15580
J. Phys. Chem. 1995,99, 15580-15586
Potential Energy Functions and Bound Rovibronic Levels of the Oz+He (X211) Complex? T. Schmelz and E.-A. Reinsch Fachbreich Chemie der Universitat Frankfurt, 0-60439 Frankfurt, Germany
P. Rosmus” Universitk de Mame-la-Vallke, F-93166 Noisy-le-Grand, France Received: April 13, 1995; In Final Form: July 18, 1995@ Three-dimensional potential energy functions have been generated for the electronic ground state of the 02+He (X2n) complex using highly correlated multireference intemally contracted configuration interaction (MRCI) and averaged coupled pair functional (ACPF) electronic wave functions. The complex is found to have a T-shaped equilibrium geometry with Re(02+***He)of 5.2363 bohr (A‘ component) and 5.1601 bohr (A” component). The ACPF calculations predict the dissociation energies of 167.5 cm-l (A’) and 194.3 cm-’ (A”) and the barriers to linearity of 50.9 cm-’ (A’), and 77.7 cm-’ (A”), respectively. The results from both MRCI and ACPF approaches have been compared. The rovibronic levels have been obtained by a variational approach which includes the rotation-vibration, electron angular momentum, and electron spin coupling effects, but neglects the nuclear spin. Vibronic states with up to six bending and two stretching quanta excited are calculated to be bound. Due to the large differences between the shapes of the A’ and A” potentials, large parity splittings (more than 1 cm-’ in excited bending levels) have been obtained. Also the influence of the vibrational excitation in the 0 2 + fragment on the large amplitude modes has been investigated.
1. Introduction
pattern. The present ab initio calculations show that at the T-shaped equilibrium structure the A’ and A” potentials differ Recent experimental advances have allowed the observation considerably and the complex is more strongly bound. Moreof van der Waals complexes involving diatomic free radicals over, in the electronic ground state of 0 2 + the spin-orbit in electronically degenerate states.’ For instance, the spectra is positive (200 cm-I); in OH it is negative (-139.2 constant of AIOH,~NeOH,3 ArCN: ArN0,5 ArBH,6 AICH,~ArNH,8 cm-I). The rotational constant in 0 2 + amounts to 1.69 cm-I; and ArAlH9 complexes have been detected and analyzed. The in OH, to 18.91 The dissociation energy of 02+He is interpretation of these experiments has often been assisted by not as large as in the linear and HeHC1+,32which have ab initio calculations.I0-l6 A large number of ionic complexes energy level structures very similar to strongly bound linear have been known,17 but only a few have been investigated by Renner-Teller molecules. The changes of the energy level high-resolution spectroscopic techniques. Impressive examples structures depending on the potential shapes and equilibrium are the recent studies of the HeHe+,I8 HeAr+,I9 N2+Hen,20 structures in open shell van der Waals complexes have been N2+Nen,21and N2OAPZ2complexes. For the Oz+He (X217) nicely interpreted by Dubemet et al.33 complex, however, neither spectroscopic nor theoretical inforRecently, we have calculated the rovibronic transitions and mation is known. Some studies have been reported concerning the parity splittings for the ArNO complexI3 from the ab initio the interaction of 0 2 + with other rare gases. For 02+Ar the potential energy functions and compared them with the available energy partitioning in collisions at low energies has been high-resolution spectra of the vibrational ground state. The reported by Scherbarth and G e r l i ~ h . The ~ ~ charge transfer absolute values of the very small energy differences agreed to reaction Kr+ 0 2 0 2 + Kr has been studied by Jarrold et within about 30-50% with experiment. Even though this al.