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Rotational perturbation between two electronically excited states of 4HeD
Volume 139, number 6
CHEMICAL PHYSICS LETTERS
ROTATIONAL PERTURBATION BETWEEN TWO ELECTRONICALLY
11 September 1987
EXCITED STATES OF 4HeD
R.L. BROOKS and B.G. NICKEL Guelph- Waterloo Program for Graduate Work in Physics, University of Guelph, Guelph, Ontario, Canada NlG 2 WI
Received 7 May 1987; in final form 4 June 1987
The rotational structure of the D ‘C + -+A *C + transition of “HeD is highly perturbed, and this is caused by the near degeneracy of the D, u= 0 and C, u= 3 vibrational levels. A perturbation analysis is presented which yields the spectroscopic constants for both of the perturbing levels, yielding the first experimental information on a vibrationally excited state of HeH.
The Rydberg molecule, helium hydride, has finally succumbed to experimental (and spectroscopic) detection during the past two years [ l-61. While known for some time to have bound, electronically excited states (though a repulsive ground state) this simplest of unlike-z neutral molecules remained undetected until its continuum emission was observed by Moller, Beland and Zimmerer following synchrotron irradiation of a helium-hydrogen gas mixture [ 11. The first discrete spectrum was observed at about the same time by Ketterle, Figger and Walther following the neutralization of an HeH+ beam by cesium or potassium vapor [ 21. Several additional experiments using the same technique followed [ 3-51, from which a good deal of information regarding the A ‘I; +, B 2H and C 2E + electronic levels were obtained. Recently, a new technique developed in this laboratory, utilizing helium gas at 4.2 K in the upper portion of a cell half tilled with solid hydrogen undergoing 15 MeV proton beam irradiation, yielded the discrete spectra of the D ‘E + +A 2E + for all four isotopic combinations [ 61. The analysis of the rotational constants from one of those four spectra ( 4HeD) was complicated by the near degeneracy of the D, Y= 0 and the C, u= 3 vibrational levels. This near degeneracy does not occur for the other isotopes, yet is so strong for 4HeD that a straightforward rotational analysis was not possible. The purpose of this Letter is to present the results of a perturbation analysis of the spectrum of 4HeD from
801 0
$-I
1
2
3
4
Internuclear Distance (A) Fig. 1. Lowest four bound potential curves (ref. [ 71) with some vibrational levels for 4HeD. Energy taken with respect to groundstate, separated-atom limit.
which the rotational constants and v. for both the D, U= 0 and the C, v= 3 levels may be extracted. No experimental information has previously been available for any vibrationally excited state of HeH. Experimental details can be found in ref. [ 61. Fig. 1 shows the ab initio potential curves of Theodorakopoulos et al. [ 71 after cubic-spline interpolation. The vibrational energy levels were calcu-
0 009-2614/87/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
503
Volume 139. number 6
lated from these using the computer code of LeRoy [ 81. The transition of interest is the D, v= O-A, v= 0 at x 18250 cm-‘, and one can readily see that the C, v= 3 level appears degenerate with the D, v= 0. Since these levels share the same symmetry, interaction via non-adiabatic terms of the Hamiltonian become possible. These non-adiabatic terms usually represent small corrections to a given energy level, but when near degeneracy occurs the corrections can shift energies by tens of wavenumbers. Under this circumstance, a two-state approximation, utilizing direct diagonalization, may be sufficient to account for the perturbation. The energy can be written using the well-known expression [ 91 E’=~{(V,+Vc)f[(Vc-VV,)2+482]1’2},
V,=V,+B,J’(J’+1)-o,(5’)2(3’+1)2,
(2)
v,=v;,+B;J’(J’+l)-D;(J’)2(J’+1)2,
(3)
v*=B;J”(J”+l)-D;;(J”)2(J”+1)2.
