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A theoretical study of the rotation-vibration energy levels for the stretching modes in C2H2+ (X̃2Πu )
JOURNAL
OF MOLECULAR
SPECTROSCOPY
141,43-48 (1990)
A Theoretical Study of the Rotation-Vibration Energy Levels for the Stretching Modes in C2 H 2’ (I? *II,) W.P. KRAEMERAND
V. SPIRKO'
Ma_x-Planck-Institute of Physics and Astrophysics, D-8046 Garching, West Germany AND
B. 0. Roos Department of Theoretical Chemistry, Chemical Center, University of Lund, S-22100 Lund, Sweden The three-dimensional potential energy surface for the stretching motions of the d211. electronic ground state of the C2H: ion is calculated at the Complete Active Space (CAS) SCF and MultiReference Configuration Interaction (MR-CI) level of theory using extended Gaussian basis sets of the general contraction type. A model Hamiltonian for pure stretching motions is applied to calculate the rovibrational energy levels. The results are in close agreement with the few available experimental data. The predicted values can therefore be expected to be reliable. o 1990 Academic Press, Inc.
I. INTRODUCTION
Recently the 584-A photoelectron spectra of rotationally cold acetylene, C2HZ, and its fully deuterated form, C2D2, were obtained with a resolution that permits a partial determination of the vibrational energies of the lowest three electronic states of the resulting ions, C2H : and C2D : ( 1) . In addition, for the 2 ‘I&, electronic ground state of the acetylene cation, the u3 band was recorded with very high resolution and was analyzed in terms of effective spectroscopic constants (2). In a recent theoretical study of the j211, state of C2H: (3), the analytic gradient technique was used at the Hartree-Fock self-consistent field (SCF) and at correlated (MCSCF and CI) levels of theory to calculate the equilibrium geometry and harmonic frequencies. From corresponding dipole moment calculations the asymmetric C-H stretching fundamental was predicted to be the strongest IR absorber. Its frequency value, obtained from the calculated harmonic SCF frequency after applying an empirical scaling factor, turns out to be very close to the experimental value. The aim of the present theoretical study was to analyze the stretching motions of C2H 2’(x2&,) going beyond the limitations of the harmonic approximation and without applying empirical corrections. The complexity of the full-dimensional problem in a four-atomic molecule was considerably reduced by using the fact that for the electronic ground state of C2H: the treatment of the high-frequency stretching motions ’ Permanent address: The Heyrovsky Institute of Physical Chemistry and Electrochemistry, Czechoslovak Academy of Sciences, 182 23 Prague, Czechoslovakia. 43
0022-2852190 $3.00 Copyright @ 1990 by Academic Press, Inc. All rights of reproduction in any form reserved.
44
KRAEMER,
SPIRKO,
AND
~00s
can be separated in a rather good approximation from the treatment of the lowfrequency bending motions. The calculation of the potential energy hypersurface was thus limited to the evaluation of the three-dimensional stretching potential. The calculations were done at the Complete Active Space (CAS) SCF (4) and the MultiReference Configuration Interaction (MR-CI) level of theory employing extended Gaussian basis sets of the general contraction type (5). Optimized CAS SCF orbitals were used as molecular orbital basis in the MR-CI treatment. Details of the ab initio calculations are presented elsewhere in a study of the energetics of several low lying electronic states of the C2Hi ion ( 6). From the ab initio calculated stretching potential the rovibrational energy levels were obtained variationally because the standard perturbation approach is known not to be quite adequate when dealing with stretching motions in which hydrogen atoms are involved (see, e.g., Refs. ( 7, 8)). II. THE
