in the present experiments was 1.5 MHz, whereas the Doppler width for the corresponding transitions in a gas sample at room temperature is 330 MHz.The 1.5 MHz linewidth arises, with comparable contribu- tions, from the finite divergence of both molecular and laser beams, from the finite time spent by the molecules in the laser radiation, and from laser instabilities. The laser used in the present work was an F-centre laser (Burleigh FCL-20) pumped by a krypton-ion laser (Spectra Physics 171). The details of the operation and performance of F-centre lasers have been well-documented in the literature8e9 and will not be discussed here. A gas cell containing the gas of interest was used for preliminary location of the desired transition and for alignment of the scanning controls of the FCL-20.An hermetically sealed, temperature stabilised confocal etalon (Burleigh CF-500 P) having a free spectral range of 150 & 0.5 MHz was used to provide frequency markers for the calibration of Stark splittings. Because of the corrosive nature of hydrogen fluoride, the molecular-beam source, gas-handling system and gas cell were all constructed from monel. Sapphire windows were used in the gas cell. The molecular beam was formed by expanding a 1.6% hydrogen fluoride in helium mixture through a 35 pm monel nozzle at room tempera- ture. The stagnation pressure in the nozzle was adjusted so as to maximise the observed strength of the R l component of the fundamental vibrational transition of hydrogen fluoride, and was ca.5 atm. The intersection point between laser and molecular beams was surrounded by FIG. 2.-Experimental spectrum showing the Stark splitting of the R1 transition of hydrogen fluoride induced by an applied field of 46 kV cm-'. The upper trace provides frequency markers at 150 MHz intervals. polished stainless-steel Stark electrodes, 4 mm apart and 5 cm in diameter. The laser radiation was polarised along the direction of the molecular beam and perpendicular to the direction of the applied electric field. In such a configuration allowed transi- tions must satisfy the selection rule AM = & l . The temporal and spatial homo- geneity of electric field were such that the components of the Stark-split spectrum were not measurably broader than was the zero-field spectrum. Fig.2 shows the Stark splitting of the R1 transition of hydrogen fluoride produced80 SUB-DOPPLER RESOLUTION SPECTROSCOPY by an applied electric field of 46 kV cm-', together with the frequency derivative of the calibrating etalon's transmission function. The FCL-20 was scanned through the spectrum by simultaneously ramping, uia piezoelectric transducers, the lengths of the laser cavity and of the intracavity mode-selecting etalon. No attempt was made to stabilise the laser cavity thermally so that the frequency scan rate varied between and during runs. However, as will be shown below, the experimental quantity of most interest is the ratio of line splittings ( y -. z)/(x - y ) , where x, y and z are as defined in fig.2. Since this quantity is ca. 2 the applied electric field was adjusted until y - z 21 300 MHz (x - y E 150 MHz) corresponding to two (and one) free spectral ranges of the calibrating etalon. The bias on this etalon was then adjusted to bring spectral peaks and calibration markers into approximate coincidence. Because of pen offsets this coincidence is not immediately apparent in fig. 2. Spectral line positions were then measured relative to the closest calibration markers. Such an approach greatly improved the precision to which measurements could be made. Experimentally it was found for the R l component of the fundamental of hydrogen fluoride that 01 - z)/(x - y ) = 2.136 & 0.009 while for the R1 component of the v3 fundamental of hydrogen cyanide ( y - z)/(x - y ) = 2.305 0.006.The errors quoted are the standard deviations obtained from an analysis of twenty separate scans of each spectrum. These errors are generated by scanning errors and drifts in the applied Stark field. ANALYSIS OF D A T A The present experiments investigate the effects of an