(4)
The spectroscopic constants B and D have their usual definitions [ lo]. The system is block-diagonal in J. 6 will be assumed to be J independent, its J dependence will be discussed below. The observed spectrum measures the transition energies, T ’ (J’, J”) given by (5)
It is a straightforward matter to perform a non-linear, least-squares fit to the observed spectrum once the J values have been assigned. Fig. 2 is the observed spectrum for 4HeD for the (D + C) 2C+ +A 2X+ transition. This spectrum, obtained using a lower proton-beam energy and beam current normalization, is a recent improvement over that of ref. [ 61. Once one realizes that an energy interval such as R( J)-P(J+Z) represents a lower level interval, which is not perturbed and can be readily extrapolated from the other isotopes, it becomes not too difficult to make the assignments 504
P5’
I 1- 0’2’ II III
I
R
I I I I I lllllll If o+ 5+ 5+
(1)
where Vi, and Vc are the energies of the unperturbed D, v=O and C, v=3 levels and 6 is the off-diagonal coupling term. If one refers the energies to the A, v= 0, J= 0 level, one can write the unperturbed energies as:
T’(J’,J”)=E’(J’)-T/,(J”).
11September 1987
CHEMICAL PHYSICS LETTERS
ioo Frequency (cm”) Fig. 2. Spectrum of 4HeD, abscissa corrected to vacuum. Instrumentalfwhm is 9 cm-‘. Transition is the (D, u=O+C, v=3)+A, v= 0. Rotational assignments of two perturbing levels given on upper scale. For the negative sequence, the band head occurs at R2-.
given in fig. 2. The notation R3 +, for example, means T+(4,3)whileP2_meansT-(1,2).Thedoublescale at the top of fig. 2 marks the spectral positions of the transitions as calculated from eq. (5) following the non-linear tit. Only three lines are observed from the alternate sequence (RO-, P 1- and P2- ), all of them close to the J value of maximum mixing of the D and C levels. The spectrum readily supports a determination of the centrifugal distortion term, D;;, of the lower A state. It barely supports, D,,, of the D state and does not support D; at all. Hence the solution is one in which there are 8 free parameters and 16 transition energies. The results are collected in table 1. 0; has been set equal to Do which seemed like a more reasonable estimate than setting it to zero, but none of the results changes outside of the quoted errors when D; is set to zero. The theoretical rotational B and centrifugal D values were generated by fitting the theoretical rotational energy levels to eq. (2) or (3) individually for the D and C potential curves. All 16 lines have been fit to better than 2 cm- ’ and 14 of them to better than 1 cm-‘. The rotational
CHEMICAL PHYSICS LETTERS
Volume 139, number 6
Table 1 Spectroscopic constants (in cm- ’ ) for 4HeD as given by fit to eq. (5) (exp) and calculated [ 81 from the theoretical potentials [ 71 (thry). Uncertainties in the last quoted figure are enclosed in parentheses A, v=O
c, v=3
v0 vo B B D D 6
- b) 22.2( 2) 22.1 0.0054(6) 0.0058
18254(3) 18299( 3) =) 18126 18608 20.4(2) 15.0(3) 15.0 20.8 0.005(Z) (0.005) c’ 0.0053 0.0056 43.4( 7)
w, d’ cJ& d)
6 -,-----\ \ ‘1,
600
2170 65
a’ 12563( 10) from C, v=O to A, v=O extrapolated from ref. [2]. b, 63370 to repulsive ground state. Using ref. [4] this yields 63047 k 485 as the experimental value. ‘) Not a free parameter; set equal to D, v=O value. d, Estimate of vibrational parameters of C potential curve based on G(3)-G(O)=5736 cm-’ (exp) and G(O)=1069 cm-’ (thry).
