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Electric dipole moment of SrF X 2Σ+ from high-precision stark effect measurements
Volume 113,number4
CHEMICAL PHYS!CS LETTERS
25 Januari 1985
ELECTRIC DIPOLE MOMENT OF SrF X 2Z+ FROM HIGH-PRECISION STARK EFFECT MEASUREMENTS W.E. ERNST, J. R&DLER, S. KINDT and T. ToRRJNG Freie Universitdt Berlin. FB Physik. Amimallee 14. D-1000 Berlin 33. West Gernimy Received 26 October 1984
The ground4ate dipole moment of SrF was measured by the molecular-beam laser-microwave double-resonance metiod.From the study oftw~vibrationalstatesthecoeffrcicntsinthenbrationalexpansion~=Cl~+ ~~(~+1/2)weredetermined as pe = 3.4676(10) D and PI = 0.0575(113) D (wirh the statistical standard deviation given in parentheses). The measured values arc compared with the results of an ionic bonding model.
l_ Intioduction The alkaline earrh monohalides represent a class of highly ionic radicals. As the structure of the aho& halides, another group of ionic molecules, can be well described by the ionic bonding models developed by Rittner [ 13 and Brumer and Karplus (“T-r(ittner” model) [2] attempts were made to apply these models to one of the :adicals, CaCl [3,4] _ The two models yielded completely different values for the electnc dipole moment and au estimate from beam deflection measurements [4] was the only experimental value for comparison with +Ae calculations. Recently, highly precise Stark effect measurements of mdrvidual hyperfiie components of ground-state rotational transitions of CaCl and CaBr were performed using the molecular-beam laser-microwave (mw) double-resonance technique [5,6] _Electric dipole moments wem derived at OS% accuracy for the first tune and similar molecular-beam laser-rf double-resonance measurements on CaF by Childs et al. [7] yielded the corresponding value for this radical- Neither the Rittner nor the T-Rittner model could reproduce all measured values- The breakdown of these models for the alkaline earth monohalides was explained iu ref- [8] and an improved model was proposed. The spectroscopic ground-state constants and dipole moments predicted by this inodel agree with the measured values within 5 to 10%. 0 009-2614/85/S 03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
The purpose of this work was to increase the number of experimentally determined dipole moments to be compared with the predictions. The largest drscrepancies between dipole moments derived from the new model and the T-Rittner model were found for the alkaline earth monofluorides [8] which can therefore provide a critical test for their validity. Strontium monofluoride was chosen for the Stark effect measurements because precrse spectroscopic constants were known for the optical B 2Z+-X 2Zf transition [9] and the ground-state rotational and hyperfme structure had been investigated in independent mw experiments by Schiitze-Pahhnann et al. [lo] and molecular-beam laser-rf double-resonance studies by Childs et al. [ll]_ In this paper we report on investigations of the Stark effect in the vibrational states u = 0 and 1 of the 22+ ground sta te of SrF. The shift of hyperfme tranitions in an electric field was measured by applying the molecular-beam laser-mw double-resonance method.
2. Experimental Details of the apparatushave been described in refs. [ 12,131. An effusive SrF beam was generated from a high-temperature reaction of SrF2 and Sr (1000 < T < 1200°C) taking place in a ceramic crucible surrounded by a resistively heated tantalum cylinder. Light from 351
Voiume 113. number 4
CHEMICAL
PHYSICS
a single frequency dye laser operating around 579 nm was split into a pump beam of 250 mW and a probe beam of 10 mW. The pump beam was used to deplete a partrcular ground-state level by inducing t~n~tions in the B %3*-X 21=+ system of SrF, while further downstream m the molecular beam the level population was monitored by the fluorescence induced by the probe laser. Between the pump and probe zones microwave (mw) radiation around 15 GHz, correspond. mg to the X ?Z+, u = O,l, N= 1 f 0 rotational transltions, was mtroduced vra a horn radiator. Repopulation of the depleted level due to mw absorption could then be observed as an increase of the laser mduced fluorescence in the probe zone_ The microwaves were amPlitude-modulated at 240 Hz and phase-sensitive detection was employed. For the electric field measurements additional Stark plates were mounted m the mw interaction region. As the mw polarization was parallel to the field vector only AMF = 0 tran~tio~s could be induced. Compared to our prevrous experiments [6,13] several improvements of the apparatus helped to mcrease the sensitivity and accuracy. In the probe zone an additional concave mirror collected fluorescence light in a direction opposite to the photomultiplier detector. thus enhancing the signal by a factor of two. New 6 cm long Stark plates mounted on quartz spacers were separated by 8 f 0.001 mm, the surface being plane to 0.003 nun. The electric field strength could then be determined with an accuracy of 0.1%. Stark voltages ranged from 0 to 800 V. The hyperfme structure in the N = 1 c- 0 rotational transition of X 22l* SrF has been reported in a microwave optical polarization spectroscopy (MOPS) study [ 141, including a level diagram. For the Stark effect measurements the following transitions were chosen, N= 1 +O,J=3/2+-l/2; (l)F= 1 +O,MF=O + 0 at zero-field positions of 15077353 MHz for v=O and 14983.949 MHz for u- 1; (2)F=2+-l,MF =O+OandIM-/=l+lat zero-field positions of 15011 567 MHz for u = 0 and 14918.565 MHz for u = 1. In each case the laser frequency was kept fmed to the appropriate hfs component of the P,
LEZ-l-ERS
25 Januari 1485
100 kHz
I
15 011 567 MHz
Fig 1. Microwave scan of the F = 2 +- 1 hyperfme component of the N = 1 CC@. J = 3f 2 + Z/2 rotational transitionof 8aSrF in zero eiectric field. Laser frequency fured to the PI (1) line of the B *E+-X *1?(0,0) band.