ROVIBRATIONAL
HAMILTONIAN
To describe the rotation-stretching dynamics of the C2H: molecular ion, rectilinear symmetry coordinates are conveniently introduced as
In these expressions rl and r2 are used for the two C-H bond distances and re for their equilibrium value, while R and R, describe correspondingly the C-C bond length. The calculation of the rovibrational energies was based on the Watson isomorphic Hamiltonian (9) expressed in terms of the above symmetry coordinates. As a further simplification, the Coriolis interaction terms were neglected because they play only a minor role in the present case. The Hamiltonian acquires then the form
where & are the momenta conjugate to the Sk for the pure vibrational part of the Hamiltonian; J is the rotational quantum number; the elements of the G matrix are G,, = GZZ= (mH,+ mc)/mHmC,~G22 = 2/mc, Glz = GzI = -dJmc (with mH and mc being the atomic masses for hydrogen and carbon, respectively); and p = ( 1/IO)p’ with
x2 mH(R,
p’ = 1 - 7
f
+ 2r,)S,
mH(R, 0
- $ [(Re -
+.b,)* -
1
s: 1
2rdmH
+
fLmcl&
STRETCHING
MODES IN &Hz
45
(?II,)
TABLE I Comparison between Calculated and Experimental Band Centers and Rotational Constants of Acetylene (in cm-’ )
9
!a VI & B(vs)
Experiment”
Ref.[‘l]
This work
3372.5 1973.5 3294.9 1.1766 1.1723
3430 1975 3330
3392 2004 3326 1.1787 1.1708
aExperimentai data quoted in Refs.[‘l] and [ll].
J?
+-WlH
I0
I +,
;
[(&
(R, + 2r,)[(R,
+
2r,)mH
+ &‘%?I
-
1
sIs2 I
+
2r,)mH + R,mc12 - ; (mf, + mc)
I
St
and I0 = i RZ ( mH + mc) + 2mHr,(R,
+ r,).
(4)
Finally, the potential energy function V was assumed to have the form I’= C FuQiQj + C Fij/sQiQiQk + i6j
where Q, = 1 - exp[-aIS,];
i
C
Fi&?iQjQkQ~,
(5)
i
Q2 = S2; Q3 = &: and al is a constant. TABLE II
Potential Energy Function Parameters for CzHt Obtained by Fitting Eq. (5) to ab initio Calculated Data Parameter
CAS SCF 1.93115 -0.17481 6.73895 2.75231 -0.22669 -9.70398 -13.34756 0.20702 9.81592 13.52624 3.99593 1.089 1.260 I.20
“Held fixed after preliminary determination.
MR-CI 2.08669
-0.23109 7.00062 2.74383 -0.07904 -11.48543 -14.04089 0.16577 24.90111 16.23864 5.00599 1.079 1.251 1.2’
46
KRAEMER,
SPIRKO, AND ROOS TABLE III
Calculated Vibrational Energies (in cm-‘)
0
Z.P.E.
0
cm-’ cm-’ cm-’ cm-’
0
0
1761 3106 3508 3066 4823 6093 6566
1644* 2472’ 3284 2273 3906 4703 5537
1864’ 3306 3715 3134d 4996 6492 6847
1801 3287 3591 3124 4924 6464 6717
1695* 2577= 3388 2300 3998 4892 5692
4102
4057
3250
4209
4163
3326
Available experimental ’ 1829 b 1651 ’ 2572 ’ 3136
0
0
18284 3127 3628 3075* 4892 6120 6694
results:
[Ref.l]. [Ref.l]. [Ref.11 [Ref.l].
III. ROTATION-VIBRATION
ENERGIES
To solve the three-dimensional vibrational problem the total wavefunction was factorized into the product $( u1)1,5(1)~)X( VS), where the individual functions $J(u1), I+!(u2), X(Q) for the symmetric CH stretch, the CC stretch, and the antisymmetric CH stretch, respectively, were obtained numerically (10) by solving the corresponding uncoupled one-dimensional Schrodinger equations. Diagonalization of the Hamiltonian (2 ) as a matrix over these basis functions yields the vibrational band centers (J = 0) and rovibrational energies (J > 0) from which the rotational and centrifugal distortion constants can be evaluated by the standard fitting procedure. Due to the symmetry of the antisymmetric CH stretching motion the Hamiltonian matrix actually factorizes into two sub-blocks depending on the parity of u3.
