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Rovibronic energy levels of HeHF+ (X 2Π)
25 Mal'Ch 1994
CHEMICAL PHYSICS LETTERS Chemical Physics Letters 220 (1994) 117-121
ELSEVIER
Rovibronic energy levels o f H e H F + ( X Thomas
2I-[)
Schmelz, Pavel Rosmus
Fachbereich Chemie der Universitiit, D-60439 Frankfurt, Germany
Received 18 January 1994
Abstract
Ab initio coupled electron pair (CEPA) potential energy functions have been used to calculate the energies of low-lying rovibronic (J~< 5/2) levels in the X 2H state of HeHF + (for Rnr+ = 1.892 bohr). The complex was found to have a linear equifibrium structure, the dissociation energy Dc has been calculated to be 1490 cm -~, and the equilibrium RH~..HF÷ distance (relative to the center of mass of HF + ) to be 4.25 bohr. The pattern of the rovibrnnic levels can be best characterized by a notation used for linear/linear Rennet-Teller states. The stretch wavenumber (II3/2 (0,0) -H3/2 ( 1,0 ) ) has been calculated to be 311.0 cm - 1, the bend wavenumber (1-13/2(0,0)--Y~2(0,1)) tO be 223.2 cm -~, and the spin-orbit splitting (II~/2(0,0)-H3/2(0,0)) to be 319.6 cm -~ .
1. Introduction
Although neutral and ionic clusters in electronically degenerate open-shell states can be produced in free jet sources, their investigation by high-resolution spectroscopic techniques has begun only recently [ 1 ]. Particularly successful examples o f such studies are the A r O H [ 2 - 4 ] and A r N O [ 5-7 ] complexes. In the case o f charged clusters an accurate spectroscopic characterization is still virtually nonexistent. The first rotationally resolved spectra have recently been detected for N ~ He [ 8 ]. In this Letter, we report theoretical calculations o f the rovibronic levels in the electronic ground state (X2I'I) o f the H e H F + ion. The neutral R g H X ( R g = H e , Ne, Ar, X = F , Cl) van der Waals complexes belong to the best experimentally investigated and theoretically interpreted weakly b o u n d species [ 9-12 ]. To date, neither theoretical nor experimeni Permanent address: Universit6 de Marne la Vall6e, F-93160 Noisy le Grand, France.
tal studies have been known for their charged counterparts. Although H F + is isoelectronic with OH, the interaction between an ion and the rare gas is stronger - the protonated rare gases are strongly b o u n d species [ 13 ] - and more oriented. Hence the pattern o f rovibronic levels in this type o f charged van der Waals complex has little in c o m m o n with that o f their neutral isoelectronic diatoms in their electronic ground states, in which the angular motion o f the diatom is best thought o f as a hindered rotation rather than a bend o f the complex.
2. R e s u l t s and d i s c u s s i o n
The potential energy functions ( P E F ) have been calculated by the size-consistent coupled electron pair (CEPA1) method [14] #1. All energies have been #l Calculated with the MOLPRO ab initio program, written by H.-J. Werner and P.J. Knowles, with comributions from J. Alm10f, R. Amos, S. Elbert, W. Meyer, E.-A. Reinsch, R. Pitzer and A. Stone.