% Ramachandran and EvaZ5have investigated the vibrational accuracy is not sufficient for high-resolution spectroscopic deactivation and, for v’ = 1 of 0 2 + colliding with Kr, have predictions, it certainly provides a reliable template for a more found substantial deviations from the Landau-Teller model, characterization of the electronically degenerate states which stimulated theoretical investigation of this p r ~ b l e m . ~ ~ , refined ~~ of the weakly bound complexes. Recently, Dim et a1.28 have calculated an ab initio potential energy function for 0 2 % (X217) with a relatively small CI 2. Electronic Structure Calculations and Potential Energy expansion. Surfaces In the linear structure the 02+He complex has 217symmetry, The three-dimensional potential energy surfaces for the A‘ which gives rise to two components, A’ and A”, in C, symmetry. and A” components of the 02+He (X2n) complex have been In such a case, the nuclear and electronic motion cannot be mapped in the geometry region 1.8 5 r 5 2.4 bohr, 3.0 5 R 5 treated separately, because of the coupling of the rotational20 bohr, and 0” 5 8 5 180” ( r is the distance in the 0 2 + vibrational and electronic and spin angular momenta. Using a fragment, R is the distance between the center of mass of 02+ variational method and potential energy functions calculated ab and He, and 6 is the angle between the vectors r and R) using initio, we have solved the problem of the coupled electronic the MRCI (multireference intemally contracted configuration and nuclear motion for J = 112 and 3/2. In the 02+He complex i n t e r a ~ t i o n ~and ~ . ~the ~ ) ACPF (averaged coupled pair functhe energy level structure is different from the cases such as t i ~ n a l ~approaches. ~ . ~ ~ ) The calculated total energies have been ArOH, in which the potential anisotropy is not strong enough corrected for the basis set superposition and size consistency to cause substantial mixing of the free-rotor states so that the errors. For oxygen we used the (14s,8p)[6s,5p] correlationperturbation theory29can be applied to explain the energy level consistent A 0 basis set of Dunning,38 which was augmented by one s-function (exp: 0.07) and one set of p-functions (exp: Dedicated in honor of Zdenek Herman. 0.05) and three optimized sets of d-functions (exp: 3.3, 1.5, Abstract published in Advance ACS Abstracts, October 1, 1995.
+
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+
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0022-365419512099-15580$09.00/0 0 1995 American Chemical Society
Potential Energy Functions of the 02+He (Xzn)Complex
J. Phys. Chem., Vol. 99, No. 42, 1995 15581
TABLE 1: MRCI Potential Energy Function of the Electronic Ground State of the Oz+***He Complex (the Calculated Energies Corrected for Basis Superposition Errors Are Given (in au) Relative to the Dissociation Asymptote) R 6 r energy R e r energy R e r energy 3.5 4.0 4.5 5.0 6.0 7.0 8.0 10.0 12.0 14.0 16.0
70 70 70 70 70 70 70 70 70 70 70
1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8
0.016920 0.003747 -0.000048 -0.000849 -0.000643 -0.000344 -0.000188 -0.000067 -0.000026 -0.000009 -0.0oooo1
2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11
0.137099 0.044435 0.002993 0.000188 -0.000440 -0.000476 -0.000386 -0.000219 -0.000 127
0 0 0 0 0
2.11 2.11 2.11 2.11 2.11
-0.oooO78 -0.000033 -0.oooO15 -0.000007 -0.000003
3.0 3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0 10.0 12.0 14.0 16.0
45 45 45 45 45 45 45 45 45 45 45 45 45 45
2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11
0.107255 0.039808 0.012285 0.002755 -0.000043 -0.000642 -0.000630 -0.000370 -0.000205 -0.000120 -0.000074 -0.000032 -0.0000 15 -0.ooooO1