TABLE IV Calculated Rotational Constants B (in cm-‘) MR-CI
CAS SCF
“CzH, + %aH : ‘=CsD: 0
0
0
10
100 0 2 0 0 0 1 101 0 2
0
0 1 1 1
1.0851” 1.0619 1.0877 1.0632 1.0784* 1.0561 1.0693 1.0547
1.0320 1.0122 1.0331 1.0129 1.0258 1.0073 1.0151 1.0057
Experimental results: ’ 1.1046 cm-’ [Ref.S]. * 1.0990 cm-’ IRef.21.
0.7851 0.7548 0.7993 0.7536 0.7793 0.7477 0.7886 0.7365
1.1028’ 1.0870 1.1019 1.0744 1.0965* 1.0842 1.0856 1.0715
1.0487 1.0353 1.0463 1.0247 1.0429 1.0326 1.0312 1.0217
0.7983 0.7758 0.8106 0.7629 0.7928 0.7737 0.7972 0.7586
STRETCHING
47
MODES IN C2H; (d211,) TABLE V
Calculated Centrifugal Distortion Constants D, X 1O6 (in cm -’ )
0 0 0 1 100 020 0 0 0 1 1 0 0 2
0
0
1 1 1 1
1.57” 5.59 2.09 7.30 1.46” 5.36 3.89 6.81
1.39 4.87 1.87 6.35 1.27 4.52 3.63 5.72
n Assumed experimental due:
0.95 3.51 1.18 5.58 0.98 4.41 1.84 7.93
1.56’ 4.82 2.16 7.79 1.290 3.70 3.61 6.42
1.37 4.20 1.98 6.78 1.12 3.14 3.27 5.52
1.44 x lo*
cm-’ [Retl].
0.98 2.98 1.32 4.65 0.89 2.90 1.82 4.99
To assess the reliability of the model used here, the rovibrational energies of the electronic ground state of acetylene CzH2 (2’Z,‘) were calculated using the quartic potential energy function of Strey and Mills ( I I ) . In the actual calculations 300 basis functions for each symmetry block were employed to converge the 10 lowest vibrational energies in each block, using a convergence threshold of 0.1 cm -I. In Table I these results are compared with experimental results and previous theoretical numbers which were obtained by a fully variational procedure ( 7). The agreement with experiment is satisfactory, the maximum discrepancy being about 30 cm-‘. In the case of the acetylene cation, Cz Hi (2211,), the potential energy function was determined by fitting the expression of Eq. (5) to the potential points calculated at the CAS SCF and the MR-CI level of theory. The fitted potential parameters are collected in Table II. The equilibrium CH and CC bond lengths obtained from the MR-CI calculations are in reasonable agreement with the corresponding bond distance values deduced from some earlier experiments ( 12). Using the fitted ab initio potential functions and applying the same computational procedure as described for acetylene, the rovibrational energies of the C2H: ion were evaluated and the results for the four lowest states of each 213parity are summarized in Tables III-IV. The MR-CI values for the symmetric CH-stretching frequency of C2D: and for the antisymmetric CHstretching frequency of C2H: are in perfect agreement with the corresponding experimental numbers, whereas the CC-stretching frequency turned out to differ from experiment by about 35 cm-‘. Comparison of the two available experimental rotation constants, Bm and BOO,,with the MR-CI results shows that the theoretical numbers differ only by about 0.2%, which is within the expected accuracy limits of the present calculations. This very satisfactory agreement of the MR-CI results with the few available experimental values indicates that the other predicted numbers are reliable and that they can be useful in future experimental studies. ACKNOWLEDGMENTS This study was initiated when one of the authors ( WPK) was visiting the Department of Theoretical Chemistry at the University of Lund in the fall of 1988. He is grateful for hospitality and financial support. We thank Dr. P. Jensen (Giessen) for critically reading the manuscript. RECEIVED:
January 19, 1990
48
mama,
SPIRKO,
AND Roos
REFERENCES 1. J. E. REUTT, L. S. WANG, J. E. POLLARD, D. J. TREVOR, Y. T. LEE, AND D. A. SHIRLEY, J. Chem.