0009-2614/94/$07.00 © 1994 Elsevier Science B.V. All fights reserved SSD10009-2614 (94) 00141-C
T. Schmelz, P. Rosmus / Chemical Physics Letters 220 (1994) 117-121
118
corrected for basis set superposition errors using the counterpoise method. The PEFs for fixed equilibrium geometry o f H F + Rrxv. = 1.892 bohr [ 15 ] have been mapped for 130 geometries; the resulting total energies are available on request. The full three-dimensional treatment would require at least four times as many geometries. In these calculations we have employed the basis sets from ref. [ 16 ], for H (8s, 4p) contracted to [4s, 3p], for He (9s, 4p, 3d) [4s, 3p, 2d] and for F (14s, 9p, 4d) [6s, 5p, 3d]. In additional calculations for geometries around the equilibrium and at the dissociation asymptote, the AO basis set has been augmented by three f functions on F and three d functions on He. The calculated dissociation energy of the complex has increased only marginally relative to the results obtained with the basis set described above. Also the calculated equilibrium geometry hardly changed. W c found itconvenient to fit the set of C E P A energies for both A' and A" component separately by an expansion in terms of reduced rotation matrix elements [7 ]- Figs. 1 and 2 present contour plots of the A' and A" PEFs. In agreement with the qualitativearguments in rcf. [ 17 ] about the
180 166 140 120
~
80 6O 40 2O 0 3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
R (boB) Fig. 2. Two-dimensional cut of the PEFs (R versus 0) of the potentialenergy function for the A" component of the 2rl stateof H e H F +, the contours are plottedin stepsof 100 c m - i.
180 160 140 120
"G
100
60 40 20 0 3.0
3.5
4.0
4.5
5,0
5,5
6.0
6.5
7.0
R (bohr) Fig. I. Two-dimensional cut of the PEFs (R versus 0) of the potentialenergy function for the A' component of the 2H state of H e H F +, the contours are plotted in steps of 100 crn- i.
splittings of the electronically degenerate states along the bending coordinates, the A' component lies energetically lower than the A" component, since the orbital of the diatom is occupied by three electrons. From Fig. 1 it follows that the A' and A" PEFs have different shapes of their repulsive parts. On the A' PEF for energies around 1200 c m - 1 the He atom can approach the ion much closer for nonlinear orientations. Both electronic components have a minimum at linear He...HF + structure with Re= 4.25 bohr (relative to the center of mass of HF + ). For the linear HF +...He structure we found a shallow barrier around R = 5.25 bohr which still lies more than 100 c m - ~below the dissociation asymptote but more than 1300 c m - i above the equilibrium He...HF + structure. The dissociation energy D e has been calculated to be 1490cm -I, i.e.cxpcctedly much larger than in the neutralH c H F complex (7.1 cm- t [l0 ] ).Fig. 3 shows a one-dimensional cut of the PEFs for the Re= 4.25 bohr along the bending coordinate. The splittingof the A' and A" components has been calculated to be rather small for angles close to 180 °.
T. $chmelz, P. Rosmus / Chemical Physics Letters 220 (1994) 117-121
119
Table 1 Calculated rovibroniclevels (J=P) of the X 2II state of HeHF+ (incm -I )
(v,, Vb)Y+
Energy
(v,, Vb)Z-
Energy
(0,1 ) ( 1,1)
590.4 850.7
(1,3)
223.2 513.6 662.0 743.5 884.7
(Vs, Vb)rll/2
Energy
(v,, Vb)H3/2
Energy
H symmetry (0,0) (0,2)/~ (1,0) (l,2)lt (0,2)K (0,4)/~ (2,0) (2,2)/~ (0,6)/~ (l,2)x
319.6 459.0 621.7 709.0 758.7 822.9 865.9 905.1 920.6 946.2
(0,0) (1,0) (0,2)~ (2,0) ( 1,2)it (0,2)r (3,0) (0,4)#
0 311.0 485.6 566.6 733.1 798.4 801.4 923.6
(V., vb)A3/2
Energy
(v., vb)As/2
Energy
587.9 864.1 902.6
(O,l) (I,I) (0,3)# (2,1)
304.3 586.0 721.9 810.2
symmetry (0,1 ) (1,1) (O,3) (2,1 )
II0
120
130
140
150
160
170
180
0(deg) Fi~ 3. One-dimensional cut of the PEFs (R versus0) of the potentialenergyfunctionsof the 211stateof HeHF + alongthe bending coordinatefor fixedR =4.25 bohr.