constants of the D 2C+ level are now consistent with the values for the same level from the other isotopes [ 61. Additionally, however, the rotational parameters for the C 2Z +, U= 3 level have also been obtained and B; compares favorably with the value calculated using the ro-vibrational energy level code of LeRoy with the potential of Theodorakopoulos et al. Fig. 3 shows a rotational energy level diagram for this system with the zero shifted to T - (0,O). Clearly the magnitude of the perturbation is large which accounts for the impossibility of describing this spectrum from a single band. Also it is clear that several lines not observed are predicted to be present. If one makes the assumption that the E ’ levels can radiatively decay only through the D, v=O part of the wavefunction (the Franck-Condon factor for the C, V= 3 levels is calculated to be less than 0. 1), the predicted spectral intensities are similar to the ones observed. However, all of our spectra show abnormal thermal distributions, discussed below, which is not surprising given the excitation mechanism, and hence line intensities are difficult to predict. For example, the missing P3- transition is predicted to be stronger than the observed P2- one, under the above assumption. Perturbation of a rotational sequence of a spec-
\
‘\
\ \ \ \\ \ \____L--5 _;_---\ \ ‘\
D, v=O
Constant (exp) (thry) (exp) (thry) (exp) (thry) (exp)
11 September 1987
600
\\
‘%
‘____--
t
400 - 4
__/---\\ ‘\
‘\
3
‘\ _-\-
‘.__-
_/--.\ ‘. *;-
-: ‘\
200 -
-._*/---m
2 1 o-
‘\
\
O
/’
--.
,A--
-:
-.c-7c
,,‘F__ -
I>,--
--c,_
-;-
,‘^.__
I
.
_<‘,,
,
,’
Fig. 3. Rotational energies of 4HeD fc. Ihe unperturbed D, v=O (left) and C, v= 3 (right) levels as well as for the perturbed (center) positive (upper) and negative (lower) levels.
trum is not uncommon [ lo] but it usually arises from an avoided crossing of two potential curves. That can be excluded for this system for two reasons. The first is that the perturbation is only observable for one of four isotopic combinations that have been studied and the second is that the ab initio potential curves of Theodorakopoulos et al. show no such avoided crossing. Furthermore the calculated position of the C, v=3, J=O level is 56 cm-’ lower than the measured position (with respect to the C, u= 0, J= 0 level), an error of only lo/b,which indicates that the shape of the potential curve is really quite good. Therefore, it seems reasonable to conclude that non-adiabatic effects, caused by kinematic coupling of nuclear and electronic motions, are responsible for the off-diagonal matrix element, 6 [ 111. Our empirical determination of S assumes that it is J independent. Since we could not obtain a better tit to our data, this assumption is, a posteriori, correct to within experimental uncertainties. Additionally, we have used the aforementioned computer code [ 81 to calculate the overlap integrals between the two relevant ro-vibrational wavefunctions for each value of J and observed only a 2Ohchange in the value from J= 0 to J= 5. These observations are consistent with 505
Volume 139, number 6
CHEMICAL PHYSICS LETTERS
other known perturbations in which 6 is J independent when AA=0 for the perturbation. Our possible explanation for the intensity anomaly of this spectrum (also observed for the other isotopes) is that the He + H (n = 2) asymptote at 82303 cm-’ above the repulsive ground state occurs somewhere between J= 5 and J= 8 for the E + levels of 4HeD. The reason for this energy uncertainty is that the position of the A potential curve, with respect to the repulsive ground state, is uncertain by 484 cm- ’ (60 meV4). The A state is known to couple strongly to the ground-state continuum and hence to predissociate, and the C and D states are symmetry-allowed to do the same. However, the cross sections fall off as the energy levels get further from threshold and the opening of a second channel is likely to cause a sudden increase in the predissociation of the upper D rotational levels which can quench their spectral intensity. This might then account for the sudden decrease in the line intensities observed for J’ 2 6. At these large J’ values, the mixing with the C state is small and such mixing should not be responsible for this intensity anomaly. A two-state perturbation analysis has been performed on the spectrum of 4HeD which has yielded the spectroscopic constants and energy positions for both the D ‘X +, u= 0 and C *C+, v= 3 vibrational levels. The measured v= 3 level lies within 1% of its calculated value (with respect to the v= 0 level) using the ab initio potential curves of Theodorakopolous et al. [ 71 indicating that at least the C curve has an accurate shape. We performed similar calculations for the other isotopic species of HeH, particularly
506
‘HeD, but the perturbation discernible.