Phys. 84,3022-3031 ( 1984). 2. M. W. CROFTON, M.-F. JAGOD,B. D. REHFUSS,AND T. OKA, J. Chem. Phys. 86,3755-3756 ( 1987). 3. T. J. LEE, J. E. RICE, AND H. F. SCHAEFER,J. Chem. Phys. 86,3051-3053 (1987). 4. B. 0. Roos, P. R. TAYLOR, AND P. E. M. SIEGBAHN,Chem. Phys. 48, 157-173 ( 1980). B. 0. Roos, Int. J. Quantum Chem. Symp. 14, 175-189 (1980). 5. R. LINDH, B. 0. Roos, AND W. P. KRAEMER, Chem. Phys. Lett. 139,407-416 (1987). 6. W. P. KRAEMER, B. 0. Roos, AND V. SPIRKO, in preparation. 7. S. CARTER AND N. C. HANDY, Comput. Phys. Commun. 51,49-71 (1988). 8. P. JENSEN,J. Chem. Sot., Faraday Trans. 2 84, I3 15- 1340 ( 1988 ); P. JENSENAND W. P. KRAEMER, J. Mol. Spectrosc. 129, 175-182 (1988). 9. J. K. G. WATSON, Mol. Phys. 19,465-487 ( 1970). 10. J. W. COOLEY, Math. Comput. 15,363-374 (1961). 11. G. STREY AND I. M. MILLS, J. Mol. Spectrosc. 59, 103-l 15 (1976). 12. J. M. HOLLAS AND T. A. SUTHERLAY, Mol. Phys. 21, 183-185 (1971).
OF MOLECULAR
SPECTROSCOPY
141,43-48 (1990)
A Theoretical Study of the Rotation-Vibration Energy Levels for the Stretching Modes in C2 H 2’ (I? *II,) W.P. KRAEMERAND
V. SPIRKO'
Ma_x-Planck-Institute of Physics and Astrophysics, D-8046 Garching, West Germany AND
B. 0. Roos Department of Theoretical Chemistry, Chemical Center, University of Lund, S-22100 Lund, Sweden The three-dimensional potential energy surface for the stretching motions of the d211. electronic ground state of the C2H: ion is calculated at the Complete Active Space (CAS) SCF and MultiReference Configuration Interaction (MR-CI) level of theory using extended Gaussian basis sets of the general contraction type. A model Hamiltonian for pure stretching motions is applied to calculate the rovibrational energy levels. The results are in close agreement with the few available experimental data. The predicted values can therefore be expected to be reliable. o 1990 Academic Press, Inc.
I. INTRODUCTION
Recently the 584-A photoelectron spectra of rotationally cold acetylene, C2HZ, and its fully deuterated form, C2D2, were obtained with a resolution that permits a partial determination of the vibrational energies of the lowest three electronic states of the resulting ions, C2H : and C2D : ( 1) . In addition, for the 2 ‘I&, electronic ground state of the acetylene cation, the u3 band was recorded with very high resolution and was analyzed in terms of effective spectroscopic constants (2). In a recent theoretical study of the j211, state of C2H: (3), the analytic gradient technique was used at the Hartree-Fock self-consistent field (SCF) and at correlated (MCSCF and CI) levels of theory to calculate the equilibrium geometry and harmonic frequencies. From corresponding dipole moment calculations the asymmetric C-H stretching fundamental was predicted to be the strongest IR absorber. Its frequency value, obtained from the calculated harmonic SCF frequency after applying an empirical scaling factor, turns out to be very close to the experimental value. The aim of the present theoretical study was to analyze the stretching motions of C2H 2’(x2&,) going beyond the limitations of the harmonic approximation and without applying empirical corrections. The complexity of the full-dimensional problem in a four-atomic molecule was considerably reduced by using the fact that for the electronic ground state of C2H: the treatment of the high-frequency stretching motions ’ Permanent address: The Heyrovsky Institute of Physical Chemistry and Electrochemistry, Czechoslovak Academy of Sciences, 182 23 Prague, Czechoslovakia. 43
0022-2852190 $3.00 Copyright @ 1990 by Academic Press, Inc. All rights of reproduction in any form reserved.