zero-field line positions were in good agreement with the ones calculated with rotational constants from ~h~~e-P~n et al. [lo] and spin-rotation and hyperfme coupling constants from Childs et al. [ 1 I]. Fig 1 shows a zero-electmc-field signal after 1 min of averagmg. The lure is still mw power broadened wluch is the main reason for the fwlun of 100 kHz. There is also residual Zeeman broadening due to imperfect magnetic sluelding of the mw and Stark interaction region.
3. Results and discussion Stark shifts in the electric field range 0 < E < 900 V/cm were measured for the mw transition N = 1 + 0. J = 312 f l/2, F= 1 c- 0. Because of the selection rule bMF = 0 no splitting of the line was observed. The data pomts with error bars for u = 0 are shown in fig. 2a. The hyperfine transition F= 2 +- 1 split into two lines with MF = 0 and IMF] = 1 the shifts of which are plotted in figs. 2b and 2c, respectively. The described experimental improvements allowed for higher fields and because of the better field homogeneity the linewidth remained at 100 k.Hz even at QOOV/cm. Compared to our CaCl[13) and CaBr [6] expe~ents the measured shifts in fig. 2 are larger by a factor of 10. Electric dipole moments are derived from the measured Stark shifts in a fit procedure described in more detail in ref. [ 133. Energies of the hfs levels in ap electric field can be calculated by diagonalization of the effective stop [ 131
Volume
113. number Stark-Shlfte V=O : N=O-1
~=3. 4963 (6) ;I
25 Janti
CiIEMICAL PHYSICS LEITBRS
4
I
I”- 5-F J=O. 5-l.
2.00
400
from the fit given in parentheses. ln the same way we obtained P = 3.5538(6) for the frost excited vibrational state u = 1. Due to the nncextalnty in the calibration of the Stark field an absolute error of s.0035 D has to be added which does not affect the relative dlfference of the values for u = 0 and 1, but only causes a shift of both values of p together. As pointed out in the experimental section the absolute error has been decreased by a factor of 5 compared to our previous measurements [6,13] _ From the two values of p the first two coefficients of the usual expansion for the y1brational dependence
5 6.00
1985
7
P=Pe+Pl(U+
l/2)
(1)
can be given as
J
Electric
Field
[V/cm1
Fig 2. Stark Shiftti versus electricfield for the microwave transitiOnSN=1~O,~=3/2~1/2,(a)F=lcO,M~=0,
@)F=Ztl,MF=O,and(c)F=2cl,M~=l_Frequency measurement errors are given a5 vertical bars for each data point. The solid lines represent the calculated he rhrfts for fi = 3.4963 D.
rr, = 3.4676(10)
+ 0.0035
& = 0.0575(10)
c _
The electric dipole moment provides mformation on the total chargedensity distribution and the bonding character of a molecular species and can therefore be used to test bon&g models. Tarring et al. [s] have shown that the Rittner ionic model [l] as well as the T-Rittner model [2] fail to explain the measured electnc dipole moments oFthe alkaline earth monohalides. A new electrostatic model has been developed [8] starting with a sun&r approach. In this approach dipole moments p+ and P- induced in the positive and negative ions are subtracted from the primary moment p. = er,, mebeing the equilibrium internuclear distance of the ionic molecule:
P=P’o
+qQ[31,2
--I(I+
1)1/41(1-l),
gc&&_ = IEI IJLIcos 8 As the rotational constants and spin-rotation and hfs interaction parameters are known from refs. [lo, 1 l] the dipole moment was the only adjustable value and could be determined by fitting the calculated Stark shifts to the experimentally observed shifts in a least-squares procedure. The solid lines in fig. 2 represent the calculated shifts for a dipole moment B = 3.4963(6) D with the statistical standard deviation
D5
-W
+N-).