3.0 3.5 4.0 4.8 5.0 5.3 5.5 6.0 7.0 8.0 9.0 10.0 12.0 14.0 16.0 18.0 20.0
90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90
2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11
0.050810 0.014823 0.003290 -0.000649 -0.000788 -0.000816 -0.000775 -0.00061 1 -0.000336 -0.000189 -0.000113 -0.oooO71 -0.00003 1 -0.0000 14 -0.OooOo7 -0.000002 -0.oooOo1
4.0 4.5 5.0 6.0 6.5 7.0 8.0 9.0 10.0 12.0 14.0 16.0 18.0 20.0
0 0 0 0 0 0 0 0 0 0 0 0 0
2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3
0.052229 0.015520 0.003753 -0.000426 -0.000496 -0.000408 -0.00023 1 -0.000133 -0.00008 1 -0.oooO34 -0.0000 15 -0.000007 -0.000003 -0.000001
3.5 4.0 4.5 5.0 5.3 6.0 7.0 8.0 14.0 16.0 20.0
70 70 70 70 70 70 70 70 70 70 70
2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3
0.018080 0.004589 0.000330 -0.000662 -0.000690 -0.000606 -0.000341 -0.0001 19 -0.000014 -0.OoooO6 -0.0oooo1
3.0 3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0 10.0 12.0 14.0 16.0 18.0
90 90 90 90 90 90 90 90 90 90 90 90 90 90 90
2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3
0.048130 0.014779 0.003444 0.000054 -0.000720 -0.0007 18 -0.000592 -0.000329 -0.000186 -0.0001 11 -0.000070 -0.000031 -0.0000 14 -0.000006 -0.000002
4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 10.0 12.0
45 45 45 45 45 45 45 45 45 45 45 45 45
2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4
0.0 13925
0.003415 0.000156 -0.000618 -0.000555 -0.000498 -0.000352 -0.000280 -0.0002 19 -0.000165 -0.000123 -0.oooO74 -0.000031
3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 8.0 9.0 10.0 12.0
70 70 70 70 70 70 70 70 70 70 70 70
2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4
0.020309 0.004835 0.000345 -0.000607 -0.000657 -0.000533 -0.000421 -0.000346 -0.000191 -0.000112 -0.oooO71 -0.000032
9.0 10.0 12.0
90 90 90
2.4 2.4 2.4
-0.000108 -0.000071 -0.000031
0 0 0 0 0 0 0 0 0 0 0
1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8
0.032893 0.009216 0.001936 -0.000049 -0.000452 -0.000349 -0.000196 -0.000113 -0.000068 -0.000026 -0.000009
3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0 10.0 12.0 14.0 16.0
45 45 45 45 45 45 45 45 45 45 45 45 45
3.0 3.5 4.0 4.5 5 .O 5.5 6.0 7.0 8.0
90 90 90 90 90 90 90 90 90
1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8
0.049498 0.014036 0.002745 -0.000349 -0.000921 -0.000835 -0.000638 -0.000340 -0.000187
3.5 4.0 5.0 5.5 6.0 6.5 7.0 8.0 9.0
0 0 0 0 0 0 0 0 0
9.0 10.0 12.0 16.0
90 90 90 90
1.8 1.8 1.8 1.8
-0.000109 -0.000066 -0.000025 -0.ooooO1
10.0 12.0 14.0 16.0 18.0
3 .O 3.5 4.0 4.5 5 .O 5.3 5.5 6.0 7.0 8.0 9.0 10.0 12.0 14.0 16.0 18.0 20.0
70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70
2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11
0.056813 0.015468 0.004362 0.000239 -0.000730 -0.000806 -0.000780 -0.000627 -0.000345 -0.000193 -0.000115 -0.oooO69 -0.000033 -0.000015 -0.000007 -0.000003 -0.000001
4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 12.0 14.0 16.0 18.0 4.0 5.0 6.0 6.5 7.0 8.0 9.0 10.0 11.0 3.5 4.0 4.5 5.0 5.4 5.5 6.0 6.5 7.0 8.0
45 45 45 45 45 45 45 45 45 45 45 45
2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4
0.013403 0.003178 0.000086 -0.000531 -0.000375 -0.000206 -0.000122 -0.000075 -0.000032 -0.000015 -0.OoooO7 -0.000002 0.056558 0.004194 -0.000417 -0.000503 -0.000419 -0.000238 -0.000134 -0.000080 -0.000051 0.014595 0.003472 -0.000086 -0.000668 -0.000707 -0.000684 -0.000522 -0.000416 -0.000326 -0.0001 84
0
0 0
0 0 0
0 0 0
90 90 90 90 90 90 90 90 90 90
A" - component 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8
0.035577 0.010341 0.002062 -0.000240 -0.000684 -0.0006 18 -0.000353 -0.000193 -0.000111 -0.oooO67 -0.oooO26 -0.000009 -0.ooooO1
4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0 10.0 12.0 14.0
0
Schmelz et al.