44
KRAEMER,
SPIRKO,
AND
~00s
can be separated in a rather good approximation from the treatment of the lowfrequency bending motions. The calculation of the potential energy hypersurface was thus limited to the evaluation of the three-dimensional stretching potential. The calculations were done at the Complete Active Space (CAS) SCF (4) and the MultiReference Configuration Interaction (MR-CI) level of theory employing extended Gaussian basis sets of the general contraction type (5). Optimized CAS SCF orbitals were used as molecular orbital basis in the MR-CI treatment. Details of the ab initio calculations are presented elsewhere in a study of the energetics of several low lying electronic states of the C2Hi ion ( 6). From the ab initio calculated stretching potential the rovibrational energy levels were obtained variationally because the standard perturbation approach is known not to be quite adequate when dealing with stretching motions in which hydrogen atoms are involved (see, e.g., Refs. ( 7, 8)). II. THE
ROVIBRATIONAL
HAMILTONIAN
To describe the rotation-stretching dynamics of the C2H: molecular ion, rectilinear symmetry coordinates are conveniently introduced as
In these expressions rl and r2 are used for the two C-H bond distances and re for their equilibrium value, while R and R, describe correspondingly the C-C bond length. The calculation of the rovibrational energies was based on the Watson isomorphic Hamiltonian (9) expressed in terms of the above symmetry coordinates. As a further simplification, the Coriolis interaction terms were neglected because they play only a minor role in the present case. The Hamiltonian acquires then the form
where & are the momenta conjugate to the Sk for the pure vibrational part of the Hamiltonian; J is the rotational quantum number; the elements of the G matrix are G,, = GZZ= (mH,+ mc)/mHmC,~G22 = 2/mc, Glz = GzI = -dJmc (with mH and mc being the atomic masses for hydrogen and carbon, respectively); and p = ( 1/IO)p’ with
x2 mH(R,
p’ = 1 - 7
f
+ 2r,)S,
mH(R, 0
- $ [(Re -
+.b,)* -
1
s: 1
2rdmH
+
fLmcl&
STRETCHING
MODES IN &Hz
45
(?II,)
TABLE I Comparison between Calculated and Experimental Band Centers and Rotational Constants of Acetylene (in cm-’ )
9
!a VI & B(vs)
Experiment”
Ref.[‘l]
This work
3372.5 1973.5 3294.9 1.1766 1.1723
3430 1975 3330
3392 2004 3326 1.1787 1.1708
aExperimentai data quoted in Refs.[‘l] and [ll].
J?
+-WlH
I0
I +,
;
[(&
(R, + 2r,)[(R,
+
2r,)mH
+ &‘%?I
-
1
sIs2 I
+
2r,)mH + R,mc12 - ; (mf, + mc)
I
St
and I0 = i RZ ( mH + mc) + 2mHr,(R,
+ r,).
(4)
Finally, the potential energy function V was assumed to have the form I’= C FuQiQj + C Fij/sQiQiQk + i6j
where Q, = 1 - exp[-aIS,];
i
C
Fi&?iQjQkQ~,
(5)
i
Q2 = S2; Q3 = &: and al is a constant. TABLE II
Potential Energy Function Parameters for CzHt Obtained by Fitting Eq. (5) to ab initio Calculated Data Parameter
CAS SCF 1.93115 -0.17481 6.73895 2.75231 -0.22669 -9.70398 -13.34756 0.20702 9.81592 13.52624 3.99593 1.089 1.260 I.20
“Held fixed after preliminary determination.