The induced moments P+ and W- are calculated from the electrrc field produced by the partner ion at the center of the ion under consideration and the appropriate polaxizablbtres (Y+and Q-. The polarizabilities (Y+ of the alkahne earth metal ions (hi+) are larger than those of the alkali ions by about one order of magnitude due to the unpaired electron outside the closedshell MB core. The single s-electron of M+ is spd hybridized m the field of the halogen ion (X-) with the center of charge pushed outside the center of mass of M*, away from the partner ion X- . In our new model [S] this charge shift is taken into account for the calcuIation of the induced moment jt+, whereas P- is determined in the same way as in ref_ [l] . The results 353
Volume 113. number 4
CHEMICAL
PHYSICS
show good agreement with the measured dipole moments of CaF [7], CaCl [S] , and CaBr [6]. Ihpole moments have been calculated for most alkaline earth monohalides [S] , and for SrF a value of 3.67 D was predicted. The T-Rittner model, on the other hand, yields -0.84 D because the corrections to the pximarymoment are largely overestunated. The Rittner model is not applicable since the resulting induced moments are even larger. Taking into account the uncertahties of the polarizabllities of Sr+ and Finserted into the model equations in ref. [Et] the agreement between prediction and measurement is surprising. It confm that the simple bonding picture of two polarkable ions fits the true charge density distribu-
LEI-I-ERS
25 Januan 1985
trate on candidates for which the different model czdculations deviate. only in t&is way can all the trends predicted by the new model be confirmed for the whole group of molecules.
Acknowledgement Financial support of the project from the Deutsche Forschungsgemeins&aft in the Sonderforschungsbereich 161 is gratefully acknowledged_
References
tion .
The vIbrationaldependence of p can be compared WitPimodel calculations when the formulae in ref. [S] are used to determine a dipole moment function JL(r) or iu(U with In this case the coefficient JIM in the vibrational expansion (1) is given by [Z]
where w,, S,, and crp;are the usual molecular constants. Calculatmg the first and second denvatives of the model dipole function of SrF according to ref. [g] and neglecting the third term in eq. (2) yields pI = O-075 D_ This result shows as good an agreement with the measurements as the T-Rittner model did for the a4W.i halides f’tf _ In the ne2.r future dipole moment measurements on alkaline earth monohalide radicals should concen-
354
[I] E-S. Rittner, 3. Chem. Whys. 19 (1951) 1030.. I21 P. Brumerand N. Karplus, J. Chem. Phys. 58 (1973) 3903. (31 L. JU~nning and H.
Martin,P&&a Scripta24 (1981)
143 z?- Da@igian. Chem. Phys Letters 88 (1982) 225 [S] W.E. Ernst, S. Kindt and T. T&ring, Phys. Rev. Le;ters Sl(1983) 979. [6] S. Kindt, W.E. Ernst and T. Tarring, Chem. Fhys. Letters 103 (1983) 241. (71 WJ. Chikk. L.S. G~odsnan, K. Nxelsen and V. Pfeufer, J. Chem. Phys. 80 (1984) 2283. [B] T. Tbxing, W.E. Ernst and S. Kjndt. J. Chem. Phys CC%%.84). to be pubJ.ished. [PI WE. Ernst and J-0. Schreder, Cbem Phys. 78 (1983) 363. [ 101 I-L-U. Schiitze-Pahlmann, Ch. Ryzleticz, J. Hoeft and T. Toning, Chem. Phys. Letters 93 (1962) 74. (111 W-J. ChiIds, L-S. Goodman and L Renhom, f. lvloL Speccry. 87 Cl9811 522. (121 W-E. Ernst and S. Kindt, AppL Phys. B31(1983) 19. (131 W.E. Ernst, S. Kindt. K_F.R. l%ir and T. T&ring, Phys_ Rev. A29 (1954) 1158. 1141 W.E. Ernst, Appl. Phys. B30 (1983) 105.