15582 J. Phys. Chem., Vol. 99, No. 42, 1995
TABLE 1 (Continued) R
0
r
energy
R
energy
r
0
energy
R
0
r
3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 12.0 14.0 16.0
90 90 90 90 90 90 90 90 90 90 90 90 90
1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8
0.046618 0.013235 0.002659 -0.000251 -0.000810 -0.000578 -0.000312 -0.000174 -0.000102 -0.oooO62 -0.000024 -0.000008 -0.000001
A‘ - component 1.8 1.8 1.8 1.8 -1.8 1.8 1.8 1.8 1.8 1.8 1.8
0.020329 0.005289 0.000657 -0.000518 -0.000663 -0.000557 -0.000314 -0.000176 -0.oooO62 -0.000024 -0.000008
70 70
2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11
0.066290 0.02 1953 0.006086 0.000989 -0.000392 -0.000591 -0.000619 -0.000543 -0.000421 -0.000316 -0.000181 -0.000109 -0.000069 -0.000030 -0.000015 -0.OooOo7 -0.0oooo2
3.0 3.5 4.0 4.5 4.8 5.0 5.5 6.0 7.0 8.0 9.0 10.0 12.0 14.0 16.0 18.0 20.0
90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90
2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11
0.047415 0.014488 0.003365 O.ooOo85 -0.000509 -0.000662 -0.000686 -0.000552 -0.000309 -0.000176 -0.000107 -0.oooO67 -0.000030 -0.000014 -0.000006 -0.000002 -0.000001
3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 12.0 14.0 16.0 18.0
90
2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30
0.047666 0.014846 0.003661 0.000252 -0.000578 -0.000531 -0.000303 -0.000174 -0.000105 -0.000067 -0.oooO29 -0.000014 -0.000006 -0.000002
3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0 10.0 12.0 14.0 16.0
45 45 45 45 45 45 45 45 45 45 45 45 45
1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8
0.049486 0.015573 0.004053 0.000526 -0.000373 -0.000493 -0.000324 -0.000183 -0.000107 -0.000064 -0.oooO24 -0.oooO08 -0.0oooo2
3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 10.0 12.0 14.0
70
3.0 3.5 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0 10.0 12.0 14.0 16.0 18.0
45 45 45 45 45 45 45 45 45 45 45 45 45 45 45
2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11 2.11
0.153813 0.056653 0.018376 0.004985 0.000783 -0.000325 -0.000502 -0.000342 -0.000195 -0.000 116 -0.000072 -0.00003 1 -0.000015 -0.oooO07 -0.000002
3.0 3.5 4.0 4.5 5.O 5.3 5.5 6.0 6.5 7.0 8.0 9.0 10.0 12.0 14.0 16.0 18.0
70 70 70 70 70 70 70 70 70
70 70 70 70 70 70 70 70 70 70
70 70 70 70 70
70
4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0 12.0 14.0 16.0 18.0 20.0
45 45 45 45 45 45 45 45 45 45 45 45 45
2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30
0.020231 0.005656 0.000997 -0.000493 -0.000345 -0.000196 -0.000117 -0.000073 -0.ooOo31 -0.000015 -0.000007 -0.000002 -0.000001
3.5 4.0 4.5 4.8 5.0 5.3 5.5 6.0 6.5 7.0 8.0 12.0 14.0 16.0
70 70 70 70 70 70 70 70 70 70 70 70 70 70
2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30 2.30
0.018405 0.006492 0.001 187 0.000060 -0.000304 -0.000536 -0.000580 -0.000525 -0.00041 1 -0.0003 13 -0.000179 -0.oooO32 -0.000014 -0.OooOo6
4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0
45 45 45 45 45 45 45 45
2.40 2.40 2.40 2.40 2.40 2.40 2.40 2.40
0.211889 0.001129 -0.000486 -0.000340 -0.000198 -0.000168 -0.000071 -0.ooOo46
3.8 4.0 5 .O 6.0 7.0 8.0 9.0 11.0
90 90 90 90 90 90 90 90
2.40 2.40 2.40 2.40 2.40 2.40 2.40 2.40
0.006788 0.003766 -0.000538 -0.000513 -0.000298 -0.000172 -0.000102 -0.000042