MR-CI 2.08669
-0.23109 7.00062 2.74383 -0.07904 -11.48543 -14.04089 0.16577 24.90111 16.23864 5.00599 1.079 1.251 1.2’
46
KRAEMER,
SPIRKO, AND ROOS TABLE III
Calculated Vibrational Energies (in cm-‘)
0
Z.P.E.
0
cm-’ cm-’ cm-’ cm-’
0
0
1761 3106 3508 3066 4823 6093 6566
1644* 2472’ 3284 2273 3906 4703 5537
1864’ 3306 3715 3134d 4996 6492 6847
1801 3287 3591 3124 4924 6464 6717
1695* 2577= 3388 2300 3998 4892 5692
4102
4057
3250
4209
4163
3326
Available experimental ’ 1829 b 1651 ’ 2572 ’ 3136
0
0
18284 3127 3628 3075* 4892 6120 6694
results:
[Ref.l]. [Ref.l]. [Ref.11 [Ref.l].
III. ROTATION-VIBRATION
ENERGIES
To solve the three-dimensional vibrational problem the total wavefunction was factorized into the product $( u1)1,5(1)~)X( VS), where the individual functions $J(u1), I+!(u2), X(Q) for the symmetric CH stretch, the CC stretch, and the antisymmetric CH stretch, respectively, were obtained numerically (10) by solving the corresponding uncoupled one-dimensional Schrodinger equations. Diagonalization of the Hamiltonian (2 ) as a matrix over these basis functions yields the vibrational band centers (J = 0) and rovibrational energies (J > 0) from which the rotational and centrifugal distortion constants can be evaluated by the standard fitting procedure. Due to the symmetry of the antisymmetric CH stretching motion the Hamiltonian matrix actually factorizes into two sub-blocks depending on the parity of u3.
TABLE IV Calculated Rotational Constants B (in cm-‘) MR-CI
CAS SCF
“CzH, + %aH : ‘=CsD: 0
0
0
10
100 0 2 0 0 0 1 101 0 2
0
0 1 1 1
1.0851” 1.0619 1.0877 1.0632 1.0784* 1.0561 1.0693 1.0547
1.0320 1.0122 1.0331 1.0129 1.0258 1.0073 1.0151 1.0057
Experimental results: ’ 1.1046 cm-’ [Ref.S]. * 1.0990 cm-’ IRef.21.
0.7851 0.7548 0.7993 0.7536 0.7793 0.7477 0.7886 0.7365
1.1028’ 1.0870 1.1019 1.0744 1.0965* 1.0842 1.0856 1.0715
1.0487 1.0353 1.0463 1.0247 1.0429 1.0326 1.0312 1.0217
0.7983 0.7758 0.8106 0.7629 0.7928 0.7737 0.7972 0.7586
STRETCHING
47
MODES IN C2H; (d211,) TABLE V
Calculated Centrifugal Distortion Constants D, X 1O6 (in cm -’ )
0 0 0 1 100 020 0 0 0 1 1 0 0 2
0
0
1 1 1 1
1.57” 5.59 2.09 7.30 1.46” 5.36 3.89 6.81
1.39 4.87 1.87 6.35 1.27 4.52 3.63 5.72
n Assumed experimental due:
0.95 3.51 1.18 5.58 0.98 4.41 1.84 7.93
1.56’ 4.82 2.16 7.79 1.290 3.70 3.61 6.42
1.37 4.20 1.98 6.78 1.12 3.14 3.27 5.52
1.44 x lo*
cm-’ [Retl].