The variational calculations were carried out using an approach closely related to the method for bound Renner-Teller systems [ 18,19 ] which includes the vibration-rotation, electronic angular m o m e n t u m and spin coupling effectsexplicitly.In contrast to this prior work, here [20] the kinetic energy operator is expressed in Jacobi rather than valence coordinates. The interested reader is referred to our earlier paper
[ 19 ] for technical details of the necessary angular momenta matrix elements and the stepwise optimization of the basis functions. This method permits full three-dimensional variational calculations to be performed on virtually any open-shell triatomic Rcnner-Teller molecule. The basis set consisted of products of 35 Morse eigenfunctions for the stretching mode and 31 Lcgendrc functions for the bending mode. The calculations were performed for total J~< 5/2 and with the experimental spin-orbit constant AS° = _ 292 c m - ~ [ 15 ] of the HF + ion. The geometry variation of both the spin-orbit constant and the electronic angular momentum expectation values were neglected. The rovibronic levels of the linear/linear RennetTeller systems can be classified according to K = A + l and P = A + l + ~ , where A represents the projections of the electronic, ! the vibrational, and 2~the spin angnlar momenta along the linear axis. For example, 2H3/2 signifies K = l, P = 3 / 2 , 2S+ 1 --2. In the present application Kcould take the values 0, l, 2, and 3. In Table l, the calculated rovibronic levels for J = P
A symmetry (0,I) (l,l) (0,3)/~
and K~<2 are given for energies up 950 c m - t above the 2I-I3/2 vibronic ground state (the zero-point vibrational correction has been calculated to be 437 c m - ~). These levels are also displayed in Fig. 4. The assignments for a given J were obtained from an inspection of the nodal structures of both RennerTeller components of the rovibronic wavefunctions and from the weights of the stepwise contracted and optimized basis functions. Whereas the vibrational assignments were more or less straightforward, the assignments of the K values is only approximate, due to the strong mixing of the K substates. Already the lowest states strongly interact. For instance, the H~/2 (0,0) state was found be coupled with the Y-+/2(0,1) state. The stretch wavenumber (1-13/2(0,0)-1"I3/2(1,0)) has been calculated to be 311.0cm -~, the bend wavenumber (Ha/2(0,0)Z ~/2 (0, ! )) to be 223.2 c m - 1, the spin-orbit split-
T. Schmelz, P. Rosmus / Chemical Physics Letters 220 (1994) 117-121
120
Table 2 Some low-lying rotational levels o f the X2]-[ state of HeHF + (e symmetry, in cm - ~)
1000 06
~'~'~--
11
800 21
33__ 04
Oa
2o
ao
~1
04
02
21
02 12
03
O3
600
--
10
01
2O
1I
0"2
01
11
O2
400 O0
10
(~, ~ )
Jto~= 1/2
Jtotat=3/2
Jtoua=5/2
(0,I)~2 (1,1)Z~2 (0,1)ZD2 (0,3)g~2 (0,0)I11/2
223.192 513.570 590.384 662.018 319.643
225.789 516.023 593.174 664.293 322.345
230.192 520.145 597.775 668.166 326.762
(1,0)Hi/2 (0,0)H3/2 (1,0)I'13/2 p(0,2)I13/2
621.669
624.120 0.0 311.021 485.630
628.256 4.522 315.253 490.083
01 Table 3 Parity splittings for some low-lying rotational levels of the X 2I'] state of HeHF + (the differences between the ( e - f ) levels in
200
em -l )
(v,, Vb)
J~,,., = 1/2
J,o,.,= 3/2
(0,1)]C ~/2 (0,0)]-[1/2 (l,1)Zi~/2 (0,3) ~ ~'/2 ( 1,1 ) ~ i-/2 (2,0)H1/2 (0,6)H1/2
-0.226 0.241 -0.083 -1.078 -0.881 --0.502 1.896
-0.621 -0.482 -0.169 -2.152 - 1.713 -- 1.077 3.247
2g~n 2g~r2Zllln 2rI~22A3:2:Alr2 Fig. 4. Pattern of the bound rovibroni¢ levels (J=P) of the zH state of HeHF+.