applied electric field upon the R1 component of parallel vibrations of linear molecules. To the precision of these experiments, their analysis must consider the second and fourth order interactions between the electric dipole moment of the molecule and the electric field, and the second-order interaction between the anistropy of the polarisability of the molecule and the electric field. The dominant interaction is the second-order dipole-applied- field interaction, the two remaining interactions providing corrections comparable to the standard deviation of the experiments.Thus, in an applied electric field, the electrical energies of the J , M levels are primarily determined by eqn (1)'O where p u and B, are, respectively, the dipole moment and rotational constant of the vibrational state u. Fig. 3 shows the situation for an R1 transition with the selection rules appropriate to the present experiments. The transitions x, y and z of fig. 2 are identified in fig. 3. Eqn (1) was used, neglecting the vibrational dependence of the dipole moment, to calculate, in terms of E2, the splitting x - y . The measured value of this splitting could then be used to calculate approximate values of the applied electric field to be used when making fourth-order corrections.Since these corrections are small, 10% accuracy in the electric field is more than sufficient. For hydrogen fluoride a field of 46 kV cm-' was used while for hydrogen cyanide the field was 7.8 kV cm-', these fields being such as to make x - y ca. 150 MHz for each molecule.T. E . GOUGH, R . E . MILLER AND G . SCOLES 81 The energy levels calculated from eqn (1) must be modified by the effects of the anistropic polarisability of the molecule according to eqn (2) 2J2 + 2J - 1 - 2M’ (2J + 3) (2J - 1) 2 &Jp,Jf = - where (al1 - axl) is the anisotropy in the polarisability of the molecule. It is not necessary to consider the vibrational dependence of this quantity because E ~ ~ , ~ is much less than ~$2.5,~.Because both E $ ~ : ~ , ~ and E : , ~ depend on E2 it is not necessary to know the applied electric field when computing the ratio (y - z)/(x - y ) . The only AM=? 1 ,- I I I I v = o I < J = l 0 -i1 - 2 2 -0 FIG. 3.-Energy levels and transitions corresponding to the experimental spectrum shown in fig. 2. unknown quantity appearing in such a calculated ratio is p1 which may thus be evalu- ated by equating the computed and experimental values for this ratio. The fourth-order contribution to the energies of the relevant states rJPM was calculated from eqn (3)” using the approximate value of E determined from the splittingx - y [(J + 1)2 - M2][(J + 212 - M’] (2J + 1) (2J + 5)(J + 1)2(2J + 3)3 [(J - 1)2 - M 2 ] [ J 2 - M2] [J’ - M2][(J + 1 - 144’1 4- (2J - 3)(2J + 1)J2(2J - 1)3 -I- (2J - 1) (2J 4- 3)(2J + 1)2J2(J 4- [J2 - M2I2 + ( 2 ~ + 3)2(2~ 3- i ) y ~ + 113 ( 2 ~ + 1 ) 2 ( 2 ~ - 1 ) ’ ~ ~ - [(J+ 1)2 - M2]’ Once again, the vibrational dependence of p and B may be neglected in making such corrections to the energy levels, In table 1 are listed the parameters taken from the literature used in the computa- tions, the uncorrected values of the excited-state dipole moments, the two corrections applied, and the final experimental values of ,ul for hydrogen fluoride and pool for hydrogen cyanide.The standard deviations on these dipole moments arise entirely from the experimental uncertainties in the ratio ( y - z)/(x - y ) . The precision of the present molecular-beam infrared measurements is comparable with that attainable from microwave spectroscopy of bulk gas samples.82 SUB-DOPPLER RESOLUTION SPECTROSCOPY We have performed experiments in which the FCL-20 was locked to the CF-5OOP etalon which was then ramped.This technique generated much more reproducible scans because of the high stability of the etalon ( < I MHz drift in 15 min). Un- fortunately a second etalon was not available to us so these scans could not be calibrated. Given such an etalon, we estimate that the precision of the present experiments could be improved by one order of magnitude. TABLE 1 .