0.98 2.98 1.32 4.65 0.89 2.90 1.82 4.99
To assess the reliability of the model used here, the rovibrational energies of the electronic ground state of acetylene CzH2 (2’Z,‘) were calculated using the quartic potential energy function of Strey and Mills ( I I ) . In the actual calculations 300 basis functions for each symmetry block were employed to converge the 10 lowest vibrational energies in each block, using a convergence threshold of 0.1 cm -I. In Table I these results are compared with experimental results and previous theoretical numbers which were obtained by a fully variational procedure ( 7). The agreement with experiment is satisfactory, the maximum discrepancy being about 30 cm-‘. In the case of the acetylene cation, Cz Hi (2211,), the potential energy function was determined by fitting the expression of Eq. (5) to the potential points calculated at the CAS SCF and the MR-CI level of theory. The fitted potential parameters are collected in Table II. The equilibrium CH and CC bond lengths obtained from the MR-CI calculations are in reasonable agreement with the corresponding bond distance values deduced from some earlier experiments ( 12). Using the fitted ab initio potential functions and applying the same computational procedure as described for acetylene, the rovibrational energies of the C2H: ion were evaluated and the results for the four lowest states of each 213parity are summarized in Tables III-IV. The MR-CI values for the symmetric CH-stretching frequency of C2D: and for the antisymmetric CHstretching frequency of C2H: are in perfect agreement with the corresponding experimental numbers, whereas the CC-stretching frequency turned out to differ from experiment by about 35 cm-‘. Comparison of the two available experimental rotation constants, Bm and BOO,,with the MR-CI results shows that the theoretical numbers differ only by about 0.2%, which is within the expected accuracy limits of the present calculations. This very satisfactory agreement of the MR-CI results with the few available experimental values indicates that the other predicted numbers are reliable and that they can be useful in future experimental studies. ACKNOWLEDGMENTS This study was initiated when one of the authors ( WPK) was visiting the Department of Theoretical Chemistry at the University of Lund in the fall of 1988. He is grateful for hospitality and financial support. We thank Dr. P. Jensen (Giessen) for critically reading the manuscript. RECEIVED:
January 19, 1990
48
mama,
SPIRKO,
AND Roos
REFERENCES 1. J. E. REUTT, L. S. WANG, J. E. POLLARD, D. J. TREVOR, Y. T. LEE, AND D. A. SHIRLEY, J. Chem.
Phys. 84,3022-3031 ( 1984). 2. M. W. CROFTON, M.-F. JAGOD,B. D. REHFUSS,AND T. OKA, J. Chem. Phys. 86,3755-3756 ( 1987). 3. T. J. LEE, J. E. RICE, AND H. F. SCHAEFER,J. Chem. Phys. 86,3051-3053 (1987). 4. B. 0. Roos, P. R. TAYLOR, AND P. E. M. SIEGBAHN,Chem. Phys. 48, 157-173 ( 1980). B. 0. Roos, Int. J. Quantum Chem. Symp. 14, 175-189 (1980). 5. R. LINDH, B. 0. Roos, AND W. P. KRAEMER, Chem. Phys. Lett. 139,407-416 (1987). 6. W. P. KRAEMER, B. 0. Roos, AND V. SPIRKO, in preparation. 7. S. CARTER AND N. C. HANDY, Comput. Phys. Commun. 51,49-71 (1988). 8. P. JENSEN,J. Chem. Sot., Faraday Trans. 2 84, I3 15- 1340 ( 1988 ); P. JENSENAND W. P. KRAEMER, J. Mol. Spectrosc. 129, 175-182 (1988). 9. J. K. G. WATSON, Mol. Phys. 19,465-487 ( 1970). 10. J. W. COOLEY, Math. Comput. 15,363-374 (1961). 11. G. STREY AND I. M. MILLS, J. Mol. Spectrosc. 59, 103-l 15 (1976). 12. J. M. HOLLAS AND T. A. SUTHERLAY, Mol. Phys. 21, 183-185 (1971).