ting IIi/2(0,0)-1[3/2 (0,0)) to be 319.6cm-L The pattern of the J = P levels resembles, for instance, those in the electronic ground state of CS~ [ 21 ] or in other similar molecules with small bend wavenumber and large spin-orbit constant. In such cases even the K = 0 state cannot be associated with one PEF only. Such a strong breakdown of the Born-Oppenheimer approximation was found for all calculated states. It should be noticed that due to the strong interaction of H F + with the rare gas, the perturbational approach successfully applied for ArOH [ 4 ] is not appropriate for H e H F +. In Table 2 some rotational levels of the rovibronic J = P states are given. The corresponding transition energies (Jto~ ~<5/2) have been calculated to lie in the range of about 2 to 5 c m - t. In Table 3 some of the calculated parity splittings in the rovibronic levels are listed. In accord with the discussion in ref. [ 22 ] these splittings are particularly large in Z levels with excited bending mode. There are several sources of errors in the ab initio calculated energies given in Tables 1-3. From previous experiences with similar CEPA potentials for ArOH [ 3 ], it is to be expected that the electronic
structure calculations probably yielded somewhat too shallow PEFs along the stretch coordinate. The coupling between the HF + stretch and the other two modes has been neglected and the H F + distance in the cluster has not been optimized. Even though this approximation is known to be well founded for many van der Waals complexes, it is less appropriate in the present case, where the ion interacts more strongly with the rare gas. It is also not known how much the spin-orbit constant will change in the cluster relative to HF +. Due to the complicated interplay of these error sources it is difficult to make an estimate for the accuracy of the present results. We believe, however, that the overall features of the rovibronic levels in the H e H F + complex are predicted correctly. It can bc regarded as a prototype of the valence isoelectronic complexes such as RgI-IX + ( R g = H e to Xe, X = F to Br), which will exhibit relatively large dissociation energies and similar spectroscopic properties o f a lin-
T. Schmelz, P. Rosmus / Chemical Physics Letters 220 (1994) I 17-121
e a r / l i n e a r R e n n e r - T e U e r system with a large s p i n o r b i t splitting a n d small b e n d i n g v i b r o n i c wavenumber.
Acknowledgement T h i s w o r k has b e e n s u p p o r t e d b y the D e u t s c h e F o r s c h u n g s g e m e i n s c h a f t , F o n d s d e r C h e m i s c h e n Ind u s t r i e a n d E u r o p e a n C o m m u n i t y grants.
References [ 1 ] M.C. Heaven, Ann. Rev. Phys. Chem. 43 (1992) 283, and references therein. [2] M.T. Berry, M.R. Brustein and M.I. Lester, J. Chem. Phys. 92 (1990) 6469. [3] C. Chakravarty, D.C. Clary, A. Degli Esposti and H.-J. Werner, J. Chem. Phys. 93 (1990) 3367. [4] W.H. Green Jr. and M.I. Lester, J. Chem. Phys. 96 (1992) 2573. [ 5 ] P.D.A. Mills, C.M. Western and B.J. Howard, J. Phys. Chem. 90 (1986) 4961.