11 September 1987
is too small to be
We would like to thank R.J. LeRoy for his computer code and suggestions and J.D. Poll for helpful discussions. We gratefully acknowledge the financial support of the Natural Science and Engineering Research Council (NSERC) of Canada.
References [ 1 ] T. Miiller, M. Beland and G. Zimmerer, Phys. Rev. Letters 55 (1985) 2145. [2] WK. Ketterle, H. Figger and H. Walther, Phys. Rev. Letters 55 (1985) 2941. [ 31 W. Ketterle, A. Dodhy and H. Walther, Chem. Phys. Letters 129 (1986) 76. [ 41 W.J. van der Zande, W. Koot, D.P. de Bruijn and C. Kubach, Phys. Rev. Letters 57 (1986) 1219. [5] J.R. Peterson and Y.K. Bae, Phys. Rev. A34 (1986) 3517. [ 61 R.L. Brooks, J.L. Hunt and J.J. Miller, Phys. Rev. Letters 58 (1987) 199. [ 71 G. Theodorakopoulos, S.C. Farantos, R.J. Buenker and S.D. Peyerimhoff, J. Phys. B17 (1984) 1453. [ 81 R.J. LeRoy, Chemical Physics Research Report CP-230R2, University of Waterloo (1985)) unpublished. [9] L. Landau and E.M. Lifshitz, Quantum mechanics, nonrelativistic theory, 2nd Ed. (Addison-Wesley, Reading, 1965). [ lo] G. Her&erg, Spectra of diatomic molecules (Van Nostrand, Princeton, 1950). [ 111 R.K. Janev, in: Advances in atomic and molecular physics, Vol. 12, eds. D.R. Bates and B. Bederson (Academic Press, New York, 1976) p. 1.
CHEMICAL PHYSICS LETTERS
ROTATIONAL PERTURBATION BETWEEN TWO ELECTRONICALLY
11 September 1987
EXCITED STATES OF 4HeD
R.L. BROOKS and B.G. NICKEL Guelph- Waterloo Program for Graduate Work in Physics, University of Guelph, Guelph, Ontario, Canada NlG 2 WI
Received 7 May 1987; in final form 4 June 1987
The rotational structure of the D ‘C + -+A *C + transition of “HeD is highly perturbed, and this is caused by the near degeneracy of the D, u= 0 and C, u= 3 vibrational levels. A perturbation analysis is presented which yields the spectroscopic constants for both of the perturbing levels, yielding the first experimental information on a vibrationally excited state of HeH.
The Rydberg molecule, helium hydride, has finally succumbed to experimental (and spectroscopic) detection during the past two years [ l-61. While known for some time to have bound, electronically excited states (though a repulsive ground state) this simplest of unlike-z neutral molecules remained undetected until its continuum emission was observed by Moller, Beland and Zimmerer following synchrotron irradiation of a helium-hydrogen gas mixture [ 11. The first discrete spectrum was observed at about the same time by Ketterle, Figger and Walther following the neutralization of an HeH+ beam by cesium or potassium vapor [ 21. Several additional experiments using the same technique followed [ 3-51, from which a good deal of information regarding the A ‘I; +, B 2H and C 2E + electronic levels were obtained. Recently, a new technique developed in this laboratory, utilizing helium gas at 4.2 K in the upper portion of a cell half tilled with solid hydrogen undergoing 15 MeV proton beam irradiation, yielded the discrete spectra of the D ‘E + +A 2E + for all four isotopic combinations [ 61. The analysis of the rotational constants from one of those four spectra ( 4HeD) was complicated by the near degeneracy of the D, Y= 0 and the C, u= 3 vibrational levels. This near degeneracy does not occur for the other isotopes, yet is so strong for 4HeD that a straightforward rotational analysis was not possible. The purpose of this Letter is to present the results of a perturbation analysis of the spectrum of 4HeD from
801 0
$-I
1
2
3
4
Internuclear Distance (A) Fig. 1. Lowest four bound potential curves (ref. [ 71) with some vibrational levels for 4HeD. Energy taken with respect to groundstate, separated-atom limit.