0.6) and two optimized sets of f-functions (exp: 1.65,O.g). For the electronic ground state of the 02+ fragment the MRCI calculations yielded with this basis set a me value smaller by 4.1 cm-’ and a re value larger by 0.006 bohr than the experimental values.30 For helium we used the Huzinaga (9s) set39 augmented by one s-function (exp: 0.055), three pfunctions (exp: 1.2,0.3,0.1), and two d-functions (exp: 0.6, 0.15). The total number of contracted Gaussians was 132. All computations were perfomed in C, symmetry. The active space in the CASSCP0q4’calculations was selected to be 6a‘-9a’ and 1a”-2a”. All nonactive orbitals were optimized. This ansatz resulted in 980 CSFs for the A’ state and 910 CSFs for the A” state. All these CSFs were taken as a reference wave function for the internally contracted CI calculations, which corresponds to about 58 x lo6 uncontracted configurations in the CI expansion. The size consistency error for the MRCI wave functions was 900 cm-I, but only about 15 cm-’ for the ACPF
90 90 90 90 90 90 90 90 90 90 90 90 90
wave functions. The computations were carried out with the MOLPRO program suite!2 The calculated energies relative to the dissociation asymptote corrected for the basis superposition errors are explicitly given in Tables 1 and 2. We have chosen to fit the sets of these energies to an expansion in terms of reduced rotation matrix element~,’~ namely, L
v(R,e)= C d
COS e) A,(R) = P~A(R)
(1)
1=0
here L is the number of Jacobi angles 8 for which the potential has been calculated. The vector A(R) can be expressed as
A(R) = P-’*B(R)
(2)
Pi,= d &(cos e,)
(3)
where
J. Phys. Chem., Vol. 99,No. 42, I995 15583
Potential Energy Functions of the 02+He (X211) Complex TABLE 2: ACPF Potential Ener Function of the Electronic Ground State of the Oy...He Complex (the Calculated Energies Corrected for Basis Superposition Errors Are Given (in au) Relative to the Dissociation Asymptote; r = 2.11 bohr) energy
R
0
A'-component
4.0 4.3 4.5 5 .O 5.5 6.0 6.5 7.0 8.0 12.0 20.0 3.5 4.0 4.5 5.0 5.3 5.5 6.0 6.5 7.0 8.0 12.0 20.0 3.0 3.3 3.5 3.8 4.0 4.3 4.5 4.6 4.7 4.8 5.0 5.2 6.0 6.5 7.0 8.0 9.0 12.0 20.0
0 0
0.044 084 6 0.021 302 2 0.012 730 6 0.002 873 5 0.001 092 9 -0.000 491 8 -0.000 509 5 -0.000 408 0 -0.000 228 9 -0.000 033 8 -0.000 000 4 0.056 153 6 0.018 156 1 0.004 861 2 0.000 704 5 -0.000 128 3 -0.000 380 4 -0.000 539 4 -0.000 465 1 -0.000 359 0 -0.000 203 2 -0.000 032 0 -0.000 004 1 0.047 561 8 0.020 721 7 0.014 264 7 0.006 192 1 0.005 511 6 0.000 798 5 -0.000 012 5
0
0 0 0 0 0 0 0 0 45 45 45 45 45 45 45 45 45 45 45 45 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90
-0.000 585 7 -0.000 727 7
-0.000 -0.000 -0.000 -0.000
321 9 182 4 109 6 030 3
A"-component
0.039 035 7 0.01 1 936 2 0.002 571 1 -0.000 151 5 -0.000 608 7 -0.000 709 6 -0.000 672 8 -0.OOO 524 0 -0.ooO 388 1 -0.ooO 213 0 -0.OOO 035.1 -0.OOO 0043 0.049 127 5 0.024 226 6 0.014 567 9
TABLE 3: Characteristics of the Potential Energy Functions barrier to method component R, (bohr) De (cm-I) linearity (cm-I) ACPF A" 5.1601 194.3 77.7 MRCI
A' A" A'
5.2363 5.1933 5.2501
-0.000 -0.000 -0.Ooo -0.000 -0.OOO -0.000 -0.000 -0.000 -0.000
50.9
73.4 48.1
156.4
calculated r. The contours show that the shorter the distance in 0 2 + , the larger the binding energy and the barrier to linearity. For the equilibrium distance of 2.1 1 bohr'in 0 2 + the dissociation energies (in cm-I) have been calculated to be 181.1 (A", MRCI), 194.3 (A", ACPF), 156.4 (A', MRCI), and 167.5 (A', ACPF), respectively. Both sets of dissociation energies are seen to represent only a lower bound for these energy differences. It is also interesting to note that the 02+He (X2n)complex is distinctly less floppy than the ArNO complex, in which the barriers to linearity have been calculated to be 22.0 and 20.5 cm-I,l3 whereas in the ion they are twice (A') or three times (A") larger (cf. Table 3).