121
[6] M.H. Alexander, J. Chem. Phys. 99 (1993) 7725. [ 7 ] T. Schmelz, P. Rosmus and M.H. Alexander, J. Phys. Chem., in press. [8] E.J. Bieske, A. Soliva, M.A. Welker and J.P. Maier, J. Chem. Phys. 93 (1990) 4477. [9] D.J. Nesbin, Chem. Rev. 88 (1988) 843, and references therein. [10] M. Lovejoy and D.J. Nesbitt, J. Chem. Phys. 93 (1990) 5387. [ 11 ] J.M. Hutson, J. Chem. Phys. 89 (1990) 4550. [ 12] H.-C. Chang, F.-M. Tao, W. Klemperer, C. Healey and J.M. Hutson, J. Chem. Phys. 99 (1993) 9337. [ 13] R. Klein and P. Rosmus, Z. Naturforsch. 39a (1984) 21. [ 14 ] W. Meyer, J. Chem. Phys. 58 ( 1973 ) 1017. [15] K.P. Huber and G. Herzberg, Constants of diatomic molecules (Van Nostrand, New York, 1979). [16] P.O. Widmark, P.-A. Malmqvist and B.O. Roos, Theoret. Chim. Acta 77 (1990) 291. [17] P.J. Dagdigdian and M.H. Alexander, J. Chem. Phys. 91 (1989) 839. [ 18] S. Carter and N.C. Handy, Mol. Phys. 52 (1984) 1367. [ 19] S. Carter, N.C. Handy, P. Rosmus and G. Chamhaud, Mol. Phys. 71 (1990) 605. [ 20 ] 1". Schmelz, Ph.D. Thesis, University of Frankfurt (1994). [21] M. Brommer and P. Rosmus, Chem. Phys. Letters 206 (1993) 540. [22] J.T. Hougen, J. Chem. Phys. 36 (1962) 519.
CHEMICAL PHYSICS LETTERS Chemical Physics Letters 220 (1994) 117-121
ELSEVIER
Rovibronic energy levels o f H e H F + ( X Thomas
2I-[)
Schmelz, Pavel Rosmus
Fachbereich Chemie der Universitiit, D-60439 Frankfurt, Germany
Received 18 January 1994
Abstract
Ab initio coupled electron pair (CEPA) potential energy functions have been used to calculate the energies of low-lying rovibronic (J~< 5/2) levels in the X 2H state of HeHF + (for Rnr+ = 1.892 bohr). The complex was found to have a linear equifibrium structure, the dissociation energy Dc has been calculated to be 1490 cm -~, and the equilibrium RH~..HF÷ distance (relative to the center of mass of HF + ) to be 4.25 bohr. The pattern of the rovibrnnic levels can be best characterized by a notation used for linear/linear Rennet-Teller states. The stretch wavenumber (II3/2 (0,0) -H3/2 ( 1,0 ) ) has been calculated to be 311.0 cm - 1, the bend wavenumber (1-13/2(0,0)--Y~2(0,1)) tO be 223.2 cm -~, and the spin-orbit splitting (II~/2(0,0)-H3/2(0,0)) to be 319.6 cm -~ .
1. Introduction
Although neutral and ionic clusters in electronically degenerate open-shell states can be produced in free jet sources, their investigation by high-resolution spectroscopic techniques has begun only recently [ 1 ]. Particularly successful examples o f such studies are the A r O H [ 2 - 4 ] and A r N O [ 5-7 ] complexes. In the case o f charged clusters an accurate spectroscopic characterization is still virtually nonexistent. The first rotationally resolved spectra have recently been detected for N ~ He [ 8 ]. In this Letter, we report theoretical calculations o f the rovibronic levels in the electronic ground state (X2I'I) o f the H e H F + ion. The neutral R g H X ( R g = H e , Ne, Ar, X = F , Cl) van der Waals complexes belong to the best experimentally investigated and theoretically interpreted weakly b o u n d species [ 9-12 ]. To date, neither theoretical nor experimeni Permanent address: Universit6 de Marne la Vall6e, F-93160 Noisy le Grand, France.
tal studies have been known for their charged counterparts. Although H F + is isoelectronic with OH, the interaction between an ion and the rare gas is stronger - the protonated rare gases are strongly b o u n d species [ 13 ] - and more oriented. Hence the pattern o f rovibronic levels in this type o f charged van der Waals complex has little in c o m m o n with that o f their neutral isoelectronic diatoms in their electronic ground states, in which the angular motion o f the diatom is best thought o f as a hindered rotation rather than a bend o f the complex.