CHEMICAL PHYS!CS LETTERS
25 Januari 1985
ELECTRIC DIPOLE MOMENT OF SrF X 2Z+ FROM HIGH-PRECISION STARK EFFECT MEASUREMENTS W.E. ERNST, J. R&DLER, S. KINDT and T. ToRRJNG Freie Universitdt Berlin. FB Physik. Amimallee 14. D-1000 Berlin 33. West Gernimy Received 26 October 1984
The ground4ate dipole moment of SrF was measured by the molecular-beam laser-microwave double-resonance metiod.From the study oftw~vibrationalstatesthecoeffrcicntsinthenbrationalexpansion~=Cl~+ ~~(~+1/2)weredetermined as pe = 3.4676(10) D and PI = 0.0575(113) D (wirh the statistical standard deviation given in parentheses). The measured values arc compared with the results of an ionic bonding model.
l_ Intioduction The alkaline earrh monohalides represent a class of highly ionic radicals. As the structure of the aho& halides, another group of ionic molecules, can be well described by the ionic bonding models developed by Rittner [ 13 and Brumer and Karplus (“T-r(ittner” model) [2] attempts were made to apply these models to one of the :adicals, CaCl [3,4] _ The two models yielded completely different values for the electnc dipole moment and au estimate from beam deflection measurements [4] was the only experimental value for comparison with +Ae calculations. Recently, highly precise Stark effect measurements of mdrvidual hyperfiie components of ground-state rotational transitions of CaCl and CaBr were performed using the molecular-beam laser-microwave (mw) double-resonance technique [5,6] _Electric dipole moments wem derived at OS% accuracy for the first tune and similar molecular-beam laser-rf double-resonance measurements on CaF by Childs et al. [7] yielded the corresponding value for this radical- Neither the Rittner nor the T-Rittner model could reproduce all measured values- The breakdown of these models for the alkaline earth monohalides was explained iu ref- [8] and an improved model was proposed. The spectroscopic ground-state constants and dipole moments predicted by this inodel agree with the measured values within 5 to 10%. 0 009-2614/85/S 03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
The purpose of this work was to increase the number of experimentally determined dipole moments to be compared with the predictions. The largest drscrepancies between dipole moments derived from the new model and the T-Rittner model were found for the alkaline earth monofluorides [8] which can therefore provide a critical test for their validity. Strontium monofluoride was chosen for the Stark effect measurements because precrse spectroscopic constants were known for the optical B 2Z+-X 2Zf transition [9] and the ground-state rotational and hyperfme structure had been investigated in independent mw experiments by Schiitze-Pahhnann et al. [lo] and molecular-beam laser-rf double-resonance studies by Childs et al. [ll]_ In this paper we report on investigations of the Stark effect in the vibrational states u = 0 and 1 of the 22+ ground sta te of SrF. The shift of hyperfme tranitions in an electric field was measured by applying the molecular-beam laser-mw double-resonance method.
2. Experimental Details of the apparatushave been described in refs. [ 12,131. An effusive SrF beam was generated from a high-temperature reaction of SrF2 and Sr (1000 < T < 1200°C) taking place in a ceramic crucible surrounded by a resistively heated tantalum cylinder. Light from 351
Voiume 113. number 4
CHEMICAL
PHYSICS
a single frequency dye laser operating around 579 nm was split into a pump beam of 250 mW and a probe beam of 10 mW. The pump beam was used to deplete a partrcular ground-state level by inducing t~n~tions in the B %3*-X 21=+ system of SrF, while further downstream m the molecular beam the level population was monitored by the fluorescence induced by the probe laser. Between the pump and probe zones microwave (mw) radiation around 15 GHz, correspond. mg to the X ?Z+, u = O,l, N= 1 f 0 rotational transltions, was mtroduced vra a horn radiator. Repopulation of the depleted level due to mw absorption could then be observed as an increase of the laser mduced fluorescence in the probe zone_ The microwaves were amPlitude-modulated at 240 Hz and phase-sensitive detection was employed. For the electric field measurements additional Stark plates were mounted m the mw interaction region. As the mw polarization was parallel to the field vector only AMF = 0 tran~tio~s could be induced. Compared to our prevrous experiments [6,13] several improvements of the apparatus helped to mcrease the sensitivity and accuracy. In the probe zone an additional concave mirror collected fluorescence light in a direction opposite to the photomultiplier detector. thus enhancing the signal by a factor of two. New 6 cm long Stark plates mounted on quartz spacers were separated by 8 f 0.001 mm, the surface being plane to 0.003 nun. The electric field strength could then be determined with an accuracy of 0.1%. Stark voltages ranged from 0 to 800 V. The hyperfme structure in the N = 1 c- 0 rotational transition of X 22l* SrF has been reported in a microwave optical polarization spectroscopy (MOPS) study [ 141, including a level diagram. For the Stark effect measurements the following transitions were chosen, N= 1 +O,J=3/2+-l/2; (l)F= 1 +O,MF=O + 0 at zero-field positions of 15077353 MHz for v=O and 14983.949 MHz for u- 1; (2)F=2+-l,MF =O+OandIM-/=l+lat zero-field positions of 15011 567 MHz for u = 0 and 14918.565 MHz for u = 1. In each case the laser frequency was kept fmed to the appropriate hfs component of the P,
LEZ-l-ERS
25 Januari 1485
100 kHz
I
15 011 567 MHz
Fig 1. Microwave scan of the F = 2 +- 1 hyperfme component of the N = 1 CC@. J = 3f 2 + Z/2 rotational transitionof 8aSrF in zero eiectric field. Laser frequency fured to the PI (1) line of the B *E+-X *1?(0,0) band.