3. Rovibronic Energies The variational calculations of the bound rovibronic energy levels of 02+He (X211) were carried out using an approach closely related to the method for bound Renner-Teller systems which includes the electron spin e x p l i ~ i t l y . ~The , ~ ~kinetic energy operator is expressed in J a ~ o b rather i ~ ~ than internal coordinates. The explicit form of the Hamiltonian is
0.003 121 7 -0.000 176 3 -0.000 430 7 -0.000 61 1 5
167.5 181.1
+ +V
H = HVR Hv
861 5 885 0 642 2 475 1 348 9 194 9 159 5 031 5 004 2
LL + p~ lr2 2 ~ 2
(6)
) +( ze A)] ae
ae2
cot
and
p l = d &,(cos 6) (4) with 8 k denoting the individual angles at which ab initio points were calculated and 3 i=O
4
ta~~h(R)][&p"R-~'] ( 5 ) i= I
The contour plots of the MRCI potential energy functions for four r ( 0 2 + ) distances and both A' and A" components are shown in Figures 1 and 2. The A" component always lies lower than the A' component, since the JC orbital of the diatom is occupied by only one electr0n.4~ The complex has a T-shaped structure (cf. Table 3) with the following calculated values of R, (in bohr): 5.1601 (A", ACPF), 5.1933 (A", MRCI), 5.2363 (A', ACPF), 5.2501 (A', MRCI), respectively. For the A" equilibrium structure the He atom lies closer to the center of mass of 0 2 + than for the A' one. The energy minima are separated by 24.7 cm-' (MRCI) or 28.8 cm-l (ACPF), respectively, which is a much larger difference than in ArNO,I3 where both components are separated by only 1.3 cm-l (here the experimental estimate of the energy difference at the minima of the electronic components is 1.5 cm-' 5 ) . The anisotropy of the A' potential is larger than that of the A" potential for all
+ + +
+
+
Here pl = m0-I mHe-' and p2 = m0-I (mHe mo)-'. Also, ll = J L S is the rotational-vibrational angular momentum, with J , L, and S designating the total, electronic, and spin angular momenta of the OzfHe complex. This method permits full three-dimensional variational calculations to be performed on virtually any open shell triatomic molecule. We refer to our earlier papef15 for technical details of the necessary angular momentum matrix elements and the stepwise optimization of the basis set. In the present application we have neglected the geometry variation of the spin-orbit constant and the electronic matrix elements of the L angular momenta. Instead the experimental value of the spin-orbit constant for the electronic ground state of the 0 2 + ion (A = 200 cm-l 30) and L, , L: = 1 have been used. As a basis set we used five Morse eigenfunctions for the 0 2 + fragment, 31 Morse eigenfunctions for the O2+He (X211) stretch coordinate, and 40 Legendre polynomials for the bending coordinate. In our approach only the total J and the parity of the wave functions are good quantum numbers. The assignments of vibrational quantum numbers have been made by inspection of the nodal properties in the plots of both vibrational components of the rovibronic wave functions. In Table 4 such assignments are
15584 J. Phys. Chem., Vol. 99, No. 42, 1995
Schmelz et al.