2. R e s u l t s and d i s c u s s i o n
The potential energy functions ( P E F ) have been calculated by the size-consistent coupled electron pair (CEPA1) method [14] #1. All energies have been #l Calculated with the MOLPRO ab initio program, written by H.-J. Werner and P.J. Knowles, with comributions from J. Alm10f, R. Amos, S. Elbert, W. Meyer, E.-A. Reinsch, R. Pitzer and A. Stone.
0009-2614/94/$07.00 © 1994 Elsevier Science B.V. All fights reserved SSD10009-2614 (94) 00141-C
T. Schmelz, P. Rosmus / Chemical Physics Letters 220 (1994) 117-121
118
corrected for basis set superposition errors using the counterpoise method. The PEFs for fixed equilibrium geometry o f H F + Rrxv. = 1.892 bohr [ 15 ] have been mapped for 130 geometries; the resulting total energies are available on request. The full three-dimensional treatment would require at least four times as many geometries. In these calculations we have employed the basis sets from ref. [ 16 ], for H (8s, 4p) contracted to [4s, 3p], for He (9s, 4p, 3d) [4s, 3p, 2d] and for F (14s, 9p, 4d) [6s, 5p, 3d]. In additional calculations for geometries around the equilibrium and at the dissociation asymptote, the AO basis set has been augmented by three f functions on F and three d functions on He. The calculated dissociation energy of the complex has increased only marginally relative to the results obtained with the basis set described above. Also the calculated equilibrium geometry hardly changed. W c found itconvenient to fit the set of C E P A energies for both A' and A" component separately by an expansion in terms of reduced rotation matrix elements [7 ]- Figs. 1 and 2 present contour plots of the A' and A" PEFs. In agreement with the qualitativearguments in rcf. [ 17 ] about the
180 166 140 120
~
80 6O 40 2O 0 3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
R (boB) Fig. 2. Two-dimensional cut of the PEFs (R versus 0) of the potentialenergy function for the A" component of the 2rl stateof H e H F +, the contours are plottedin stepsof 100 c m - i.
180 160 140 120
"G
100
60 40 20 0 3.0
3.5
4.0
4.5
5,0
5,5
6.0
6.5
7.0
R (bohr) Fig. I. Two-dimensional cut of the PEFs (R versus 0) of the potentialenergy function for the A' component of the 2H state of H e H F +, the contours are plotted in steps of 100 crn- i.
splittings of the electronically degenerate states along the bending coordinates, the A' component lies energetically lower than the A" component, since the orbital of the diatom is occupied by three electrons. From Fig. 1 it follows that the A' and A" PEFs have different shapes of their repulsive parts. On the A' PEF for energies around 1200 c m - 1 the He atom can approach the ion much closer for nonlinear orientations. Both electronic components have a minimum at linear He...HF + structure with Re= 4.25 bohr (relative to the center of mass of HF + ). For the linear HF +...He structure we found a shallow barrier around R = 5.25 bohr which still lies more than 100 c m - ~below the dissociation asymptote but more than 1300 c m - i above the equilibrium He...HF + structure. The dissociation energy D e has been calculated to be 1490cm -I, i.e.cxpcctedly much larger than in the neutralH c H F complex (7.1 cm- t [l0 ] ).Fig. 3 shows a one-dimensional cut of the PEFs for the Re= 4.25 bohr along the bending coordinate. The splittingof the A' and A" components has been calculated to be rather small for angles close to 180 °.