which the rotational constants and v. for both the D, U= 0 and the C, v= 3 levels may be extracted. No experimental information has previously been available for any vibrationally excited state of HeH. Experimental details can be found in ref. [ 61. Fig. 1 shows the ab initio potential curves of Theodorakopoulos et al. [ 71 after cubic-spline interpolation. The vibrational energy levels were calcu-
0 009-2614/87/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
503
Volume 139. number 6
lated from these using the computer code of LeRoy [ 81. The transition of interest is the D, v= O-A, v= 0 at x 18250 cm-‘, and one can readily see that the C, v= 3 level appears degenerate with the D, v= 0. Since these levels share the same symmetry, interaction via non-adiabatic terms of the Hamiltonian become possible. These non-adiabatic terms usually represent small corrections to a given energy level, but when near degeneracy occurs the corrections can shift energies by tens of wavenumbers. Under this circumstance, a two-state approximation, utilizing direct diagonalization, may be sufficient to account for the perturbation. The energy can be written using the well-known expression [ 91 E’=~{(V,+Vc)f[(Vc-VV,)2+482]1’2},
V,=V,+B,J’(J’+1)-o,(5’)2(3’+1)2,
(2)
v,=v;,+B;J’(J’+l)-D;(J’)2(J’+1)2,
(3)
v*=B;J”(J”+l)-D;;(J”)2(J”+1)2.
(4)
The spectroscopic constants B and D have their usual definitions [ lo]. The system is block-diagonal in J. 6 will be assumed to be J independent, its J dependence will be discussed below. The observed spectrum measures the transition energies, T ’ (J’, J”) given by (5)
It is a straightforward matter to perform a non-linear, least-squares fit to the observed spectrum once the J values have been assigned. Fig. 2 is the observed spectrum for 4HeD for the (D + C) 2C+ +A 2X+ transition. This spectrum, obtained using a lower proton-beam energy and beam current normalization, is a recent improvement over that of ref. [ 61. Once one realizes that an energy interval such as R( J)-P(J+Z) represents a lower level interval, which is not perturbed and can be readily extrapolated from the other isotopes, it becomes not too difficult to make the assignments 504
P5’
I 1- 0’2’ II III
I
R
I I I I I lllllll If o+ 5+ 5+
(1)
where Vi, and Vc are the energies of the unperturbed D, v=O and C, v=3 levels and 6 is the off-diagonal coupling term. If one refers the energies to the A, v= 0, J= 0 level, one can write the unperturbed energies as:
T’(J’,J”)=E’(J’)-T/,(J”).
11September 1987
CHEMICAL PHYSICS LETTERS
ioo Frequency (cm”) Fig. 2. Spectrum of 4HeD, abscissa corrected to vacuum. Instrumentalfwhm is 9 cm-‘. Transition is the (D, u=O+C, v=3)+A, v= 0. Rotational assignments of two perturbing levels given on upper scale. For the negative sequence, the band head occurs at R2-.