180
180
160
160
140
140 120
120
s? 100
aa
jW 8
Y
w
0
100
a m
60
Bo
40
40
20
20
0
r( 2’)
6 f
6
7
1.8 bohr
8
9
1
0
1
1
1
0
2
5
6
8
7
9
1
0
1
1
1
2
R(bohr)
180 160 140
@
120
h
a
p
100
a
- 8 0 60 40 20
n -4
r(o2’)
6 E
6
2.3 bohr
7
8
9
10
11
12
r(O2+) = 2.4 bohr
R(bohr)
R(bohr)
Figure 1. Contour plots of the A“ OZ+He potential energy functions for four different 402’) distances. The contours are equally spaced in 10 cm-’ intervals. TABLE 4: Calculated Rovibronic Levels cm-lY of OZ+He (X211) a v,
ACPF- MRCI- MRCIACPF(2D) (2D) (3D) a u1 (2D)
= 1/2, e, in MRCI(2D)
MRCI(3D)
~~~
0.P 1 1 85.401 80.887 0.0 0.351 0.345 1 1 87.864 82.507 0 0 1 0 22.650 22.318 5 0 90.083 85.749 1 0 23.362 23.012 0 2 95.217 89.530 2 0 38.290 38.001 5 0 96.409 91.017 40.472 40.155 0 2 98.030 91.744 2 0 50.274 50.001 2 1 106.259 99.459 3 0 54.163 54.142 2 1 106.689 99.589 3 0 0 1 63.138 62.979 6 0 109.229 103.709 0 1 64.032 63.881 1 2 111.254 104.655 4 0 66.960 68.993 1 2 113.656 106.953 4 0 71.432 71.291 3 1 114.859 108.384 0 Only the levels with the parity (- 1)J-I” = 1 are given. 1 level of 0 2 + at 1870.5 cm-’ was selected as reference. 0
0
0.0 0.325 23.438 24.061 40.753 42.803 53.062 57.215 67.091 68.170 70.256 74.597
80.643 82.299 85.493 89.452 91.466 91.589 99.455 99.498 104.312 104.356 106.911 108.009 The u‘ =
given for all levels (J = 1/2, e) which have been calculated to be bound. Comparison is made between the ACPF and MRCI results for r ( 0 2 + ) fixed at the equilibrium distance of 2.1 1 bohr and with a three-dimensional calculation using the MRCI potential. In the latter case the rovibronic levels are given relative to v’ = 1 of 0 2 + at the calculated value of 1870.5 cm-I. In the isolated fragment the MRCI value has been calculated to be 1872.25 cm-’ (experimental value is 1876.25 cm-’ 37). Hence, the interaction with He lowers the v‘ = 1 only by a couple of wavenumbers. Both two-dimensional potential pairs yield two bound stretching, six bound bending, and two bound combination levels (cf. Table 4). The stretching wavenumber has been calculated to be 67.09 cm-’ (MRCI) and 63.13 cm-I (ACPF), the bending wavenumber, 23.43 cm-l (MRCI) and
TABLE 5: Calculated Rovibronic Levels (J = 3/2,e, in cm-lY of OZ+He Or%) vb
us
0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
MRCIQD)
Ub
0.981 4 1.804 4 3.924 1 4.857 1 23.548 1 1 24.978 4 1 28.025 4 1 1 28.844 0 39.094 1 2 2 42.876 0 46.589 0 2 2 47.797 0 5 1.943 5 3 3 57.516 1 61.748 2 3 0 64.947 5 0 1 65.120 5 0 65.628 2 3 0 1 68.234 2 0 1 69.375 a Only the levels with the parity (-l)’-”z
us
MRCIQD)
0 0
69.530 75.254 79.923 83.455 83.950 85.512 88.216 88.649 89.721 9 1.272 93.733 94.816 98.328 98.958 99.982 100.906 104.980 105.255 106.988
1 1 1
0 0 2 1 2 2 2 0 2 1
0 0 1 1
= -1 are given.