zero-field line positions were in good agreement with the ones calculated with rotational constants from ~h~~e-P~n et al. [lo] and spin-rotation and hyperfme coupling constants from Childs et al. [ 1 I]. Fig 1 shows a zero-electmc-field signal after 1 min of averagmg. The lure is still mw power broadened wluch is the main reason for the fwlun of 100 kHz. There is also residual Zeeman broadening due to imperfect magnetic sluelding of the mw and Stark interaction region.
3. Results and discussion Stark shifts in the electric field range 0 < E < 900 V/cm were measured for the mw transition N = 1 + 0. J = 312 f l/2, F= 1 c- 0. Because of the selection rule bMF = 0 no splitting of the line was observed. The data pomts with error bars for u = 0 are shown in fig. 2a. The hyperfine transition F= 2 +- 1 split into two lines with MF = 0 and IMF] = 1 the shifts of which are plotted in figs. 2b and 2c, respectively. The described experimental improvements allowed for higher fields and because of the better field homogeneity the linewidth remained at 100 k.Hz even at QOOV/cm. Compared to our CaCl[13) and CaBr [6] expe~ents the measured shifts in fig. 2 are larger by a factor of 10. Electric dipole moments are derived from the measured Stark shifts in a fit procedure described in more detail in ref. [ 133. Energies of the hfs levels in ap electric field can be calculated by diagonalization of the effective stop [ 131
Volume
113. number Stark-Shlfte V=O : N=O-1
~=3. 4963 (6) ;I
25 Janti
CiIEMICAL PHYSICS LEITBRS
4
I
I”- 5-F J=O. 5-l.
2.00
400
from the fit given in parentheses. ln the same way we obtained P = 3.5538(6) for the frost excited vibrational state u = 1. Due to the nncextalnty in the calibration of the Stark field an absolute error of s.0035 D has to be added which does not affect the relative dlfference of the values for u = 0 and 1, but only causes a shift of both values of p together. As pointed out in the experimental section the absolute error has been decreased by a factor of 5 compared to our previous measurements [6,13] _ From the two values of p the first two coefficients of the usual expansion for the y1brational dependence
5 6.00
1985
7
P=Pe+Pl(U+
l/2)
(1)
can be given as
J
Electric
Field
[V/cm1
Fig 2. Stark Shiftti versus electricfield for the microwave transitiOnSN=1~O,~=3/2~1/2,(a)F=lcO,M~=0,
@)F=Ztl,MF=O,and(c)F=2cl,M~=l_Frequency measurement errors are given a5 vertical bars for each data point. The solid lines represent the calculated he rhrfts for fi = 3.4963 D.
rr, = 3.4676(10)
+ 0.0035
& = 0.0575(10)
c _
The electric dipole moment provides mformation on the total chargedensity distribution and the bonding character of a molecular species and can therefore be used to test bon&g models. Tarring et al. [s] have shown that the Rittner ionic model [l] as well as the T-Rittner model [2] fail to explain the measured electnc dipole moments oFthe alkaline earth monohalides. A new electrostatic model has been developed [8] starting with a sun&r approach. In this approach dipole moments p+ and P- induced in the positive and negative ions are subtracted from the primary moment p. = er,, mebeing the equilibrium internuclear distance of the ionic molecule:
P=P’o
+qQ[31,2
--I(I+
1)1/41(1-l),
gc&&_ = IEI IJLIcos 8 As the rotational constants and spin-rotation and hfs interaction parameters are known from refs. [lo, 1 l] the dipole moment was the only adjustable value and could be determined by fitting the calculated Stark shifts to the experimentally observed shifts in a least-squares procedure. The solid lines in fig. 2 represent the calculated shifts for a dipole moment B = 3.4963(6) D with the statistical standard deviation
D5
-W
+N-).