given in fig. 2. The notation R3 +, for example, means T+(4,3)whileP2_meansT-(1,2).Thedoublescale at the top of fig. 2 marks the spectral positions of the transitions as calculated from eq. (5) following the non-linear tit. Only three lines are observed from the alternate sequence (RO-, P 1- and P2- ), all of them close to the J value of maximum mixing of the D and C levels. The spectrum readily supports a determination of the centrifugal distortion term, D;;, of the lower A state. It barely supports, D,,, of the D state and does not support D; at all. Hence the solution is one in which there are 8 free parameters and 16 transition energies. The results are collected in table 1. 0; has been set equal to Do which seemed like a more reasonable estimate than setting it to zero, but none of the results changes outside of the quoted errors when D; is set to zero. The theoretical rotational B and centrifugal D values were generated by fitting the theoretical rotational energy levels to eq. (2) or (3) individually for the D and C potential curves. All 16 lines have been fit to better than 2 cm- ’ and 14 of them to better than 1 cm-‘. The rotational
CHEMICAL PHYSICS LETTERS
Volume 139, number 6
Table 1 Spectroscopic constants (in cm- ’ ) for 4HeD as given by fit to eq. (5) (exp) and calculated [ 81 from the theoretical potentials [ 71 (thry). Uncertainties in the last quoted figure are enclosed in parentheses A, v=O
c, v=3
v0 vo B B D D 6
- b) 22.2( 2) 22.1 0.0054(6) 0.0058
18254(3) 18299( 3) =) 18126 18608 20.4(2) 15.0(3) 15.0 20.8 0.005(Z) (0.005) c’ 0.0053 0.0056 43.4( 7)
w, d’ cJ& d)
6 -,-----\ \ ‘1,
600
2170 65
a’ 12563( 10) from C, v=O to A, v=O extrapolated from ref. [2]. b, 63370 to repulsive ground state. Using ref. [4] this yields 63047 k 485 as the experimental value. ‘) Not a free parameter; set equal to D, v=O value. d, Estimate of vibrational parameters of C potential curve based on G(3)-G(O)=5736 cm-’ (exp) and G(O)=1069 cm-’ (thry).
constants of the D 2C+ level are now consistent with the values for the same level from the other isotopes [ 61. Additionally, however, the rotational parameters for the C 2Z +, U= 3 level have also been obtained and B; compares favorably with the value calculated using the ro-vibrational energy level code of LeRoy with the potential of Theodorakopoulos et al. Fig. 3 shows a rotational energy level diagram for this system with the zero shifted to T - (0,O). Clearly the magnitude of the perturbation is large which accounts for the impossibility of describing this spectrum from a single band. Also it is clear that several lines not observed are predicted to be present. If one makes the assumption that the E ’ levels can radiatively decay only through the D, v=O part of the wavefunction (the Franck-Condon factor for the C, V= 3 levels is calculated to be less than 0. 1), the predicted spectral intensities are similar to the ones observed. However, all of our spectra show abnormal thermal distributions, discussed below, which is not surprising given the excitation mechanism, and hence line intensities are difficult to predict. For example, the missing P3- transition is predicted to be stronger than the observed P2- one, under the above assumption. Perturbation of a rotational sequence of a spec-
\
‘\
\ \ \ \\ \ \____L--5 _;_---\ \ ‘\
D, v=O
Constant (exp) (thry) (exp) (thry) (exp) (thry) (exp)
11 September 1987
600
\\
‘%
‘____--
t
400 - 4
__/---\\ ‘\
‘\
3
‘\ _-\-
‘.__-
_/--.\ ‘. *;-
-: ‘\
200 -
-._*/---m
2 1 o-
‘\
\
O
/’
--.
,A--
-:
-.c-7c
,,‘F__ -
I>,--
--c,_
-;-
,‘^.__
I
.
_<‘,,
,
,’
Fig. 3. Rotational energies of 4HeD fc. Ihe unperturbed D, v=O (left) and C, v= 3 (right) levels as well as for the perturbed (center) positive (upper) and negative (lower) levels.