T. $chmelz, P. Rosmus / Chemical Physics Letters 220 (1994) 117-121
119
Table 1 Calculated rovibroniclevels (J=P) of the X 2II state of HeHF+ (incm -I )
(v,, Vb)Y+
Energy
(v,, Vb)Z-
Energy
(0,1 ) ( 1,1)
590.4 850.7
(1,3)
223.2 513.6 662.0 743.5 884.7
(Vs, Vb)rll/2
Energy
(v,, Vb)H3/2
Energy
H symmetry (0,0) (0,2)/~ (1,0) (l,2)lt (0,2)K (0,4)/~ (2,0) (2,2)/~ (0,6)/~ (l,2)x
319.6 459.0 621.7 709.0 758.7 822.9 865.9 905.1 920.6 946.2
(0,0) (1,0) (0,2)~ (2,0) ( 1,2)it (0,2)r (3,0) (0,4)#
0 311.0 485.6 566.6 733.1 798.4 801.4 923.6
(V., vb)A3/2
Energy
(v., vb)As/2
Energy
587.9 864.1 902.6
(O,l) (I,I) (0,3)# (2,1)
304.3 586.0 721.9 810.2
symmetry (0,1 ) (1,1) (O,3) (2,1 )
II0
120
130
140
150
160
170
180
0(deg) Fi~ 3. One-dimensional cut of the PEFs (R versus0) of the potentialenergyfunctionsof the 211stateof HeHF + alongthe bending coordinatefor fixedR =4.25 bohr.
The variational calculations were carried out using an approach closely related to the method for bound Renner-Teller systems [ 18,19 ] which includes the vibration-rotation, electronic angular m o m e n t u m and spin coupling effectsexplicitly.In contrast to this prior work, here [20] the kinetic energy operator is expressed in Jacobi rather than valence coordinates. The interested reader is referred to our earlier paper
[ 19 ] for technical details of the necessary angular momenta matrix elements and the stepwise optimization of the basis functions. This method permits full three-dimensional variational calculations to be performed on virtually any open-shell triatomic Rcnner-Teller molecule. The basis set consisted of products of 35 Morse eigenfunctions for the stretching mode and 31 Lcgendrc functions for the bending mode. The calculations were performed for total J~< 5/2 and with the experimental spin-orbit constant AS° = _ 292 c m - ~ [ 15 ] of the HF + ion. The geometry variation of both the spin-orbit constant and the electronic angular momentum expectation values were neglected. The rovibronic levels of the linear/linear RennetTeller systems can be classified according to K = A + l and P = A + l + ~ , where A represents the projections of the electronic, ! the vibrational, and 2~the spin angnlar momenta along the linear axis. For example, 2H3/2 signifies K = l, P = 3 / 2 , 2S+ 1 --2. In the present application Kcould take the values 0, l, 2, and 3. In Table l, the calculated rovibronic levels for J = P
A symmetry (0,I) (l,l) (0,3)/~
and K~<2 are given for energies up 950 c m - t above the 2I-I3/2 vibronic ground state (the zero-point vibrational correction has been calculated to be 437 c m - ~). These levels are also displayed in Fig. 4. The assignments for a given J were obtained from an inspection of the nodal structures of both RennerTeller components of the rovibronic wavefunctions and from the weights of the stepwise contracted and optimized basis functions. Whereas the vibrational assignments were more or less straightforward, the assignments of the K values is only approximate, due to the strong mixing of the K substates. Already the lowest states strongly interact. For instance, the H~/2 (0,0) state was found be coupled with the Y-+/2(0,1) state. The stretch wavenumber (1-13/2(0,0)-1"I3/2(1,0)) has been calculated to be 311.0cm -~, the bend wavenumber (Ha/2(0,0)Z ~/2 (0, ! )) to be 223.2 c m - 1, the spin-orbit split-
T. Schmelz, P. Rosmus / Chemical Physics Letters 220 (1994) 117-121
120
Table 2 Some low-lying rotational levels o f the X2]-[ state of HeHF + (e symmetry, in cm - ~)
1000 06
~'~'~--
11
800 21
33__ 04
Oa
2o
ao
~1
04
02
21