The induced moments P+ and W- are calculated from the electrrc field produced by the partner ion at the center of the ion under consideration and the appropriate polaxizablbtres (Y+and Q-. The polarizabilities (Y+ of the alkahne earth metal ions (hi+) are larger than those of the alkali ions by about one order of magnitude due to the unpaired electron outside the closedshell MB core. The single s-electron of M+ is spd hybridized m the field of the halogen ion (X-) with the center of charge pushed outside the center of mass of M*, away from the partner ion X- . In our new model [S] this charge shift is taken into account for the calcuIation of the induced moment jt+, whereas P- is determined in the same way as in ref_ [l] . The results 353
Volume 113. number 4
CHEMICAL
PHYSICS
show good agreement with the measured dipole moments of CaF [7], CaCl [S] , and CaBr [6]. Ihpole moments have been calculated for most alkaline earth monohalides [S] , and for SrF a value of 3.67 D was predicted. The T-Rittner model, on the other hand, yields -0.84 D because the corrections to the pximarymoment are largely overestunated. The Rittner model is not applicable since the resulting induced moments are even larger. Taking into account the uncertahties of the polarizabllities of Sr+ and Finserted into the model equations in ref. [Et] the agreement between prediction and measurement is surprising. It confm that the simple bonding picture of two polarkable ions fits the true charge density distribu-
LEI-I-ERS
25 Januan 1985
trate on candidates for which the different model czdculations deviate. only in t&is way can all the trends predicted by the new model be confirmed for the whole group of molecules.
Acknowledgement Financial support of the project from the Deutsche Forschungsgemeins&aft in the Sonderforschungsbereich 161 is gratefully acknowledged_
References
tion .
The vIbrationaldependence of p can be compared WitPimodel calculations when the formulae in ref. [S] are used to determine a dipole moment function JL(r) or iu(U with In this case the coefficient JIM in the vibrational expansion (1) is given by [Z]
where w,, S,, and crp;are the usual molecular constants. Calculatmg the first and second denvatives of the model dipole function of SrF according to ref. [g] and neglecting the third term in eq. (2) yields pI = O-075 D_ This result shows as good an agreement with the measurements as the T-Rittner model did for the a4W.i halides f’tf _ In the ne2.r future dipole moment measurements on alkaline earth monohalide radicals should concen-
354
[I] E-S. Rittner, 3. Chem. Whys. 19 (1951) 1030.. I21 P. Brumerand N. Karplus, J. Chem. Phys. 58 (1973) 3903. (31 L. JU~nning and H.
Martin,P&&a Scripta24 (1981)
143 z?- Da@igian. Chem. Phys Letters 88 (1982) 225 [S] W.E. Ernst, S. Kindt and T. T&ring, Phys. Rev. Le;ters Sl(1983) 979. [6] S. Kindt, W.E. Ernst and T. Tarring, Chem. Fhys. Letters 103 (1983) 241. (71 WJ. Chikk. L.S. G~odsnan, K. Nxelsen and V. Pfeufer, J. Chem. Phys. 80 (1984) 2283. [B] T. Tbxing, W.E. Ernst and S. Kjndt. J. Chem. Phys CC%%.84). to be pubJ.ished. [PI WE. Ernst and J-0. Schreder, Cbem Phys. 78 (1983) 363. [ 101 I-L-U. Schiitze-Pahlmann, Ch. Ryzleticz, J. Hoeft and T. Toning, Chem. Phys. Letters 93 (1962) 74. (111 W-J. ChiIds, L-S. Goodman and L Renhom, f. lvloL Speccry. 87 Cl9811 522. (121 W-E. Ernst and S. Kindt, AppL Phys. B31(1983) 19. (131 W.E. Ernst, S. Kindt. K_F.R. l%ir and T. T&ring, Phys_ Rev. A29 (1954) 1158. 1141 W.E. Ernst, Appl. Phys. B30 (1983) 105.