trum is not uncommon [ lo] but it usually arises from an avoided crossing of two potential curves. That can be excluded for this system for two reasons. The first is that the perturbation is only observable for one of four isotopic combinations that have been studied and the second is that the ab initio potential curves of Theodorakopoulos et al. show no such avoided crossing. Furthermore the calculated position of the C, v=3, J=O level is 56 cm-’ lower than the measured position (with respect to the C, u= 0, J= 0 level), an error of only lo/b,which indicates that the shape of the potential curve is really quite good. Therefore, it seems reasonable to conclude that non-adiabatic effects, caused by kinematic coupling of nuclear and electronic motions, are responsible for the off-diagonal matrix element, 6 [ 111. Our empirical determination of S assumes that it is J independent. Since we could not obtain a better tit to our data, this assumption is, a posteriori, correct to within experimental uncertainties. Additionally, we have used the aforementioned computer code [ 81 to calculate the overlap integrals between the two relevant ro-vibrational wavefunctions for each value of J and observed only a 2Ohchange in the value from J= 0 to J= 5. These observations are consistent with 505
Volume 139, number 6
CHEMICAL PHYSICS LETTERS
other known perturbations in which 6 is J independent when AA=0 for the perturbation. Our possible explanation for the intensity anomaly of this spectrum (also observed for the other isotopes) is that the He + H (n = 2) asymptote at 82303 cm-’ above the repulsive ground state occurs somewhere between J= 5 and J= 8 for the E + levels of 4HeD. The reason for this energy uncertainty is that the position of the A potential curve, with respect to the repulsive ground state, is uncertain by 484 cm- ’ (60 meV4). The A state is known to couple strongly to the ground-state continuum and hence to predissociate, and the C and D states are symmetry-allowed to do the same. However, the cross sections fall off as the energy levels get further from threshold and the opening of a second channel is likely to cause a sudden increase in the predissociation of the upper D rotational levels which can quench their spectral intensity. This might then account for the sudden decrease in the line intensities observed for J’ 2 6. At these large J’ values, the mixing with the C state is small and such mixing should not be responsible for this intensity anomaly. A two-state perturbation analysis has been performed on the spectrum of 4HeD which has yielded the spectroscopic constants and energy positions for both the D ‘X +, u= 0 and C *C+, v= 3 vibrational levels. The measured v= 3 level lies within 1% of its calculated value (with respect to the v= 0 level) using the ab initio potential curves of Theodorakopolous et al. [ 71 indicating that at least the C curve has an accurate shape. We performed similar calculations for the other isotopic species of HeH, particularly
506
‘HeD, but the perturbation discernible.
11 September 1987
is too small to be
We would like to thank R.J. LeRoy for his computer code and suggestions and J.D. Poll for helpful discussions. We gratefully acknowledge the financial support of the Natural Science and Engineering Research Council (NSERC) of Canada.
References [ 1 ] T. Miiller, M. Beland and G. Zimmerer, Phys. Rev. Letters 55 (1985) 2145. [2] WK. Ketterle, H. Figger and H. Walther, Phys. Rev. Letters 55 (1985) 2941. [ 31 W. Ketterle, A. Dodhy and H. Walther, Chem. Phys. Letters 129 (1986) 76. [ 41 W.J. van der Zande, W. Koot, D.P. de Bruijn and C. Kubach, Phys. Rev. Letters 57 (1986) 1219. [5] J.R. Peterson and Y.K. Bae, Phys. Rev. A34 (1986) 3517. [ 61 R.L. Brooks, J.L. Hunt and J.J. Miller, Phys. Rev. Letters 58 (1987) 199. [ 71 G. Theodorakopoulos, S.C. Farantos, R.J. Buenker and S.D. Peyerimhoff, J. Phys. B17 (1984) 1453. [ 81 R.J. LeRoy, Chemical Physics Research Report CP-230R2, University of Waterloo (1985)) unpublished. [9] L. Landau and E.M. Lifshitz, Quantum mechanics, nonrelativistic theory, 2nd Ed. (Addison-Wesley, Reading, 1965). [ lo] G. Her&erg, Spectra of diatomic molecules (Van Nostrand, Princeton, 1950). [ 111 R.K. Janev, in: Advances in atomic and molecular physics, Vol. 12, eds. D.R. Bates and B. Bederson (Academic Press, New York, 1976) p. 1.