22.65 cm-’ (ACPF). The three-dimensional MRCI stretching wavenumber for v‘ = 1 lies about 4 cm-I lower than for the vibrational ground state of the diatom. The interaction of 0 2 + with He removes the 2j 1 degeneracy of the rotational levels in the diatom. In contrast to A I O H , ~even ~ for J = 112, 02+He exhibits all of the bound triatomic bending modes. Each J = 112 level splits into two states with different parity. For J = 3/2 similar data are given in Table 5 . Due to strong coupling, we could not attribute, other than vibrational quantum numbers,
+
Potential Energy Functions of the 02+He (X2n) Complex
-4
5
6
r(Oz+) = 1.8 bohr
7
8
9
10
11
J. Phys. Chem., Vol. 99, No. 42, 1995 15585
12
-4
R(bohr)
"4
r(02+) = 2.3 bohr
5
6
7
r(02+) = 2.11 bohr
5
6
r(O2+) = 2.4 bohr
R(bohr)
8
9
10
9
10
11
12
R(bohr)
7
8
11
12
R(bohr)
Figure 2. Contour plots of the A' 02+He potential energy functions for fc)ur different r(02+) distances. The contours are equally spaced in 10 cm-' intervals.
TABLE 6: Calculated MRCI Parity Splittings (e-f, in cm-lY us J = 112 J = 312 J = 112 (u'= 1 in 0 2 ' ) ~
0 0 0 0
0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 1 1
0
1
2 2
2 2 4 4 4 4 0
a
-0.1947 -0.3366 -0.3408 0.3974 -1.1492 1.M58 -0.0511 -0.1794
0.1399 -0.3747 -0.0169 0.1257 -0.2720 0.3152 -0.2723 0.1819 -1.1264 1.1154 -0.8613 0.6305 -0.0188 -0.1598 -0.1076 -0.4916
-0.1893 -0.3232 -0.3417 0.3996 -1.1647 1.4283 -0.0674 -0.1838
by the vibrational excitation of the 0 2 + fragment (cf. Table 4). In ArNO, the theoretical values agreed to within about 30% of the available experiments. So far, the largest differences between the A' and A" potential shapes were found for N2+(A211)He.47In this case, the parity splitting of - 1.65 cm-' (e-f) has been calculated already in the vibronic ground state.46
4. Conclusions The electronic ground state of the 02+He complex has been characterized by its potential energy functions and bound rovibronic levels using high-quality ab initio electronic structure calculations and variational calculations of the bound rovibronic levels, which included the coupling of the rotational-vibrational, electron, and electron spin angular momenta. Since neither experimental nor theoretical studies of this complex aie so far known, the present theoretical work should aid future spectroscopic investigations of this ion-complex.
The parity e corresponds to (-1)5-"2, and parity f corresponds to
(- 1)J+l i 2 .
J and parity to these states. The calculated parity splittings are given in Table 6. In the present case they are mainly caused by the Renner-Teller coupling effects. By comparison with ArNO (X2n), the differences between both functions in the O2+He (X2rZ) complex are much larger and lead to large parity splittings. In ArNO the theoretical and experimental ~plittings'~ are smaller than 0.1 cm-l, whereas in 02+He the splittings have been calculated to lie between 0.2 and 1.4 cm-I. They are smaller in the stretching levels than in the bending levels and increase strongly with increasing bending quantum numbers. Only small changes in the large amplitude modes are caused
Acknowledgment. This work has been supported by the Deutsche Forschungsgemeinschaft and Fonds der Chemischen Industrie. References and Notes (1) Heaven, M. C. Phys. Chem. 1993, 97, 8567.
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