-EXPERIMENTAL DIPOLE MOMENTS OF VIBRATIONALLY EXCITED HYDROGEN FLUORIDE AND HYDROGEN CYANIDE AND THE MOLECULAR CONSTANTS USED IN THEIR EVALUATION hydrogen ff uoride hydrogen cyanide Bo B1 PO 0111 - 011 E Pl (uncorrected) correction for eqn (2) correction for eqn (3) P I 20.5602 cm-' l 9 19.7862 cm-' l 9 1.826 18 D2 0.220 i 0.006 A3 46 kV cm-' 1.8716 D +0.0006 D -0.0003 D 1.872 i 0.003 D Boo0 Boo 1 Po00 c( - E Po0 1 (uncorrected) correction for eqn (2) correction for eqn (3) Po0 1 1.4782 cm-l *O 1.4678 cm-' 2o 2.98459 D 7.8 kV cm-I 3.0199 D 1.0 A3 21 +0.0001 D -0.0078 D 3.012 & 0.002 D DISCUSSION (a) EXPERIMENTAL DIPOLE-MOMENT FUNCTION OF HYDROGEN FLUORIDE In the past decade, a considerable amount of effort has been expended in obtaining an accurate parameterization of the dipole-moment function of hydrogen fluoride.The most common expression of the dipole moment in terms of the internuclear separation has been that of a truncated Taylor series expansion about the equilibrium separation.Making use of such an expansion, the dipole moment in a vibrational state u can be written12*13 where pe is the equilibrium dipole moment and Ml and M2 are the first and second derivatives of ,u with respect to the deviation from the equilibrium separation. Eqn (4) is obtained by neglecting higher-order terms in BJco,. This ratio for hydrogen fluoride is 0.005 1. In fig. 4 are plotted the present value for p l and the molecular-beam electric resonance value for p0. Extrapolation to (u + +) = 0 gives a value of pe = 1.803 & 0.002 D, which should be compared with the literature value of pe = 1.7965 D.' This latter value lies well outside the range of the results of the present experiment, and was obtained by assuming that the difference in p o between hydrogen fluoride and deuterium fluoride is entirely due to their difference in zero-point energies, both species being assumed to share the same electronic potential well.Making this Born-Oppenheimer approxima- tion allows pe to be evaluated by an extrapolation through the two v = 0 levels to the Eqn (4) predicts that p" should plot linearly against (u + 3).T. E . GOUGH, R. E . MILLER AND G . SCOLES 83 minimum of the potential well. by using an effective (u + 4)eff which is found by solving This extrapolation may be cast in the form of fig. 4 The extrapolation is shown in fig. 4 as a dotted line. If the Born-Oppenheimer approximation holds, the dashed and full lines in fig. 4 should coincide. This is clearly not the case, as might be anticipated from the work of Kaiser3 on hydrogen and deuterium chlorides. Kaiser showed that the plot of pc, against (v + +)eff for deuterium chloride lay approximately D below the corresponding plot for hydro- gen chloride. From the solid line in fig.4 the anticipated dipole moment at FIG. 4.-Experimentally 1.79' I I I 0.5 1 .o 1.5 ( 0 + + > c r f . pfotted against (v + +)cff. measured dipole moments for (0) hydrogen and (0) deuterium fluorides (v + +)eff = 0.36, equivalent to the vibrational ground state of deuterium fluoride, is 1.820 D, D greater than the experimental value for deuterium fluoride of 1.8 18 8 1 D.l The implication is clear that the appropriate plot for deuterium fluoride should lie This is in accord with calculations by Schlier l4 which suggest that the largest perturbation resulting from the breakdown of the Born-Oppenheimer approximation is independent of vibrational quantum number.Sileo and Cool have made extensive measurements of infrared emission intensities for hydrogen fluoride and deuterium fluoride. However, because the infrared intensity measurements give no information on pe, these authors adopted the literature value, which we have now found to be inaccurate, in order to complete their character- isation of the dipole moment function. The quantity (po - pe) from infrared intensity measurements was found to be 0.023 D while the presumably more accurate molecular-beam electric resonance value for (po - pe) was 0.030 D. In table 2 we show predicted values for po and p1 of hydrogen fluoride obtained from the Sileo and Cool derivatives in conjunction with the literature and present values for pe.The excellent agreement between experiment and predictions based on the present value for pe strongly supports adoption of this new value, and shows that Sileo and Cool's intensity measurements are more reliable than previously appeared to be the case. below the plot for hydrogen fluoride.84 SUB-DOPPLER RESOLUTION SPECTROSCOPY TABLE 2.