02 12
03
O3
600
--
10
01
2O
1I
0"2
01
11
O2
400 O0
10
(~, ~ )
Jto~= 1/2
Jtotat=3/2
Jtoua=5/2
(0,I)~2 (1,1)Z~2 (0,1)ZD2 (0,3)g~2 (0,0)I11/2
223.192 513.570 590.384 662.018 319.643
225.789 516.023 593.174 664.293 322.345
230.192 520.145 597.775 668.166 326.762
(1,0)Hi/2 (0,0)H3/2 (1,0)I'13/2 p(0,2)I13/2
621.669
624.120 0.0 311.021 485.630
628.256 4.522 315.253 490.083
01 Table 3 Parity splittings for some low-lying rotational levels of the X 2I'] state of HeHF + (the differences between the ( e - f ) levels in
200
em -l )
(v,, Vb)
J~,,., = 1/2
J,o,.,= 3/2
(0,1)]C ~/2 (0,0)]-[1/2 (l,1)Zi~/2 (0,3) ~ ~'/2 ( 1,1 ) ~ i-/2 (2,0)H1/2 (0,6)H1/2
-0.226 0.241 -0.083 -1.078 -0.881 --0.502 1.896
-0.621 -0.482 -0.169 -2.152 - 1.713 -- 1.077 3.247
2g~n 2g~r2Zllln 2rI~22A3:2:Alr2 Fig. 4. Pattern of the bound rovibroni¢ levels (J=P) of the zH state of HeHF+.
ting IIi/2(0,0)-1[3/2 (0,0)) to be 319.6cm-L The pattern of the J = P levels resembles, for instance, those in the electronic ground state of CS~ [ 21 ] or in other similar molecules with small bend wavenumber and large spin-orbit constant. In such cases even the K = 0 state cannot be associated with one PEF only. Such a strong breakdown of the Born-Oppenheimer approximation was found for all calculated states. It should be noticed that due to the strong interaction of H F + with the rare gas, the perturbational approach successfully applied for ArOH [ 4 ] is not appropriate for H e H F +. In Table 2 some rotational levels of the rovibronic J = P states are given. The corresponding transition energies (Jto~ ~<5/2) have been calculated to lie in the range of about 2 to 5 c m - t. In Table 3 some of the calculated parity splittings in the rovibronic levels are listed. In accord with the discussion in ref. [ 22 ] these splittings are particularly large in Z levels with excited bending mode. There are several sources of errors in the ab initio calculated energies given in Tables 1-3. From previous experiences with similar CEPA potentials for ArOH [ 3 ], it is to be expected that the electronic
structure calculations probably yielded somewhat too shallow PEFs along the stretch coordinate. The coupling between the HF + stretch and the other two modes has been neglected and the H F + distance in the cluster has not been optimized. Even though this approximation is known to be well founded for many van der Waals complexes, it is less appropriate in the present case, where the ion interacts more strongly with the rare gas. It is also not known how much the spin-orbit constant will change in the cluster relative to HF +. Due to the complicated interplay of these error sources it is difficult to make an estimate for the accuracy of the present results. We believe, however, that the overall features of the rovibronic levels in the H e H F + complex are predicted correctly. It can bc regarded as a prototype of the valence isoelectronic complexes such as RgI-IX + ( R g = H e to Xe, X = F to Br), which will exhibit relatively large dissociation energies and similar spectroscopic properties o f a lin-
T. Schmelz, P. Rosmus / Chemical Physics Letters 220 (1994) I 17-121
e a r / l i n e a r R e n n e r - T e U e r system with a large s p i n o r b i t splitting a n d small b e n d i n g v i b r o n i c wavenumber.
Acknowledgement T h i s w o r k has b e e n s u p p o r t e d b y the D e u t s c h e F o r s c h u n g s g e m e i n s c h a f t , F o n d s d e r C h e m i s c h e n Ind u s t r i e a n d E u r o p e a n C o m m u n i t y grants.
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