-EXPERIMENTAL DIPOLE MOMENTS OF HYDROGEN FLUORIDE, AND VALUES CALCULATED FROM INFRARED EMISSION INTENSITIES USING THE LITERATURE AND PRESENT VALUES OF ,!fe experiment calculated calculated P e P 1.796" 1.803 POID 1.826" 1.819 1.826 P I D 1.872 1.865 1.872 Ref. (1) and (2); present work. (b) COMPARISON WITH a priori CALCULATION The a priori calculation of molecular electric dipole moments has proved to be a difficult task.In recent years, however, considerable progress has been made so that, at least for small molecules, agreement between experiment and theory can be considered satisfactory. Because of the availability of Sileo and Cool's extensive infrared emission intensity data, hydrogen fluoride has been the subject of many theoretical calculations of dipole moment functions. Recently Werner and R o ~ m u s ' ~ have performed several ab initio calculations of the dipole moment functions of hydrogen fluoride, comparing their results to those of previous calculation^^^*^^ and of e~periment.~ Werner and Rosmus concluded that MCSCF and SCEP/CEPA calculations predict spectroscopic constants and dipole-moment functions with com- parable accuracy, each method giving deviations of a few percent.In such a context our revised value of pe is of little significance, although its inclusion may cause one to favour the MCSCF method. In an earlier paper Werner and Rosmusls reported SCEP/CEPA calculations using a slightly different basic set. These calculations agreed with the experimental dipole-moment function of hydrogen fluoride to within 0.3% for the first few vibra- tional states. Incorporating the present experimental correction to p e improves the agreement to 0.1 % which is somewhat better than our experimental uncertainties. This agreement, although impressive, is of course dependent upon the choice of basis set. This situation is not satisfactory for ab initio theoreticians but does suggest the possibility of optimising a basis set to obtain agreement with available experi- mental data, and then using this basis set to predict dipole-moment functions in less experimentally accessible regions. This work was carried out with the financial support of the Natural Sciences and The experiments were performed with the Engineering Research Council of Canada.assistance of D. A. Gravel. J. S. Muenter and W. Klemperer, J . Chem. Phys., 1970,52,6033. J. S. Muenter, J. Chem. Phys., 1972, 56, 5409. E. W. Kaiser, J . Chem. Phys., 1970, 53, 1686. R. N. Sileo and T. A. Cool, J . Chetn. Phys., 1976, 65, 1 17. T. E. Gough, R. E. Miller and G. Scoles, Appl. Phys. Lett., 1977, 30, 338. T. E. Gough, R. E. Miller and G. Scoles, J . Mol. Spectrosc., 1978, 72, 124. ' T. E. Gough, R. E. Miller and G. Scoles, J . Chem. Phys., 1978, 69, 1588. L. F. Mollenauer and D. H. Olson, J . Appl. Phys., 1975, 46, 3019. R. Beigang, G. Liffin and H. Welling, J . Opt. Soc. Am., 1978, 68, 636. lo C. H. Townes and A. L. Schawlow, Microwave Spectroscopy (McGraw Hill, New York, 1955). l1 J. H. Scharpen, J. S. Muenter and V. W. Laurie, J. Chem. Phys., 1967,46, 2431. l2 A. D. Buckingham, J. Chem. Phys., 1962, 36, 3096. l3 R. M. Herman and S. Short, J . Chem. Phys., 1968, 48, 1266. l4 C. Schlier, Fortschr. Phys., 1961, 9, 455.T. E . GOUGH, R. E . MILLER AND G . SCOLES l5 H. J. Werner and P. Rosmus, J. Chem. Phys., 1980,73, 2319. l6 W. Meyer and P. Rosmus, J. Chem. Phys., 1975, 63, 2356. l7 R. D. Amos, Mol, Phys., 1978, 35, 1765. H. J. Werner and P. Rosmus, J. Mol. Strucf., 1980, 60, 405. l9 D. V. Webb and K. N. Rao, J. Mol. Spectrosc., 1968, 28, 121. 2o D. H. Rank, D. P. Eastman, B. S. Rao and I. A. Wiggins, J. Opt. Suc. Am., 1961, 51, 929. 21 G . Tomasevich, Ph.D Thesis (Harvard University, 1972). 85